Technical Note: A novel approach to estimation of time-variable ...

Atmos Chem Phys 11 10305ndash10315 2011wwwatmos-chem-physnet11103052011doi105194acp-11-10305-2011copy Author(s) 2011 CC Attribution 30 License

AtmosphericChemistry

and Physics

Technical Note A novel approach to estimation of time-variablesurface sources and sinks of carbon dioxide using empiricalorthogonal functions and the Kalman filter

R Zhuravlev1 B Khattatov2 B Kiryushov1 and S Maksyutov3

1Central Aerological Observatory Dolgoprudny Russia2Fusion Numerics International Boulder CO USA3Center for Global Environmental Research National Institute for Environmental Studies Tsukuba Japan

Received 19 November 2010 ndash Published in Atmos Chem Phys Discuss 18 January 2011Revised 23 September 2011 ndash Accepted 25 September 2011 ndash Published 17 October 2011

Abstract In this work we propose an approach to solvinga source estimation problem based on representation of car-bon dioxide surface emissions as a linear combination of a fi-nite number of pre-computed empirical orthogonal functions(EOFs) We used National Institute for Environmental Stud-ies (NIES) transport model for computing response functionsand Kalman filter for estimating carbon dioxide emissionsOur approach produces results similar to these of other mod-els participating in the TransCom3 experiment

Using the EOFs we can estimate surface fluxes at higherspatial resolution while keeping the dimensionality of theproblem comparable with that in the regions approach Thisalso allows us to avoid potentially artificial sharp gradientsin the fluxes in between pre-defined regions EOF resultsgenerally match observations more closely given the sameerror structure as the traditional method

Additionally the proposed approach does not require addi-tional effort of defining independent self-contained emissionregions

1 Introduction

It is well known that greenhouse gases and in particu-lar greenhouse gases of anthropogenic origin influence theEarth climate to a great extend Accurate estimates ofstrengths and spatial and temporal variability of the surface

Correspondence toR Zhuravlev(ruslanzhuravlevgmailcom)

sources and sinks of greenhouse gases are thus of great inter-est to both the scientific community and the policy makersCarbon dioxide (CO2) is the most important greenhousegas of anthropogenic origin that affects radiative balance ofthe atmosphere and eventually the climate Observations ofCO2 concentrations in the atmosphere demonstrated short-time variability and spatial patterns reflecting influence oftime-variable strengths of regional surface sources and sinksof carbon dioxide

The objective of this research is to study advantages andshortcomings of traditional methods of solving the inverseproblem of estimating spatial and temporal variability of sur-face fluxes and compare them with the EOF approach Whilewe do not wish to position our approach as superior wewould like to propose it as an alternative that seems to beworth exploring further Traditional approaches to solvinginverse flux estimation problem include dividing the Earthrsquossurface by a number of non-overlapping regions and estimat-ing the strengths of sources and sinks for each one of themOne of the most well-known and successful experiments fol-lowing this approach was TransCom 3 (T3 Gurney et al2000) which used 22 distinct regions 11 for land surfaceand 11 for ocean surface Advantage of this technique is thatthe problem is mathematically over-determined because thenumber of unknowns is much less than number of availableobservations In subsequent work of Patra et al (2005a b2006) the number of the regions has been increased to 64In both cases monthly mean CO2 surface emissions havebeen successfully estimated using monthly averaged groundbased observations of carbon dioxide concentrations

Published by Copernicus Publications on behalf of the European Geosciences Union

10306 R Zhuravlev et al Estimation surface fluxes

Recently (Feng et al 2009) 144 distinct regions havebeen used and the time scale of carbon dioxide variabilitywas reduced to 8 days using satellite observations of CO2In each region distribution of CO2 emission is forced to besmooth so the resulting emission fields will be piecewise-smooth because of regions were demarcated by hard bound-aries Refining the results would necessitate increasing thenumber of regions and thus increasing computational re-quirements which might prove to be impractical Alsolumping small basis regions into larger combined regionsmay lead to aggregation errors because observed CO2 fieldsare sensitive to the distribution of sources and sinks withinlarge basis regions (Kaminski et al 2001)

Additionally it is reasonable to assume that at least insome cases there is a correlation in emission strengths be-tween the regions It is usually proposed that such correla-tion is negligible and grid-points within regions are perfectlycorrelated in space but in reality this might not be accurate

