Tree Physiology 6, 807 817 006 Hero Publishig Victoria, Caada Parameter sesitivity ad ucertaity of the forest carbo flux model FORUG: a Mote Carlo aalysis HANS VERBEECK, 1,3,4 ROELAND SAMSON, 1 FREDERIK VERDONCK ad RAOUL LEMEUR 1 1 Laboratory of Plat Ecology, Ghet Uiversity, Coupure Liks 653, 9000 Ghet, Belgium Departmet of Applied Mathematics, Biometrics ad Process Cotrol, Ghet Uiversity, Coupure Liks 653, 9000 Ghet, Belgium 3 Curret address: Research Group of Plat ad Vegetatio Ecology, Departmet of Biology, Uiversity of Atwerp,Uiversiteitsplei 1, 610 Atwerp, Belgium 4 Correspodig author (Has.Verbeeck@ua.ac.be) Received April 13, 005; accepted September 14, 005; published olie March 1, 006 Summary The Mote Carlo techique ca be used to propagate iput variable ucertaity ad parameter ucertaity through a model to determie output ucertaity. However, to carry out Mote Carlo simulatios, the ucertaity distributios or the probability desity fuctios (PDFs) of the model parameters ad iput variables must be kow. This remais oe of the bottleecks i curret ucertaity research i forest carbo flux modelig. Because forest carbo flux models ivolve may parameters, we questioed whether it is ecessary to take ito accout all parameters i the ucertaity aalysis. A sesitivity aalysis ca determie the parameters cotributig most to the overall model output ucertaity. This paper illustrates the usefuless of the Mote Carlo simulatio techique for rakig parameters for sesitivity ad ucertaity i process-based forest flux models. The ucertaity of the output (et ecosystem exchage, NEE) of the FORUG model was estimated for the Hesse beech forest (1997). Based o the arbitrary ucertaity of te key parameters, a stadard deviatio of 0.88 Mg C ha 1 year 1 NEE was foud which is equal to 4% of the mea value of NEE. Sesitivity aalysis showed that the overall output ucertaity of the FORUG model ca largely be determied by accoutig for the ucertaity of oly a few key parameters. The results led to the idetificatio of the key FORUG parameters ad to the recommedatio for a process-based descriptio of the soil respiratio process i the FORUG model. Keywords: least square liearizatio, photosythesis, probability desity fuctio, soil respiratio. Itroductio I the cotext of a chagig climate, the primary objectives of forest modelig studies are to predict how forests will respod to climate chage ad to ehace our curret kowledge of the ecophysiological processes affected by climate chage. Process-based models provide a opportuity to achieve both objectives. However, these models usually predict discrete outputs ad provide o assessmet of the reliability of the output. Gerter (1987) cocluded that a assessmet of the variace or ucertaity of model outputs is desirable ad useful: (1) to gauge the reliability ad precisio of predictios; () to calculate cofidece itervals; (3) to statistically test hypotheses whe experimets are performed with the model; ad (4) to weight outputs used as a auxiliary source of iformatio i combiatio with field-sample estimates. The Mote Carlo techique ca be used to estimate model output ucertaity. Moreover, the mai disadvatage of the Mote Carlo techique, which is the log computatioal time (Gerter et al. 1996), has dimiished i importace as the computatioal capacity of computers has icreased. However, to carry out Mote Carlo simulatios, the ucertaity distributios or the probability desity fuctios (PDFs) of the model parameters must be kow. The PDFs ca be estimated based o experimetal data, as has bee doe for forest growth ad allocatio models. For example, MacFarlae et al. (000) estimated the PDFs of 14 physiological or morphological parameters of the Pipestem model based o published data. Mäkelä (1988) described this type of aalysis for a forest growth model based o the fuctioal balace ad the pipe model theory. For more complex ecophysiological flux models, however, the estimatio of PDFs is time cosumig ad so teds to severely limit the frequecy with which ucertaity aalyses are coducted (MacFarlae et al. 000), eve though estimatio of the PDFs of parameters based o expert kowledge rather tha experimetal data is better tha failig to coduct ucertaity aalyses. Parameters ca be raked for ucertaity by the Mote Carlo techique i combiatio with a multiple liear regressio. This sesitivity aalysis estimates the ucertaity cotributio of all parameters to the overall output ucertaity. This catalogue of error sources is also called the error budget (Gerter et al. 1996). Parysow et al. (000) applied this method to a process growth model based o the pipe model This paper was preseted at the iteratioal coferece o "Modelig Forest Productio," which was held April 19 1, 004 at the Uiversity of Natural Resources ad Applied Life Scieces, Viea, Austria. Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018
