chapter 7 review of breastfeeding assessment tools

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Advances in Space Research 41 (2008) 20–35

On uncertainties in carbon flux modelling and remotely senseddata assimilation: The Brasschaat pixel case

Willem W. Verstraeten a,*, Frank Veroustraete b, Walter Heyns b,Tom Van Roey b, Jan Feyen c

a Geomatics Engineering, Katholieke Universiteit Leuven (K.U.Leuven), Celestijnenlaan 200E, BE-3001 Heverlee, Flanders, Belgiumb Centre for Remote Sensing and Earth Observation Processes, Flemish Institute for Technological Research (VITO), Boeretang 200,

BE-2400 Mol, Flanders, Belgiumc Laboratory for Soil and Water Management, Katholieke Universiteit Leuven (K.U.Leuven), Geo-institute, Celestijnenlaan 200E,

BE-3001 Heverlee, Flanders, Belgium

Received 30 September 2006; received in revised form 10 August 2007; accepted 16 August 2007

Abstract

Uncertainty on carbon fluxes is determined by the uncertainties of ecosystem model structure, data and model parameter uncertaintiesand the temporal and spatial inaccuracy of the input data retrieval. The objective of this paper is to understand the error propagationand uncertainty of evaporative fraction (EF), soil moisture content (SMC) and water limited net ecosystem productivity (NEP). In thisrespect, C-Fix and spaceborne remote sensing are used for the ‘Brasschaat’ pixel. A simple model based on error theory and a Monte-Carlo approach are used. Different error scenarios are implemented to assess input uncertainty on EF, SMC and NEP as estimated withC-Fix.

The minimum and maximum relative errors on the time averaged EF values from the simple error modelling approach amount 11%and 54%. Applying the Monte-Carlo approach, the minimum and maximum relative errors on EF are 8% and 34%, respectively. Theminimum and maximum relative errors on the averaged SMC from the simple error approach are 4% and 18%, respectively. Fromthe Monte-Carlo approach, the minimum and maximum relative errors on SMC are 4% and 12%. The minimum and maximum absoluteerrors on daily NEP (of 0.22 gC m�2 d�1) estimated from the simple error approach are 1.28 and 4.51 gC m�2 d�1. From the Monte-Car-lo approach, the minimum and maximum absolute errors are 0.86 and 1.85 gC m�2 d�1.

The simple error modelling and Monte-Carlo approach lead to error estimates of the same order of magnitude, though the error val-ues derived from the Monte-Carlo approach are lower. For ecosystem carbon fluxes, both error assessment approaches lead to largedifferences. The complexity of the model and hence the correlation between model parameters might be responsible for this.

Conclusively, the contribution of the error on soil respiration produces the largest uncertainty on NEP. This carbon flux is the mostdifficult one to measure and model. Improvement in NEP flux estimation can only be expected when more insight is gained in the processof soil respiration and hence, better (sub-)models are obtained.� 2007 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Spaceborne remote sensing; Water limitation; Ecosystem carbon fluxes; Uncertainty; Error propagation; Monte-Carlo approach

1. Introduction

Knowledge of the spatial and temporal behaviour of netecosystem carbon uptake is crucial in the framework of

0273-1177/$30 � 2007 COSPAR. Published by Elsevier Ltd. All rights reserv

doi:10.1016/j.asr.2007.08.021

* Corresponding author.E-mail address: [email protected] (W.W. Verstraeten).

environmental conservation, global climate change, thereduction of greenhouse gas emissions (Kyoto protocol,1997) and crop production. The processes of carbon fixa-tion and release are primarily driven by solar radiation,ambient temperature and plant water availability. It iscommon knowledge that water limitation in ecosystemsquite significantly determines net ecosystem productivity(NEP), for example water stress decreases carbon uptake

ed.

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W.W. Verstraeten et al. / Advances in Space Research 41 (2008) 20–35 21

by vegetation. On the other hand, in extremely dry soils,heterotrophic respiration is reduced so that a decrease incarbon uptake by photosynthesis may be (over-)compen-sated (Verstraeten et al., 2007). Another extreme situationis water saturation. Water saturated soils have a reducedsoil fauna and flora activity (hence soil respiration) dueto oxygen depletion. Clearly, knowledge about water avail-ability is crucial to assess carbon fluxes at the ecosystemlevel. Decision-makers require knowledge of how uncer-tainty and potential errors propagate and affect estimatesof carbon fluxes as well.

Since no single model is a true representation of bio-geo-physical processes, all model based estimates of bio-geo-physical variables are subject to uncertainty (Beven andFreer, 2001; Beven, 2006). Due to parameter interactionsand non-linearity it is likely that more than one parameterset fulfils the condition of being the best fit for estimatedand observed data. Additionally, in many ecosystem mod-els, system variables may not be accessible to accurateobservation given the limitations of current measurementtechniques. Hence, errors arise due to model structureand algorithm boundary conditions on the one hand andobservation errors with which the model is compared orfed (Beven, 2001) on the other hand. Model boundary con-ditions cannot be defined with absolute accuracy and theobservational data for model calibration, evaluation andmodel input are not error free. In addition, some modelparameters cannot be measured directly. Hence, broadranges encompassing expected parameter values can beidentified to characterize the relative uncertainty in param-eter measurement (McCabe et al., 2005).

In this paper, a case study is presented using time seriesof METEOSAT pixel data for the Brasschaat EURO-FLUX site, on the error propagation and related uncer-tainty of evaporative fraction (EF), soil moisture content(SMC) and water limited NEP. Error propagation andconsequently, the related uncertainties on EF, SMC andNEP, is modelled with assimilated data from optical andthermal coarse resolution sensors. Two methods character-izing error propagation and variable uncertainty areapplied:

i. A simple error propagation (EP) model, based on errortheory, and

ii. An adopted generalised likelihood uncertainty estima-tion (GLUE, Beven and Binley, 1992) approach withemphasis on a Monte-Carlo (MC) technique.

Spaceborne EF and SMC are assimilated in the Produc-tion Efficiency Model C-Fix which estimates ecosystem car-bon fluxes using earth observation data. The ecosystemcarbon fluxes are gross primary productivity (GPP), netprimary productivity (NPP), soil respiration (SR) and netecosystem productivity (NEP). To illustrate the uncertaintyrelated to EF and SMC inputs and C-Fix structure andinput uncertainty, five error scenarios are defined basedon both the error propagation and Monte-Carlo

approaches. The focus of this paper resides on the resultsof the application of the two error approaches, cited above,which assess the uncertainties on EF, SMC and NEP. Datafrom the EUROFLUX site of Brasschaat for 1997 are usedin this context. The abbreviations, acronyms and symbolsused in this paper are listed in Table 1.

2. Uncertainty assessment methods

In modelling, the incentive to look for an optimal repre-sentation of reality, is a strong requirement. Traditionally,research should lead to a realistic description of processes.In this context, the assumption that a single physicallybased parameter set is obtained after model calibration,is widely dispersed. Although physically based modellingmight suggest that objective results are obtained, non-rig-orous choices are always imminent (Beven and Freer,2001; Beven, 2006), e.g.:

i. The choice of the model to be considered (which sys-tem processes must be included?);

ii. The choice of parameter value ranges;iii. The choice of input data to run the model and their

absolute errors andiv. The choice of tolerance criteria applied in the model

considered.

