Saarela et al Forest Ecosystems (2020) 743 httpsdoiorg101186s40663-020-00245-0

RESEARCH Open Access

Mapping aboveground biomass and itsprediction uncertainty using LiDAR and fielddata accounting for tree-level allometric andLiDAR model errorsSvetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm Alex Appiah Mensah Soumlren HolmMats Nilsson Jonas Fridman and Goumlran Staringhl

Abstract

Background The increasing availability of remotely sensed data has recently challenged the traditional way ofperforming forest inventories and induced an interest in model-based inference Like traditional design-basedinference model-based inference allows for regional estimates of totals and means but in addition for wall-to-wallmapping of forest characteristics Recently Light Detection and Ranging (LiDAR)-based maps of forest attributes havebeen developed in many countries and been well received by users due to their accurate spatial representation offorest resources However the correspondence between such mapping and model-based inference is seldomappreciated In this study we applied hierarchical model-based inference to produce aboveground biomass maps aswell as maps of the corresponding prediction uncertainties with the same spatial resolution Further an estimator ofmean biomass at regional level and its uncertainty was developed to demonstrate how mapping and regional levelassessment can be combined within the framework of model-based inference

Results Through a new version of hierarchical model-based estimation allowing models to be nonlinear weaccounted for uncertainties in both the individual tree-level biomass models and the models linking plot levelbiomass predictions with LiDAR metrics In a 5005 km2 large study area in south-central Sweden the predictedaboveground biomass at the level of 18 mtimes18 mmap units was found to range between 9 and 447 Mgmiddothaminus1 Thecorresponding root mean square errors ranged between 10 and 162 Mgmiddothaminus1 For the entire study region the meanaboveground biomass was 55 Mgmiddothaminus1 and the corresponding relative root mean square error 8 At this level 75of the mean square error was due to the uncertainty associated with tree-level models

Conclusions Through the proposed method it is possible to link mapping and estimation within the framework ofmodel-based inference Uncertainties in both tree-level biomass models and models linking plot level biomass withLiDAR data are accounted for both for the uncertainty maps and the overall estimates The development ofhierarchical model-based inference to handle nonlinear models was an important prerequisite for the study

Keywords Aboveground biomass assessment Forest mapping Gauss-Newton Regression HierarchicalModel-Based inference LiDAR maps National Forest Inventory Uncertainty estimation Uncertainty map

Correspondence svetlanasaarelasluseFaculty of Forest Sciences Swedish University of Agricultural Sciences SLUSkogsmarksgraumlnd 17 SE-90183 Umearing Sweden

copy The Author(s) 2020Open Access This article is licensed under a Creative Commons Attribution 40 International License whichpermits use sharing adaptation distribution and reproduction in any medium or format as long as you give appropriate creditto the original author(s) and the source provide a link to the Creative Commons licence and indicate if changes were made Theimages or other third party material in this article are included in the articlersquos Creative Commons licence unless indicatedotherwise in a credit line to the material If material is not included in the articlersquos Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitted use you will need to obtain permission directly from thecopyright holder To view a copy of this licence visit httpcreativecommonsorglicensesby40

Saarela et al Forest Ecosystems (2020) 743 Page 2 of 17

IntroductionThe interest in forest inventories is increasing due to thekey role of forests in global carbon cycles and thus inthe climate change discourse (eg Bellassen and Luyssaert2014) For a long time regional and national forest inven-tories have been based on field surveys of sparse networksof sample plots (eg Tomppo et al 2010 McRoberts et al2009) Such surveys rely on sampling theory and design-based inference (eg Gregoire and Valentine 2008) Anadvantage is that estimates of population parameters suchas mean and total biomass and the corresponding uncer-tainties can be obtained without making any assumptionsabout the population studied Examples of national for-est inventories of this kind are given in Tomppo et al(2008a 2010) and Fridman et al (2014) Currently theyare the backbones of national forest statistics and interna-tional reporting for compiling regional and global forestresources assessments (eg Forest Europe 2015)Some disadvantages of traditional forest inventories are

that they are expensive to carry out in areas with poorroad infrastructure and they provide only limited spatialinformation about the forest resources Use of remotelysensed (RS) data is a means to overcome these disadvan-tages and an increasing number of studies suggests thatRS data can be used in several ways to improve the effi-ciency of forest inventories and add new dimensions to theresults For example RS data can be used for stratificationin the design phase of forest inventories (eg Katila andTomppo 2002McRoberts et al 2002 Haakana et al 2019)unequal probability sampling (eg Saarela et al 2015a) orbalanced sampling (eg Grafstrom et al 2017a) The lat-ter design was implemented in the Swedish NFI in 2018for the temporary plots (Grafstrom et al 2017b) Gregoireet al (2011) and Gobakken et al (2012) have shown howRS data can be applied in the estimation phase to improvethe precision of estimators through model-assisted esti-mation Franco-Lopez et al (2001) Tomppo et al (2008b)and recently Esteban et al (2019) have shown how RSdata can be combined with field sample data to providemaps of forest resources as a complement to estimates ofmeans and totals of the variables of interestWhen these types of maps are developed RS and field

data are used in regression analysis or other types ofmachine learning algorithms to predict the variable ofinterest for each map element based on the RS data usingan explicit or implicit model relationship between fieldand RS data (eg Andersen et al 2005 Hudak et al2008) Other examples are Wulder et al (2008) whoused regression analysis to map predicted forest biomassover a large spatial domain in central SaskatchewanCanada using Landsat Thematic Mapper (TM) multi-spectral data at a spatial resolution of 30 mtimes30 m andZald et al (2016) who applied the random forest algo-rithm to map forest attributes using a combination of

airborne Light Detection and Ranging (LiDAR) sam-ple data and wall-to-wall multispectral data derivedfrom Landsat TM and the Enhanced Thematic Map-per Plus (ETM+) imagery over a large spatial domain inSaskatchewan Canada An increasing number of studiesof this kind address forest mapping not only at local andnational scales but also at regional and global scales Forexample Saatchi et al (2011) used a fusion of wall-to-wall spatial data frommultiple sensors and sampled forestheight data collected by the Geoscience Laser AltimeterSystem (GLAS) onboard the Ice Cloud and land Eleva-tion Satellite-1 (ICESat-1) to construct a global map offorest carbon stocks at a 1-km spatial resolution Hansenet al (2003) compiled a global map of percent tree coverthrough a supervised regression tree algorithm usingMODIS data at a 500-m spatial resolution and in theGEDI project (Dubayah et al 2014 Qi and Dubayah 2016Qi et al 2019 Patterson et al 2019) a space-borne laser isapplied for mapping biomass with almost global coverageat a 1-km spatial resolutionFollowing the introduction of LiDAR-assisted forest

inventories (eg Nelson et al 1988 Naeligsset 2002) for-est resource maps of this kind have become accurateenough to support management decisions at local levelranging from the scale of individual map units to standsand aggregates of forest stands Examples are harvest-ing decisions at the level of stands and valuation at thelevel of forest properties (eg Nilsson et al 2003) Thussparse samples of field data can now be used for sev-eral purposes eg they can be used to provide traditionalforest statistics as well as maps of forest resources andenvironmental conditions when combined with RS data(eg Nilsson et al 2017)However it is far from trivial how maps of forest

resources should be used for producing reliable foreststatistics For example McRoberts et al (2018) point outthat land cover statistics will be subject to systematicerrors if classifications for map units are merely summedacross a study area Further Gregoire et al (2016) identifyseveral problems related to how uncertainties are typicallycomputed and reported in LiDAR-supported studies Forexample in many cases uncertainties are merely assessedintuitively based on subjectively selected sets of map unitswhere predicted values are compared with the corre-sponding field measurements It is not straightforward touse data from such comparisons for estimating variancesor mean square errors of estimated means and totals foran entire study areaOne way of using RS auxiliary data (eg in the form

of forest maps) in a statistically rigorous way for produc-ing forest statistics is to apply model-assisted estimation(eg Gregoire et al 2011 Saarela et al 2015a) With thisdesign-based inference technique model predictions aresubtracted from field reference values for a probability

Saarela et al Forest Ecosystems (2020) 743 Page 3 of 17

sample of map units The mean of the observed devia-tions is added to the mean of the model predictions acrossall map units to obtain an estimate of the mean of thevariable of interest While this approach has a poten-tial to make cost-efficient use of RS data that correlatestrongly with study variables of interest a drawback ofmodel-assisted estimation is that it requires a fairly largeprobability sample of field data (Staringhl et al 2016)An alternative approach to using RS auxiliary data in a

statistically rigorous manner for producing forest statis-tics is to apply model-based inference (eg Magnussen2015) Model-based inference relies on an assumption ofa model that is assumed to generate random values foreach map unit based on the auxiliary data available forthe map units This model is sometimes referred to asa superpopulation model from which the actual popu-lation is realized (Cassel et al 1977 Saumlrndal et al 1978Gregoire 1998) Since the individual values of the popula-tion elements are random variables so are the populationmeans and totals While many concepts and estimationprocedures related to model-based inference tend to bemore complicated than design-based inference (eg Gre-goire 1998) model-based inference also has advantages(eg Magnussen 2015) eg with regard to combiningforest mapping and compiling forest statistics since thesame basic procedures can be applied for both purposesMoreover model-based inference can be used for pro-ducing maps of uncertainties for the variables of interestalthough this opportunity seems to be seldom exploitedAn example is Esteban et al (2019) who presented amap of uncertainties for forest attribute predictions usingthe random forest algorithm The uncertainties were esti-mated through bootstrapping Such uncertainty maps areimportant for assessing to what extent the map infor-mation is accurate enough for making different kinds ofdecisionsInterestingly uncertainty estimation under model-

based inference is fairly straightforward both at the levelof individual map units and at the level of totals andmeans across large study areas (eg Staringhl et al 2016) Forsmall aggregates of map units it is more complicated dueto the need for information about how strongly modelerrors are correlated across space (McRoberts et al 2018)Still many different techniques have been presented forsmall area estimation (eg Breidenbach and Astrup 2012Magnussen et al 2014)Conventional model-based inference utilizes auxiliary

information from all map units and a single model forthe relationship between auxiliary data and the variableof interest However in many important applications sev-eral models need to be combined in a hierarchical mannersuch as when biomass models are first developed for sin-gle trees and then applied to predict aggregate plot levelbiomass which is used as training data for developing a

model linking RS data with plot level biomass In suchcases it is not trivial how uncertainties should be esti-mated Saarela et al (2016) developed a method calledhierarchical model-based (HMB) estimation and showedthat neglecting the uncertainty associated with one of thetwomodels involvedmight lead to severe underestimationof the uncertainty The method was further developed inSaarela et al (2018) although both studies were restrictedto use of linear models