808 VERBEECK, SAMSON, VERDONCK AND LEMEUR theory ad the self-thiig rule ad cocluded that the method provides a efficiet strategy for buildig error budgets of process models with may model iputs. Most complex forest models have may parameters. The goal of our study was to idetify, by sesitivity aalysis, the parameters of the FORUG model a process-based forest flux model cotributig most to overall output (et ecosystem exchage (NEE)) ucertaity, ad thereby determie if it is ecessary to estimate the ucertaity of all parameters of a model to determie overall output ucertaity. We made o attempt to verify the validity of the FORUG model, although we recogize that the characterizatio of errors ad the cotributio of each model parameter to the total error are key aspects of the evaluatio of forest growth models (Vaclay ad Skovsgaard 1997). Materials ad methods The FORUG model The FORUG model is a multi-layer process-based model that simulates CO ad H O exchage betwee forest stads ad the atmosphere. Mai model outputs are NEE, total ecosystem respiratio (TER), gross primary productio (GPP) ad evapotraspiratio. We focused o NEE as output because et exchage of carbo betwee forests ad atmosphere determies the role of forests i the global carbo cycle (Law et al. 001). Oe uderstory ad three upperstory caopy layers were cosidered. A radiatio module calculates the available direct ad diffuse photosythetic active radiatio (PAR) i each vegetatio layer (Spitters 1986, Spitters et al. 1986). I each layer, itercepted PAR is calculated for the sulit ad shaded leaf fractio (Lemeur 1973). This itercepted PAR drives the photosythesis submodel. Photosythesis ad stomatal coductace are calculated accordig to Farquhar et al. (1980) ad Ball et al. (1987). Photosythesis ad leaf respiratio parameters are temperature depedet as described by Medly et al. (00) ad de Pury ad Farquhar (1997). Soil respiratio is calculated based o a simple expoetial fuctio of soil temperature as described by Graier et al. (00). Woody biomass respiratio is calculated with a temperature fuctio accordig to Ceschia et al. (00). To simulate NEE, the FORUG model uses 54 parameters. All relevat equatios are listed i the Appedix ad Table A1. For a detailed model descriptio, see Samso (001) ad Booe et al. (00). Data The data used i this modelig study are the meteorological ad flux data of the beech (Fagus sylvatica L.) site i Hesse (Frace, 48 40 N, 7 05 E, 300 m above sea level) measured withi the framework of the EUROFLUX project (Mocrieff et al. 1997, Valetii 1999, Aubiet et al. 000). The Hesse forest is 30 to 35 years old ad has bee described by Graier et al. (000, 00). Ucertaity The overall ucertaity of ay model is a combiatio of three sources of ucertaity (Beck 1987): ucertaity of the iput variables (e.g., measuremet errors), ucertaity of the model parameters (e.g., lack of iformatio to calibrate all parameters) ad ucertaity of the model structure (e.g., uderlyig equatios ad assumptios). We focused o the secod source of ucertaity. The Mote Carlo techique is a umerical techique to calculate output ucertaity of a model. The Mote Carlo approach was developed by Staislaw Ulam ad Joh vo Neuma to simulate probabilistic evets for military purposes i 1946 (Frey ad Li 001). The method, which has bee described extesively (e.g., Hammersley ad Morto 1964, Vose 1996, Culle ad Frey 1999), is robust ad relatively easy to implemet. Whereas error propagatio methods ca be difficult to implemet whe the ucertaity distributios or PDFs are ot ormally distributed, the Mote Carlo techique ca hadle differet distributio types ad ca always be implemeted i a relatively straight-forward maer (Gerter et al. 1996). To use the Mote Carlo techique, a PDF is eeded for each model parameter ad iput variable that is cosidered to be ucertai. To start the Mote Carlo aalysis, oe radom sample from the PDF of each parameter ad iput variable is selected ad the set of samples is etered ito the determiistic model. The model is the solved as for ay determiistic aalysis. The model output variables are stored ad the process is repeated util a specified umber of model simulatios is completed. Istead of obtaiig a discrete umber for the model output as i a determiistic simulatio, a set of output samples is obtaied (Culle ad Frey 1999). After a sufficietly large umber of simulatios, the distributio fuctio of the output ca be determied. We made o distictio betwee the ucertaity ad the atural variability of the parameters. Ucertaity of a parameter ca usually be reduced by collectig more iformatio about that parameter. Natural variability is a characteristic of a parameter that caot be reduced by collectig more iformatio. To make this distictio, a secod-order Mote Carlo aalysis should be applied (Culle ad Frey 1999). Most of the FORUG parameters are oly ucertai. Although some of the parameters of the FORUG model are both ucertai ad variable, the ucertaity of these parameters is domiat. Therefore the distictio betwee ucertaity ad variability of the FORUG model parameters was cosidered irrelevat ad all FORUG parameters were cosidered as ucertai. The ucertaity of the measured iput variables (e.g., temperature, icomig radiatio) was ot take ito accout because this ucertaity is maily due to relatively small measuremet errors. I Mote Carlo simulatios, the PDF of the parameters ad iput variables are required. If eough data are available, iput PDFs ca be determied based o kowledge ad measuremets. Ufortuately, iformatio about distributios of parameters is ofte uavailable ad as a result, iput distributios TREE PHYSIOLOGY VOLUME 6, 006 Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018