An error propagation model implies that all inputs areindependent (not correlated). This premise allows errorpropagation to be calculated straightforwardly by estimat-ing the impact of model input variable absolute errors onmodel output errors. An absolute error is estimated as astandard deviation of a variable, not as the 95% confidenceinterval. Given a functional relationship of the shape,

y ¼ f ðxÞ ð1Þ

the sensitivity of the dependent variable y to a genericparameter or variable x in f(x) can be determined as:

Dy ¼ ofox� Dx ð2Þ

Another way of assessing uncertainty, different from the er-ror propagation method is based on a Monte-Carlo (MC)approach. Multiple parameter sets can be defined usingMonte-Carlo sampling to randomly extract parameter setswith pre-defined ranges. The model is run with all sets ofparameter values. Subsequently, the model is run with eachset of parameter values. Assuming an existing confidence inthe model, it is acceptable to assume a number of model re-sults to reflect actual land surface observations, with multi-ple simulations. The generalised likelihood uncertaintyestimation (GLUE) methodology is based on the Monte-Carlo approach to sample parameter sets. More detailsabout the GLUE methodology are provided by Franksand Beven (1997), Beven and Freer (2001) and McCabeet al. (2005). The basic principle of GLUE is the rejectionof the idea of an optimum parameter set or model structure

Table 1Abbreviations, acronyms and symbols used throughout the text

a, b Empirical coefficients of the NDVI–fAPAR relationshipaH, aL Slopes of the line of high/low LST’s as a function of albedo [K]Ad Autotrophic respiratory fraction [–]Allo Allometric factor [–]ATI Apparent thermal inertiabH, bL Intercepts of the line of high/low LST’s as a function of albedo [K]c Climatic efficiency [–]c5, c6 Empirical coefficients weighting water limitation in RUE originating from SMC or from the atmosphere [–]C Solar correction factorCO2fert Normalised CO2 fertilisation factor,{0:1} [–]EF Evaporative fraction [–]EP Error propagation (simple error modelling approach)fAPAR Fraction of absorbed photosynthetic active radiation {0:1} [–]Fa Stomatal regulating factor controlled by atmospheric changes [–]Fs Stomatal regulating factor controlled by soil moisture availability [–]GLUE Generalised likelihood uncertainty estimationGPP Gross primary productivity [gC m�2 d�1]H Sensible heat flux [W m�2]:LST Land surface temperature [K]LSTd,n Daytime (subscript d) or night time (subscript n) land surface temperature [K]MC Monte-Carlo approachME Model efficiency [–]n Amount of data (number) [–]NDVI Normalised difference vegetation index [–]NDVItoc NDVI at the top of the canopy {�1:1} [–]NEP Net ecosystem productivity [gC m�2 d�1]NEP b�1

y Component of the heterotrophic respiration rate coefficient [gC m�2 a�1]NPP Net primary productivity [gC m�2 d�1]Oi The ith measured or observed valueO The average of the observationsPAR Absorbed photosynthetic active radiation [MJ m�2 d�1]Pi The ith simulated or predicted valuep(Tc) Normalised temperature dependency factor at canopy level {0:1} [–]Rh Heterotrophic respiration [gC m�2 d�1](R)RMSE (Relative) root means square errorRUE Radiation use efficiency [gC MJ(APAR)�1]SAS Soil aeration stress [–]Sg,d Daily incoming global solar radiation [MJ m�2 d�1]SMC Soil moisture content [m3 m�3]SMCmax Maximum soil moisture content [m3 m�3]SMCmin Minimal soil moisture content [m3 m�3]SMSI Soil moisture saturation index [–]SR Soil respiration [gC m�2 d�1]SRF Soil stress respiration factor [–]SSS Soil stress strength [–]Tc(t) Air temperature measured in the canopy at time t [�C]Ts(t) Soil temperature measured at time t [�C]a0 Albedo [–]kE Latent heat flux [W m�2]h Volumetric soil moisture content [m3 m�3]hres, hsat Volumetric residual and saturate soil moisture content [m3 m�3]hmax, hmin Volumetric maximum and minimum soil moisture content [m3 m�3]Da Absolute error of the slope of the fAPAR–NDVI relationDhopt Absolute error of the optimal volumetric soil moisture content [m3 m�3]Dhwp Absolute error of the volumetric soil moisture content at wilting point [m3 m�3]

22 W.W. Verstraeten et al. / Advances in Space Research 41 (2008) 20–35

in favour of the concept of ‘equifinality’ or equivalence ofmodel structures and parameter sets. This implies thatthe relative model performance in terms of a likelihoodmeasure is evaluated.

Multiple simulations are run considering multiple sets ofparameter values. These sets are established, using theirassumed respective probability distributions, in the MC

framework. The simulations of the MC realisations aresubsequently weighted by the likelihood measures. Then,all simulations with a value above a relevant thresholdare given a positive weight in the prediction and areselected for further consideration. Simulations below therelevant threshold lead to parameter sets considered to benon-behavioural and hence not predictive. A probability


distribution of the parameter sets is obtained by rescalingthe behavioural parameter sets so that their sum equalsone. Distribution type assumptions are not made whendetermining the prediction limits, only the available sampleof predictions is used. The errors of input data, non-linear-ity and parameter interactions are implicitly treated by thelikelihood measure, since this measure reflects the modelresponse for a given set of parameters and inputs. Theerror structure observed within the evaluation period isassumed to be similar to future model versions. The GLUEprocedure involves the following steps (Beven and Binley,1992; Campling et al., 2002):

i. Definition of a sampling sub-space for each parame-ter included in the analysis;

ii. Sampling of the parameter space;iii. Selection of a likelihood measure for model

performance;iv. Selection of acceptable (behavioural) simulations andv. Upgrade of prior information by means of sample

data to produce posterior probabilities.

Since information on the distribution of most of theparameters of the EF model is scarce, a uniform distribu-tion of the parameter space, from which parameter setsare selected, is used in this study. For each parameter ofthe EF and SMC model, 5000 (EF) and 1000 (SMC) valuesare randomly chosen from a uniform distribution. Theamount of parameter values is limited to accommodatereasonable computing times. Only 1000 numbers were cho-sen for the SMC case, since the model is more complex andthus more computing time is required. For each parameterset the models are run with, confidence limits of the modeloutputs are computed. To evaluate model performance ofEF and SMC the likelihood measure for the parameter setsselected is model efficiency (ME):

ME ¼ 1�Pn

i¼1ðOi � P iÞ2PðOi � OÞ2

ð3Þ

In Eq. (3):Oi is the ith measured or observed value;Pi is ith simulated or predicted value;n is the total amount of available measurements in thetime interval considered;O is the average value of the observations.

