Chapter 8

TREE AND STAND BIOMASS

1 INTRODUCTION

Until recently, forest mensuration has emphasized the estimation of the totaland utilizable volume rather than weight, partly because timber is usually soldon a volume basis and partly because the volume of standing trees can be esti-mated more easily than their weight. In many countries and regions there is anincreasing need to express the productivity of forests in terms of weight, moreparticularly in those plantation forests which are managed for the productionof pulpwood and mining timber or when by-products, for example bark forthe production of tannins, are involved. A similar situation arises when treesare planted or natural forests are managed to produce wood for energy, sincemass rather than volume is a yardstick to quantify the production of wood forenergy. Other reasons for the increased interest in forest biomass, initiated inthe early 1960s, was the necessity to measure biological productivity in termsof dry weight of the organic matter, and the oil crisis, which induced a greateremphasis on the utilization of wood as a renewable natural resource.

Because of these developments, the need arose to develop sampling meth-ods and to construct functions and tables which give the estimated oven-drybiomass of trees as a function either of dbh, or dbh as well as height. Wheneverthe tree is converted into cellulose products, the oven-dry weight of its mer-chantable part expresses its value more adequately then its green weight. Forthis reason biomass tables were constructed, which give the average oven-dryinstead of green weight as a function of one or more than one tree characteristic.

The first extensive sampling studies to estimate the weight of the above-ground tree components were carried out much earlier, primarily because of thenecessity to measure the biological productivity of tree species. These studies,undertaken in Pinus strobus in Switzerland (Burger 1929), were followed bysimilar studies in Larix decidua (Burger 1945) and Picea abies (Burger 1953).Concurrently, ecologists and physiologists became increasingly interested in

183

184 Tree and Stand Biomass

this research direction and made a contribution towards the development ofmore efficient sampling methods to estimate the quantity of foliage, eitherexpressed in terms of their oven-dry mass or in terms of the leaf surface areaper hectare. The latter was converted into the leaf-area index, defined as theratio of leaf-surface area over ground area. Studies conducted by Kittredge(1944) and Ovington (1957) emphasized their importance within the frame-work of ecological research. Studies in Danish beech forests, conducted byMøller (1946), which were preceded by those carried out by Boysen–Jensen(1932), were primarily undertaken to establish the relationship between degree,as well as kind of thinning and yield in terms of mass instead of volume. Theyhad a considerable impact on thinning research in Europe and the USA. Thestudies were also conducive for the initiation of sampling studies for the con-struction of biomass tables and functions.

An evaluation of the extensive world literature about this subject shows thatthe majority of the sampling studies estimate the oven-dry weight of each bio-mass component separately. In many of these studies, root biomass is ignored,probably because of the prohibitively high cost of estimating root biomass suf-ficiently accurately. The estimation of biomass components in relation to treesize requires that they are oven-dried separately. The drying temperature is nor-mally around 70◦ C, but in Pinus radiata, Forrest (1968) recorded a 2% weightloss by increasing the temperature from 70◦ C to 105◦ C. In Picea mariana aweight loss of 3% was observed by increasing the drying temperature from65◦ C to 103◦ C (Barney et al. 1978).

2 BIOMASS COMPONENTS

The total aboveground green weight of young trees is conveniently measuredby felling sample trees and weighing the entire tree. This method, which wasapplied by Young et al. (1976), ensures that no sampling errors are involved todetermine the green weight of the entire tree. This is feasible for young treesbut prohibitively expensive for mature trees. Methods were therefore developedto estimate biomass by sampling. The following notes serve as a general guide-line for sampling. In practical situations it is usually necessary to modify theproposed procedure.

2.1 Branches

Two-stage sampling is an efficient method to estimate the branchwood weightof the single tree. The diameter at the base of the branch (preferably at a

Biomass Components 185

fixed distance of 4 or 5 cm from the main stem) is measured on all branches.A subsample of branches is drawn at random to estimate either the green or theoven-dry branchwood weight of the single branch. The observed weights aresubsequently regressed on branch diameter or branch basal area.

In biomass studies in Pinus radiata, branch length was a statistically sig-nificant additional predictor variable but the moderate increase in R2 didnot justify its inclusion into the prediction equation (van Laar 1973). Otherresearchers included branch position to improve the prediction model.