UNCERTAINTY AND SENSITIVITY OF THE FORUG MODEL 809 are ofte estimated based o a expert guess. Beside the PDFs, the miimum umber of simulatios has to be determied which depeds o the model structure ad the statistic of iterest. Our statistic of iterest was the variace of the FORUG model output. Because forest models ca be complex ad ca require substatial computatioal resources, it may be importat to reduce the overall requiremets for calculatio. It is importat that each simulatio of a aalysis is a feasible sceario. To exclude impossible parameter combiatios, correlatios betwee the differet ucertai parameters should be take ito accout. This is possible whe the correlatio coefficiets are kow. Burmaster ad Aderso (1994) state that the presece of moderate to strog correlatios will have little effect o the cetral portios of the output distributios, but may have larger effects o the tails of the output distributios. Therefore, whe there is iterest i the tails of the distributios, correlatios should be take ito accout. Several techiques exist to simulate correlatios i Mote Carlo aalysis (Ima ad Coover 198, Vose 1996, Culle ad Frey 1999). Sesitivity aalysis to rak for sesitivity ad ucertaity It is difficult to determie the probability desity fuctio of all parameters i a process-based forest model. For models with a large umber of parameters (the FORUG model has 54 parameters), we ca questio whether it is ecessary to take all parameters ito accout i the ucertaity aalysis. To determie the parameters cotributig most to the output ucertaity, a sesitivity aalysis ca be coducted to rak the parameters. Oe drawback of the Mote Carlo techique is that a combied output ucertaity is calculated. This meas that it is impossible to determie the cotributio of each parameter to the overall output ucertaity. A possible solutio is to use the least square liearizatio (LSL; Lei ad Schillig 1996) which splits output ucertaity ito its sources ad ca be coducted o the results of a Mote Carlo aalysis. The LSL is a multiple regressio betwee the parameter deviatio from the mea ad the output. All parameters are varied at the same time, whereas some sesitivity aalysis methods perturb oly oe parameter at a time. By usig the LSL, the cotributio of each parameter to the overall output ucertaity is estimated. Parameters cotributig little to the output ucertaity ca be excluded. The LSL method i combiatio with Mote Carlo aalysis has the advatage of beig able to simultaeously: (1) rak parameters accordig to their importace i ifluecig output ucertaity; () predict output ucertaity as a fuctio of ucertaity i model iput variables ad parameters; (3) partitio the error cotributio of the model iput variables ad parameters i terms of output variace; ad (4) provide the foudatio for the optimal reductio of output ucertaity or cost associated with additioal data collectio (Parysow et al. 000). Moreover, as Saltelli et al. (000) stated, sesitivity estimators such as stadardized regressio coefficiets are easy to implemet, relatively iexpesive ad ituitive. Equatios 1 ad provide some mathematical backgroud to the LSL method which is i essece a multiple regressio betwee the parameter deviatio from the mea ad the output. Cosider a variable, y, that depeds o a umber of idepedet variables, v 1, v,, v. The variatio of y as a fuctio of small variatios i idepedet variables ca be expressed as: Δy = Δv1 + Δv +... + Δv (1) If y is cosidered as y 1 +Δ y, the: y = y + Δv1 + Δv +... + Δv () 1 The LSL coducted o the Mote Carlo simulatio results ca be expressed as follows: Δv i is defied as the differece betwee v i, the radom chose sample of parameter i ad m Vi, the mea value of parameter i of all the radom samples. The value of Δv i is assumed to be equal to δ Vi, the true ucertaity of parameter i ad V i,true is the true value of parameter i. Δv = v m δ = v V (3) i i vi vi i i,true Whe m Mote Carlo simulatios are carried out, Δv i for each parameter ad the model output y are calculated for each simulatio. Next, a multi-liear regressio o the obtaied dataset is performed. The Δv i values are cosidered as idepedet variables ad the output y is the depedet variable. This gives the followig regressio equatio: y w Δv + w Δv +... + wδv + b (4) 1 1 The regressio coefficiets (w i ) are estimated by miimizig the sum of squared errors. Comparig this with Equatio, it ca bee see that the coefficiets w 1, w,,w are estimates of the partial derivatives of y with respect to v i ad b is a estimate of the value of y at default parameter values (i.e., whe Δv i = 0 for all i). If the ucertaities of the idepedet parameters are statistically idepedet, the overall variace of the model output ( σ δ y ) ca be calculated as: σ δy w i σδ (5) i = 1 vi where σ δ vi is the variace of the calculated differece δ Vi. Based o the regressio coefficiets ad the variatios of the parameter ucertaities, the sesitivity coefficiet of each parameter i (S Vi ) ca be approximated as: Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018 TREE PHYSIOLOGY ONLINE at http://heropublishig.com