Selecting ME as a measure of likelihood is intrinsicallysubjective. Indeed, different likelihood measures will leadto different likelihoods. Model efficiency is the ratio of datamean square error on observed data variance. Hence, MEis a measure of the correlation between observed and sim-ulated data and should optimally be equal to unity. If0 < ME < 1, then the modelling results are acceptable. IfME = 0, then the average of the observed values is as goodas that of the simulated values. If ME < 0, the average ofthe observed values leads to a better prediction than the

model itself. The model should then be rejected. Hence,only those simulations are considered for further analysis,having a ME value between 0 and 1. Behavioural modelsimulations of EF and SMC are selected when ME is largerthan 0.20 (for EF) and 0.50 (for SMC). The previous MEvalue selection is chosen to ensure enough behavioural sim-ulations are left in the analysis since for EF, 5000 valueswere chosen for each input parameter and for SMC only1000 values for each model input parameter. In this studywe limit the GLUE approach for EF and SMC to step iv.Step v, i.e., the upgrade of prior information to produce aposteriori probabilities, is not performed by abstention ofappropriate prior information.

3. Remote sensing models estimating fully water limitedecosystem carbon fluxes

3.1. The Product Efficiency Model C-Fix: summary

C-Fix has been applied on the local (Veroustraete et al.,2004; Verstraeten et al., 2006a) as well as regional (Verou-straete et al., 2002; Chhabra and Dhadwall, 2004; Lu et al.,2005; Verstraeten et al., 2007) scales and has the potentialto be applied at the global scale as well. The fully water lim-ited C-Fix model has been validated based on Europeanforest data by Verstraeten et al. (2006a).

In C-Fix, the evolution of radiation absorption effi-ciency in the PAR (photosynthetic active radiation) band(or fAPAR) for vegetation, is inferred from space observa-tions using the NDVI (Normalized Difference VegetationIndex) and from radiation use efficiency (RUE). RUE isthe integrated efficiency of photosynthetic metabolism,e.g., the conversion efficiency of absorbed PAR into drymatter or fixed carbon. fAPAR is estimated from theNDVI using a linear relationship according to Myneniand Williams (1994). Daily net ecosystem productivity(NEP) is the balance of daily gross carbon uptake by pho-tosynthesis (GPP) reduced by autotrophic (vegetation) res-piration and reduced as well by a soil respiratory flux (SR)originating from the decomposition of soil organic matterand root respiration. Full water limitation of carbonuptake and release of ecosystems can be associated withtwo carbon flux levels:

(i) The GPP level. Water availability for photosynthesisand evapotranspiration is crucial for these processes.Moreover in the Product Efficiency Model approach,RUE is an integral efficiency of photosyntheticmetabolism and is represented by one value. RUEdepends on water availability.

(ii) The soil respiration level. Soil moisture affects soilecology and biological soil life profoundly.

A detailed description of the C-Fix model is found inVerstraeten et al. (2006a).

Daily net ecosystem carbon flux (gC m�2 d�1) isestimated as:


NEPd ¼ ð1� allo � AdðT cÞÞ �GPPd � ½SRF � RhðT sÞþ ðð1� alloÞ � AdðT cÞÞ �GPPd� ð4Þ

wherein

GPPd ¼ pðT cÞ � CO2fert �RUEwl � f APAR � c � Sg;d ð5Þf APAR ¼ a �NDVItoc þ b ð6ÞRUEwl ¼ bRUEmin þ ðc6 � F s þ c5 � F aÞ

� ðRUEmax �RUEminÞc ð7ÞSRF ¼ ½SRmin þ ð1� SASÞ � ðSSSÞ � ðSRmax � SRminÞ� ð8Þ

Rh;d ¼ ks;y � QT atm;d

1010 ð9Þ

ks;y ¼P365

d¼1NEPd

b

P365d¼1Q

T atm;d10

10

ð10Þ

In Eqs. (3)–(10):NEPd is daily net ecosystem productivity [gC m�2 d�1];GPPd is daily gross primary productivity [gC m�2 d�1];allo is the allometric factor dividing the autotrophic car-bon release in an above soil (leaves) and within soil part(roots) [–];Ad is the autotrophic respiratory fraction (computedaccording to Goward and Dye, 1987) [–];Tc and Ts are canopy and soil temperatures, respectively[�C];p(Tc) is a normalised temperature dependency factor{0:1} [–] (defined according to Veroustraete et al., 1994);CO2fert is a normalised CO2 fertilisation factor (definedaccording to Veroustraete et al., 1994);RUEwl is RUE taking full water limitation into account(Verstraeten et al., 2006a) [gC MJ(APAR)�1];RUEmin and RUEmax represent minimum and maxi-mum RUE [gC MJ(APAR)�1];fAPAR is the fraction of absorbed PAR (PhotosyntheticActive Radiation), or the efficiency of PAR absorptionby a plant canopy {0:1} [–];NDVItoc is the NDVI at the top of a canopy {�1:1} [–];Sg,d is daily incoming Global Solar Radiation[MJ m�2 d�1];c is climatic efficiency and equals 0.48 (McCree, 1972) [–];Rh is heterotrophic respiration (computed according toVeroustraete et al., 2004) [gC m�2 d�1];ks,y is a heterotrophic respiration coefficient estimatedfrom Eq. (10) on a yearly basis (subscript y).Q10 is the relative increase of the respiration flux for a10 K increase in temperature (Q10 is 1.5 according toMaisongrande et al., 1995);b is a parameter which can be obtained from calibrationwith yearly NEP tower flux measurements;Fs is a stomatal regulation factor controlled by soilmoisture availability [–];Fa is the stomatal regulation factor for atmosphericwater vapour changes [–];SRF is a soil respiration factor under soil water stressconditions [–];

SAS is a soil aeration stress dependent on soil moisturecontent [–];SSS is a soil strength stress dependent on soil moisturecontent [–];SRmax and SRmin are minimal, respectively, maximalsoil respiration factors (between 0 and 1) [–];c5 and c6 are empirical coefficients [–] reflecting the rela-tive importance of water limitation of photosynthesis forsoil and atmosphere.

Ambient temperature is an input for C-Fix, determiningthe value of the normalised temperature dependency factor,the CO2 fertilisation factor and the autotrophic and hetero-trophic respiration rates (Veroustraete et al., 2002, 2004).The canopy compartment is assumed to act as a heat buf-fer, hence canopy photosynthesis and respiration are deter-mined by canopy temperature, which takes values differentfrom atmospheric temperature. When estimating the het-erotrophic respiration of soils, it is clearly incorrect to takeatmospheric temperature as a proxy for soil temperature. Asimple heat transfer model is used, analogous to the semi-empirical modelling approach of Wagner et al. (1999, 2003)and Ceballos et al. (2005), to estimate both canopy and soiltemperatures from atmospheric temperature as describedby Verstraeten et al. (2006a). Water limitation is includedto estimate NEP using:

(i) The stomatal regulating factor (Fs) controlled by soilmoisture availability (SMC);

(ii) The stomatal regulation factor (Fa) for atmosphericwater vapour changes (EF);

(iii) Soil aeration stress (SAS) which depends on soilmoisture content (SMC) and

(iv) Soil strength stress (SSS) which also depends onSMC.