The estimated weights per tree are subsequently regressed on dbh or on dbhas well as height to obtain a regression estimator for this biomass component forthe entire stand. A direct measurement of the oven-dry weight eliminates thenecessity to convert the green weight into the oven-dry weight. For practicalreasons it is sometimes preferred to measure the branchwood weight as greenweight. In that case, a subsample of branches of known green weight is drawnand oven-dried to estimate the ratio oven-dry over green weight. In P. radiatathis ratio was about 0.45 for live and 0.80 for dead branches (Satoo 1982).In consequence, the conversion of green into oven-dry weight should be carriedout for live and dead branches separately.

2.2 Foliage

In young trees all leaves or needles are removed from the tree, dried andweighed, but sampling is required when mature trees are involved. The needlesor leaves, for example, from a 25% random sample of branches, are removedand their green or oven-dry weight is determined and recorded for each branchseparately. For reasons of cost-efficiency the same branches will be used formeasuring the branch-wood and foliage weight. The foliage weight may bemeasured as green weight, in which case subsampling is required to convertgreen weight into oven-dry weight. In regions with hot dry summers and wetwinters, the ratio green weight to oven-dry weight shows a seasonal trend.Ratio estimates obtained during summer cannot be used when sampling con-tinues during winter. The regression model used to estimate the branch-woodweight from branch diameter can also be used to estimate leaf weight.

2.3 Stemwood weight

Sampling usually proceeds in two stages. In stage 1 the volume of the felledsample tree is determined by measuring the diameter at the midpoint of 1 msections, with Smalian’s formula being used to obtain the volume of each stemsection. This produces an estimate of the stem volume, although it may be

186 Tree and Stand Biomass

negatively biased if too few stem sections are used. In stage 2 wood discs of apredetermined thickness are extracted, preferably at the midpoint or else at thethin end of each stem section. Since the disc volume is proportional to the vol-ume of the stem section, this method implies self-weighting and their volume,as well as green weight is determined. A subsample is drawn for drying and toconvert green weight into oven-dry weight. The use of a single-conversion fac-tor erroneously assumes that the dry weight over green weight ratio is unrelatedto the position within the stem. This simplification could produce biased esti-mates, since the ratio tends to decrease with increasing height above the baseof the tree. In many regions this ratio might also reveal a seasonal trend.

2.4 Bark weight

The previously described sampling procedure is repeated to estimate the oven-dry bark weight. It can be combined with the measurement of stemwood weightby measuring the bark thickness of each stem disc from four directions underan angle of 90◦, or by removing the bark from each disc to measure its weight.Alternatively bark-thickness functions, such as those constructed by Deetlefs(1957), can be used to estimate the bark volume which is subsequently multi-plied by a single oven-dry/green-weight ratio. This method will be applied ifprediction errors associated with bark thickness equations are negligible.

2.5 Root weight

The estimation of the weight of roots requires a complete excavation of the rootsystem, which in practice is virtually impossible. Some kind of subsamplingmay be necessary to estimate the weight of the root sections which remain inthe soil. The green weight of all roots together is determined and a subsampleis drawn to convert green weight into dry weight.

The above sampling methods serve as general guidelines but are modifiedin specific situations. In a biomass study in Pinus radiata, carried out by Forrest(1969), the breast height diameter and height of all trees within a given sampleplot were measured. After grouping the trees in five-diameter classes, two treeswere selected at random from each class. The oven-dry weight of each bole wasdetermined in its entirety. Discs were cut and their bark removed to estimate theratio bark to total weight. Foliage and branch weight were obtained by dryingand weighing all branches and needles. This was done for each age stratumseparately. Spank (1982) conducted sampling studies to estimate the crownand needle biomass of Pinus sylvestris. Samples were taken to estimate theneedle biomass from the biomass of the branchlets. The crown and branchlet

Biomass Components 187

biomass per tree were regressed on tree diameter and tree height, with a log-arithmic transformation of the dependent and predictor variables. Ranasingheet al. (1991) conducted biomass studies on two sites in Eucalyptus camaldu-lensis plantations of different ages. The mean tree within each sample plot wasfelled to determine the green weight of the leaves, large branches, bole, bark,and roots. Subsamples were dried at 70◦ during 48 h to estimate the conversionfactor from green to oven-dry weight.