810 VERBEECK, SAMSON, VERDONCK AND LEMEUR S v i wi σ δ vi = 100% (6) σ δ y Depedig o the scale of the parameter variatio, differet variats of the sesitivity aalysis ca be coducted (Frey ad Patil 00). We used two variats of the sesitivity aalysis. First, sesitivity of the model output to a ifiitesimal small variatio of all 54 parameters was assessed because o proper PDFs are available for all parameters. A ifiitesimal variatio is approximated by usig uiform distributios with a maximum deviatio of 1% for all 54 parameters. This aalysis assessed the impact of the parameter values o the output without the effect of their idividual distributios. This method is helpful for screeig the most importat parameters whe o iformatio o their ucertaity is available, which was the mai goal of this study. This variat of sesitivity aalysis is used to rak the parameters of the FORUG model for sesitivity. Secod, simulatios were ru i which parameters were assiged probability distributios ad the effect of variace i the parameters o the output distributio was assessed. These distributios were based o expert kowledge. This secod variat of sesitivity aalysis was used to test the effect of arbitrary PDFs o the rakig of the parameters of the FORUG model. This rakig is called rakig for ucertaity. For both variatios of the sesitivity aalysis, correlatio betwee parameters was ot take ito accout. All parameter combiatios of the FORUG model were checked systematically based o the available iformatio. This exercise led to the coclusio that some of the parameters were expected to be correlated. For a few parameters, relatioships are available i the literature. For example the relatioship betwee maximum electro trasport rate (J max ) ad maximum carboxylatio rate (V cmax ) has bee described i several papers, e.g., Wullschleger (1993), Niiemets et al. (1998) ad Medly et al. (1999). However, correlatio was expected to have o major ifluece o the first sesitivity aalysis because oly small deviatios with a maximum of 1% were used. Because the secod sesitivity aalysis was used oly to test the effect of arbitrary PDFs ad give the curret poor kowledge of PDFs, icludig the PDFs of J max ad V cmax, it was cosidered irrelevat to take correlatios ito accout. Results Figure 1. The variace of the FORUG output et ecosystem exchage (NEE; Mg C ha 1 year 1 ) for the Hesse forest i 1997 plotted as a fuctio of the umber of Mote Carlo simulatios. For these simulatios, all 54 ucertai parameters were take i accout. Number of simulatios The variace was selected to represet the output ucertaity. I Figure 1, the variace of the simulated NEE is plotted after every simulatio. Figure 1 shows that after 000 simulatios the variace of the model output coverged. Thus, 000 simulatios are sufficiet to predict the output variace of the FORUG model. This compares well to the 048 simulatios ecessary to obtai good approximatios of variaces foud by Gerter et al. (1996). Ruig oly 000 simulatios istead of 10000 (a commoly used umber of simulatios) results i a 80% reductio i calculatio time. Rakig parameters for sesitivity For this aalysis, a uiform distributio with a maximum deviatio of 1% was attributed to all 54 ucertai parameters. Based o 000 Mote Carlo simulatios (Figure 1), the cotributio of all 54 parameters to the overall output ucertaity was calculated accordig to Equatios 3 6. Table 1 shows that 96% of the output ucertaity is caused by the ucertaity of the 10 most importat parameters, i.e., the other 44 parameters determied oly 4% of the output ucertaity. More tha 70% of the overall output ucertaity is determied by the two coefficiets (a soil ad b soil ) of the soil respiratio model. Two importat parameters determie the photosythesis process: the iitial quatum yield (α F ) ad the maximum carboxylatio rate (V cmax ). Table 1 also shows that the light extictio coefficiet for diffuse radiatio (k d ) plays a importat role. Other sesitive parameters appearig i the list are associated with the photosythetic process: the activatio eergy of the temperature depedece of V cmax (E av ) ad the Michaelis-Mete costat for carboxylatio (K c ). The rest of the list cotais less sesitive parameters: the Michaelis Mete costat for oxygeatio (K o ), the CO compesatio poit (Γ ) ad the partial pressure of oxyge i the air (O). The other 44 parameters do ot appear i the list ad have a idividual cotributio that is less tha 0.94%. Rakig parameters for ucertaity Because of lack of iformatio of the differet parameter values of the FORUG model, the PDFs of the parameters could ot be estimated based o experimetal data. Therefore, i cotrast with the rakig for sesitivity, a wider rage ad sometimes differet type of distributio was attributed to all 54 ucertai parameters for this aalysis. The choice of TREE PHYSIOLOGY VOLUME 6, 006 Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018