The approaches to estimate EF and SMC applying earthobservation data, are described in the following twosections.

3.2. The evaporative fraction model generating input for C-

Fix

Evaporative fraction (EF) is the ratio of the surfacelatent heat flux over the sum of the surface latent and sen-sible heat fluxes. A simple method to obtain EF is by com-bining albedo (a0) and land surface temperature (LST0) (Suand Menenti, 1999; Su et al., 1999; Roerink et al., 2000).Detailed descriptions of this method as well as its resultscan be found in Verstraeten et al. (2005b).

Surface wetness affects both surface reflectance and landsurface temperature (LST). Basically, linear relationshipsbetween LST and albedo are obtained using cold and wet(evaporation controlled), and warm and dry pixels (radia-tion controlled). For each image these relationships are cal-culated based on a LST-Albedo scatterogram:


K ¼ kEH þ kE

� LSTH � LST0

LSTH � LSTkE

� aH � a0 þ bH � LST0

ðaH � akEÞ � a0 þ ðbH � bkEÞð11Þ

In Eq. (11):H is the sensible heat flux [W m�2];kE is the latent heat flux [W m�2];LSTH is land surface temperature for dry pixels [K];LSTkE is land surface temperature for wet pixels [K];LST0 is land surface temperature [K];a0 is surface albedo [–];aH, akE are the slopes of the lines of, respectively, high andlow temperatures as a function of surface albedo [K];bH, bkE are the intercepts of the lines of, respectively,high and low temperatures as a function of surfacealbebo [K].

3.3. Generating input for C-Fix, applying the soil moisture

content model

A method to estimate SMC with optical and thermalspectral information from METEOSAT imagery basedon thermal inertia is presented and validated for Euro-pean forests by Verstraeten et al. (2006b). Thermal inertiais a body property of materials and describes their resis-tance to temperature variations. The approach of Mitraand Majumdar (2004) to infer apparent thermal inertia(ATI) has been adopted because of its straightforwardapplicability in routine calculations. ATI is estimatedusing measurements of surface spectral albedo (a0) anda range of diurnal land surface temperature variations,D LST0 (LST0,d–LST0,n). From ATI time series, minimaand maxima ATI values are extracted and recombinedinto a soil moisture saturation index (SMSI) as expressedin Eq. (10):

SMSI0ðtÞ ¼ATIðtÞ �ATImin

ATImax �ATImin

¼CðtÞ � 1�a0ðtÞ

DLST0ðtÞ

� �� C � 1�a0

DLST0

� �min

C � 1�a0

DLST0

� �max� C � 1�a0

DLST0

� �min

ð12Þ

In Eq. (12):ATImin, ATImax, ATI(t) are the minimum, maximumand apparent thermal inertia [K�1] at time t [day];C is a solar correction factor (a function of latitude andsolar declination) [–];a0 is the broadband albedo [–];DLST0 is the difference between day and night bright-ness temperatures [K];SMSI0(t) is a remotely sensed surface moisture satura-tion index at time t [–].

The soil moisture saturation index (SMSI) is the degreefor which a soil reaches its maximal water content and isexpressed as in Eq. (13):

SMSIðtÞ ¼ hðtÞ � hres

hsat � hres

ð13Þ

Since saturated as well as residual soil moisture content arequantified in laboratory conditions, soil moisture values infield conditions are unlikely to equal extremely high or lowvalues. Hence, hsat and hres are substituted by a maximaland minimal SMC (hmax and hmin), respectively. To convertsoil surface SMSI0(t) to the SMSI(t) of a 1 m soil profile, aMarkov type filter is applied based on a two layer waterbalance equation (the surface layer and the reservoir be-low). According to Wagner et al. (1999) and Ceballoset al. (2005), a temporal autocorrelation function is added.This autocorrelation function stems from Delworth andManabe (1993) and Vinnikov et al. (1996). They have con-cluded that the spectrum of temporal variations of SMCcorresponds to a first-order Markov process with a charac-teristic time length of decay. The SMC of the 1 m soil pro-file can then be estimated using Eq. (14):

hðtÞ ¼ SMSIpðtÞ � ðhmax � hminÞ þ hmin

¼P

tSMSI0ðtiÞ � e�t�ti

Tð ÞP

te� t�ti

Tð Þ� ðhmax � hminÞ þ hmin ð14Þ

In Eqs. (13) and (14):SMSI0 is the soil moisture saturation index for the sur-face layer derived from remote sensing [–];SMSIp(t) is the soil moisture saturation index for a 1 msoil profile at time t [–];ti is the time at which a discrete change of SMSI0 takesplace [day];T is the time constant or characteristic time length [day]related to a specific quantum of SMSIp change occurringat a specific moment t in time, after a discrete change ofSMSI0 at time t whereby t > ti. T depends essentially onclimate and soil type;ht is the volumetric soil moisture content at a time t

[m3 m�3];hres, hsat, hmax and hmin are respectively volumetric resid-ual, saturated, maximum and minimum soil moisturecontents [m3 m�3].

3.4. Scenario’s to quantify uncertainty

Table 2 shows the assumed absolute errors of the modelparameters to estimate EF, SMC and NEP for five errorcases. The best (1), a medium (2) and a worst case scenario(3) as well as a best case scenario with varying errors of aH

and akE for the EF model, hmax and hmin for the SMCmodel and DNEP b�1

y (component of the heterotrophic res-piration rate coefficient, see also Eq. (10)), Dhwp, Dhopt andDa (slope of the fAPAR–NDVI relationship) for the C-Fix

Table 2Absolute error values for the model parameters for five error case scenarios

Absolute errors Case Product

1 2 3 4 5

DLST0,d [K] 1.0 2.0 4.0 1.0 1.0 EF, SMCDLST0,n [K] 1.0 2.0 4.0 1.0 1.0 SMCDa0 [–] 0.01 0.05 0.10 0.01 0.01 EF, SMCDaH,L [–] 0.10 0.15 0.20 0.15 0.20 EFDbH,L [K] 1.0 2.0 4.0 1.0 1.0 EFDhmax [m3 m�3] 0.010 0.050 0.100 0.050 0.100 SMCDhmin [m3 m�3] 0.005 0.010 0.020 0.010 0.020 SMCDNDVI [–] 0.01 0.1 0.2 0.01 0.01 NEPDRUE [gC MJ�1 APAR] 0.03 0.3 0.6 0.03 0.03 NEPDNEPyb�1

y [gC m�2 d�1] 13.2 132 264 132 264 NEPDhwp [m3 m�3] 0.001 0.01 0.05 0.01 0.05 NEPDhopt [m3 m�3] 0.003 0.01 0.05 0.01 0.05 NEPDaa [–] 0.015 0.075 0.15 0.075 0.15 NEP

In the right column the end product is given (EF, SMC or NEP).a Da is the slope of the fAPAR–NDVI relation.


model (scenario 4 and 5). In the best and worst case sce-nario’s, it is assumed that the absolute errors of the selectedparameter are respectively the lowest and highest error val-ues. For the medium case scenario the absolute errorboundaries are located between the best and worst case sce-nario error values. In scenario 4(5), aH and akE for the EFmodel, hmax and hmin for the SMC model and DNEP b�1

y ,Dhwp, Dhopt and Da for the C-Fix model, are the same asin scenario 2(3). NEP b�1

y is used in Eqs. (9) and (10) toderive the heterotrophic respiration component of the eco-system carbon balance. DNEP b�1

y stands for the error inthe NEP b�1

y value. The other parameters take the sameerror values as in scenario 1. Table 2 reads as follows.For scenario 1, the assumed absolute error of daily LST0

is 1.0 K, e.g., the actual LST value boundary is ±1.0 K.Analogous for the other Table 2 entries.