2.6 Pooling of data

The construction of general biomass tables and equations requires a large sam-ple representing the entire range of tree sizes, ages, sites, and silvicultural treat-ments. It is feasible to rationalize the previously described sampling proceduresby pooling some observations obtained from single trees. In order to estimatethe ratio oven-dry/green weight of branchwood, foliage, stem wood, and bark,for example, it can be expected that the between-trees and within-trees vari-ability of these ratios, within a given stand, might be of the same order of mag-nitude. Pooling the ratios observed for single trees seems justified but mightproduce slightly biased estimates of the biomass of trees in different stands, forexample because of the effect of site. When regressing tree biomass estimateson dbh and other tree characteristics, the assumption of uncorrelated residualswill then be violated, although not necessarily seriously.

2.7 Randomized branch and importance sampling

In order to eliminate the necessity to determine the weight of the entire tree,Valentine et al. (1984) compared randomized branch sampling (RBS) with theclosely related importance sampling (IS), which is based on an earlier concept,introduced by Jessen (1955), to obtain estimates for the aboveground biomassof individual trees. RBS selects a path, which is defined as a series of connectedbranch segments or internodes. Within the context of RBS a branch is definedas the entire stem system which develops either from a terminal or from a lateralbud. A segment is defined as a part of a branch between two successive intern-odes. In RBS no distinction is made between main stem and branch. The firstsegment extends between the base of the tree and the first node. At this pointthe diameter and length of the two branches (the diameter of the branch and thatof the main stem) are measured. A selection probability which is proportionalto d2l is calculated. with l being the length of the segment between the nodeand the tip of the branch and main stem respectively. If the two selection prob-abilities are 0.30 and 0.70 respectively and a random number is drawn which

188 Tree and Stand Biomass

is greater than 0.70, the second branch is selected. Its associated probability ofselection is equal to 0.70. When three branches (segments) occur at the firstnode, with selection probabilities of 0.20, 0.35, and 0.45 and a random num-ber equal of 0.60 is drawn, the second branch will be selected with a selectionprobability of 0.35, since 0.60 > (0.20+0.35). After having selected the third,and possibly last branch, for example with a conditional probability of 0.45, thelatter is weighed and the unconditional probability of selection is obtained bymultiplying conditional probabilities 1∗0.70∗0.35∗0.45 = 0.1103. The weightis divided by 1/0.1103 to obtain the estimated weight of the entire tree.

Importance sampling is a related method which eliminates the necessity todetermine the weight of heavy segments of a path. Weighing is limited to asingle stem disc within each path. The procedure starts with the measurementof the diameter at various points along the bole. These are used to estimatethe timber volume of the path. A random number is drawn from a uniformdistribution which extends between 0 and 1. In order to determine the positionof sampling it is multiplied by the estimated volume. The weight is determinedand converted into weight per unit thickness of the disc. This value is multipliedby an expansion factor to obtain the estimated total weight.

3 TREE-LEVEL REGRESSION MODELS

Several models have been proposed and used to estimate the aboveground bio-mass components and the total tree biomass from tree characteristics. Manymodels are based on the assumed allometric relation between biomass (y) andthe tree or branch characteristic (x)

ln y = b0 + b1 ln x

Allometry deals with the relationship between the relationship between thegrowth rates of two organs of an individual. The assumption that the relativegrowth rate of x is a fixed proportion of that of y can be stated as follows:

1y

· dydt

= k · 1x

dxdt

where k represents a proportionality coefficient. The terms of the equation canbe rearranged as follows:

dydt

= yx

· k · dxdt

which represents a differential equation. It has been shown (Batscheler 1975)that the solution of this equation is:

y = c.xk

Tree-Level Regression Models 189

or

ln y = ln c + k ln x

Although the allometric model produces a satisfactory fit, the resultant esti-mates are biased. Several formulae have been proposed to correct for bias.Baskerville (1972) approximated the regression estimate corrected for bias asfollows:

yadj. = eln bo+b1 ln x+M Serror/2

where M Serror = mean square for error, obtained from the regression analysiswith the transformed variables. Using Baskerville’s correction for bias, Wiantet al. (1979) estimated bias as follows:

Bias = eM Serror/2−1

eM Serror/2

A more complex correction formula was presented by Finney (1941). Yandle(1981) compared Finney’s estimator with Baskerville’s function. For small sam-ples and a large-error mean square, Baskerville’s function performed betterthan Finney’s estimator. This was confirmed in similar studies carried out byLee (1982). Snowdon (1991) proposed to multiply the estimate obtained fromthe log–log regression of biomass on dbh by a correction factor C, obtainedfrom the observed and estimated biomass of the n sampling units. In a latersimulation study Snowdon (1992) compared this method with the ratio of meanmethods, based on basal area as auxiliary variable. This was done for simplerandom sampling, for sampling with selection probability proportional to size(PPS) and for sampling with selection PPS for the first sampling unit beingdrawn and at random for the others. The estimates obtained by combining PPSsampling with the simple ratio method of adjustment produced highly biasedresults.

One disadvantage of the log–log model is that the sum of the estimatedbiomass components differs from the regression estimate obtained from theequation with total biomass as dependent variable. A second problem is theestimation of confidence intervals. Meyer (1944) suggested to adjust the vari-ance formula

s2y = ey2+M Serror/2

and to use the corrected variance for obtaining confidence intervals for the esti-mated biomass.

Campbell et al. (1985) examined the allometric relationship between treebiomass and breast height diameter (model 1) and that between tree bio-mass and the combined variable d2h (model 2). An analysis of covariance

190 Tree and Stand Biomass

for model 1, with region as additional variable and dbh as covariate, did notindicate significant differences amongst intercepts but those between slopeswere significant, whereas the analysis based on model 2 disclosed significantdifferences amongst slopes. It was suggested that the d/h ratio might be a use-ful additional variable for predicting the biomass of trees growing in differentregions, but the study did not disclose which of the two models performedbetter.

Ruark et al. (1987) introduced the concept VAR (variable allometric rela-tionship) for estimating the tree biomass from a tree-size characteristic x. It washypothesized that the allometric ratio b1 of the equation

y = b0xb1

is a function of x. The proposed equation

y = b0xb1eb2x

is linearizable and produces the equation

ln y = c0 + c1 ln x + c2x

The model performed satisfactorily if the relationship between x and y differsfor different ages of the organism. For some biomass components it produced asignificantly better fit than the constant allometric model. Geron et al. (1988)tested the usefulness of Ruark’s variable allometric ratios for predicting thefoliar biomass of different tree species. In Populus tremuloides, it compensatedthe negative bias associated with the constant allometric ratio model, whichoverpredicted foliage biomass in the lower and upper ends of the diameter dis-tribution. In a study with a similar objective, Crow (1980) compared the per-formance of the allometric model ln y = b0 + b1 ln x with others, for example:

w = b0db1

w = b0db1 hb2

combined with different weighting functions. The weighted nonlinear equationsand the allometric model performed equally well. Clark (1990) predicted bio-mass components of planted southern pines from models which used either d2

or d2h, or d2h4 (h4 = height to 4 in. top) or sawlog merchantable heightas predictor and log(biomass) as target variable. Schlaegel (1982) estimatedbiomass components of Acer negundo from fitted regression equations withd2h as predictor variable. In order to ensure homoscedasticity, the data wereweighted inversely proportional to d2h. Payandeh (1981) discussed regressionmodels for biomass prediction equations

Tree-Level Regression Models 191

W = b0 + b1d (1)

W = b0db1 (2)

W = eb0+b1 ln d (3)

W = b0 + b1d2 + b2h + b3d2h (4)

W = b0db1 hb2 (5)

The performance of the models was based on an index of fit, which is identi-cal with R2 in models with no constraints on the parameter b0. The models 1and 2 performed equally well and were superior to the others. In biomass stud-ies, based on a sample of size 38, which were carried out in the beech forestsof Romania, Decei (1981) used a second-degree equation to estimate variousaboveground biomass components as well as root biomass from breast heightdiameter.

Landis et al. (1975) regressed bole, stem bark, branchwood, and foliagebiomass, as well as the total aboveground biomass on breast height diameter.The equation W = b0 + b1d2 was preferred before those with logtransformed-dependent and predictor variables, because of nonnormality of the distributionof residuals, generated by this transformation. Mitchell et al. (1981) developedregression equations with stem, branch, and foliage dry weight of conifers astarget variables and dbh as predictor, after weighting the sampling observa-tions proportional to the inverse of dbh. Apparently there was no evidence ofa nonlinear relationship between dbh and weight. In South African Eucalyptusplantations, Schonau et al. (1981) regressed the biomass components on thesquared dbh.