Another computational approach relies on the adjoint ofthe forward transport model (Kaminski et al 1999) Theyestimated a coarse grid of fluxes at 8 latitude and 10 lon-gitude in monthly time scales The problem was under-determined and a solution was found by gathering a priori in-formation on surface fluxes In this case influence from eachgrid box could be estimated but it could be computationallyexpensive due to increasing the number of unknowns andcreating adjoint versions of forward models is not straight-forward In addition the correlation between neighboringcells is still unknown and the results are likely to be in-fluenced by misspecifications of such correlations Roden-beck (2003) performed a grid-scale inversion assuming dif-ferent de-correlation length over the land and the ocean us-ing the same spatial resolution as Kaminski (1999) For thisapproach reality lies somewhere between the two extremesperfectly correlated or completely uncorrelated fluxes at agrid scale

The main idea of our approach is to use empirical or-thogonal functions (EOFs) in place of distinct geographi-cal regions Use of the EOFs as a tool to reduce degreesof freedom in inverse modeling has become a widespreadpractice in geophysics (eg Wikle and Cressie 1999) It isalso mentioned (Desbiens et al 2007) that use of EOF ininversion is similar to truncated SVD technique by Hansen(1987 1998) We propose representing geographic distribu-tion of surface emissions of carbon dioxide as a linear com-bination of a number of pre-computed empirical orthogonalfunctions This combination contains information about cli-matological spatial variability of the emissions as well as sta-tistical correlations between different grid-points This ap-proach would yield smooth surface fluxes on a global scaleand it does not require additional research for defining in-dependent self-contained emission regions Since as shownlater in this manuscript a relatively small number of EOFsis needed to accurately represent the CO2 surface emissionsfor a number of sources the estimation problem becomes

fairly inexpensive computationally Practical applications ofthe derived EOFs can also be envisioned in a framework ofthe geostatistical inverse modeling (Michalak et al 2004)which requires a set of the global flux patterns to approxi-mate optimal flux field

2 Methodology

While solving the inverse problem we utilize the traditionalBayesian approach We attempt to optimize the cost functionconsisting a sum of the two parts a disagreement betweenthe desired solution and the priori estimation and a disagree-ment between the desired solution and the observations For-mally the cost function is given by the following expression

F(x) =1

2(y minusHx)T Rminus1(y minusHx)+

1

2(x minusxp)

T Bminus1(x minusxp) (1)

herey ndash is a vector of observations of sizen xp ndash is an prioriestimate for the surface fluxes of sizem x is a vector of un-known parameters (to be estimated) of sizem H ndash is a modeloperator (a matrix of dimensionsntimesm) transforming vectorx to observationsy R ndash is a representativeness error ma-trix of sizentimesn andB ndash is the background error covariancematrix of sizemtimesm

As mentioned above the main idea of our approach is touse empirical orthogonal functions as the basis of decompo-sition for the emission sources This would hopefully resultin a more accurate representation of the spatial distributionof the emission sources while minimizing the computationalrequirements on the inverse problem solution Formally foreach kind of sources we have

f luxres= f luxpr+

Nsumi=1

αiEOF i (2)

wheref luxresndash is a result field of flux in units g mminus2 dayminus1f luxpr ndash is a pre-set ldquopresubtractedrdquo fields (Fig 1) used inthe experiment T3 in units g mminus2 dayminus1 EOF i ndash empiricalorthogonal functionsαi ndash are the corresponding basis expan-sion coefficients andN ndash is the number of the EOFs Theinverse problem task essentially involves to a determinationof the corresponding weights for each of the basis function(EOFs)

An advantage of our approach is that we are implicitlyusing information about correlation between different grid-points that could emit or absorb a trace gas in questions anddo so for each emission type This information is embeddedin the computed spatial distribution of the EOFs No poten-tially subjective a priori data (for example regarding correla-tion lengths in land and ocean fluxes) is being used in the cal-culations which may or may not be advantageous Clearlyif external information of good quality regarding correla-tions of emission between different grid-ponts is available it

Atmos Chem Phys 11 10305ndash10315 2011 wwwatmos-chem-physnet11103052011

R Zhuravlev et al Estimation surface fluxes 10307

Fig 1 A fixed ldquopresubtractedrdquo average emission fields in units of g mminus2 dayminus1 from T3 experiment for three different emission sourcesfossil fuel (top figures year data) biosphere exchange (middle figures monthly data) ocean exchange (bottom figures monthly data)

would be advantageous to use it in solving the inverse prob-lem Yet such data could be difficult to obtain and wouldlikely not be of similar quality for different regions of theworld thus introducing an additional bias to the solution