4. Datasets

To analyse the uncertainty of water limited ecosystemcarbon fluxes estimated with C-Fix, the following datasets(for 1997) are used: a meteorological time series dataset oftemperature, incoming radiation, NEP flux and otherin situ data from the Brasschaat EUROFLUX site in Flan-ders (BE2). The fAPAR dataset from Gond et al. (1999) isused. It is recalculated to NDVI values based on NOAA/AVHRR data since C-Fix is NDVI driven (Verstraetenet al., 2006a). Optical and thermal METEOSAT imageryis applied to derive albedo and land surface temperature(LST). From these remote sensing products EF and SMCare estimated as proposed in Verstraeten et al. (2005b,2006b). The EUROFLUX sites, operational during 1997,are all located in European forested areas as can be verifiedin Valentini et al. (2000), the website http://www.fluxnet.ornl.gov/fluxnet/EUROFLUX/index.htm and numerousother authors (among them Wilson et al., 2002a,b; Dolmanet al., 1998, 2003). The Brasschaat site can be characterized

as a Pinus sylvestris stand located on a soil with a coarsesoil texture. Its altitude is 10 m above sea level, and itsgeo-location is 51�18 0N latitude and 04�31 0E longitude.

5. Results and discussion

5.1. Remarks on uncertainty

The error estimated with the error propagation method(Eq. (1)) is computed as a total differential (function ofmore than one independent variable). This total differentialimplies that its variables are mutually independent. Theerror estimated with Eq. (1) is assumed to represent the fullerror range wherein the actual value is located. Expresseddifferently, no assumptions are made about the type ofprobability distribution a variable takes. When definingthe error scenarios (Table 2), it is assumed that the absoluteerror of the model input parameters, encompass all possi-ble values within valid boundaries.

Opposite to error theory based on total differentiation, aMonte-Carlo (MC) approach, allows a priori assumptionsto be made on the type of statistical distribution a variableor parameter can elicit. Moreover, with the MC approach,no assumption of independence of parameters or variablesis made. With the MC approach, the actual probability dis-tributions are not assumed to be known, e.g., uniform dis-tributions are chosen. Moreover, when applying a uniformdistribution one closely approaches the absolute errorrange of the error propagation approach. Typically, a uni-form distribution makes all sample values to have equalweights. In the MC approach, as implemented in thisstudy, when producing random parameter sets and whenrunning the C-Fix model, the error on the output is repre-sented by half the 95% confidence interval, thereby assum-ing that the output values are normally distributed. As arule of thumb the error interval can be recalculated to

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the 68% or 99% confidence intervals as one and three unitsof the standard deviation, respectively.

A last remark to be made is that, notwithstanding theresults of this error analysis the real error of the model out-puts is unknown, since the exact error on the inputs is notwell known either. Therefore, multiple error cases (differenterror scenarios) are applied.

5.2. The uncertainty of EF retrieval

Using the five error scenarios as outlined in Table 2, themaximum, minimum, average and standard deviations ofthe time series of the absolute errors of EF are given inTable 3. These errors are computed by applying the EPand MC approaches for a METEOSAT image time series

Table 3Maximum, minimum, average and standard deviation of the absolute errors onand the Monte-Carlo approach (MC) (right) for the Brasschaat times series u

Case Error intervals EPEF

Max Min Average S

1 0.19 0.00 0.07 02 0.42 0.00 0.17 03 0.83 0.00 0.33 04 0.34 0.00 0.09 05 0.85 0.00 0.11 0

Average observed EF from METEOSAT is 0.62.

EP Case 1

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EF [-

]

Error interval Spaceborne Observed

Case 3

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0.80

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EF [-

]

Error interval Spaceborne Observed

Fig. 1. The uncertainty of EF for the time series derived with the iMETEOSMonte-Carlo approach for case 1 and 3 of the five error scenarios (Table 2). Tconfidence interval is indicated for the MC approach.

for the growing season of 1997 using the ‘Brasschaat pixel’(Gond et al., 1999).

Fig. 1 illustrates the uncertainty of the time series of EFfor the 1997 growing season for the EP and MCapproaches. It is shown for case 1 and 3 of the five errorscenarios of Table 2.

The tabulated error analysis values of the EP approachindicates that the average absolute error more or less dou-bles from case 1 to 2, and from case 2 to 3. Error magni-tudes are comparable for case 1, 4 and 5, being anindication, when determining EF, for error insensitivityof the parameters akE and aH. This is corroborated withthe results of the MC error analysis approach. The abso-lute errors obtained with the MC approach are lower thanthose computed with the EP approach. The magnitude of

the EF computed with the simple error propagation approach (EP) (left)sing the five error scenarios given in Table 2

Error intervals MCEF

D Max Min Average SD

.02 0.14 0.00 0.05 0.02

.05 0.22 0.00 0.11 0.03

.09 0.50 0.01 0.21 0.06

.05 0.18 0.00 0.06 0.02

.07 0.20 0.00 0.06 0.02

MC

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[-]

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EF [-

]


AT-Chain (growing season of 1997) using the error propagation and thehe absolute error is given on the time series of the EP approach, the 95%

0.00

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ME

[-]

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0.75

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2900 2920 2940 2960 2980 3000bH [10K]

ME

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Behavioural Non-behavioural

Fig. 2. Dot plots for a parameter range in the x-axis versus modelefficiency in the y-axis. These model simulations result from the Monte-Carlo approach for coefficients aH and bH (scaled with a factor 10).


the MC average absolute error estimates is approximately23% to 47% of the magnitude of the EP error estimates.

Similar to the GLUE approach dot plots – i.e., plots fora parameter value range in the x-axis versus model effi-ciency in the y-axis – dot plots can be constructed as illus-trated in Fig. 2. They confirm that a large value range ofthe EF parameters (aH, akE, bH, bkE) leads to high modelefficiency values. This suggests that there is no identifiableunique parameter set that produces the highest model effi-ciency value. When the cumulative distribution functionsof the behavioural and non-behavioural model simulationsfor the parameters aH, akE, LST and albedo are compared,the discriminative power of the statistical parametricKolmogrov–Smirnov test is used. The test reveals thatthe cumulative distribution functions are not significantlydifferent and hence, the parameters used to estimate EFare rather insensitive to the errors on the model inputs.