UNCERTAINTY AND SENSITIVITY OF THE FORUG MODEL 811 Table 1. Results of the rakig for sesitivity. The cotributio (%) of the ucertai parameters to the overall ucertaity of the FORUG model output et ecosystem exchage (NEE). Results are based o 000 Mote Carlo simulatios for the year 1997 for the Hesse forest i Frace. Parameter Descriptio Process Cotributio to overall ucertaity (%) b soil Coefficiet Soil respiratio 54.16 a soil Coefficiet Soil respiratio 16.03 α F Iitial quatum yield Photosythesis 6.36 V cmax Maximum carboxylatio Photosythesis 6.18 rate k d Extictio coefficiet Light extictio 3.71 for diffuse radiatio E av Activatio eergy of Photosythesis 3.54 temperature depedece of V cmax K c Michaelis-Mete cos- Photosythesis 3.3 tat for the carboxylatio Γ CO compesatio poit Photosythesis.70 K o Michaelis-Mete Photosythesis 0.98 costat for oxygeatio O Partial pressure of oxyge Photosythesis 0.94 i the air the distributios was based o expert kowledge. Although there are eough published data from measuremets o seedligs to fit a logormal distributio for J max ad V cmax (Jarvis 1999, Medly et al. 1999, Levy ad McKay 003), the PDFs of these parameters were also based o expert kowledge because our study was coducted i a more tha 30-year-old beech stad. Normal ad logormal distributios are widely adopted PDFs for modelig ucertai parameters. Because these distributios are ubouded at two sides ad oe side, respectively, they are iappropriate for bouded parameters (Wu ad Tsag 004). To exclude radom values that caot appear i the eviromet, trucated distributios were useful. Therefore, a uiform or triagular distributio was used for all parameters (see Appedix, Table A). A uiform distributio was attributed to a parameter with a miimum ad a maximum boudary, but without iformatio about the ucertaity. Maximum deviatios of 5, 10 ad 0% were used. A triagular distributio was attributed to parameters that were thought to have a higher probability close to the mea value. It is easier to estimate miimum ad maximum boudaries of a parameter (which has to be doe for uiform ad triagular PDFs) tha to estimate the variatio of a PDF (which is eeded for ormal ad logormal PDFs). A maximum deviatio of 10% was chose for most parameters (see Appedix, Table A) which is a commo deviatio that correspods with other carbo flux studies, e.g., Hirsch et al. (004). The triagular probability distributio fuctio was also used by Paul et al. (003) who applied Mote Carlo simulatios to the carbo accoutig model GRC3. A triagular distributio with a maximum deviatio of 10% meas that the parameter has a probability of oe at the mea value ad a probability of zero at ± 10% of the mea value. The cotributio to the overall ucertaity of the te most importat parameters is show i Table. Compared with the rakig for sesitivity (Table 1), the order of importace chaged. Parameter H has appeared i the list. This is the deactivatio eergy of the temperature depedece of J max. O the other had, parameter O (the partial pressure of oxyge i the air) has disappeared. Whe the cotributios are summed, more the 97% of the overall ucertaity appears to be due to these te parameters. Agai soil respiratio coefficiets appear high i the list cotributig more tha 63% of the output ucertaity. The light extictio coefficiet for diffuse radiatio (k d ) gaied i importace. As expected, the remaider of the list comprises parameters describig the photosythetic process. Output ucertaity The distributio of the simulated NEE for the year 1997 for the Hesse forest is show i Figure. This distributio is based o 000 Mote Carlo simulatios. The ucertaity aalysis accouted for oly the te parameters appearig i Table. The mea simulated output value is 3.63 Mg C ha 1 year 1 (a egative value of NEE idicates a et carbo uptake by the ecosystem). The stadard deviatio of the output distributio is 0.88 mg C ha 1 year 1. The variace is 0.77. I the Euroflux project, a total NEE value of.83 Mg C ha 1 year 1 was mea- Table. Results of the rakig for ucertaity. The cotributio (%) of the ucertai parameters to the overall ucertaity of the FORUG model output et ecosystem exchage (NEE). Results are based o 000 Mote Carlo simulatios for the year 1997 for the Hesse forest i Frace. Parameter Descriptio Process Cotributio to overall ucertaity (%) b soil Coefficiet Soil respiratio 49.67 k d Extictio coefficiet Light extictio 14.91 for diffuse radiatio a soil Coefficiet Soil respiratio 13.66 α F Iitial quatum yield Photosythesis 5.4 V cmax Maximum carboxylatio Photosythesis 4.51 rate K c Michaelis-Mete co- Photosythesis 3.17 stat for the carboxylatio E av Activatio eergy of temp- Photosythesis.80 erature depedece of V cmax Γ CO compesatio poit Photosythesis 1.84 H De-activatio eergy of Photosythesis 1.0 temperature depedece of J max K o Michaelis-Mete co- Photosythesis 0.55 stat for oxygeatio Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018 TREE PHYSIOLOGY ONLINE at http://heropublishig.com
81 VERBEECK, SAMSON, VERDONCK AND LEMEUR Figure. Distributio of the total et ecosystem exchage (NEE; Mg Cha 1 year 1 ) i 1997 for the Hesse forest i Frace. The calculatio of this distributio is based o 000 Mote Carlo simulatios. Oly the ucertaity of te key parameters was take ito accout. The black bar above shows the measured (Euroflux) NEE value ad does ot represet a frequecy. sured for the year 1997. The measured value of the NEE differs from the mea simulated value, but falls withi the rage of oe stadard deviatio of the output distributio. Discussio The te key parameters foud i both rakigs correspod to critical parameters foud i the literature. Before discussig the ecological meaig of the sesitivity aalysis ad the output ucertaity, we emphasize two limitatios of the aalyses. First, the PDFs used for the rakig for ucertaity are arbitrarily chose based o expert kowledge. This secod variat of the sesitivity aalysis based o