In this investigation we attempt to show that using a ratherlimited set of parameters (basis coefficients of EOF expan-sion) it is possible to quantify different surface sources andsinks of CO2 at a fairly high spatial resolution and withoutthe use of the regional source approach

21 Determination of EOF

As in the T3 experiment we consider separately three kindsof surface CO2 sources burning fossil fuel biosphere ex-changes and ocean exchanges Therefore it is necessary to

compute EOFs for each of them except the fossil fuel emis-sions (which are fixed in TransCom3 experiment)

For computing the EOFs we need spatial and time vari-able statistics of the surface emissions This dataset has beenobtained from the CarbonTracker web site (httpwwwesrlnoaagovgmdccggcarbontracker) While this data set is al-ready a result of numerical inversion yet in the absence ofa desirable quantity of observations falling back on a vari-ability of a set of well-established inversion results using itseems to be a reasonable approach CarbonTracker dataset(release 2009) provides global surface fluxes from 2000 to2008 at 3-hour time intervals for four kinds of surface emis-sions burning fossil fuels biosphere exchanges ocean ex-changes and fires Due to the spin-up effect we removed year2000 from our analysis The spatial resolution of these fields

wwwatmos-chem-physnet11103052011 Atmos Chem Phys 11 10305ndash10315 2011


Fig 2 Error of decomposition (y-axis) between original emissionfields and those restored depends on number of most significantEOFs (x-axis)

is one by one degree hence the resolution of the EOFs willbe the same

Since the transport model (as is described in Sect 22) hasspatial resolution of 25 degrees we should be estimating thefluxes at similar resolution Therefore all fields should beinterpolated to a 25 degree grid

Since in our investigation we are interested in monthlyemissions (as in experiment T3) we need to average the 3-hour fields to make them comparable with the monthly-meanemissions Therefore we get a set of 96 monthly-mean fields(8 years times 12 months) for each emission type Notethat we are not attempting to optimize the anthropogenicemissions as they are considered fixed in the T3 experimentsetup

Similarly to T3 experiment prior to computing the EOFsmean T3 emission fields were subtracted from the Carbon-Tracker time-dependent monthly emission fields In ourpresent setup due to using T3 mean fluxes we likely havebias as part of EOF set But we cannot predict if we have anegative effect on the following optimization The 2-D fieldsof deviations of surface emissions from the mean are avail-able at monthly intervals

After that each 2-D field is re-shaped as a vector of size144times72 = 10 368 (25times25 degree) and is added as columnvector to the matrix representing the surface sourcessinks

We apply SVD decomposition to the matrix (size 10 368rows per 96 columns) and obtain the resulting 96 empiricalorthogonal functions We use a standard procedure from [9]

For the next step it is desirable to determine the minimumnumber of EOFs needed to reasonably accurately representCO2 surface emissions for each source in order to reduce thedimensionality of the inverse problem In order to do sowe compute relative error of the EOF-restored fluxes with

respect to original fluxes at 25 degree resolution as follows

error =

∣∣f luxorigminusf luxEOF

∣∣∣∣f luxorig

∣∣ middot100 (3)

heref luxorig ndash original emission fields from CarbonTrackerf luxEOF ndash emission fields reconstructed from the EOFsAfterwards the obtained error fields are averaged in spaceand time Figure 2 shows averaged relative error be-tween monthly averaged emission fields from CarbonTrackerproject and fields obtained after decomposing on EOFs Itappears that in order to be able to represent emission sourcesabout 5 accuracy one needs to use 30 EOFs for the bio-spheric sources and 40 EOFs for the ocean sources (more de-tails on the ability of the EOF decomposition to capture spa-tial and temporal variability of the underlying fields are givenin Appendix A) Thus the highest possible dimensionality ofthe inversion problem for one year is (30 + 40)middot12 + 4 = 844In these calculation 4 accounts for the a priori fixed averageemission fields two for the anthropogenic emissions one forthe biosphere and one for the ocean Since the EOFs them-selves are obtained the re-analysis data it is reasonable to as-sume that the number of the EOFs needed to adequately rep-resent the emission fields will be similar to the number of thedegrees of freedom used in the CarbonTracker frameworkCarbonTracker uses 30 regions for the ocean which compa-rable to the number in our simulations For the biosphereCarbonTracker employs surface emission regions similar tothose utilized in the T3 experiment yet introduces 19 addi-tional ecosystems but not in all geographical regions As aresult they have about 125 degrees of freedom in their covari-ance matrix It means that CarbonTracker would be ldquoover-solvingrdquo system due to as we suppose spatial correlationsbetween some of these regions and ecosystems