5.3. Uncertainty of SMC retrieval

The uncertainty of SMC time series derived withMETEOSAT imagery for the 1997 growing season, apply-ing the EP and MC approaches with the five error scenar-ios as presented in Table 2 is summarised in Table 4.

The EP approach results in Table 4 show that the ATIaverage absolute error doubles with each error case (dou-bling the error on LST from 1 to 2, and from 2 to 3, andfor albedo from 0.01 to 0.05, and from 0.05 to 0.10). Botherror analysis approaches lead to similar results, thoughthe absolute errors on ATI are generally smaller with the

MC approach. For the SMSI0 parameter the average abso-lute error almost doubles with the EP approach (case 1 to3: 0.08–0.14–0.28). This is opposite to the MC approach(0.08–0.12–0.18). Similar observations can be made forthe absolute errors of SMSIp time series in case of theapplication of the EP approach. Here, a large differencebetween the two error approaches can be observed. Withthe MC approach, a smoothing of the errors occurs forSMSIp (average error is 0.03). The absolute error of SMSIp

derived with the EP approach, roughly doubles for case 1to 2, and 2 to 3. For cases 1 and 2, both error approacheslead to comparable results. Finally, the absolute error ofthe SMC values originating from the two error approachesis similar. The average errors also double for case 1 over 2to 3 for the EP and MC approaches.

To illustrate the uncertainty of ATI, SMSI0, SMSIp andSMC time series derived with METEOSAT imagery for the1997 growing season, the 95% confidence interval is givenfor the MC approach and for the four time series men-tioned for error scenario case 1 in Fig. 3.

Table 4 clarifies that the parameters SMCmin andSMCmax have the highest impact on the error on estimatedSMC, since the average absolute error does not changemuch between error cases 2 and 4 (errors on LST andalbedo are halved, while the errors of SMCmin and SMCmax

remain unchanged) and between error cases 3 and 5 (sameas in case 2 and 4, but with different error values). Theimpact of a 4 K LST error on the absolute error of SMCis not highly different from a 1 K error on LST.

To summarise, the model error sensitivity decreasesfrom ATI over SMSI0 and SMSIp till it reaches its mini-mum with SMC.

The cumulative error distribution functions of thebehavioural and non-behavioural model simulations forthe parameter SMCmin indicate a low sensitivity of thisparameter when simulating SMC (not significantly differentaccording to the Kolmogrov–Smirnov test). On the otherhand, a statistically significant difference is observed forthe parameter SMCmax. When starting from the SMSIp

time series and when implementing 10,000 different valuesfor both the SMCmin and SMCmax parameters, the Kolmo-grov–Smirnov test indicates a significant difference for bothparameters. Both types of error model simulations are eval-uated comparing the remotely sensed based SMC time ser-ies with forest plot scale observations obtained at theBrasschaat site (Verstraeten et al., 2005a).

5.4. Uncertainty of NEP retrieval

The uncertainty of the ecosystem carbon fluxes derivedwith C-Fix for the Brasschaat site (1997 growing season)with the EP and MC approach using five error scenarios(Table 2) is listed in Table 5.

The uncertainties of GPP, NPP, SR and NEP time ser-ies, estimated with C-Fix (1997 growing season), for case 1using the EP and MC approaches is illustrated in Figs. 4and 5, respectively.

Table 4The maximum, minimum, average and standard deviation of the absolute errors on ATI, SMSI0, SMSIp, and SMC computed using the EP (left) and MC(right) approaches with the Brasschaat time series and for the five error scenarios of Table 2

Case Error intervals EP Error intervals MCATI ATI

Max [–] Min [–] Average [–] SD [–] Max [–] Min [–] Average [–] SD [–]

1 0.00 0.06 0.14 0.00 0.419 0.001 0.047 0.0762 0.00 0.12 0.28 0.00 0.489 0.002 0.102 0.1353 0.00 0.24 0.57 0.00 0.572 0.004 0.203 0.1984 / / / / / / / /5 / / / / / / / /

SMSI0 SMSI0

1 0.00 0.07 0.17 0.00 0.317 0.000 0.075 0.0752 0.00 0.14 0.32 0.00 0.447 0.000 0.123 0.1433 0.00 0.27 0.63 0.00 0.472 0.000 0.185 0.1774 / / / / / / / /5 / / / / / / / /

SMSIp SMSIp

1 0.01 0.02 0.02 0.00 0.041 0.011 0.025 0.0072 0.02 0.04 0.03 0.00 0.039 0.012 0.026 0.0083 0.02 0.09 0.05 0.00 0.051 0.027 0.031 0.0084 / / / / / / / /5 / / / / / / / /

SMC SMC

Max [m3 m�3] Min [m3 m�3] Average [m3 m�3] SD [m3 m�3] Max [m3 m�3] Min [m3 m�3] Average [m3 m�3] SD [m3 m�3]

1 0.01 0.01 0.00 0.00 0.011 0.005 0.008 0.0012 0.01 0.01 0.01 0.00 0.026 0.011 0.014 0.0023 0.02 0.03 0.04 0.00 0.059 0.020 0.024 0.0064 0.00 0.01 0.00 0.00 0.025 0.011 0.013 0.0025 0.01 0.02 0.00 0.00 0.051 0.020 0.024 0.005

Average observed SMC from METEOSAT is 0.20 [m3 m�3].

0.000.200.400.600.801.001.20

60 84 100

126

158

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288

Julian day [-]

TI [K

-1]


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0 [-]

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p [-]

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SMC

[m3 m

-3]


a b

c d

Fig. 3. Uncertainty of the ATI, SMSI0, SMSIp and SMC time series derived from METEOSAT imagery (growing season of 1997) for the Monte-Carloapproach for error scenario case 1 (Table 2). The 95% confidence interval is given in grey.


Table 5The maximum, minimum, average and standard deviation of the absolute errors on the daily GPP, NPP, SR and NEP values computed using the EP (left)and MC (right) approaches for the Brasschaat times series and for the five error scenarios of Table 2

C-Flux Case Error intervals EP Error intervals MC

Max Min Average SD Max Min Average SD

GPP 1 6.14 0.00 0.96 1.24 4.98 0.00 0.93 1.052 13.06 0.00 1.91 2.56 6.43 0.00 1.20 1.323 26.14 0.00 3.92 5.20 9.34 0.00 1.73 1.894 6.56 0.00 1.07 1.35 5.24 0.00 0.94 1.065 16.71 0.00 2.74 3.49 5.63 0.00 1.01 1.15

NPP 1 6.93 0.00 1.09 1.42 1.62 0.00 0.44 0.412 14.39 0.00 2.09 2.83 1.64 0.00 0.44 0.413 28.67 0.00 4.25 5.72 3.03 0.00 0.88 0.794 7.42 0.00 1.19 1.54 1.61 0.00 0.44 0.415 19.38 0.00 2.98 3.85 1.73 0.00 0.47 0.44

SR 1 2.16 0.25 0.54 0.29 0.41 0.15 0.28 0.062 2.33 0.29 0.60 0.31 0.61 0.22 0.42 0.083 3.09 0.54 1.04 0.40 1.58 0.44 0.85 0.194 2.18 0.27 0.57 0.29 0.58 0.20 0.41 0.095 2.72 0.54 0.97 0.32 1.16 0.37 0.77 0.18

NEP 1 6.87 0.25 1.28 1.32 4.21 0.14 0.86 0.812 14.12 0.29 2.25 2.68 5.52 0.22 1.13 1.013 28.08 0.54 4.51 5.42 7.39 0.43 1.85 1.374 7.43 0.27 1.39 1.43 4.40 0.21 0.95 0.815 18.87 0.54 3.28 3.59 4.70 0.37 1.24 0.82

The daily average GPP, NPP, SR and NEP are 3.10, 1.83, 1.80, 0.22 gC m�2 d�1, respectively.