ObjectivesThe objective of this study was to clarify the link betweenproducing forest statistics for large areas through model-based inference and mapping of forest resources provid-ing high spatial resolution maps of aboveground biomass(AGB) and the corresponding uncertainties in terms ofroot mean square errors (RMSE) This link may be obvi-ous since model-based inference requires predictionsfor each map unit but it is seldom acknowledged inresearch studies Data were available from single treesharvested and measured for developing allometric treebiomass models field plots from the Swedish NationalForest Inventory (NFI) and wall-to-wall airborne LiDARdata acquired in 2009 ndash 2011 The map units were18times18 meters large HMB inference was applied throughan individual tree biomass model and a biomass modellinking LiDAR data with aggregate plot-level biomassAn important part of the study was to further developthe HMB inference theory to incorporate nonlinearmodels

Material andMethodOverviewForest attribute maps are usually based on wall-to-wallRS data In our example we used airborne wall-to-wallLiDAR data collected on a national scale in SwedenRegression analysis was employed to regress the forestattribute plot-level AGB on LiDAR metrics The AGB-LiDAR regression model was trained on field plot datafrom the Swedish NFI The plot-level AGB value is asum of tree-level AGB predictions using field measure-ments of height and diameter at breast height (DBH)ie tree-level AGB was regressed on tree height andDBH To train the tree-level regression models here-after referred to as AGB allometric models datasets col-lected by Marklund (1987 1988) were used Therefore inour estimation procedure there are two modelling stepsinvolved the AGB allometric models and the (plot-level)AGB-LiDAR modelHMB inference was employed to estimate the RMSE

and the relative RMSE (RelRMSE) accounting for uncer-tainties due to both modelling steps New theory wasdeveloped for incorporating nonlinear models in theHMB framework Through the new theory the previous

Saarela et al Forest Ecosystems (2020) 743 Page 4 of 17

derivations of HMB estimators Saarela et al (2016 2018)could be simplified A graphical overview of the study isshown in Fig 1

Study areaOur study area was a rectangular 5005 km2 large blocklocated in south-central Sweden The forest area wasapproximately 3909 km2 ie 78 of the total area Agri-cultural lands and urban areas were masked out usingland-use maps available through the Swedish NationalMapping Agency (Lantmaumlteriet 2019) The study area wastessellated into 18 mtimes18 m grid-cells (map units) Themap unit size (324 m2) is approximately equal to the areasize of the NFI circular field plots described in ldquoSwedishNFI data and AGB-LiDAR regression modelrdquo section Thelocations of the study area and the clusters of NFI plots areshown in Fig 2

Swedish national airborne LiDAR survey dataIn 2009 the Swedish National Mapping Agency initiateda national airborne LiDAR survey to obtain a new digitalelevation model (DEM) with 2 mtimes2 m spatial resolutionBy 2015 almost all forest lands in Sweden were scanned(Nilsson et al 2017) The scanning altitude was between1700 m and 2300 m and the pulse density about 05-1pulsesmiddotmminus2For each map unit and NFI field plot LiDAR met-

rics were derived using the Fusion software (McGaughey2012) following the area-based approach proposed by

Naeligsset (2002) from the height distribution of laser returnsbetween 15 m and 50 m height above ground Theupper and lower height thresholds were set to excludenon-vegetation returns (eg Lindberg et al 2012) Asregressors in our AGB-LiDARmodel we used two LiDARmetrics the laser height 80 percentile (hp80) and the veg-etation ratio (vr) which is a ratio between the numberof first returns above 15 m and the total number of firstreturns The choice of these variables follows findings inNilsson et al (2017)

Swedish NFI data and AGB-LiDAR regression modelPlot-level field data were obtained from the Swedish NFIwhich applies a systematic design of clusters with 4ndash12plots We utilized data collected during 2009ndash2011 from504 permanent plots located in the study area and in theneighbourhood of the study area (Fig 2) The plot radiuswas 10 m A criterion for the plot selection was that thesame LiDAR instrument as the one used to collect LiDARover the study area had been applied Information on indi-vidual tree locations species and DBH was available foreach tree in all plots However tree height was availableonly for a subsample of the trees (Fridman et al 2014)Table 1 provides descriptive statistics of the NFI data sep-arated for each species on trees with and without treeheight measurementsFigure 3 shows a histogram of plot-level AGB aggre-

gated from the tree-level AGB predictions and converted

Fig 1 Study overview

Saarela et al Forest Ecosystems (2020) 743 Page 5 of 17

Fig 2 The study area and the clusters of NFI plots applied in the study

to tonnes per hectare (Mgmiddothaminus1) The tree-level AGBvalues were predicted using the AGB allometric modelsdescribed in ldquoTree-level data and AGB allometric modelsrdquosectionThe plot-level AGB-LiDAR model denoted as the G

model in our study has a model form similar to Nils-son et al (2017) However whereas Nilsson et al (2017)used a standard square root transformation model weapplied a similar nonlinear model with additive errorsincluding the same LiDAR metrics (see Table 2) We

employed the restricted maximum likelihood estimatorsavailable in the R package nlme (Pinheiro et al 2016)through the gnls() R function To overcome problemsdue to heteroskedasticity we fitted a random error vari-ance model to estimate the individual error variance forevery predicted plot-level AGB value Table 2 presents theestimatedmodel parameters Figure 4 shows the standard-ized residuals for plot-level AGB predictions from LiDARmetrics displayed versus predicted plot-level AGBs usingLiDAR metrics and (lower) LiDAR-based predictions of

Table 1 NFI tree-level data descriptive statistics separated for each tree species on trees with and without height measurements

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch and other 172 400 1934 10010 1245 500 1687 3200 632

deciduous 2088 100 1144 4150 611 ndash ndash ndash ndash

Norway Spruce544 410 2276 7020 1119 320 1788 4320 688

4265 100 1515 4750 693 ndash ndash ndash ndash

Scots Pine555 430 2290 5790 1006 380 1672 3310 569

4841 100 1574 4740 630 ndash ndash ndash ndash

Saarela et al Forest Ecosystems (2020) 743 Page 6 of 17

Fig 3 A histogram of plot-level AGB (aggregated tree-level AGB predictions converted to tonnes per hectare) (Mgmiddothaminus1)

plot-level AGB displayed versus AGBs obtained from fieldmeasurements ie the AGBs obtained from aggregat-ing tree-level AGB predictions to plot level Althoughthe Swedish NFI applies clusters of plots the long dis-tance between plots in a cluster implied that we treatedeach plot as an independent observation in the regressionanalysis (cf Nilsson et al 2017)In Table 2 yGi denotes predicted plot-level AGB

(Mgmiddothaminus1) using LiDAR metrics from the ith NFI plot α0α1 and α2 are AGB-LiDAR model parameters to be esti-mated V(υi) is individual random error variance whichwas modelled through a nonlinear power model withthree parameters denoted σ 2 δ0 and δ1

Tree-level data and AGB allometric modelsThe tree-level data used for developing AGB allomet-ric models were collected across Sweden by Marklund(19871988) We used data collected only from the centraland southern parts of Sweden (Table 3)Since heights were not available for all trees on the NFI

plots (Fridman et al 2014) we developed two AGB allo-

metric model types for each tree species one with DBH(cm) and height (m) as regressors (type I) and the otherwith DBH (cm) only (type II) In our study we used AGBallometric model forms by similar to Repola (2008) forbirch and Repola (2009) for pine and spruce Table 4presents the models forms by speciesIn Table 4 yk is a tree-level AGB (kg) for the kth tree

based on data from Marklund (1987 1988) dtrk = 2 +15 times DBHk is a transformation of DBH (cm) to approx-imate the stump diameter (Laasasenaho 1982) hk is thetree height (m) and β0 β1 and β2 are AGB allometricmodel parameters to be estimatedThe R function gnls() (Pinheiro et al 2016) was used

to fit model parameters accounting for heteroskedasticityTable 5 presents the estimated model parametersIn Table 5 yk is a tree-level predicted AGB (kg) using

the corresponding allometric model for the kth tree V(εk)is individual random error variance modelled through anonlinear power model with two parameters ω2 and γ Figure 5 shows residual scatterplots and Fig 6 presents

scatterplots for AGB versus predicted AGB at tree level

Table 2 Estimated AGB-LiDAR model parameters

Model Model form Model parameter estimates

AGB-LiDAR (G) yi = (α0 + α1hp80i + α2vri)2 + υiα0 α1 α2

253 027 127times10minus3

V(υi) σ 2(δ0 + yδ1Gi )2 σ 2

δ0 δ1

078 150 074

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 3 of 17

sample of map units The mean of the observed devia-tions is added to the mean of the model predictions acrossall map units to obtain an estimate of the mean of thevariable of interest While this approach has a poten-tial to make cost-efficient use of RS data that correlatestrongly with study variables of interest a drawback ofmodel-assisted estimation is that it requires a fairly largeprobability sample of field data (Staringhl et al 2016)An alternative approach to using RS auxiliary data in a