22 Transport model

The transport model used in this work has been developed atthe National Institute for Environmental Studies (Maksyutovet al 2008) The model effectively converts CO2 surfaceemissions (a derivable parameter) to the volume concentra-tions of CO2 (an observable parameter)

We use pressure level ECMWF operational analyses at 12-hour time step and 25 degrees horizontal resolution in modelsimulations (ECMWF 1999 Courtier et al 1998) Whilethe same horizontal resolution is used in the model the gridlayout is different from meteorological dataset

The model is designed to handle constant surface emissionfields and seasonally changing emissions in the form of 12monthly average fields per year The monthly average emis-sions are interpolated linearly to daily values so that on the15th of each month the emission rate is equal to the monthlyaverage for that month as provided by emission inventoryfiles The emission inventory fields have higher resolution(1times1 degree) than the model grid (25times25 degrees) so the



Fig 3 Results of simulation experiment for biosphere(a) pseudotruth(b) recovered fluxes(c) difference in percentage between truth andrecovered fluxes

input dataset is mapped to a model grid by computing theoverlap area of each input data cell to all model grid data cellThis assures that the global total emission flux is conservedduring interpolation

In the described work the model was run on a gridof 25times25 degrees with 15 vertical levels using a semi-Lagrangian transport scheme As such we can reasonableconstrain surface fluxes only for that model resolution in-stead of 1times1 degree In this case the maximum dimensional-ity of the state vector for the grid-scale inversion would havebeen (144middot72)middot12 = 124 416 (for one year) It is clear that us-ing the given number of observations the grid-scale inverseproblem would have been significantly under-determinedBut this problem can be solved by introduction some addi-tional information (eg correlation length) and proper firstguess In case of EOF approach as described in Sect 21 forthe same spatial resolution we have the dimensionality of thestate vector only (30 + 40)middot12 = 840 That means reduction ofthe under-determined degree and in case of a large numberof observations per month it could be over-determined

23 Inversion method

In our approach all non-observed parameters are estimatedsimultaneously at each solution step using all available ob-servations This technique is known as a ldquobatch moderdquo in-version In order to minimize the cost function (Eq 1) we

utilize Kalman filter (KF) (Eqs (4) (5) and (6))

x = xp+K(y minusH[xp]) (4)

K = BHT (HBHT+R)minus1 (5)

A = (I minusKH )B (6)

herex is the posteriori vector of EOF expansion coefficientsxp is a priori vector of EOF expansion coefficientsy is ob-servation vectorH is observation operator that describes therelationship between the state vector and the observationsKis the Kalman gain matrix that determines the adjustment tothe a priori based on the difference between model and ob-servations and their uncertaintiesR is the observation errorcovariance matrix andB andA are the a priori and poste-riori error covariance matrices for state vectorI is identitymatrix

KF allow us to estimate coefficients of the EOF expan-sion and errors of the resulting emissions using ground-based observations of CO2 concentrations This techniqueis a powerful and widespread method for inverse problemKalman filter applications for trace gases have been de-scribed in detail in Verlaan and Heemink (1995 1996) Cohnand Todling (1995) Pham (1998) Zhang et al (1999) Haneaet al (2004) We would like to point out that in our case wedescribe Kalman Filter in conjunction with EOF approach



Fig 4 Results of simulation experiment for ocean(a) pseudotruth(b) recovered fluxes(c) difference in percentage between truth andrecovered fluxes

24 Simulation experiment

In order to verify the proper functionality of our inverse sys-tem we performed experiments with simulated data sets

First we selected a limited number of EOFs (correspond-ing to the Sect 21) and reasonable a priori set of EOF ex-pansion coefficients as the state vector which is to be re-trieved Monthly-mean surface sources of CO2 for 2 typesof emissions (biosphere and ocean) obtained from Carbon-Tracker 2001ndash2008 data have been de-composed using theset of EOF As a priori values for the expansion coefficientswe use monthly averages We added 30 Gaussian noiseto the simulated data for January for the biosphere sourcesand 15 to the ocean sources for the ldquotruerdquo values The for-ward model has been run for a period of one year using thesesources Monthly mean simulated data have been computedat the locations of the 75 stations used in the T3 experiment(described in Sect 35) for a total of 900 simulated observa-tions