In summary, Table 5 illustrates that the average abso-lute error on GPP, computed for the EP approach,increases with a factor two from error case 1 to 2 and case2 to 3. The absolute error roughly doubles for case 4, and isof the same order of magnitude for case 1 and 5. Since theaverage absolute error of EF does not significantly changebetween error cases 1, 4 and 5, and since the average abso-lute error of SMC changes only marginally between errorcases 1 and 4, it is suggested that the highest impact onGPP originates from the error on SMC and much less fromthe combined errors of optimal SMC, SMC at wilting pointor the two parameters of the fAPAR–NDVI relationship.Concerning the flux NPP, a similar behaviour of the erroris observed with the distinction that the error offset ishigher than that for GPP. In contrast to the errors ofGPP and NPP, the error of soil respiration (SR) approxi-mately doubles for case 2 to 3 and for case 4 to 5.

The error of SR seems to cancel out looking at the errorcases 1, 2 and 4. The average errors for these cases have thesame order of magnitude and elicit the same trend as theerrors of SMC in Table 4. This suggests that the error ofSMC influences the error of SR quite importantly. Appar-ently, the impact of the NEP b�1

y error seems to make thedifference.

Error cases 3 and 5 give similar results; hence the errorsof the inputs in Table 2 and the errors of the time seriesinputs EF and SMC seem to compensate for the error onSR. The absolute errors of NEP roughly double for case1 to 2, case 2 to 3, and case 4 to 5. Error cases 1 and 4are similar, eliciting the same trends as for the error case

results of EF and SMC (Tables 3 and 4, respectively). Thisobservation suggests that the critical error on optimal andwilting point SMC and the slope of the fAPAR–NDVIrelationship seems to take error values between thoseobtained for case 4 and 5.

The impact of the different error cases on the absoluteerrors of GPP and NPP when applying the MC approachis not strong, except for error case 3 (Table 5). For case3 the average absolute error of NPP almost doubles, ascompared to cases 1 and 4. The same is true for the abso-lute error on NPP for cases 1, 2, 4 and 5. According to theMC approach, error cases 1, 2, 4 and 5 are similar. Hence,only error accumulation (case 3) seems to impact on theerror of NPP. This is opposite to the impact of the fiveerror cases on the absolute error of SR. For this lastparameter the error values are similar for cases 3 and 5,and for cases 2 and 4. The smaller error values of SMCand EF inputs in case 5 are slightly compensated by the lar-ger errors on the NEP b�1

y parameter and SMC at wiltingpoint and for optimal conditions. Here contrasts appearwith the EP approach. The absolute error of SR for theMC approach is one-half to one-third of that of the EPapproach. This suggests that a significant correlation existsbetween model and input parameters. For example, whenconsidering the NEP flux error, the error induced byNEP b�1

y makes a significant difference since the errors onthe other parameters and time series inputs are smallerand have an only small effect on the NEP error.

The results of the EP approach suggest that ecosystemcarbon fluxes are very sensitive to soil moisture parameters

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a b

c d

Fig. 4. The uncertainty of GPP (a), NPP (b), SR (c) and NEP (d) time series derived from C-Fix (growing season of 1997) using the simple errorpropagation approach for case 1 of Table 2.


(such as SMC at field capacity, saturation and wiltingpoint) and the parameter NEP b�1

y (for the SR and NEPfluxes). Moreover, for the uncertainty analysis of SMCretrieved from remote sensing the outcome is that theSMC time series estimation is subject to the impacts ofSMCmin as well as SMCmax. Note that maximum and min-imum SMC’s are linked to SMC at wilting point and fieldcapacity.

Comparing both error estimation approaches (see Table5, and Figs. 4 and 5), the behaviours of the absolute aver-age errors of GPP and NPP are slightly different. Theerrors are smaller for the MC approach except for errorcase 1 (and 4) while the error of GPP is not much differentbetween both methods. Concerning SR and NEP fluxes,the error for case 5 is smaller when compared to case 3for the MC approach as opposed to the EP approach.

In general, the small errors observed with the MCapproach, can be understood by the correlation betweensome parameters located in the EF and SMC time series.The impact of the errors of the fAPAR and EF inputs wereanalysed by Verstraeten et al. (2006a). The analysisrevealed that the impact of the fAPAR input error on

GPP and NEP is more pronounced than the impact ofthe EF input error.

5.5. Uncertainty of EF, SMC and NEP as evidenced by the

literature

Jiang and Islam (2003) applied different methods to cal-culate EF (with AVHRR imagery of the Great Plains, USAJuly 1997) and reported RMSE values between 0.12 and0.48 with a minimal RRMSE of 14.3%. Bastiaanssenet al. (1998b), validating the SEBAL model, report differ-ences between tower and remotely sensed EF, RMSE esti-mates, to range from 0.10 to 0.20.

Concerning SMC, reported errors are between 0.002 and0.05 m3 m�3 or 0.5% and 10% (Narayan and Lakshmi,2005). De Jeu (2003) report coefficients of variation of22% to 37% for retrieved SMC (standard deviationbetween 0.02 and 0.12 m3 m�3). De Ridder (2000) reportsa relative error of approximately 20%. Although not allperfectly comparable, these error estimates indicate theaccuracy and uncertainty of remote sensing derived EFand SMC.

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[gC

m-2

d-1

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a b

c d

Fig. 5. The uncertainty of GPP (a), NPP (b), SR (c) and NEP (d) time series derived from C-Fix (growing season of 1997) using the Monte-Carloapproach for case 1 of Table 2. The 95% confidence interval is given.


When running 17,000 simulations with the forest ecosys-tem model CASTANEA (Dufrene et al., 2005; Davi et al.,2005), and assuming a ±10% error on 17 out of the 150parameters used in the model, Dufrene et al. (2005) reportan error (coefficient of variation) of approximately 30% onNEP (standard deviation = 120 gC m�2 a�1), of 10% onGPP (standard deviation = 151 gC m�2 a�1), of 10% onautotrophic respiration (standard deviation = 75 gC m�2 a�1),of 2% on heterotrophic respiration (standard deviation =5 gC m�2 a�1) and a coefficient of variation of 6% onevapotranspiration.