these PDFs was coducted to see if these PDFs would give ew iformatio about the key model parameters. The rakig for ucertaity, however, did ot give ay additioal iformatio about the key parameters compared with the rakig for sesitivity. Therefore, because proper PDFs are ot kow, the discussio focuses oly o the key parameters foud by rakig for sesitivity. The oly coclusio that ca be draw from this rakig for ucertaity is the eed for experimetal data to quatify these PDFs. However, such experimetal data are ot easy to obtai. Secod, the mea values for each parameter were assumed for both rakigs, represetig a particular sceario. The major coclusio that ca be draw is that the te most importat parameters determie more tha 90% of the output ucertaity. However, the aalysis does ot suggest that the mea output value is determied by these te parameters aloe. A sesitivity aalysis o the same model usig differet mea parameters may give differet results. Soil respiratio parameters The high rakig of the soil respiratio parameters a soil ad b soil is oteworthy. The simple soil respiratio model used i the FORUG model is depedet oly o soil temperature (see Appedix, model descriptio). The high sesitivity of the model output ucertaity for soil respiratio could be expected because flux measuremets give a mea aual soil respiratio that is 69% of the total ecosystem respiratio i Europea forests (Jasses et al. 001). The correspodece with fidigs of eddy covariace measuremets (Valetii et al. 000) ca be cosidered as a verificatio of the FORUG model. High sesitivity for soil parameters ad processes has also bee foud for other carbo flux models, e.g., EFIMOD (Komarov et al. 003). The high sesitivity for the idividual parameters of soil respiratio poits to the importace of the soil respiratio process, but it also partly reflects the fact that this soil respiratio model uses oly two parameters. This is a small umber of parameters compared with, for example, the photosythesis model. Use of a more complex soil respiratio model will likely ot decrease the cotributio of the soil respiratio process, but it will decrease the sesitivity to each idividual parameter. Epro et al. (1999) have already recommeded that soil water cotet be icorporated i further developmet of predictive models of NEE because summer drought may occur at irregular itervals i Wester Europe. Clearly, a more complex soil respiratio model which is process-based ad ot empirical like the curret model is eeded to predict the soil respiratio process more accurately. The model should differetiate betwee heterotrophic ad autotrophic respiratio ad should be based o several evirometal factors icludig temperature, soil water cotet ad soil orgaic matter. I additio, soil respiratio has a year-roud ifluece o NEE i cotrast to, e.g., the photosythetic process. Durig witer, soil respiratio is almost the oly factor ifluecig carbo exchage betwee the forest stad ad the atmosphere which is aother reaso for the high rakig of the soil respiratio parameters. The impact of the soil parameters also depeds o the type of ecosystem. For example, i tropical forest, where relatively small amouts of carbo are passed to the soil pools ad turover times are fast, soil submodel parameters have a smaller effect o the predicted NEE (Hirsch et al. 004). Photosythesis parameters The high rakig of the parameters that determie the photosythetic process directly ad idirectly reflects the fact that this process drives carbo uptake. Leuig et al. (1998) report a high sesitivity to the quatum yield of photosythesis. This also correspods with the results of Mäkelä (1988) where parameters related with shadig ad potetial photosythesis were foud to be critical. I forest ecosystems, most leaves are ot light saturated; therefore, a small chage i quatum yield has a importat impact o overall caopy CO uptake. However, maximum carboxylatio rate (V cmax ) has almost the same cotributio as quatum yield (Table 1), idicatig that these TREE PHYSIOLOGY VOLUME 6, 006 Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018
UNCERTAINTY AND SENSITIVITY OF THE FORUG MODEL 813 two parameters are most importat i determiig overall caopy photosythesis. Parameters V cmax ad α F determie A (c) ad A (j), respectively (Equatios A10 ad A11). I cotrast, some photosythetic parameters have oly a small impact o NEE. For example, dark respiratio rate (R d ) has a cotributio of oly 0.35%. Crow architecture The importace of light regime as a drivig factor for photosythesis is idicated by the high rakig of k d. This is because, i cotrast to direct radiatio, diffuse radiatio is itercepted by sulit, as well as by shaded, leaves. Moreover, diffuse radiatio affects the ecosystem every day, eve whe it is cloudy, whereas the direct radiatio compoet is preset oly o suy days. Although leaf area idex (L) has a idirect ifluece o the output through the extictio coefficiet for diffuse radiatio (Equatios A A4), L itself cotributes oly 0.75%. Rakig for sesitivity versus rakig for ucertaity Whe both rakigs are compared, differeces are relatively small (Tables 1 ad ) because for most parameters the same triagular PDFs with a maximum deviatio of 10% are used to rak for ucertaity. Because the expert kowledge about most parameters is quite