The error of observations was set to be 03 for all sta-tions An a priori error for the coefficients of the EOF expan-sion was set to be 30 for the biosphere and 15 for theocean The inversion for all parameters and time points wasperformed in one iteration of the Kalman filter

25 Comparison with regional inversion (TransCom3experiment level 2)

To compare our approach with regional methodology we re-peated the T3 experiment Level 2 (cyclo-stationary problem)as described by Gurney et al (2004) and obtained monthlysources and sinks of CO2 In our project we used NationalInstitute for Environmental Studies Transport Model (NIESTM see Sect 22) Although T3 experiment uses the av-eraged observation for 1992ndash1996 period and the set of pre-computed EOF belong to 2001ndash2008 period we made the as-sumption that since TransCom3 Level 2 is a cyclo-stationaryexperiment its results would adequately describe the maindegrees of variability (EOFs) that would be approximatelyindependent of temporal differences

The seasonal inversion in T3 (cyclo-stationary problem)framework consist of a 3-year forward simulation (365 daysper year) containing 4 ldquopresubtractedrdquo tracers 2 fields forfossil fuel biosphere and ocean and 22 CO2 tracers (11 ter-restrial 11 oceanic) For each month observation from 75ground-based stations are used for inversion

In accordance with T3 Level 2 protocol we used NIES TMmodel to compute response functions for each pre-computedEOF representing CO2 emissions for each month at 3-yearintervals Then all response functions were collected intoone model matrix for inversion Observations of CO2 con-centrations along with the covariance matrix were taken fromTransCom3 data set



Fig 5 Mean annual fluxes for biosphere and ocean in units g mminus2 dayminus1 Column 1 represents results for region approach column 2represents results for EOF approach



Fig 6 Mean fluxes for T3 regions with uncertainties(a) region 1(b) region 6(c) region 12(d) region 20 Blue ndash prior for EOF red ndashestimated with EOF green ndash estimated with regions (with NIES TM)

Fig 7 Averaged error RMS error and systematic error betweenobservations and model simulations of CO2 distributions for differ-ent transport models and experiments with NIES model using EOFapproach (last case) described in the text

Since all unknown parameter are estimated in one iter-ation of our method all observations (75 stationstimes 12months = 900) represent one observational vector Thus thesize of the observation error covariance matrix is 900times900Similarly as in the T3 experiment we estimate correctionsto only biospheric and oceanic fluxes

Therefore the state vector for each month consists of 70EOFs (30 for the biosphere and 40 for the ocean) The fi-nal size of the state vector is then 70times12 + 4 = 844 where4 refers to the number of ldquopresubtractedrdquo fluxes Thus thesize of the background error covariance matrix is 844times844Priori values for state vector is the same as described in

Sect 24 and as the errors we used the standard deviationsfor each coefficient

3 Results

First we will present results of the simulation inversion ex-periment described in Sect 24

Figures 3 and 4 display results of simulation experimentfor the biosphere and ocean emissions Figures 3c and 4cdemonstrate the relative error between the ldquotruerdquo (simulated)emission field and the restored emission field As one canconclude from examining these plots the error of the restoredemission strengths is about 5

Figure 5 presents results of the inversion for the traditionalregion-based approach for January and July as well as forthe results of the EOF approach described here Clearlythe overall distributions of the emission fields are similar inshape giving some confidence in the validity of the EOF ap-proach yet noticeable quantitative differences are present

We will now describe the results of estimating regionalemission sources After determining spatial distribution ofthe global sources we can attempt to estimate surface fluxesusing the regions defined in the T3 runs for comparison

We obtained monthly-mean fluxes in Gt yrminus1 for each ofthe regions as defined in the T3 protocol

Figure 6(andashd) shows four regions two for the land andtwo for the ocean emissions Each panel demonstrates a pri-ori fluxes for EOF approach inversion results using the EOFapproach and inversion results using the regional approachwith the same NIES transport model for 22 regions



As one can see the reduction of the uncertainties over thesurface is evident from the plots Note that the dimensional-ity of the problem is roughly the same or comparable in thecase of the EOF and the regional approach by order of magni-tude For the ocean regions we can observe a slight decreasein the estimation error In all likelihood this is related tothe relatively low number of over-the-ocean observations Insummary we conclude that our inversion results reproducethe expected annual cycle

As another control we have used averaged error RMS er-ror and systematic error between observations and modelsimulations of CO2 distributions for different transport mod-els and experiments with NIES model using regional andEOF approaches These results are presented in figure 7 Par-ticularly interesting are results in the last two columns asthey represent inversion errors for the same model but dif-ferent inversion approaches As one can see our approachindicates a smaller RMS error and a smaller systematic erroras compared with the regions approach