statistically rigorous manner for producing forest statis-tics is to apply model-based inference (eg Magnussen2015) Model-based inference relies on an assumption ofa model that is assumed to generate random values foreach map unit based on the auxiliary data available forthe map units This model is sometimes referred to asa superpopulation model from which the actual popu-lation is realized (Cassel et al 1977 Saumlrndal et al 1978Gregoire 1998) Since the individual values of the popula-tion elements are random variables so are the populationmeans and totals While many concepts and estimationprocedures related to model-based inference tend to bemore complicated than design-based inference (eg Gre-goire 1998) model-based inference also has advantages(eg Magnussen 2015) eg with regard to combiningforest mapping and compiling forest statistics since thesame basic procedures can be applied for both purposesMoreover model-based inference can be used for pro-ducing maps of uncertainties for the variables of interestalthough this opportunity seems to be seldom exploitedAn example is Esteban et al (2019) who presented amap of uncertainties for forest attribute predictions usingthe random forest algorithm The uncertainties were esti-mated through bootstrapping Such uncertainty maps areimportant for assessing to what extent the map infor-mation is accurate enough for making different kinds ofdecisionsInterestingly uncertainty estimation under model-

based inference is fairly straightforward both at the levelof individual map units and at the level of totals andmeans across large study areas (eg Staringhl et al 2016) Forsmall aggregates of map units it is more complicated dueto the need for information about how strongly modelerrors are correlated across space (McRoberts et al 2018)Still many different techniques have been presented forsmall area estimation (eg Breidenbach and Astrup 2012Magnussen et al 2014)Conventional model-based inference utilizes auxiliary

information from all map units and a single model forthe relationship between auxiliary data and the variableof interest However in many important applications sev-eral models need to be combined in a hierarchical mannersuch as when biomass models are first developed for sin-gle trees and then applied to predict aggregate plot levelbiomass which is used as training data for developing a

model linking RS data with plot level biomass In suchcases it is not trivial how uncertainties should be esti-mated Saarela et al (2016) developed a method calledhierarchical model-based (HMB) estimation and showedthat neglecting the uncertainty associated with one of thetwomodels involvedmight lead to severe underestimationof the uncertainty The method was further developed inSaarela et al (2018) although both studies were restrictedto use of linear models

ObjectivesThe objective of this study was to clarify the link betweenproducing forest statistics for large areas through model-based inference and mapping of forest resources provid-ing high spatial resolution maps of aboveground biomass(AGB) and the corresponding uncertainties in terms ofroot mean square errors (RMSE) This link may be obvi-ous since model-based inference requires predictionsfor each map unit but it is seldom acknowledged inresearch studies Data were available from single treesharvested and measured for developing allometric treebiomass models field plots from the Swedish NationalForest Inventory (NFI) and wall-to-wall airborne LiDARdata acquired in 2009 ndash 2011 The map units were18times18 meters large HMB inference was applied throughan individual tree biomass model and a biomass modellinking LiDAR data with aggregate plot-level biomassAn important part of the study was to further developthe HMB inference theory to incorporate nonlinearmodels

Material andMethodOverviewForest attribute maps are usually based on wall-to-wallRS data In our example we used airborne wall-to-wallLiDAR data collected on a national scale in SwedenRegression analysis was employed to regress the forestattribute plot-level AGB on LiDAR metrics The AGB-LiDAR regression model was trained on field plot datafrom the Swedish NFI The plot-level AGB value is asum of tree-level AGB predictions using field measure-ments of height and diameter at breast height (DBH)ie tree-level AGB was regressed on tree height andDBH To train the tree-level regression models here-after referred to as AGB allometric models datasets col-lected by Marklund (1987 1988) were used Therefore inour estimation procedure there are two modelling stepsinvolved the AGB allometric models and the (plot-level)AGB-LiDAR modelHMB inference was employed to estimate the RMSE

and the relative RMSE (RelRMSE) accounting for uncer-tainties due to both modelling steps New theory wasdeveloped for incorporating nonlinear models in theHMB framework Through the new theory the previous

Saarela et al Forest Ecosystems (2020) 743 Page 4 of 17

derivations of HMB estimators Saarela et al (2016 2018)could be simplified A graphical overview of the study isshown in Fig 1

Study areaOur study area was a rectangular 5005 km2 large blocklocated in south-central Sweden The forest area wasapproximately 3909 km2 ie 78 of the total area Agri-cultural lands and urban areas were masked out usingland-use maps available through the Swedish NationalMapping Agency (Lantmaumlteriet 2019) The study area wastessellated into 18 mtimes18 m grid-cells (map units) Themap unit size (324 m2) is approximately equal to the areasize of the NFI circular field plots described in ldquoSwedishNFI data and AGB-LiDAR regression modelrdquo section Thelocations of the study area and the clusters of NFI plots areshown in Fig 2

Swedish national airborne LiDAR survey dataIn 2009 the Swedish National Mapping Agency initiateda national airborne LiDAR survey to obtain a new digitalelevation model (DEM) with 2 mtimes2 m spatial resolutionBy 2015 almost all forest lands in Sweden were scanned(Nilsson et al 2017) The scanning altitude was between1700 m and 2300 m and the pulse density about 05-1pulsesmiddotmminus2For each map unit and NFI field plot LiDAR met-

rics were derived using the Fusion software (McGaughey2012) following the area-based approach proposed by

Naeligsset (2002) from the height distribution of laser returnsbetween 15 m and 50 m height above ground Theupper and lower height thresholds were set to excludenon-vegetation returns (eg Lindberg et al 2012) Asregressors in our AGB-LiDARmodel we used two LiDARmetrics the laser height 80 percentile (hp80) and the veg-etation ratio (vr) which is a ratio between the numberof first returns above 15 m and the total number of firstreturns The choice of these variables follows findings inNilsson et al (2017)

Swedish NFI data and AGB-LiDAR regression modelPlot-level field data were obtained from the Swedish NFIwhich applies a systematic design of clusters with 4ndash12plots We utilized data collected during 2009ndash2011 from504 permanent plots located in the study area and in theneighbourhood of the study area (Fig 2) The plot radiuswas 10 m A criterion for the plot selection was that thesame LiDAR instrument as the one used to collect LiDARover the study area had been applied Information on indi-vidual tree locations species and DBH was available foreach tree in all plots However tree height was availableonly for a subsample of the trees (Fridman et al 2014)Table 1 provides descriptive statistics of the NFI data sep-arated for each species on trees with and without treeheight measurementsFigure 3 shows a histogram of plot-level AGB aggre-

gated from the tree-level AGB predictions and converted

Fig 1 Study overview

Saarela et al Forest Ecosystems (2020) 743 Page 5 of 17

Fig 2 The study area and the clusters of NFI plots applied in the study

to tonnes per hectare (Mgmiddothaminus1) The tree-level AGBvalues were predicted using the AGB allometric modelsdescribed in ldquoTree-level data and AGB allometric modelsrdquosectionThe plot-level AGB-LiDAR model denoted as the G

model in our study has a model form similar to Nils-son et al (2017) However whereas Nilsson et al (2017)used a standard square root transformation model weapplied a similar nonlinear model with additive errorsincluding the same LiDAR metrics (see Table 2) We

employed the restricted maximum likelihood estimatorsavailable in the R package nlme (Pinheiro et al 2016)through the gnls() R function To overcome problemsdue to heteroskedasticity we fitted a random error vari-ance model to estimate the individual error variance forevery predicted plot-level AGB value Table 2 presents theestimatedmodel parameters Figure 4 shows the standard-ized residuals for plot-level AGB predictions from LiDARmetrics displayed versus predicted plot-level AGBs usingLiDAR metrics and (lower) LiDAR-based predictions of

Table 1 NFI tree-level data descriptive statistics separated for each tree species on trees with and without height measurements

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch and other 172 400 1934 10010 1245 500 1687 3200 632

deciduous 2088 100 1144 4150 611 ndash ndash ndash ndash

Norway Spruce544 410 2276 7020 1119 320 1788 4320 688

4265 100 1515 4750 693 ndash ndash ndash ndash

Scots Pine555 430 2290 5790 1006 380 1672 3310 569

4841 100 1574 4740 630 ndash ndash ndash ndash

Saarela et al Forest Ecosystems (2020) 743 Page 6 of 17

Fig 3 A histogram of plot-level AGB (aggregated tree-level AGB predictions converted to tonnes per hectare) (Mgmiddothaminus1)

plot-level AGB displayed versus AGBs obtained from fieldmeasurements ie the AGBs obtained from aggregat-ing tree-level AGB predictions to plot level Althoughthe Swedish NFI applies clusters of plots the long dis-tance between plots in a cluster implied that we treatedeach plot as an independent observation in the regressionanalysis (cf Nilsson et al 2017)In Table 2 yGi denotes predicted plot-level AGB

(Mgmiddothaminus1) using LiDAR metrics from the ith NFI plot α0α1 and α2 are AGB-LiDAR model parameters to be esti-mated V(υi) is individual random error variance whichwas modelled through a nonlinear power model withthree parameters denoted σ 2 δ0 and δ1