For the Brasschaat site, C-Fix standard deviations of129 gC m�2 a�1 are obtained for NEP, 102 gC m�2 a�1

for GPP and 30 gC m�2 a�1 for SR. These values havethe same order of magnitude as those reported by Dufreneet al. (2005). When multiplying the standard deviationsreported by Dufrene et al. (2005) with a factor 2, toapproach the 95% confidence interval of a Normal distri-bution (assuming that the distribution of the carbon fluxesis normal), values of 302, 240 and 70 gC m�2 a�1 areobtained for respectively NEP, GPP and SR. These values

are in the same order of magnitude as those listed in Table5 (when rescaled to an annual basis).

The absolute errors of the ecosystem carbon fluxesobtained with C-Fix and derived with both the EP andMC approaches, are slightly higher than the values sug-gested by Dufrene et al. (2005). Possibly, the very lowerrors on evapotranpiration (coefficient of variation of6% with a standard deviation of 33 mm a�1) and transpi-ration (coefficient of variation of 12% with a standarddeviation of 37 mm a�1) can offer an explanation for this.The error on remote sensing derived EF is at least 8%,hence the error on evapotranpiration should implicitlybe higher. Kite and Droogers (2000), report that anuncertainty on evapotranpiration of approximately 50%is to be expected. Verstraeten et al. (2005b) calculatedthe error on evapotranpiration to be at least 27%. Hence,a coefficient of variation of 6% might be an optimistic,best case value.

The error on SMC derived from remote sensing is atleast 7%. When quantifying the errors of C-Fix outputs,it was assumed that input time series like EF, SMC and


NDVI and all the model parameters have a minimal error.Dufrene et al. (2005) does not report input errors to betaken into account.

6. Lessons learned from the error analysis: summary and

conclusions

Model uncertainty results from a combination of con-ceptual and functional model attributes like model struc-ture, data inputs as well as model parameters andvariables. Model output uncertainty is determined by theuncertainty of the input measurements.

The objective of this paper is to assess error propagationand uncertainties, as a result of it, when modelling evapo-rative fraction (EF), soil moisture content (SMC) andwater limited NEP from optical and thermal spatiallycoarse resolution earth observation data. This uncertaintystudy was conducted using a 1997 METEOSAT time seriesof ‘the Brasschaat pixel’. Its aim is to demonstrate the errorpropagation in modelling approaches using earth observeddata assimilation. The central question of this study was:What is the error on the C-Fix model output given theerror on the input?

Error propagation was modelled using a simplified butmathematically explicit approach of basic error theoryand secondly, a Monte-Carlo approach. Subsequently,remotely sensed EF and SMC are inputs of C-Fix, appliedto estimate ecosystem carbon mass fluxes and their uncer-tainties. To illustrate the uncertainties of EF and SMC aswell as the uncertainties related to C-Fix model structureand input, different error scenarios are defined coveringthese issues.

The EF average absolute error based on the simple errorapproach (EP) is 0.07% or 11%. When using the Monte-Carlo approach (MC), the EF average absolute error is0.05% or 8%. The SMC average absolute error based onthe EP and MC approaches is 0.01% or 4%. For the EPapproach, the NEP minimal average absolute error is1.28 gC m�2 d�1 while an error value of 0.86 gC m�2 d�1

is obtained for the MC approach.From this error analysis the following conclusions can

be drawn:

i. The error estimates from the simple error and Monte-Carlo approaches for EF and SMC are of the sameorder of magnitude, though the Monte-Carloapproach tends to result into lower error values.Generally, the error on these hydrological parameterestimates might reach 50%, though errors of less than20% are possible.

ii. For the ecosystem carbon fluxes, both error assessmentapproaches give large differences. The Monte-Carloapproach results in lower errors. From GPP, to NPP,SR and finally NEP, the errors evolve from fractions(error smaller than the parameter value) towards fac-tors of the average daily value (errors are multiples ofthe parameter value).

iii. Structurally complex models (mostly the physicallybased ones) tend to elicit model parameter inter-corre-lation. Hence, the Monte-Carlo approach seems moreadapted to perform error estimation for models witha more complex mathematical structure.

iv. Concerning the estimation of NEP, the contribution ofthe error on soil respiration produces the largest uncer-tainty on NEP. In turn, this carbon flux is the most dif-ficult flux to measure and model since complexprocesses are involved (Davidson and Janssens,2006). Davidson and Janssens (2006) published a thor-ough discussion on the temperature sensitivity of soilorganic matter decomposition and list environmentalconstraints for this process (e.g., the physical andchemical interactions of soil organic matter withdecomposition enzymes, the effects of drought, floodsand frost periods). Hence, large improvements inNEP flux estimations can only be expected when moreinsight is gained in the process of soil respiration, sothat better models can be developed. From our analysismodelled SMC and hence its related model parameters,influence the uncertainty of SR estimates the most.Another parameter with a major impact of its uncer-tainty on SR and thus NEP fluxes, is parameter NEPb�1 which is mainly based on the calibration withyearly NEP tower flux measurements.

v. The GPP and NPP fluxes can be reasonably well esti-mated, although there is room for improvement. Forexample on the estimation of radiation use efficiencyor autotrophic respiration.

vi. Concerning the remote sensing based input of the C-Fix carbon flux ecosystem model, more efforts in theestimation of temporal and spatial fAPAR valuesshould improve the accuracy of the carbon fluxes, espe-cially GPP (Verstraeten et al., 2006a). Once moreinsight is gained in the complex process of soil respira-tion, remote sensing can grow into a major factor in theestimation of spatially and temporally distributedparameters and as a result of this, of the modelling ofecosystem carbon fluxes as well.

The effect of the uncertainty of meteorological data onecosystem carbon fluxes estimated with C-Fix has not beeninvestigated thoroughly yet, though the importance ofmeteorology as a driving factor in vegetation and soil pro-cesses is well documented. Complex models with highamounts of parameters, show a high degree of parameterinter-correlations. As a result, a Monte-Carlo approachto estimate model uncertainty is more appropriate whenmodelling ecosystem carbon fluxes. Therefore in this stageof research, a more comprehensive uncertainty study is rec-ommended wherein more parameter sets are involved aswell as a priori information used to reduce model outputvariable uncertainty.

Finally, as discussed in Verstraeten et al. (2006a), C-Fixdoes not compete with dynamic global vegetation andstand scale models (e.g., Cramer et al., 2001; Rasse et al.,


2001; Krinner et al., 2005). It is useful to compare C-Fixwith these models, but clearly the added value of C-Fix isits capacity for spatially explicit, regional and globalparameter estimation. The assimilation of remotely senseddata results in a better spatial accuracy of ecosystem car-bon fluxes estimates.

Acknowledgements

The authors thank the Flemish Institute for Technolog-ical Research (VITO) for the scholarship and financial sup-port for this study, as well as the support offered by theGLOVEG contract (VG/00/01). The authors acknowledgethe work accomplished in the EUROFLUX project, whichresulted in unique validation datasets. The authors alsoacknowledge the work performed and comments as wellas suggestions made by the ASR referees.

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chapter 7 review of breastfeeding assessment tools

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