4 Conclusions

We presented an alternative inversion method to the tradi-tional approach that uses discrete geographical spatial re-gions We instead relied on a pre-computed set of the em-pirical orthogonal functions that capture spatial variabilityof the CO2 sources and performed the inversion with regardto the EOF expansion coefficients The procedure of com-puting such EOFs has been described for the CarbonTrackerreanalysis data We also described determination of the op-timal number of the EOFs for more accurate determinationof the surface sources and sinks of CO2 A simulated ob-servational system experiment has been performed to verifythat the proposed inversion system captures variability of thederived parameters as demonstrated by figures 3ndash4 We re-peated TransCom3 Level 2 experiment Our system demon-strated successful retrieval of the spatial distribution of themonthly mean sources similar to the regional approach of 22regions We also derived reasonable seasonal variability ofthe sources as compared to the published T3 experiment re-sults

As next steps in developing our approach we would like tocalculate EOFs using the global biosphere and ocean modelsseparately That will be the simplification of these modelsand its parameters We also consider increasing the numberof observations using space-borne observations in additionto the ground-based data hopefully achieving a significantlymore advanced inversion accuracy and precision

Fig A1 Relative error in percent between RMS error for spatialvariability of original fields and RMS error for spatial variabilityof fields restored with EOFs for two kinds of sources depends onmonth

Fig A2 Relative error in percent between RMS error for temporalvariability of original fields and RMS error for temporal variabil-ity of fields restored with EOFs for two kinds of sources (top ndashbiosphere bottom ndash ocean)



Appendix A

Investigation of the ability of EOFs to capture timeand space variability of surface sources of CO2

Two numerical experiments have been performed in orderto assess how well EOF decomposition captures spatial andtemporal variability of the emission fieldsExperiment 1 Spatial variability

12 monthly mean original CarbonTracker emission fields forthe biosphere and oceans have been used to compute aver-age root-mean-squared deviations for each monthRMSorigSimilar procedure has been performed using fields obtainedfrom EOF expansion using 30 EOFs for biosphere and40 EOFs for the oceans (RMSEOF) A relative error be-tween the two has been computed as (RMSorigminusRMSEOF)middot100 RMSorig Results are shown in Fig A1 As one cansee EOF decomposed fields show degree of spatial variabil-ity similar to that of the original fields

Experiment 2 Temporal variability

Similar to Experiment 1 procedure has been followed in Ex-periment 2 except that instead of computing spatial root-mean-squared deviations for each month we computed tem-poral root-mean-squared deviations at each grid point Thuswe obtained two 2-D fields of RMS deviations and the com-puted relative difference between them Results are shown inFig A2 which demonstrates that temporal variability com-puted from original fields and that computed from EOF-restored fields are generally in agreement with the notableexception of Sahara region for the biosphere This is possi-bly due to missing data in that region in the CarbonTrackerdata

AcknowledgementsThis study has been supported by GOSATproject at NIES Japan CarbonTracker 2008 results providedby NOAA ESRL Boulder Colorado USA from the website athttpcarbontrackernoaagov TransCom project data was madeavailable due to support from the National Science Foundation(OCE-9900310) the National Oceanic and Atmospheric Adminis-tration (NA67RJ0152 Amend 30) and the International GeosphereBiosphere ProgramGlobal Analysis Interpretation and ModelingProject

Edited by M Heimann

References

Courtier P Andersson E Heckley W Pailleux J VasiljevicD Hamrud M Hollingsworth A Rabier F and Fisher MThe ECMWF implementation of three dimensional variationalassimilation (3D-Var) I Formulation Q J Roy Meteorol Soc124 1783ndash1807 1998

Desbiens R Aoki T and Yokota T Optimization of GOSATAtmospheric Retrieval of CO2 in Presence of Atmospheric Parti-cles Using Empirical Orthogonal Function Representation EOStransactions American Geophysical Union Fall Meeting 2007abstract A13D-1506 2007

ECMWF ECMWFWCRP Level III-A Global Atmospheric DataArchive ECMWF available online athttpwwwecmwfint1999

Gurney K Law R Rayner P and Denning A S TransCom3 Experimental Protocol Department of Atmospheric ScienceColorado State University USA Paper No 707 2000