Tree-level data and AGB allometric modelsThe tree-level data used for developing AGB allomet-ric models were collected across Sweden by Marklund(19871988) We used data collected only from the centraland southern parts of Sweden (Table 3)Since heights were not available for all trees on the NFI

plots (Fridman et al 2014) we developed two AGB allo-

metric model types for each tree species one with DBH(cm) and height (m) as regressors (type I) and the otherwith DBH (cm) only (type II) In our study we used AGBallometric model forms by similar to Repola (2008) forbirch and Repola (2009) for pine and spruce Table 4presents the models forms by speciesIn Table 4 yk is a tree-level AGB (kg) for the kth tree

based on data from Marklund (1987 1988) dtrk = 2 +15 times DBHk is a transformation of DBH (cm) to approx-imate the stump diameter (Laasasenaho 1982) hk is thetree height (m) and β0 β1 and β2 are AGB allometricmodel parameters to be estimatedThe R function gnls() (Pinheiro et al 2016) was used

to fit model parameters accounting for heteroskedasticityTable 5 presents the estimated model parametersIn Table 5 yk is a tree-level predicted AGB (kg) using

the corresponding allometric model for the kth tree V(εk)is individual random error variance modelled through anonlinear power model with two parameters ω2 and γ Figure 5 shows residual scatterplots and Fig 6 presents

scatterplots for AGB versus predicted AGB at tree level

Table 2 Estimated AGB-LiDAR model parameters

Model Model form Model parameter estimates

AGB-LiDAR (G) yi = (α0 + α1hp80i + α2vri)2 + υiα0 α1 α2

253 027 127times10minus3

V(υi) σ 2(δ0 + yδ1Gi )2 σ 2

δ0 δ1

078 150 074

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 4 of 17

derivations of HMB estimators Saarela et al (2016 2018)could be simplified A graphical overview of the study isshown in Fig 1

Study areaOur study area was a rectangular 5005 km2 large blocklocated in south-central Sweden The forest area wasapproximately 3909 km2 ie 78 of the total area Agri-cultural lands and urban areas were masked out usingland-use maps available through the Swedish NationalMapping Agency (Lantmaumlteriet 2019) The study area wastessellated into 18 mtimes18 m grid-cells (map units) Themap unit size (324 m2) is approximately equal to the areasize of the NFI circular field plots described in ldquoSwedishNFI data and AGB-LiDAR regression modelrdquo section Thelocations of the study area and the clusters of NFI plots areshown in Fig 2

Swedish national airborne LiDAR survey dataIn 2009 the Swedish National Mapping Agency initiateda national airborne LiDAR survey to obtain a new digitalelevation model (DEM) with 2 mtimes2 m spatial resolutionBy 2015 almost all forest lands in Sweden were scanned(Nilsson et al 2017) The scanning altitude was between1700 m and 2300 m and the pulse density about 05-1pulsesmiddotmminus2For each map unit and NFI field plot LiDAR met-

rics were derived using the Fusion software (McGaughey2012) following the area-based approach proposed by

Naeligsset (2002) from the height distribution of laser returnsbetween 15 m and 50 m height above ground Theupper and lower height thresholds were set to excludenon-vegetation returns (eg Lindberg et al 2012) Asregressors in our AGB-LiDARmodel we used two LiDARmetrics the laser height 80 percentile (hp80) and the veg-etation ratio (vr) which is a ratio between the numberof first returns above 15 m and the total number of firstreturns The choice of these variables follows findings inNilsson et al (2017)

Swedish NFI data and AGB-LiDAR regression modelPlot-level field data were obtained from the Swedish NFIwhich applies a systematic design of clusters with 4ndash12plots We utilized data collected during 2009ndash2011 from504 permanent plots located in the study area and in theneighbourhood of the study area (Fig 2) The plot radiuswas 10 m A criterion for the plot selection was that thesame LiDAR instrument as the one used to collect LiDARover the study area had been applied Information on indi-vidual tree locations species and DBH was available foreach tree in all plots However tree height was availableonly for a subsample of the trees (Fridman et al 2014)Table 1 provides descriptive statistics of the NFI data sep-arated for each species on trees with and without treeheight measurementsFigure 3 shows a histogram of plot-level AGB aggre-

gated from the tree-level AGB predictions and converted

Fig 1 Study overview

Saarela et al Forest Ecosystems (2020) 743 Page 5 of 17

Fig 2 The study area and the clusters of NFI plots applied in the study

to tonnes per hectare (Mgmiddothaminus1) The tree-level AGBvalues were predicted using the AGB allometric modelsdescribed in ldquoTree-level data and AGB allometric modelsrdquosectionThe plot-level AGB-LiDAR model denoted as the G

model in our study has a model form similar to Nils-son et al (2017) However whereas Nilsson et al (2017)used a standard square root transformation model weapplied a similar nonlinear model with additive errorsincluding the same LiDAR metrics (see Table 2) We

employed the restricted maximum likelihood estimatorsavailable in the R package nlme (Pinheiro et al 2016)through the gnls() R function To overcome problemsdue to heteroskedasticity we fitted a random error vari-ance model to estimate the individual error variance forevery predicted plot-level AGB value Table 2 presents theestimatedmodel parameters Figure 4 shows the standard-ized residuals for plot-level AGB predictions from LiDARmetrics displayed versus predicted plot-level AGBs usingLiDAR metrics and (lower) LiDAR-based predictions of

Table 1 NFI tree-level data descriptive statistics separated for each tree species on trees with and without height measurements

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch and other 172 400 1934 10010 1245 500 1687 3200 632

deciduous 2088 100 1144 4150 611 ndash ndash ndash ndash

Norway Spruce544 410 2276 7020 1119 320 1788 4320 688

4265 100 1515 4750 693 ndash ndash ndash ndash

Scots Pine555 430 2290 5790 1006 380 1672 3310 569

4841 100 1574 4740 630 ndash ndash ndash ndash

Saarela et al Forest Ecosystems (2020) 743 Page 6 of 17

Fig 3 A histogram of plot-level AGB (aggregated tree-level AGB predictions converted to tonnes per hectare) (Mgmiddothaminus1)

plot-level AGB displayed versus AGBs obtained from fieldmeasurements ie the AGBs obtained from aggregat-ing tree-level AGB predictions to plot level Althoughthe Swedish NFI applies clusters of plots the long dis-tance between plots in a cluster implied that we treatedeach plot as an independent observation in the regressionanalysis (cf Nilsson et al 2017)In Table 2 yGi denotes predicted plot-level AGB

(Mgmiddothaminus1) using LiDAR metrics from the ith NFI plot α0α1 and α2 are AGB-LiDAR model parameters to be esti-mated V(υi) is individual random error variance whichwas modelled through a nonlinear power model withthree parameters denoted σ 2 δ0 and δ1

Tree-level data and AGB allometric modelsThe tree-level data used for developing AGB allomet-ric models were collected across Sweden by Marklund(19871988) We used data collected only from the centraland southern parts of Sweden (Table 3)Since heights were not available for all trees on the NFI

plots (Fridman et al 2014) we developed two AGB allo-

metric model types for each tree species one with DBH(cm) and height (m) as regressors (type I) and the otherwith DBH (cm) only (type II) In our study we used AGBallometric model forms by similar to Repola (2008) forbirch and Repola (2009) for pine and spruce Table 4presents the models forms by speciesIn Table 4 yk is a tree-level AGB (kg) for the kth tree

based on data from Marklund (1987 1988) dtrk = 2 +15 times DBHk is a transformation of DBH (cm) to approx-imate the stump diameter (Laasasenaho 1982) hk is thetree height (m) and β0 β1 and β2 are AGB allometricmodel parameters to be estimatedThe R function gnls() (Pinheiro et al 2016) was used

to fit model parameters accounting for heteroskedasticityTable 5 presents the estimated model parametersIn Table 5 yk is a tree-level predicted AGB (kg) using

the corresponding allometric model for the kth tree V(εk)is individual random error variance modelled through anonlinear power model with two parameters ω2 and γ Figure 5 shows residual scatterplots and Fig 6 presents

scatterplots for AGB versus predicted AGB at tree level

Table 2 Estimated AGB-LiDAR model parameters

Model Model form Model parameter estimates

AGB-LiDAR (G) yi = (α0 + α1hp80i + α2vri)2 + υiα0 α1 α2

253 027 127times10minus3

V(υi) σ 2(δ0 + yδ1Gi )2 σ 2

δ0 δ1

078 150 074

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 5 of 17

Fig 2 The study area and the clusters of NFI plots applied in the study

to tonnes per hectare (Mgmiddothaminus1) The tree-level AGBvalues were predicted using the AGB allometric modelsdescribed in ldquoTree-level data and AGB allometric modelsrdquosectionThe plot-level AGB-LiDAR model denoted as the G

model in our study has a model form similar to Nils-son et al (2017) However whereas Nilsson et al (2017)used a standard square root transformation model weapplied a similar nonlinear model with additive errorsincluding the same LiDAR metrics (see Table 2) We

employed the restricted maximum likelihood estimatorsavailable in the R package nlme (Pinheiro et al 2016)through the gnls() R function To overcome problemsdue to heteroskedasticity we fitted a random error vari-ance model to estimate the individual error variance forevery predicted plot-level AGB value Table 2 presents theestimatedmodel parameters Figure 4 shows the standard-ized residuals for plot-level AGB predictions from LiDARmetrics displayed versus predicted plot-level AGBs usingLiDAR metrics and (lower) LiDAR-based predictions of

Table 1 NFI tree-level data descriptive statistics separated for each tree species on trees with and without height measurements

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch and other 172 400 1934 10010 1245 500 1687 3200 632

deciduous 2088 100 1144 4150 611 ndash ndash ndash ndash

Norway Spruce544 410 2276 7020 1119 320 1788 4320 688

4265 100 1515 4750 693 ndash ndash ndash ndash

Scots Pine555 430 2290 5790 1006 380 1672 3310 569

4841 100 1574 4740 630 ndash ndash ndash ndash

Saarela et al Forest Ecosystems (2020) 743 Page 6 of 17

Fig 3 A histogram of plot-level AGB (aggregated tree-level AGB predictions converted to tonnes per hectare) (Mgmiddothaminus1)

plot-level AGB displayed versus AGBs obtained from fieldmeasurements ie the AGBs obtained from aggregat-ing tree-level AGB predictions to plot level Althoughthe Swedish NFI applies clusters of plots the long dis-tance between plots in a cluster implied that we treatedeach plot as an independent observation in the regressionanalysis (cf Nilsson et al 2017)In Table 2 yGi denotes predicted plot-level AGB

(Mgmiddothaminus1) using LiDAR metrics from the ith NFI plot α0α1 and α2 are AGB-LiDAR model parameters to be esti-mated V(υi) is individual random error variance whichwas modelled through a nonlinear power model withthree parameters denoted σ 2 δ0 and δ1