Alemdag et al. (1981) tested the performance of four equations for estimat-ing stemwood, stem bark, live branches, twigs, leaves, and the total biomassof some hardwood species. One model used d2, h and d2h as regressors, andcontained an intercept, a second equation used the same predictor variablesbut with zero-intercept, a third function was based on d2h as predictor (withintercept), and the last one represented the concurrent zero-intercept model.The first and last model performed equally well in terms of R2.

Madgwick (1984) investigated the amount of bias when estimating theaboveground biomass per unit area with the basal area ratio method, theunweighted regression of biomass on squared diameter approach and the log–log regression of biomass on dbh. Five different adjustment factors to correctestimates, based on the allometric model tended to produce biased estimates.The best results were obtained with the basal area ratio method, which is a

192 Tree and Stand Biomass

ratio of means estimator, requiring the multiplication of the observed biomassof the n sample trees by the ratio of plot basal area over sample trees basalarea.

Snowdon (1986) conducted simulation studies to devise a sampling strat-egy for estimating the biomass of crown components of individual treesof Pinus radiata. The efficiency of simple random sampling was comparedwith various versions of stratified random sampling and ratio estimators.Not surprisingly, stratified random sampling produced better results thansimple random sampling. In a subsequent regression analysis branch posi-tion and branch diameter were used as predictor variables and branch-wood biomass as target variable. Both independent variables were enteredas a linear, quadratic, and cubic term, together with the linear interactionterm.

4 ADDITIVITY OF BIOMASS COMPONENTS

Kozak (1970) proposed to fit identical models for estimating the total tree bio-mass and biomass components, in order to ensure additivity of biomass com-ponents, which implies that the sum of these estimates is equal to that obtainedfrom the total-biomass equation. However, the question arises whether differentsubsets of independent variables should be used to estimate different biomasscomponents. Krumlik (1974) compared several models for predicting biomasscomponents. Linear models with d2h as predictor variable were suitable to esti-mate stem biomass, d2h∗crown width to estimate the biomass of big branches,basal area was more suitable to predict bark biomass, d∗crown length for esti-mating the twig and foliage biomass. Models based on the same predictor-variables, but with a logtransformed-dependent variable, were also tested andperformed satisfactorily.

Example 8.1 Regression equations were fitted to data from a biomass studyin P. radiata (Forrest 1969). The equations to predict total biomass are shownin Table 8-1.The adjusted R2-values based on stepwise elimination of variables and thecorresponding values of �(w − west)

2 and the associated ranks are given inTable 8-2.The two criteria being used produce different ranks. Model (2) is superior interms of R2 but is of rank 3 when expressed in terms of total squared error.Models (3) and (4) with d and d2 as predictors perform well in terms of sumof squared deviations but correspond with ranks 3 and 4 when expressed interms of R2. Models (5) and (6) are the poorest performers independently of

Additivity of Biomass Components 193

Table 8-1. Regression models for predicting biomass in P. radiata

Eq. Dependent variable Predictor variable Weight

(1) ln(mass) ln(d) unit weight(2) Predictor ln(d), ln(h) unit weight(3) mass d, d2 unit weight(4) mass d, d2 wi = 1/d2

(5) mass d2h unit weight(6) mass d2h wi = 1/d2

Table 8-2. Regression statistics and ranks for predicting total biomass

Eq. R2 Rank �(w − west)2 Rank Eq. R2 Rank �(w − west)

2 Rank

(1) 0.978 2 33.18 4 (4) 0.977 3 26.59 1(2) 0.979 1 30.64 3 (5) 0.909 6 59.66 5(3) 0.960 4 26.62 2 (6) 0.949 5 60.78 6

Table 8-3. Regression statistics for predicting stem biomass

Eq. R2 Rank �(w − west)2 Rank Eq. R2 Rank �(w − west)

2 Rank

(1) 0.954 2 14.84 6 (4) 0.950 3 9.01 3(2) 0.968 1 10.52 5 (5) 0.899 6 8.65 2(3) 0.905 5 8.19 1 (6) 0.949 4 9.97 4

the criterium being used. Weighting produces the best results when using R2 ascriterium, but not necessarily when the performance of the model is based ontotal squared error.