Gurney K R Law R M Denning A S Rayner P JPac B C Baker D Bousquet P Bruhwiler L ChenY H Ciais P Fung I Heimann M John J MakiT Maksyutov S Peylin P Prather M and Taguchi STranscom 3 inversion intercomparison Model mean results forthe estimation of seasonal carbon sources and sinksrdquo GlobalBiogeochem Cycles 18 GB1010httpdxdoiorg1010292003GB002111doi1010292003GB002111 2004

Hansen P C The truncated SVD as a method for regularizationBIT 27 534ndash553 1987

Hansen P C Rank-Deficient and Discrete Ill-Posed ProblemsNumerical Aspects of Linear Inversionrdquo SIAM Monogr onMath Modeling and Computation 4 247 pp 1998

Kalman R E A new approach to linear filtering and predictionproblems J Basic Eng 82 35ndash45 1960

Kaminski T Heimann M and Giering R A coarse grid three-dimensional global inverse model of atmospheric transport 2Inversion of the transport of CO2 in 1980srsquo J Geophys Res104 18555ndash18582 1999

Kaminski T Rayner P J Heimann M and Enting I G On ag-gregation errors in atmospheric transport inversionsrdquo JGeophysRes 106 4703ndash4715 2001

Maksyutov S Patra P K Onishi R Saeki T and NakazawaT NIESFRCGC global atmospheric tracer transport model de-scription validation and surface sources and sinks inversionrdquo JEarth Simulator 9 3ndash18 2008

Michalak A M Bruhwiler L and Tans P P A geostatisti-cal approach to surface flux estimation of atmospheric tracegasesrdquo J Geophys Res 109 D14109httpdxdoiorg1010292003JD004422doi1010292003JD004422 2004

Patra P K Ishizawa M Maksyutov S Nakazawa T and InoueG Role of biomass burning and climate anomalies on land-atmosphere carbon fluxes based on inverse modelling of atmo-spheric CO2 Global Biogeochem Cy 19 GB3005httpdxdoiorg1010292004GB002258doi1010292004GB0022582005a

Patra P K Maksyutov S Ishizawa M Nakazawa T UkitaJ and Takahashi T Interannual and decadal changes in thesea-air CO2 flux from atmospheric CO2 inverse modellingGlobal Biogeochem Cy 19 GB4013httpdxdoiorg1010292004GB002257doi1010292004GB002257 2005b

Patra P K Mikaloff-Fletcher S E Ishijima K MaksyutovS and Nakazawa T Comparison of CO2 fluxes estimatedusing atmospheric and oceanic inversions and role of fluxesand their interannual variability in simulating atmospheric CO2concentrations Atmos Chem Phys Discuss 6 6801ndash6823httpdxdoiorg105194acpd-6-6801-2006doi105194acpd-6-6801-2006 2006



Peters et al rdquoAn atmospheric perspective on North American car-bon dioxide exchange CarbonTrackerrdquo PNAS November 272007 vol 104 no 48 18925-18930 2007

Press W Teukolsky S Vetterling W and Flannery B Numeri-cal Recipes in FORTRAN The Art of Scientific Computing Sec-ond Edition 1995

Rodenbeck C Houweling S Gloor M and HeimannM CO2 flux history 1982ndash2001 inferred from atmosphericdata using a global iversion of atmospheric transportrdquo At-mos Chem Phys 3 1919ndash1964httpdxdoiorg105194acp-3-1919-2003doi105194acp-3-1919-2003 2003

Wikle C K and Cressie N A dimension reduced approach tospace-time Kalman filtering Biometrika 86 815ndash829 1999



Recently (Feng et al 2009) 144 distinct regions havebeen used and the time scale of carbon dioxide variabilitywas reduced to 8 days using satellite observations of CO2In each region distribution of CO2 emission is forced to besmooth so the resulting emission fields will be piecewise-smooth because of regions were demarcated by hard bound-aries Refining the results would necessitate increasing thenumber of regions and thus increasing computational re-quirements which might prove to be impractical Alsolumping small basis regions into larger combined regionsmay lead to aggregation errors because observed CO2 fieldsare sensitive to the distribution of sources and sinks withinlarge basis regions (Kaminski et al 2001)

Additionally it is reasonable to assume that at least insome cases there is a correlation in emission strengths be-tween the regions It is usually proposed that such correla-tion is negligible and grid-points within regions are perfectlycorrelated in space but in reality this might not be accurate