Tree-level data and AGB allometric modelsThe tree-level data used for developing AGB allomet-ric models were collected across Sweden by Marklund(19871988) We used data collected only from the centraland southern parts of Sweden (Table 3)Since heights were not available for all trees on the NFI

plots (Fridman et al 2014) we developed two AGB allo-

metric model types for each tree species one with DBH(cm) and height (m) as regressors (type I) and the otherwith DBH (cm) only (type II) In our study we used AGBallometric model forms by similar to Repola (2008) forbirch and Repola (2009) for pine and spruce Table 4presents the models forms by speciesIn Table 4 yk is a tree-level AGB (kg) for the kth tree

based on data from Marklund (1987 1988) dtrk = 2 +15 times DBHk is a transformation of DBH (cm) to approx-imate the stump diameter (Laasasenaho 1982) hk is thetree height (m) and β0 β1 and β2 are AGB allometricmodel parameters to be estimatedThe R function gnls() (Pinheiro et al 2016) was used

to fit model parameters accounting for heteroskedasticityTable 5 presents the estimated model parametersIn Table 5 yk is a tree-level predicted AGB (kg) using

the corresponding allometric model for the kth tree V(εk)is individual random error variance modelled through anonlinear power model with two parameters ω2 and γ Figure 5 shows residual scatterplots and Fig 6 presents

scatterplots for AGB versus predicted AGB at tree level

Table 2 Estimated AGB-LiDAR model parameters

Model Model form Model parameter estimates

AGB-LiDAR (G) yi = (α0 + α1hp80i + α2vri)2 + υiα0 α1 α2

253 027 127times10minus3

V(υi) σ 2(δ0 + yδ1Gi )2 σ 2

δ0 δ1

078 150 074

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 6 of 17

Fig 3 A histogram of plot-level AGB (aggregated tree-level AGB predictions converted to tonnes per hectare) (Mgmiddothaminus1)

plot-level AGB displayed versus AGBs obtained from fieldmeasurements ie the AGBs obtained from aggregat-ing tree-level AGB predictions to plot level Althoughthe Swedish NFI applies clusters of plots the long dis-tance between plots in a cluster implied that we treatedeach plot as an independent observation in the regressionanalysis (cf Nilsson et al 2017)In Table 2 yGi denotes predicted plot-level AGB

(Mgmiddothaminus1) using LiDAR metrics from the ith NFI plot α0α1 and α2 are AGB-LiDAR model parameters to be esti-mated V(υi) is individual random error variance whichwas modelled through a nonlinear power model withthree parameters denoted σ 2 δ0 and δ1

Tree-level data and AGB allometric modelsThe tree-level data used for developing AGB allomet-ric models were collected across Sweden by Marklund(19871988) We used data collected only from the centraland southern parts of Sweden (Table 3)Since heights were not available for all trees on the NFI

plots (Fridman et al 2014) we developed two AGB allo-

metric model types for each tree species one with DBH(cm) and height (m) as regressors (type I) and the otherwith DBH (cm) only (type II) In our study we used AGBallometric model forms by similar to Repola (2008) forbirch and Repola (2009) for pine and spruce Table 4presents the models forms by speciesIn Table 4 yk is a tree-level AGB (kg) for the kth tree

based on data from Marklund (1987 1988) dtrk = 2 +15 times DBHk is a transformation of DBH (cm) to approx-imate the stump diameter (Laasasenaho 1982) hk is thetree height (m) and β0 β1 and β2 are AGB allometricmodel parameters to be estimatedThe R function gnls() (Pinheiro et al 2016) was used

to fit model parameters accounting for heteroskedasticityTable 5 presents the estimated model parametersIn Table 5 yk is a tree-level predicted AGB (kg) using

the corresponding allometric model for the kth tree V(εk)is individual random error variance modelled through anonlinear power model with two parameters ω2 and γ Figure 5 shows residual scatterplots and Fig 6 presents

scatterplots for AGB versus predicted AGB at tree level

Table 2 Estimated AGB-LiDAR model parameters

Model Model form Model parameter estimates

AGB-LiDAR (G) yi = (α0 + α1hp80i + α2vri)2 + υiα0 α1 α2

253 027 127times10minus3

V(υi) σ 2(δ0 + yδ1Gi )2 σ 2

δ0 δ1

078 150 074

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 7 of 17

Fig 4 AGB-LiDAR model scatterplots at the NFI plot level

Generalized hierarchical model-based estimation withnonlinear modelsHMB inference is based on the model-based inferencephilosophy (Saarela et al 2018) The target population(of map units with AGB as the population attribute inour example) is seen as a random realization shaped by

a superpopulation model involving some auxiliary infor-mation If there is more than one superpopulation modelinvolved and if the estimation procedure employs themodels in a hierarchical order we can speak of HMBinference (Saarela et al 2018) We denote the first super-population model linked to the AGB allometric models

Table 3 Tree-level data descriptive statistics (Marklund 1987 1988)

SpeciesNumber of DBH (cm) Height (m)

trees min mean max sd min mean max sd

Birch 158 020 1222 3680 799 140 1123 2480 532

Norway Spruce 401 040 1633 6340 1071 130 1292 3520 749

Scots Pine 337 050 1933 4890 1023 130 1374 2830 654

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 8 of 17

Table 4 AGB allometric model forms by species

Species Model type Model form Reference

BirchI yk = exp

(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+22)

) + εk(cf Repola 2008 Eq 11 p 612)

II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

Norway I yk = exp(

β0 + β1dtrk

(dtrk+20) + β2 ln hk) + εk

(cf Repola 2009 Eq 17 p 633)Spruce II yk = exp

(

β0 + β1dtrk

(dtrk+20)

) + εk

Scots I yk = exp(

β0 + β1dtrk

(dtrk+12) + β2hk

(hk+20)

) + εk(cf Repola 2009 Eq 9 p 631)

Pine II yk = exp(

β0 + β1dtrk

(dtrk+12)

) + εk

(as will be shown in more detail below) as F

Model F y = f(xβ) + ε ε sim N(0) (1)

The model describes the relationship between AGB (y)and p field-measured variables at plot level ie x =(1 x1 x2 xp) The dataset used to estimate the modelparameters is denoted SAGB predictions for the NFI plots are denoted yFSa ie

a vector of predicted AGB using the F model Here Sadenotes the 504 NFI field plots for which LiDAR met-rics also available The dataset Sa was used to train theAGB-LiDAR model described in ldquoSwedish NFI data andAGB-LiDAR regression modelrdquo section The model is thesecond model in our modelling chain and its generalform is

Model G y = g(zα) + υυ sim N(0) (2)

Here AGB (y) is regressed on q LiDAR metrics z =(1 z1 z2 zq) However instead of the unknown actualAGB (y) the predicted AGB (yFSa ) is used in the analysis toestimate the model parameters in the Gmodel (Eq (2))The estimated parameters αSa are then used to predict

AGB for each map unit using wall-to-wall LiDAR data forthe target population U ie

yGi = g(zi αSa) (3)

where yGi is the predicted AGB using the Gmodel for themap unit i and zi is a (q + 1)-length vector of LiDARmetrics for the map unitUnder the assumption that yGi is model-unbiased the

RMSE of the predicted AGB is (Cassel et al 1977 Eq 34p 94)

RMSE(yGi) =radic

ziCov(αSa)zᵀi + V(υi) (4)

where zi is a (q + 1)-length vector of partial derivativesof g(zi αSa) with respect to αSa and V(υi) is the varianceof the random error υi By replacing Cov(αSa) and V(υi)with the corresponding estimators we obtain the RMSEestimator

RMSE(yGi) =radic

zi Cov(αSa)zᵀi + V(υi) (5)

From Eq (5) it can be seen that the RMSE estimatorconsists of two components the estimated uncertaintydue to the estimated model parameters zi Cov(αSa)zᵀi andthe estimated uncertainty due to the individual randomerrors V(υi) We now focus further on the Cov(αSa)estimator

Covariancematrix of estimatedmodel parameters whenapplying twomodels in hierarchical orderThe covariance matrix of estimated model parametersαSa is a core component model-based inference (eg

Table 5 Estimated parameters in the AGB allometric models

AGB allometric models V(εk)

Species Model type β0 β1 β2 Model form ω2 γ

BirchI ndash351 1074 270

ω2y2γk013 080

II ndash365 1261 ndash 015 083

Norway I ndash171 929 050ω2y2γk

014 079

Spruce II ndash157 1145 ndash 012 084

Scots I ndash371 1063 263ω2y2γk

073 064

Pine II ndash426 1306 ndash 103 065

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 9 of 17

Fig 5 Standardized residuals versus predicted tree-level AGB

McRoberts 2006 Staringhl et al 2011 2016 Saarela et al2015b) Since the αSa estimator is dependent on the AGBpredictions yFSa Cov(αSa) can be expressed conditionallyon yFSa using the law of total covariance (eg Rudary2009)

Cov(αSa) = E[Cov(αSa |yFSa)]+Cov(E[ αSa |yFSa ] )(6)

This is a new way of deriving HMB estimators com-pared to Saarela et al (2016) and Saarela et al (2018) inwhich cases only linear models could be applied The newapproach through conditional covariance simplifies thetheoretical procedure and allows use of nonlinear modelswithin the HMB estimation framework Details are givenin Additional file 2It can be observed that the first term on the right-

side part of (6) can be expressed as the model-based

generalized nonlinear least squares (GNLS) covarianceof estimated model parameters (eg Davidson andMacKinnon 1993) conditionally on yFSa ie

E[ Cov(αSa |yFSa)]= (˜ZᵀSa

minus1Sa

˜ZSa)minus1 (7)