limited, oly a few parameters were attributed PDFs with a differet shape or rage. Use of more differetiated PDFs would result i larger differeces betwee the rakigs for sesitivity ad ucertaity ad would make the rakig for ucertaity more valuable. Output ucertaity The aalysis of output ucertaity resulted i a stadard deviatio of 0.88 Mg C ha 1 year 1 which is 4% of the mea value of NEE. Because the ucertaity we calculated is based o arbitrarily chose PDFs based o expert kowledge, the estimated ucertaity has o absolute value ad is largely subjective. It oly idicates the effect of the chose PDFs o the output ucertaity. Ucertaities with the same order of magitude ca be foud i the literature. For example, Hirsch et al. (004) foud a ucertaity of 35% of the mea value of the et carbo flux simulated with the CARLUC (3PG) model for the Brazilia Amazo. Our ucertaity aalysis took accout of oly the te key parameters that explaied more tha 90% of the output ucertaity (Figure ). More accurate parameter distributios of these key parameters (Tables 1 ad ) ad detailed kowledge of their correspodig processes will lead to a decrease i the overall ucertaity of the FORUG model output. I coclusio, the stadard deviatio of the ucertaity of the output (NEE) of the FORUG model for the year 1997 for the Hesse beech forest, based o arbitrary PDFs of te key parameters, was calculated as 0.88 Mg C ha 1 year 1, which is 4% of the mea value of NEE. Future research o the FORUG model should focus o a better descriptio of poorly described key processes. I particular, the expoetial soil respiratio model eeds revisio. More accurate estimates of the key parameters (Tables 1 ad ) are ecessary to make more reliable FORUG predictios. Determiatio of the parameter PDFs ad the correlatios betwee the differet ucertai parameters remai bottleecks i ucertaity aalysis for complex forest flux models. Because of lack of iformatio about the key parameters, the fial ucertaity aalysis of the FORUG model is based o simple PDFs of the key parameters ad correlatio was ot take ito accout. Future research should attempt to obtai proper parameter PDFs ad to describe correlatios where preset. We coclude that sesitivity aalysis allows efficiet error resource allocatio which ca help focus future research o the key parameters. The Mote Carlo simulatio techique is a useful tool for rakig the ucertaity of parameters of process-based forest flux models. Ackowledgmets This research was supported by the CASTEC research project (carbo sequestratio potetial i differet Belgia terrestrial ecosystems: quatificatio ad strategic exploratio), fuded by the Belgia Federal Sciece Policy Office (Grat EV/0/1). The authors thak Peter Varolleghem for valuable commets o a previous versio of the mauscript. Refereces Aubiet, M., A. Grelle, A. Ibrom et al. 000. Estimates of the aual et carbo ad water exchage of Europea forests: the EURO- FLUX methodology. Adv. Ecol. Res. 30:113 175. Baldocchi, D. 1994. A aalytical solutio for coupled leaf photosythesis ad stomatal coductace models. Tree Physiol. 14: 1069 1079. Ball, J.T, I.E. Woodrow ad J.A. Berry. 1987. A model predictig stomatal coductace ad its cotributio to the cotrol of photosythesis uder differet evirometal coditios. I Progress i Photosythesis Research, Vol. IV. Ed. I. Biggis. Martius Nijhoff, Dordrecht, pp 1 4. Beck, M.B. 1987. Water quality modelig: a review of the aalysis of ucertaity. 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814 VERBEECK, SAMSON, VERDONCK AND LEMEUR Frey, H.C. ad S. Li. 001. Quatificatio of variability ad ucertaity i statioary gas-fuelled iteral combustio egie NOx ad total orgaic compouds emissio factors. Proc. A. Meetig Air ad Waste Maagemet Assoc., Orlado, Florida, USA. Abstract No. 695. Frey, H.C. ad S.R. Patil. 00. Idetificatio ad review of sesitivity aalysis methods. Risk Aal. :553 578. Gerter, G. 1987. Approximatig precisio i simulatio projectios: a efficiet alterative to Mote Carlo methods. For. Sci. 33: 30 39. Gerter, G., P. Parysow ad B. Gua. 1996. Projectio variace partitioig of a coceptual forest growth model with orthogoal polyomials. For. Sci. 4:474 486. Graier, A., E. Ceschia, C. Damesi et al. 000. The carbo balace of a youg beech forest. Fuct. Ecol. 14:31 35. Graier, A., K. Pilegaard ad N.O. Jese. 00. Similar et ecosystem exchage of beech stads located i Frace ad Demark. Agric. For. Meteorol. 114:75 8. Hammersley, J.M. ad K.W. Morto. 1964. Mote Carlo methods. 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UNCERTAINTY AND SENSITIVITY OF THE FORUG MODEL 815 Appedix: Descriptio of the FORUG model PAR iterceptio model (Lemeur 1973) For the direct ad diffuse compoet of the PAR radiatio, the peetrated irradiace ca be writte as: I ( L) = ( I ρ ) I exp ( k L) (A1) b b o,b b I ( L) = ( I ρ ) I exp ( k L) (A) d d o, d d The shaded fractio of a leaf itercepts oly diffuse radiatio: I = k ( I ρ ) I exp ( k L) (A3) shade d d o, d d Sulit leaves itercept diffuse ad direct radiatio: Isu = Ishade + kb Io, d (A4) Biochemical photosythesis model (Farquhar et al. 1980, de Pury ad Farquhar 1997) V = V f( D ) (A5) α cmax cmax F = α f( D ) (A6) F D f( D ) = 365 365 b (Wag 1996) J J su shade ( L) = ( L) = α 1 + J α 1 + J 1 b α F, LIsu F, L su max, L b exp b ( L) [ I ( L) ] α F, LIshade F, L shade max, L 365 D 1 365 b 3 1 b 3 (A7) (A8) 0.5 ( L) [ I ( L) ] (A9) 05. C C i s A = Cs (A14) g b A = Ca (A15) g b RH gs = g0 + g 1 A (Ball et al. 1987) (A16) C s The above equatios are aalytically solved as described by Baldocchi (1994). Temperature depedecies T Kc Kc exp 59.4( 1 5) = ( 5) 10 98R( T + 73) (de Pury ad Farquhar 1997) 1 36( T 5) Ko = Ko( 5) 10 exp 98R( T + 73) (de Pury ad Farquhar 1997) (A17) (A18) Γ = Γ ( 5 ) + 0. 