4.1 Stem biomass

The same data set was also used to compare and test the same set of regressorvariables. The results are given in Table 8-3.

The criterium being used has a considerable impact on the ranking results.Eq. (1) was the best performer when based on R2 but performed poorly whentotal squared error was used as quality criterium. The weighted models (4) and(6) were superior to unweighted equations, when based on R2 but the oppositewas true when total squared error was used as criterium for goodness of fit.

194 Tree and Stand Biomass

4.2 Branch and needle biomass

The previous models were applied to estimate the sum of branch and leaf bio-mass. The results are shown in Table 8-4.Models (5) and (6) were the poorest performers, independently of the criteriumbeing used. Eqs. (1) and (2) were superior in terms of R2 but Eq. (1) did notperform well in terms of total squared error. The weighted models (4) and (6)were superior to unweighted models (3) and (5), when based on R2 but notwhen based on total squared error was used as criterium.

4.3 Root biomass

The regression statistics for predicting root biomass are given in Table 8-5.The evaluation of the root biomass models reveals the same inconsistency asthat observed for other biomass components and shows similar trends. Eqs. (4)and (6) are superior to Eqs. (3) and (5) when assessed in terms of R2 but weight-ing had no influence on the total squared error. Model (6) was the poorest per-former in terms of R2 but the best model when based on total squared error.

The criteria R2 and total-squared error present an overall evaluation of thegoodness of fit. Additional information about the performance of the models isobtained by calculating the observed and estimated biomass within the lower,middle, and upper range of diameters. In the present case the calculations werecarried out for the trees below the 10th and above the 90th percentile of thediameter distribution and for 10 trees around the median. In the present case

Table 8-4. Regression statistics for predicting the sum of branch and needle biomass

Eq. R2 Rank �(w − west)2 Rank Eq. R2 Rank �(w − west)

2 Rank

(1) 0.944 2 13.61 4 (4) 0.941 3 13.55 3(2) 0.946 1 13.08 1 (5) 0.807 6 30.35 5(3) 0.916 4 13.40 2 (6) 0.879 5 30.64 6

Table 8-5. Regression statistics for predicting root biomass

Eq. R2 Rank �(w − west)2 Rank Eq. R2 Rank �(w − west)

2 Rank

(1) 0.919 3 2.91 4 (4) 0.922 2 2.79 2(2) 0.923 1 2.80 3 (5) 0.829 6 3.47 5(3) 0.866 5 2.75 1 (6) 0.904 4 3.48 6

Additivity of Biomass Components 195

these calculations were carried out for each of the four biomass components,with estimates being obtained from the equations

total biomass = exp(a0 + a1 ln(d) + a2 ln(h)) (1)

total biomass = b0 + b1d + b2d2(weights proportional to 1/d2) (2)

stem biomass = exp(a0 + a1 ln(d) + a2 ln(h)) (3)

stem biomass = b0 + b1d + b2d2(weights proportional to 1/d2) (4)

branches + foliage = exp(a0 + a1 ln(d) + a2 ln(h)) (5)

branches + foliage = b0 + b1d + b2d2(weights proportional to 1/d2) (6)

root biomass = exp(a0 + a1 ln(d) + a2 ln(h)) (7)

root biomass = b0 + b1d + b2d2(weighting proportional to 1/d2) (8)

The results are given in Table 8-6.In addition to the above 6 models (Example 8.1), an equation was tested

with ln(biomass) as well as ln(stem biomass), and ln(leaves + branches) asdependent, ln(d), ln(h), and ln(crown volume) as predictors. To predict totalbiomass, crown volume was significant in presence of ln(d), but ln(h) was non-significant. R2 however increased only marginally from 0.979 to 0.980 whenln(h) was replaced by ln(crown volume). The addition of crown volume did notimprove the accuracy of the prediction of stem biomass and that of branch +leaf biomass and in both cases dropped out as predictor variable. The estima-tion of ln(root biomass) from ln(d), ln(h) and ln(C L) produced an R2 of 0.928.In this case crown dimension improved the precision of root-biomass estimates.