The ˜ZSa is a matrix of partial derivatives of gSa(z αSa)with respect to αSa based on dataset Sa and Sa is thecovariancematrix of individual errors (υSa) for the datasetSa In our study the form of the matrix is diagonal eachelement is estimated as shown in Table 2 for V(υi) Thisfollows since NFI plots are located far from each other andthus the spatial autocorrelation is negligible (hence theoff-diagonal elements of Sa are zero) As in the previousHMB study (Saarela et al 2018) the uncertainty due tothe estimated Sa was not accounted for following com-mon practice in this type of studies (eg Davidson andMacKinnon 1993 Melville et al 2015)

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 10 of 17

Fig 6 Predicted tree-level AGB versus measured tree-level AGB

The second term on the right-side of expression (6) is the core expression in HMB inference and shows howuncertainty due to the predicted AGB using the F model yFSa is propagated through the estimation

Cov(E[ αSa |yFSa ] ) =(˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1 (8)

Thus the covariance of the estimated model parameters αSa can be expressed as

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa Cov(yFSa)

minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(9)

This is a general form of the covariance if the model G is linear then ˜ZSa will become ZSa ie a matrix of predictorvariables with a column of units (eg Davidson and MacKinnon 1993)By replacing Cov(yFSa) with its estimator we obtain an approximately unbiased estimator of Cov(αSa) ie

Cov(αSa) = (˜ZᵀSa

minus1Sa

˜ZSa)minus1

+ (˜ZᵀSa

minus1Sa

˜ZSa)minus1

˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1

(10)

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 11 of 17

where

Cov(yFSa) = ˜XSa Cov(βS)˜XᵀSa (11)

with ˜XSa being a matrix of partial derivatives of fSa(xβS) with respect to βS based on the dataset Sa evidently if themodel F is linear then ˜XSa will be the XSa matrix of predictor variables The covariance matrix of estimated β wasestimated using the GNLS estimator (eg Davidson and MacKinnon 1993)

Cov(βS) = (˜XᵀSminus1

S˜XS)

minus1 (12)

where S is a covariance matrix of individual errors εS over the dataset S The S matrix is a diagonal matrix eachelement is estimated using the model form presented in Table 5 for V(εk) This follows since Marklund (1987 1988) col-lected tree-level data to avoid spatial autocorrelation For similar reasons as in Eq (10) we do not account for uncertaintydue to estimating SDetailed derivations of (6) (9) and (10) are presented in Additional file 2

Aggregation of tree-level AGB predictions to plot levelSo far the plot level model F has been presented in a generic way We now show how it was based on an aggregation oftree-level AGB predictions and how this aggregation affects the uncertaintyIndividual tree-level predictions of AGB were summed to plot-level values [kg] and converted to tonnes per hectare

values (Mgmiddothaminus1) ie

yFi = 10 timesti

sum

k=1

1RefAreaki

ylowastki (13)

where ylowastki is the predicted tree-level AGB value using one of the six allometric models (predictions based on a tree-level

model are denoted with lowast) for the kth tree from the ith NFI plot 10RefAreaki

is a factor converting AGB expressed as kg perplot to Mg per hectare ldquoRefArea is the NFI plot reference area for different values of DBH (Fridman et al 2014) ti isthe number of trees on the ith NFI plot Through Eq (13) the plot-level AGB predictions yFSa = yF1 yF2 yFM overthe Sa dataset were generated (504 NFI plots)Since several models for individual tree AGB were developed and applied in the study there was a need to distinguish

between different models in the formulas For this purpose we introduced a star notation with one () or two () starsattached to different variables parameters and datasets to generically distinguish between them The covariance matrixestimator of estimated β is then

Cov(βSlowast βSlowastlowast)=(˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1˜XᵀSlowastminus1

Slowast Cov(εSlowast εSlowastlowast)minus1Slowastlowast˜XSlowastlowast(˜Xᵀ

Slowastlowastminus1Slowastlowast˜XSlowastlowast)minus1 (14)

If model (lowast) is the same as (lowastlowast) the right-side part in the expression Eq (14) is reduced to (˜XᵀSlowastminus1

Slowast ˜XSlowast)minus1 whichcorresponds to (12) Since the model fitting was performed independently for each model the covariance betweenmodel errors (Cov(εSlowast εSlowastlowast)) from different models is zero which simplifies the estimation procedureThe estimated covariance matrix Cov(yFSa) of AGB predictions using the F model over the Sa dataset of NFI plots is

a quadratic matrix Its dimension is given by the number of NFI field plots (ie 504times504) each entity of the matrix wasestimated as

100 timesti

sum

k=1

tjsum

l=1

1RefAreaki

xlowastki

Cov(βSlowast βSlowastlowast )xlowastlowastᵀlj

1RefArealj

(15)

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 12 of 17

where ti and tj is the number of trees in the ith and the jth NFI plots xki is a (p + 1)-length vector of partial derivativesof the f (xlowast

ki βSlowast) model with respect to βSlowast for the kth tree on the ith NFI plot xlowastlowast

lj is a (p + 1)-length vector of partialderivatives of the f (xlowastlowast

lj βSlowastlowast) function with respect toβSlowastlowast for the lth tree on the jth NFI plot

Summary of applied estimatorsIn summary the target population mean was estimated as

μ = 1N

Nsum

i=1yGi (16)

where N is the number of map unitsThe corresponding RMSE estimator was

RMSE(μ) asymp 1N

radic

radic

radic

radic

Nsum

i=1

Nsum

j=1zi Cov(αSa)zᵀj (17)

In our study the target population U was large ie the target population mean y = 1N

sumNi=1 yi is approximately equal

to the superpopulation mean y asymp μ and thus instead of predicting the random population mean we estimated the fixedsuperpopulation mean (eg Staringhl et al 2016) This implies that MSE(μ) asymp V(μ)The relative RMSE (RelRMSE) was estimated as

RelRMSE(μ) asymp 100 timesRMSE(μ)

μ (18)

And the relative contribution of allometric model ucertanity was estimated as

RelAllometryUncert(μ)=100timessumN

i=1sumN

j=1 zi(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀjsumN

i=1sumN

j=1 zi Cov(αSa)zᵀj

(19)

LiDAR data available for the entire study area were used to predict AGB for each map unit Equation (5) makes itpossible to estimate RMSE for every AGB prediction and thus to create a map of estimated uncertainty Additionallyto these two maps we produced a map of relative RMSE (RelRMSE)

RelRMSE(yGi) = 100 times RMSE(yGi)

yGi (20)

And a map showing the relative contribution of allometric model uncertainty to the total MSE (RelAllometryUncert)ie

RelAllometryUncert(yGi)=100timeszi

(

(˜ZᵀSa

minus1Sa

˜ZSa)minus1˜ZᵀSa

minus1Sa

Cov(yFSa)minus1Sa

˜ZSa(˜ZᵀSa

minus1Sa

˜ZSa)minus1)

zᵀizi Cov(αSa)zᵀi + V(υi)

(21)

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 13 of 17

ResultsThe mean and total AGB in the study area were estimatedthrough HMB estimation thus combining tree level allo-metric modelling aggregation of tree level predictionsto NFI plot level developing AGB-LiDAR models fromthe aggregated predictions and applying the AGB-LiDARmodels to all non-masked map units in the study areaThe resulting estimates and uncertainty estimates are pre-sented in Table 6 In the table the proportion of the totalMSE that is due to the tree-level allometric model uncer-tainty is also shown It can be observed that it accounts fora large portion about 75 of the total MSEAs an add-on to these basic estimates for the study area

HMB also allows for mapping of AGB and the RMSEof the AGB estimates at the level of 18 mtimes18 m mapunits In Fig 7 maps of predicted AGB estimated RMSEestimated RelRMSE and the relative contribution of allo-metric modelling variance to the total MSE are shownfor a small portion of the study area The map prod-ucts for the entire study area are available via the linkSupplementary Material1It can be noted that RelRMSE was mostly large at small

predicted AGB values which is normal due to the denom-inator being a small number Exploring this relationshipfurther in Fig 8 the RMSE is displayed vs predicted AGB(left) the relative RMSE vs predicted AGB (right) andthe proportion of the total MSE due to the allometricmodelling (below) The proportion of the total MSE dueto the allometric modelling reached a minimum whenthe AGB predictions were about 55 tonnes per hectarecorresponding to the estimated population mean

DiscussionIn this study we have demonstrated the correspondencebetween mapping and formal model-based estimation ofaboveground forest biomass for a large study area insouth-central Sweden While a large number of RS-basedstudies focus on biomass mapping (eg Wallerman etal 2010 Santoro et al 2012 Nilsson et al 2017) fewerstudies address statistically sound estimation for largeareas (eg McRoberts 2010 Staringhl et al 2011 Saarela etal 2015a Gregoire et al 2016) and even fewer stud-ies the link between mapping and large-area estimationalthough eg Esteban et al (2019) is an exceptionIn our study we show how the integration of map-

ping and estimation can be further elaborated throughmapping not only the biomass as such but also of itsuncertainty Uncertainties from two modelling steps wereaccounted for ie both the uncertainty due to AGB allo-metric models at tree level and due to the model linkingplot-level biomass with LiDAR metrics the AGB-LiDAR