Another computational approach relies on the adjoint ofthe forward transport model (Kaminski et al 1999) Theyestimated a coarse grid of fluxes at 8 latitude and 10 lon-gitude in monthly time scales The problem was under-determined and a solution was found by gathering a priori in-formation on surface fluxes In this case influence from eachgrid box could be estimated but it could be computationallyexpensive due to increasing the number of unknowns andcreating adjoint versions of forward models is not straight-forward In addition the correlation between neighboringcells is still unknown and the results are likely to be in-fluenced by misspecifications of such correlations Roden-beck (2003) performed a grid-scale inversion assuming dif-ferent de-correlation length over the land and the ocean us-ing the same spatial resolution as Kaminski (1999) For thisapproach reality lies somewhere between the two extremesperfectly correlated or completely uncorrelated fluxes at agrid scale

The main idea of our approach is to use empirical or-thogonal functions (EOFs) in place of distinct geographi-cal regions Use of the EOFs as a tool to reduce degreesof freedom in inverse modeling has become a widespreadpractice in geophysics (eg Wikle and Cressie 1999) It isalso mentioned (Desbiens et al 2007) that use of EOF ininversion is similar to truncated SVD technique by Hansen(1987 1998) We propose representing geographic distribu-tion of surface emissions of carbon dioxide as a linear com-bination of a number of pre-computed empirical orthogonalfunctions This combination contains information about cli-matological spatial variability of the emissions as well as sta-tistical correlations between different grid-points This ap-proach would yield smooth surface fluxes on a global scaleand it does not require additional research for defining in-dependent self-contained emission regions Since as shownlater in this manuscript a relatively small number of EOFsis needed to accurately represent the CO2 surface emissionsfor a number of sources the estimation problem becomes

fairly inexpensive computationally Practical applications ofthe derived EOFs can also be envisioned in a framework ofthe geostatistical inverse modeling (Michalak et al 2004)which requires a set of the global flux patterns to approxi-mate optimal flux field

2 Methodology

While solving the inverse problem we utilize the traditionalBayesian approach We attempt to optimize the cost functionconsisting a sum of the two parts a disagreement betweenthe desired solution and the priori estimation and a disagree-ment between the desired solution and the observations For-mally the cost function is given by the following expression

F(x) =1

2(y minusHx)T Rminus1(y minusHx)+

1

2(x minusxp)

T Bminus1(x minusxp) (1)

herey ndash is a vector of observations of sizen xp ndash is an prioriestimate for the surface fluxes of sizem x is a vector of un-known parameters (to be estimated) of sizem H ndash is a modeloperator (a matrix of dimensionsntimesm) transforming vectorx to observationsy R ndash is a representativeness error ma-trix of sizentimesn andB ndash is the background error covariancematrix of sizemtimesm

As mentioned above the main idea of our approach is touse empirical orthogonal functions as the basis of decompo-sition for the emission sources This would hopefully resultin a more accurate representation of the spatial distributionof the emission sources while minimizing the computationalrequirements on the inverse problem solution Formally foreach kind of sources we have

f luxres= f luxpr+

Nsumi=1

αiEOF i (2)

wheref luxresndash is a result field of flux in units g mminus2 dayminus1f luxpr ndash is a pre-set ldquopresubtractedrdquo fields (Fig 1) used inthe experiment T3 in units g mminus2 dayminus1 EOF i ndash empiricalorthogonal functionsαi ndash are the corresponding basis expan-sion coefficients andN ndash is the number of the EOFs Theinverse problem task essentially involves to a determinationof the corresponding weights for each of the basis function(EOFs)

An advantage of our approach is that we are implicitlyusing information about correlation between different grid-points that could emit or absorb a trace gas in questions anddo so for each emission type This information is embeddedin the computed spatial distribution of the EOFs No poten-tially subjective a priori data (for example regarding correla-tion lengths in land and ocean fluxes) is being used in the cal-culations which may or may not be advantageous Clearlyif external information of good quality regarding correla-tions of emission between different grid-ponts is available it





















error =



∣∣ middot100 (3)


22 Transport model









23 Inversion method





























3 Results











4 Conclusions







Appendix A







Edited by M Heimann

References










































error =



∣∣ middot100 (3)


22 Transport model









23 Inversion method





























3 Results











4 Conclusions







Appendix A







Edited by M Heimann

References



























23 Inversion method





























3 Results











4 Conclusions







Appendix A







Edited by M Heimann

References











































3 Results











4 Conclusions







Appendix A







Edited by M Heimann

References


























4 Conclusions







Appendix A







Edited by M Heimann

References
























Appendix A







Edited by M Heimann

References























Technical Note: A novel approach to estimation of time-variable ...

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