Additivity

In order to illustrate nonadditivity of biomass estimates for the logarithmicmodel equations were fitted with ln(biomass) as dependent and ln(d) as well as

Table 8-6. Estimates in the lower, middle, and upper ranges

Biomass componentTotal Stem Fol. + branches Root

Middle range Eq. (1) 96.7 106.5 106.9 94.5Eq. (2) 98.3 103.0 105.9 97.2

Upper range Eq. (1) 117.6 107.6 109.7 116.6Eq. (2) 94.3 96.7 98.6 99.7

Lower range Eq. (1) 101.6 94.2 97.1 99.6Eq. (2) 102.6 104.7 44.1 43.4

196 Tree and Stand Biomass

Table 8-7. Regression coefficients for equations to predict the biomass components stem,leaves, and branches

Component Logarithmic model (1) Linear model (2)b0 b1 b2 c0 c1

Stems −3.598 2.156 0.5147 3.758 0.01206Leaves −4.968 2.542 0.2265 −0.056 0.00575Branches −4.686 3.148 −0.4425 −0.812 0.00975

ln(h) predictor variables and biomass as dependent and d2h as predictor vari-able. The calculations were carried out for stem, foliage, and for live branchesas dependent variables. The parameter estimates are given in Table 8-7.

5 DUMMY-VARIABLES FOR TREE SPECIES

Jacobs et al. (1980) developed prediction equations for ten tree species. In orderto harmonize tree biomass tables dummy-variables were introduced which rep-resented biomass components, based on stem-biomass categories demarcatedby specified top diameters. The regression analysis did not give evidence ofnonparalellism, but the relationship between the intercept parameter and topdiameter could be expressed by a second-degree equation. Crow et al. (1980)applied a two-variable linear model, with a logtransformed-dependent variableand a logtransformed (d2h) as predictor variable, to estimate the biomass oftropical forest trees and found that habitat as dummy-variable was not influ-ential in presence of dbh as predictor variable. In sampling studies conductedby Brown et al. (1989) however, equations for estimating the total abovegroundbiomass of tropical trees were developed, with a model using dbh and d2 as pre-dictor variables. This model was used for trees from the dry life zone whereasan equation with ln(d2h∗SG) with SG = specific gravity for trees was appliedto those growing in the dry life zone.

The objective of many sampling studies has been to estimate biomass com-ponents such as branch weight and needle weight. Hepp et al. (1982) developedbranch, tree, and stand level equations to estimate the branch and needle weightof loblolly pine. The branch model was

ln Wbr = b0 + b1 ln di + b2 ln h · hi + b3 ln A

Dummy-Variables for Tree Species 197

where hi = branch height is aboveground, di = branch basal diameter,h = tree height and A = age. The tree model was

ln Wcr = b0 + b1d + b2 ln CR

where Wcr = crown weight and CR = crown ratio. The stand level model was

ln W = b0 + b1 A + b2 ln G

where W = weight per hectare and G = basal area per hectareMadgwick et al. (1974) applied the following equation for predicting branch

weight

ln W = b0 + b1 ln d2l + b2 RH + b3 RH2

with l = branch length, RH = relative height above ground of the samplebranch, W = branch weight. Branch weight estimates in different crown sec-tions, however, were not unbiased. Ek (1979) proposed a model for estimatingbranch and foliage weight as a function of branch diameter at 25 mm from thestem. Starting point for the model development was the allometric relationshipbetween branch diameter and the target variable. Several models which tookcognizance of the effect of height aboveground level of the i th branch and ofspacing, the latter either expressed in terms of distance between the trees orby the height over dbh ratio, were tested for their predictive ability. The rela-tive height aboveground was less influential than spacing in predicting branchbiomass, but the variable which expressed branch position was significant inpresence of the spacing variable. The following model was more suitable toestimate foliage mass

y = b1db2i (h − hi )

b3

but the addition of a spacing variable served no useful purpose. The analysissuggested the possibility of pooling data from stands of different ages and spac-ings. Piene (1983) established a close linear relationship between needle lengthand weight per 1000 needles, as well as between-needle density (expressed asthe number of needles per centimeter shoot length) as dependent and the recip-rocal of needle length as predictor variable. The relevant regression equationswere combined to estimate the needle biomass for a branch with estimation ofneedle biomass required the measurement of shoot lengths, needle length, andneedle density. The author determined sampling positions within the shoot andwithin the crown, which should produce unbiased estimates.

Several studies were undertaken to compare the efficiency of different sam-pling methods. Attiwill et al. (1968) compared four methods to estimate thebiomass per unit area

198 Tree and Stand Biomass

• From the average of four weight measurements of trees with a dbh equal tothe mean stand diameter.