1It is a lsquotifrsquo file which can be opened in any GIS software or in R StatisticalEnvironment

model We showed that the relative uncertainty (Rel-RMSE) was largest at small biomass values but decreasedrather quickly and remained more or less constant atbiomass predictions for biomass values of 75 Mg middot haminus1and greater Compared to Esteban et al (2019) who stud-ied biomass change in Norway and Spain our estimateduncertainties appear to be larger most likely as a resultof our inclusion of two modelling steps in the uncertaintyestimationAn interesting pattern was observed when the pro-

portion of the total MSE due to the allometric modeluncertainty was displayed versus predicted AGB (Fig 8)The pattern was U-shaped with large contributions fromthe allometric modelling at small and large biomass pre-dictions but with rather limited contribution around theaverage predicted value The likely reason for this patternis that regression models tend to be precise around meanobserved values but less precise towards the extremeswith regard to the observations (Davidson and MacKin-non 1993) Thus this pattern is likely due to the pattern oftree-level data from the tree biomass dataset in Marklund(1987 1988) combined with the AGB-LiDAR modelAnother interesting pattern is visible in the scatterplots

of Fig 8 The point clouds appear to be forked witha large number of observations at the lower and upperextremes A likely reason for this is the combined effectof species-specific models and the two allometric modeltypes developed for each species Obviously the AGB allo-metric models using only DBH as a predictor variable willresult in larger prediction uncertainty than models basedon both tree height and DBHThe development of a theoretical framework for incor-

porating nonlinear models into HMB estimation wasan important part of the study which required a newapproach for developing HMB estimators compared toprevious studies Saarela et al (2016 2018) With the pro-posed decomposition of the covariance matrix estimatorany nonlinear models could be applied in HMB in casethe objective is to estimate the variance of an estimator ofthe expected value of the superpopulation mean or totalHowever in case the objective is to estimate the MSE ofthe actual superpopulation mean or total or the MSE forpredictions at the level of map units only nonlinear mod-els with additive error terms can be applied Since ourobjective was to construct uncertainty maps at the level ofindividual map units a nonlinear model with an additiveerror term was selectedThe new framework makes it possible to trace uncer-

tainties due to several modelling steps In our casetwo modeling steps were included However the covari-ance decomposition approach makes it straightforwardto derive variance estimators also for cases with threeor more modelling steps Thus the new HMB frame-work would also allow for targeted improvement of

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 14 of 17

Table 6 Estimated population mean and total and corresponding uncertainties and the allometry uncertainty contribution to overallMSE

Estimated populationEstimate RMSE RelRMSE

Relative contribution of

parameter allometric model uncertainty

Population mean 5502 Mgmiddothaminus1 411 Mgmiddothaminus1 749 7491

Population total 2151times106 Mg 161times106 Mg 749 7491

AGB maps since it can be evaluated what mod-elling step(s) would be most beneficial to improvefrom the point of view of obtaining small overalluncertaintyMaps of stand characteristics such as biomass and vol-

ume constructed using LiDAR data can be expected tobe increasingly demanded for planning forestry opera-tions in the future Our results indicate that the type of

models involved provides accurate results in terms of Rel-RMSE in old and middle-aged stands ie stands withintermediate and large AGB values However in youngforests (small biomass values) the uncertainties were largesuggesting that other types of data should be applied Infuture applications it might be possible to bring addi-tional modelling details into the HMB framework makingit possible to distinguish uncertainties not only between

Fig 7Map products

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 15 of 17

Fig 8 Predicted AGB versus RMSE relative RMSE and the allometry uncertainty contribution the black solid vertical line marks the estimatedpopulation mean (55 Mg middot haminus1)

different estimated biomass levels but also between dif-ferent species and site conditions This would require newtypes of data such as tree species data derived from mul-tispectral LiDAR (Axelsson et al 2018) The uncertaintymaps would then provide additional information to forestplannersAn interesting next step regarding uncertainty mapping

would be to move from map units to stands This wouldrequire aggregation of map units through segmentation(eg Olofsson and Holmgren 2014) Stand level uncer-tainty maps would also require information about the spa-tial autocorrelation of model errors within stands whichwould be a challenge However several approaches for thistype of small area estimation are available (eg Breiden-bach and Astrup 2012 Magnussen et al 2014) and shouldbe evaluated for application in the HMB frameworkA final remark concerns the difference between hierar-

chical model-based inference and similar techniques suchas conventional model-based inference (eg McRoberts2006) hybrid inference (eg Staringhl et al 2011) and design-based inference through model-assisted estimation (egGregoire et al 2011) A large number of studies per-formed during the last decades focus on assessment ofbiomass and carbon and related uncertainties applying

techniques that may appear similar to the techniqueapplied in this study Examples include Petersson et al(2017) McRoberts and Westfall (2016) and McRoberts etal (2016) In the latter study Monte-Carlo simulation isapplied to assess the overall uncertainty in growing stockvolume estimation schemes involving several modellingsteps

ConclusionsIn this study a new approach to developing HMB esti-mators was derived making it possible to apply non-linear models as well as combinations of linear andnonlinear models in the HMB estimation frameworkThe estimators were applied to a study area in south-central Sweden and it was shown that the relativeuncertainties in the biomass predictions were largestwhen the predicted biomass was small Further it wasdemonstrated how biomass mapping uncertainty map-ping and estimation of the mean biomass across alarge area could be combined through HMB inferenceUncertainty maps of the kind presented in this studyallow for judgments about whether or not the forestattribute map is accurate enough for the intended decisionmaking

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 16 of 17

Supplementary informationSupplementary information accompanies this paper athttpsdoiorg101186s40663-020-00245-0

Additional file 1 Supplementary material

Additional file 2 Appendix

AcknowledgementsThe authors are thankful to the two anonymous Reviewers whose commentshelped to improve the clarity of the article

Authorsrsquo contributionsConceptualization Svetlana SaarelaData curation Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah MatsNilsson and Jonas FridmanFunding acquisition Svetlana SaarelaInvestigation Svetlana Saarela Andreacute Waumlstlund Emma Holmstroumlm AlexAppiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and Goumlran StaringhlMethodology Svetlana Saarela Soumlren Holm and Goumlran StaringhlProject administration Svetlana SaarelaSoftware Svetlana Saarela Andreacute Waumlstlund Alex Appiah Mensah and MatsNilssonWriting ndash original draft Svetlana SaarelaWriting ndash review amp editing Svetlana Saarela Andreacute Waumlstlund Emma HolmstroumlmAlex Appiah Mensah Soumlren Holm Mats Nilsson Jonas Fridman and GoumlranStaringhl The author(s) read and approved the final manuscript

FundingFunding was provided by the Swedish NFI Development Foundation and theSwedish Kempe Foundation (SMK-1847) The computations were performedon resources provided by the Swedish National Infrastructure for Computing(SNIC) at the High Performance Computing Center North (HPC2N)

Availability of data andmaterialsThe data are available upon a reasonable request to the Authors

Ethics approval and consent to participateNot applicable

Consent for publicationNot applicable

Competing interestsThe authors declare no conflict of interest

Received 10 October 2019 Accepted 24 April 2020

ReferencesAndersen H-E McGaughey RJ Reutebuch SE (2005) Estimating forest canopy

fuel parameters using LIDAR data Remote Sens Environ 94(4)441ndash449Axelsson A Lindberg E Olsson H (2018) Exploring multispectral ALS data for

tree species classification Remote Sens 10(2)183Bellassen V Luyssaert S (2014) Carbon sequestration Managing forests in

uncertain times Nat News 506(7487)153Breidenbach J Astrup R (2012) Small area estimation of forest attributes in the

Norwegian National Forest Inventory Eur J For Res 131(4)1255ndash1267Cassel C-M Saumlrndal CE Wretman JH (1977) Foundations of inference in survey

sampling Wiley httpsdoiorg1023072287794Davidson R MacKinnon JG (1993) Estimation and inference in econometrics

Oxford University PressDubayah R Goetz S Blair JB Fatoyinbo T Hansen M Healey SP Hofton M

Hurtt G Kellner J Luthcke S Swatantran A (2014) The global ecosystemdynamics investigation AGU Fall Meeting Abstracts

Esteban J McRoberts RE Fernaacutendez-Landa A Tomeacute JL Naeligsset E (2019)Estimating forest volume and biomass and their changes using randomforests and remotely sensed data Remote Sens 11(16)1944

Forest Europe (2015) State of Europersquos forests 2015 Status and trends insustainable forest management in Europe

Franco-Lopez H Ek AR Bauer ME (2001) Estimation and mapping of foreststand density volume and cover type using the k-nearest neighborsmethod Remote Sens Environ 77(3)251ndash274

Fridman J Holm S Nilsson M Nilsson P Ringvall A Staringhl G (2014) AdaptingNational Forest Inventories to changing requirements mdash the case of theSwedish National Forest Inventory at the turn of the 20th century SilvFenn 481ndash29

Gobakken T Naeligsset E Nelson R Bollandsarings OM Gregoire TG Staringhl G Holm SOslashrka HO Astrup R (2012) Estimating biomass in Hedmark County Norwayusing national forest inventory field plots and airborne laser scanningRemote Sens Environ 123(0)443ndash456

Grafstroumlm A Schnell S Saarela S Hubbell S Condit R (2017a) The continuouspopulation approach to forest inventories and use of information in thedesign Environmetrics 28(8) httpsdoiorg101002env2480

Grafstroumlm A Zhao X Nylander M Petersson H (2017b) A new samplingstrategy for forest inventories applied to the temporary clusters of theSwedish national forest inventory Can J For Res 47(9)1161ndash1167

Gregoire TG (1998) Design-based and model-based inference in surveysampling appreciating the difference Can J For Res 28(10)1429ndash1447

Gregoire TG Naeligsset E McRoberts RE Staringhl G Andersen H-E Gobakken T EneNelson R (2016) Statistical rigor in LiDAR-assisted estimation ofaboveground forest biomass Remote Sens Environ 17398ndash108

Gregoire TG Staringhl G Naeligsset E Gobakken T Nelson R Holm S (2011)Model-assisted estimation of biomass in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)83ndash95

Gregoire TG Valentine HT (2008) Sampling strategies for natural resources andthe environment Vol 1 CRC Press httpsdoiorg1012019780203498880

Haakana H Heikkinen J Katila M Kangas A (2019) Efficiency of post-stratificationfor a large-scale forest inventorymdashcase Finnish NFI Ann For Sci 76(1)9

Hansen MC DeFries RS Townshend JR Carroll M DiMiceli C Sohlberg RA(2003) Global percent tree cover at a spatial resolution of 500 meters Firstresults of the MODIS vegetation continuous fields algorithm Earth Interac7(10)1ndash15