188( T 5) + 0. 0036( T 5) (A19) (de Pury ad Farquhar 1997) V cmax = V cmax (Medly et al. 00) Eav( T 5) ( 5) exp 98R( T + 73) Eaj( T 5) Jmax = Jmax( 5) exp 98R( T + 73) 1 + exp 98 S H 98R S( T + 73) H 1 + exp R( T + 73) (Medly et al. 00) (A0) (A1) V A () c = ( Ci Γ ) R C + K cmax J( Ci Γ ) A() j = R 4( C + Γ ) i i d d (A10) (A11) Dark respiratio R d ΔHaR( T 5) = Rd( 5) exp 98R( T + 73) (de Pury ad Farquhar 1997) Soil respiratio (A) O K = Kc 1 + (A1) K o { } A = mi A ( c), A ( j) (A13) R = a exp( b T ) (Graier et al. 00) (A3) soil soil soil soil Woody biomass respiratio (( T 15)/10) stem = stem,15 10 R R Q (Ceschia et al. 00) (A4) Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018 TREE PHYSIOLOGY ONLINE at http://heropublishig.com
816 VERBEECK, SAMSON, VERDONCK AND LEMEUR Table A1: Parameters ad variables used i the Equatios A1 A9. Symbol Parameter or variable Uits a soil Regressio parameter µmol CO m soil s 1 A Net photosythetic rate µmol CO m s 1 A (c) Carboxylatio-limited rate of A µmol CO m s 1 A (j) Electro trasport-limited rate of A µmol CO m s 1 b 1 Parameter (150.0) o uit b Parameter (10.0) o uit b 3 Parameter (0.) o uit b soil Regressio parameter o uit C a CO cocetratio of the atmosphere µmol CO mol 1 air C i Itercellular CO cocetratio µmol CO mol 1 air C s CO cocetratio at the leaf surface µmol CO mol 1 air D Day of the year o uit E aj Activatio eergy for J max kj mol 1 E av Activatio eergy for V cmax kj mol 1 g b Leaf lamiar boudary layer coductace for CO ms 1 g l Coductace slope o uit g o Coductace itercept m s 1 g s Stomatal coductace for CO ms 1 H Deactivatio eergy for J max kj mol 1 I(L) PAR itesity at dowward cumulative LAI L µmol photos m s 1 I o PAR itesity above the caopy µmol photos m s 1 J Electro trasport rate µmol electros m s 1 J max Maximum electro trasport rate µmol electros m s 1 J max (5) J max at 5 C µmol electros m s 1 k d Extictio coefficiet for diffuse radiatio o uit K Effective Michaelis-Mete costat of Rubisco Pa K c Michaelis-Mete costat of carboxylatio Pa K c (5) K c at 5 C Pa K o Michaelis-Mete costat of oxygeatio Pa K o (5) K o at 5 C Pa L Leaf area idex m leaf area m soil area O Partial pressure of oxyge i the air kpa Q 10 Temperature respose factor o uit R Gas costat (8.314) J mol 1 K 1 R d Dark respiratio rate µmol CO m s 1 R d (5) R d at 5 C µmol CO m s 1 R soil Soil respiratio rate µmol CO m soil s 1 R stem Total stem ad brach respiratio rate µmol CO m wood s 1 R stem,15 R stem at 15 C µmol CO m wood s 1 S Electro-trasport temperature respose parameter J K 1 mol 1 T Temperature C T soil Soil temperature C T stem Stem temperature C V cmax Maximal carbo assimilatio rate µmol CO m s 1 V cmax ( 5) V cmax at 5 C µmol CO m s 1 V cmax Peak value of V cmax durig the growig seaso µmol CO m s 1 α F Quatum yield µmol CO µmol 1 photos α F Peak value of α F durig the growig seaso µmol CO µmol 1 photos ΔH ar Activatio eergy for R d kj mol 1 Γ CO -compesatio poit Pa Γ(5) Γ at 5 C Pa ρ b Reflectio coefficiet for direct PAR o uit ρ d Reflectio coefficiet for diffuse PAR o uit Subscripts b Beam (direct) radiatio d Diffuse radiatio shade Shaded leaf fractio su Sulit leaf fractio Abbreviatio RH Relative humidity o uit TREE PHYSIOLOGY VOLUME 6, 006 Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018
UNCERTAINTY AND SENSITIVITY OF THE FORUG MODEL 817 Table A: Mea values ad probability desity fuctios (PDFs) of the parameters used i Equatios A1 A4. Parameter Value PDF (% deviatio) a soil 0.436 µmol CO m soil s 1 (Graier et al. 00) Triagular (10) b soil 0.156 ( ) (Graier et al. 00) Triagular (10) E aj 37.0 (kj mol 1 ) (de Pury ad Farquhar 1997) Triagular (10) E av 64.8 (kj mol 1 ) (de Pury ad Farquhar 1997) Triagular (10) g l 9.50 ( ) (Harley ad Baldocchi 1995) Uiform (10) g o 17.5 ( ) (Harley ad Baldocchi 1995) Uiform (10) H 0 (de Pury ad Farquhar 1997) Triagular (10) J max (5) 11.6; 79.6; 64.4 (Jarvis 1999) Triagular (10) k d 0.7 (Samso 001) Triagular (0) K c (5) 40.4 (de Pury ad Farquhar 1997) Triagular (10) Ko(5) 4.8 10 3 (de Pury ad Farquhar 1997) Triagular (10) L max 5.6 (Graier et al. 000) Triagular (10) O 1 (de Pury ad Farquhar 1997) Triagular (5) Q 10 1.33 (Ceschia et al. 00) Triagular (10) R d (5) 0.4 (Harley ad Baldocchi 1995) Triagular (10) R stem,15 0.39 (Ceschia et al. 00) Triagular (10) S 710 (de Pury ad Farquhar 1997) Triagular (10) V cmax (5) 49.4; 3.34; 6.16 (Jarvis 1999) Triagular (10) α F 0.4 (Harley ad Baldocchi 1995) Triagular (10) ΔH ar 66.4 (de Pury ad Farquhar 1997) Triagular (10) Γ(5) 3.69 (de Pury ad Farquhar 1997) Triagular (10) ρ d 0.09 (de Pury ad Farquhar 1997) Triagular (5) Top, middle ad lower caopy layer, respectively. Dowloaded from https://academic.oup.com/treephys/article-abstract/6/6/807/1647778 o 10 February 018 TREE PHYSIOLOGY ONLINE at http://heropublishig.com