• From the average weight of four trees in each of five dbh classes, to be mul-tiplied by the stem number in these classes.

• From weight estimates obtained from regression equations based on a sampleof size 20.

• From the expected weight of the tree with the mean dbh, to be estimatedfrom the regression equation and to be multiplied by the number of trees.

Since the third method was assumed to produce the most accurate estimates,those based on the others were compared with this method. The negative biaswas −8.5%, −3.1%, and −11.6% for the first, second, and fourth methodrespectively. Williams et al. (1991) investigated the relationship betweensite index and biomass production within four drainage classes in even-agedspruce–fir stands in Maine. Stand age and site index were used as predictorvariables for biomass production per hectare. The regression model was

y = b0 + b1 A + b2SI · A2 + b3SI · A3

It was found however that site index was a poor predictor for biomass yield.

6 RATIO ESTIMATORS AND CLUSTER SAMPLING

The purpose of the previously described sampling procedures is to obtain themost accurate estimate of the biomass components at the lowest possible cost.The ultimate goal of sampling for biomass, however, is to obtain estimates perunit area. If a random sample of n trees is selected from a population con-taining N trees, the sample mean, i.e., the mean of the n biomass estimates,multiplied by the ratio N/n, represents un unbiased estimate of the populationtotal. This assumes that no measurement bias is involved in the determinationof the biomass components and that sampling for biomass, within the selectedsample trees, produces unbiased biomass estimates per sample tree. However,for given fixed cost, a more efficient estimate can be obtained by making useof ratio estimators (Chapter 10) than would be possible by simple random sam-pling. It requires the measurement of an auxiliary variable x , which can be mea-sured quickly and cheaply. The assumption is that the relationship between thetarget variable y and this variable x can be expressed by the equation y = b.x ,which represents a zero intercept regression equation and implies that y = 0for x = 0. In order to be useful the population total for the ancillary variablemust be known. For practical reasons the latter will be measured in sampleplots with biomass being calculated for each plot. The biomass of the stand isobtained by multiplying the mean plot biomass by an expansion factor. Cunia

Ratio Estimators and Cluster Sampling 199

(1981) suggested the ancillary variable x = d2h. This means that all diameterswithin a given population should be measured and a height curve fitted to obtainthe estimated heights for given dbh. Alternatively basal area, which is equiv-alent with squared dbh could be used as auxiliary variable. In both instancesthe assumption of a zero intercept of the regression line which reflects the rela-tionship between basal area and biomass does not hold true, since tree biomasstends to a value which is equal to the expected biomass of a tree which is 1.3 mhigh as breast height basal area tends to zero.

In cluster sampling groups of trees, usually all those within a given sampleplot, are selected at random, their biomass is determined either by a completemeasurement or possibly by subsampling and the tree biomass (or biomasscomponent) for the cluster total represents the subject variable. This methodensures that the residuals of the model are distributed independently. In biomassstudies based on linear regression methods, Briggs et al. (1982) investigated theimplications of cluster sampling. Because of heteroscedasticity ordinary leastsquares estimates were replaced by weighted least squares estimates and thedependent and predictor variables expressed the sample totals for each clus-ter. The efficiency of cluster sampling was investigated by Cunia et al. (1980,1981). The ratio of means produced the best, the mean of ratios estimator thepoorest estimates. Kotimaki et al. (1981) compared various auxiliary variableswithin the framework of cluster sampling to estimate biomass components.Snowdon (1992) conducted simulation studies in which three sampling strate-gies were involved• Simple random sampling• Sampling with probability of a given sample proportional to the sum of the

sizes• Sampling with probability of selecting each tree proportional to its sizeThese strategies were combined with two ratio estimators. RATIO1 was a ratioof means estimator, with cross-sectional area as auxiliary variable, RATIO2defined the auxiliary variable as db1 , where b1 represented the regression coef-ficient of the log–log regression of biomass on dbh. The results were evaluatedin terms of precision as well as accuracy. RATIO1 tended to perform betterwhen estimating total stem biomass, RATIO2 was superior to estimate otherbiomass components, but invariably PPS sampling outperformed other strate-gies in terms of precision as well as accuracy, usually in combination withRATIO2. The average bias was highest by combining the simple ratio estima-tor with PPS sampling.