Hudak AT Crookston NL Evans JS Hall DE Falkowski MJ (2008) Nearestneighbor imputation of species-level plot-scale forest structure attributesfrom LiDAR data Remote Sens Environ 112(5)2232ndash2245

Katila M Tomppo E (2002) Stratification by ancillary data in multisource forestinventories employing k-nearest-neighbour estimation Can J For Res32(9)1548ndash1561

Ku HH (1966) Notes on the use of propagation of error formulas J Res Natl BurStand 70(4)263ndash273

Laasasenaho J (1982) Taper curve and volume functions for pine spruce andbirch [Pinus sylvestris Picea abies Betula pendula Betula pubescens] TheFinnish Forest Research Institute

Lantmaumlteriet (2019) Lantmauml httpswwwlantmaterietsesv Accessed 15 Feb2019

Lindberg E Olofsson K Holmgren J Olsson H (2012) Estimation of 3Dvegetation structure from waveform and discrete return airborne laserscanning data Remote Sens Environ 118151ndash161

Magnussen S (2015) Arguments for a model-dependent inference For Int JFor Res 88(3)317ndash325

Magnussen S Mandallaz D Breidenbach J Lanz A Ginzler C (2014) Nationalforest inventories in the service of small area estimation of stem volumeCan J For Res 44(9)1079ndash1090

Marklund LG (1987) Biomass functions for Norway spruce (Picea abies (L)Karst) in Sweden Vol 43 of Report Swedish University of AgriculturalSciences Department of Forest Survey

Marklund LG (1988) Biomassafunktioner foumlr tall gran och bjoumlrk i Sverige VolRapport of 45 Sveriges lantbruksuniversitet Institutionen foumlr skogstaxering

McGaughey R (2012) FUSIONLDV Software for LIDAR Data Analysis andVisualization Version 301 httpforsyscfrwashingtonedufusionfusionlatesthtml Accessed 24 Aug 2017

McRoberts RE (2006) A model-based approach to estimating forest areaRemote Sens Environ 103(1)56ndash66

McRoberts RE (2010) Probability- and model-based approaches to inferencefor proportion forest using satellite imagery as ancillary data Remote SensEnviron 114(5)1017ndash1025

McRoberts RE Chen Q Domke GM Staringhl G Saarela S Westfall JA (2016) Hybridestimators for mean aboveground carbon per unit area Forest Ecol Manag37844ndash56

Saarela et al Forest Ecosystems (2020) 743 Page 17 of 17

McRoberts RE Naeligsset E Gobakken T Chirici G Condeacutes S Hou Z Saarela SChen Q Staringhl G Walters BF (2018) Assessing components of themodel-based mean square error estimator for remote sensing assistedforest applications Can J For Res 48(6)642ndash649

McRoberts RE Tomppo E Schadauer K Vidal C Staringhl G Chirici G Lanz ACienciala E Winter S Smith WB (2009) Harmonizing national forestinventories J For 107(4)179ndash187

McRoberts RE Wendt DG Nelson MD Hansen MH (2002) Using a land coverclassification based on satellite imagery to improve the precision of forestinventory area estimates Remote Sens Environ 81(1)36ndash44

McRoberts RE Westfall JA (2016) Propagating uncertainty through individualtree volume model predictions to large-area volume estimates Ann ForSci 73(3)625ndash633

Melville G Welsh A Stone C (2015) Improving the efficiency and precision oftree counts in pine plantations using airborne LiDAR data andflexible-radius plots model-based and design-based approaches J AgricBiol Environ Stat 20(2)229ndash257

Naeligsset E (2002) Predicting forest stand characteristics with airborne scanninglaser using a practical two-stage procedure and field data Remote SensEnviron 80(1)88ndash99

Nelson R Krabill W Tonelli J (1988) Estimating forest biomass and volumeusing airborne laser data Remote Sens Environ 24(2)247ndash267

Nilsson M Folving S Kennedy P Puumalainen J Chirici G Corona P MarchettiM Olsson H Ricotta C Ringvall A Staringhl G Tomppo E (2003) Combiningremote sensing and field data for deriving unbiased estimates of forestparameters over large regions In Corona P Kohl M Marchetti M (eds)Advances in forest inventory for sustainable forest management andbiodiversity monitoring Springer pp 19ndash32

Nilsson M Nordkvist K Jonzeacuten J Lindgren N Axensten P Wallerman J EgberthM Larsson S Nilsson L Eriksson J Olsson H (2017) A nationwide forestattribute map of Sweden predicted using airborne laser scanning data andfield data from the National Forest Inventory Remote Sens Environ194447ndash454

Olofsson K Holmgren J (2014) Forest stand delineation from lidar point-cloudsusing local maxima of the crown height model and region merging of thecorresponding Voronoi cells Remote Sens Lett 5(3)268ndash276

Patterson PL Healey SP Staringhl G Saarela S Holm S Andersen H-E Dubayah RODuncanson L Hancock S Armston J Kellner JR Cohen WB Yang Z (2019)Statistical properties of hybrid estimators proposed for GEDImdashNASArsquosGlobal Ecosystem Dynamics Investigation Environ Res Lett 14(6)065007

Petersson H Breidenbach J Ellison D Holm S Muszta A Lundblad M Staringhl GR(2017) Assessing uncertainty sample size trade-offs in the developmentand application of carbon stock models For Sci 63(4)402ndash412

Pinheiro J Bates D DebRoy S Sarkar D (2016) R Core Team nlme Linear andNonlinear Mixed Effects Models R package version 31-127 httpCRANR-projectorgpackage=nlme

Qi W Dubayah RO (2016) Combining Tandem-X InSAR and simulated GEDIlidar observations for forest structure mapping Remote Sens Environ187253ndash266

Qi W Saarela S Armston J Staringhl G Dubayah RO (2019) Forest biomassestimation over three distinct forest types using TanDEM-X InSAR data andsimulated GEDI lidar data Remote Sens Environ 232111283

Repola J (2008) Biomass equations for birch in finland Silv Fenn 42(4)605ndash624Repola J (2009) Biomass equations for scots pine and norway spruce in finland

Silv Fenn 43(4)625ndash647Rudary MR (2009) On Predictive Linear Gaussian Models PhD thesis Ann

Arbor MI USASaarela S Grafstroumlm A Staringhl G Kangas A Holopainen M Tuominen S

Nordkvist K Hyyppauml J (2015a) Model-assisted estimation of growing stockvolume using different combinations of LiDAR and Landsat data asauxiliary information Remote Sens Environ 158431ndash440

Saarela S Holm S Grafstroumlm A Schnell S Naeligsset E Gregoire TG Nelson RFStaringhl G (2016) Hierarchical model-based inference for forest inventoryutilizing three sources of information Ann For Sci 73(4)895ndash910

Saarela S Holm S Healey SP Andersen H-E Petersson H Prentius W PattersonPL Naeligsset E Gregoire TG Staringhl G (2018) Generalized Hierarchicalmodel-based estimation for aboveground biomass assessment using GEDIand Landsat data Remote Sens 10(11)1832

Saarela S Schnell S Grafstroumlm A Tuominen S Nordkvist K Hyyppauml J Kangas AStaringhl G (2015b) Effects of sample size and model form on the accuracy ofmodel-based estimators of growing stock volume Can J For Res451524ndash1534

Saatchi SS Harris NL Brown S Lefsky M Mitchard ET Salas W Zutta BRBuermann W Lewis SL Hagen S Petrova S White L Silman M Morel A(2011) Benchmark map of forest carbon stocks in tropical regions acrossthree continents PNAS 108(24)9899ndash9904

Santoro M Pantze A Fransson JE Dahlgren J Persson A (2012) Nation-wideclear-cut mapping in Sweden using ALOS PALSAR strip images RemoteSens 4(6)1693ndash1715

Saumlrndal CE Thomsen I Hoem JM Lindley DV Barndorff-Nielsen O Dalenius T(1978) Design-based and model-based inference in survey sampling [withdiscussion and reply] Scand J Stat27ndash52

Staringhl G Holm S Gregoire TG Gobakken T Naeligsset E Nelson R (2011)Model-based inference for biomass estimation in a LiDAR sample survey inHedmark County Norway Can J For Res 41(1)96ndash107

Staringhl G Saarela S Schnell S Holm S Breidenbach J Healey SP Patterson PLMagnussen S Naeligsset E McRoberts RE Gregoire TG (2016) Use of modelsin large-area forest surveys comparing model-assisted model-based andhybrid estimation Forest Ecosyst 3(5)1ndash11

Tomppo E Gschwantner T Lawrence M McRoberts R Gabler K Schadauer KVidal C Lanz A Staringhl G Cienciala E Chirici G Winter S Bastrup-Birk ATomter S Kandler G McCormick M (2010) National Forest InventoriesPathways for Common Reporting Springer

Tomppo E Haakana M Katila M Peraumlsaari J (2008a) Multi-source NationalForest Inventory Methods and Applications Vol 18 Springer New York

Tomppo E Olsson H Staringhl G Nilsson M Hagner O Katila M (2008b) Combiningnational forest inventory field plots and remote sensing data for forestdatabases Remote Sens Environ 112(5)1982ndash1999

Wallerman J Fransson JE Bohlin J Reese H Olsson H (2010) Forest mappingusing 3D data from SPOT-5 HRS and ZI DMC 2010 IEEE InternationalGeoscience and Remote Sensing Symposium IEEE

Wulder M White J Fournier R Luther J Magnussen S (2008) Spatially explicitlarge area biomass estimation three approaches using forest inventoryand remotely sensed imagery in a GIS Sensors 8(1)529ndash560

Zald HS Wulder MA White JC Hilker T Hermosilla T Hobart GW Coops NC(2016) Integrating Landsat pixel composites and change metrics with lidarplots to predictively map forest structure and aboveground biomass inSaskatchewan Canada Remote Sens Environ 176188ndash201