PRELES (PREdict with LESs – or – PREdict Light-use efficiency, Evapotranspiration and Soil water) is an ecosystem model of intermediate complexity developed by Peltoniemi et al. (2015a), in which the dependent variables, GPP (P, gCm-2day-1), evapotranspiration, ET (E, mm) and soil water (θ, mm), are interlinked. The model works at daily time-step and requires minimal input data. The climatic driving variables are daily mean temperature (T, ∘C), vapour pressure deficit (D, kPa), precipitation (R, mm) and photosynthetic photon flux density (PPFD, Φ, µmolm-2day-1). The only stand structural information is the fraction of absorbed PPFD (faPPFD), estimated using the Beer–Lambert law as faPPFD faPPFD=1-exp⁡-kL where L is the leaf area index (m2m-2) and k the extinction coefficient.

A detailed description of the PRELES can be found in Peltoniemi et al. (2015a); herein we briefly outline model structure and provide all the equations.

Water storage (θ) consists of three pools: intercepted water (θsurf) (mainly on canopy surfaces), snow/ice (θsnow) and soil water storage (θsoil). All components are described by simple bucket models. θ=θsurf+θsnow+θsoilθsurf=θsurf+R1-Esurf-Fsurfθsnow=θsnow+R0-Esnow-Mθsoil=θsoil+Fsurf+M-F-Esoil where R1 is rainfall and R0 is snowfall. Precipitation is assumed to be snow when air temperature is below 0 ∘C. M is snowmelt, Fsurf is drainage from canopy surface and F is drainage from the soil. Ecosystem evapotranspiration is given by the sum of evaporation and transpiration from the three water storage components, i.e. canopy, snow and soil (Eq. 6). E=Esurf+Esnow+Esoil We assume that the canopy intercepts precipitation up to a maximum, θsurf, max. Fsurf=min⁡(0,θsurf, max-θsurf) When precipitation exceeds this limit the additional water reaches the soil and accumulates in the soil water storage (θsoil) up to the field capacity of soil. Additional water drains away from the system with a fix time constant (τF). We used τF=3 days derived from soil water measurements at a boreal forest site on mineral soil (Peltoniemi et al., 2015a). F=θsoil-θFCτF Snow water storage θsnow accumulates when T<0 ∘C and melts at a rate m when T>0 ∘C (Kuusisto, 1984). M=mT,T>00,T<0 The GPP-submodel is a modification of Prelued (Mäkelä et al., 2008). The photosynthetic production is related to the light use efficiency of the stand (β) and the absorbed photosynthetic photon flux density. P=βΦfaPPFD Light use efficiency is given by the potential light use efficiency (βP) multiplied by an array of modifiers (fi) varying between 0 and 1 that accounts for the environmental conditions. β=βP∏fi The saturation of photosynthetic production with high PPFD is expressed by the light modifier fL, that follows the rectangular-hyperbola photosynthesis model (Mäkelä et al., 2008). fL=(γΦ+1)-1 Temperature impacts photosynthesis using a modifier for temperature acclimation (fS) (Mäkelä et al., 2004, 2008). fS=min⁡(S/Smax,1)Sk=max⁡(X-X0,0)X=Xt-1+1τ(T-Xt-1),whereX1=T1 Smax (∘C) is the minimum temperature threshold at which canopy photosynthesis is not limited by temperature (i.e., fS=1 for T≥Smax). S (∘C) is the state of acclimation that depends on the limit (X0) above which fS is higher than 0 and on the a priori estimate for the state of acclimation (X). X is calculated using a first-order dynamic delay model influenced by the daily air temperature T (∘C) and the value of X during the previous day (Xt-1). The parameter τ, expressed in days, represents the speed of response of the current acclimation status to changes in T.

Plant water stress (fDW,P) reduces photosynthesis and it can be caused by vapour pressure deficit of the atmosphere (fD) and soil water availability (fW,P). We assumed that only the most limiting factor between fD and fW,P reduces photosynthesis (Landsberg and Waring, 1997). fDW,P=min⁡(fD,fW,P) where fD affects GPP through an exponential relationships: fD=eκD fW,P depends on the relative extractable water W fWP,k=min⁡1,Wk/ρPW=θ-θWPθFC-θWP where θsoil is water stored in the soil, θWP is the wilting point and θFC is the field capacity, and ρP is the threshold of W (relative extractable water) below which P is reduced linearly.

Evapotranspiration is calculated by means of a simple empirical equation that requires minimal input data, but links the predicting variables P, E and θsoil. E=αPDλfW,PνD+χ(1-faPPFD)φfW,E where α and χ are empirical parameters; λ is a parameter that relates evapotranspiration and vapour pressure deficit (Medlyn et al., 2011b; Peltoniemi et al., 2015a). E is influenced by the soil water modifier, but fW,P is raised to the power ν since the response of P and E to drought is different. The E is also affected by soil drought through the fW,E modifier; fW,E follows the same equation of fW,P (Eq. 19) but it has its own threshold ρE.

Stand-scale net ecosystem exchange (NEE) of CO2, evapotranspiration and meteorological data from ten boreal coniferous forest sites located in Finland and Sweden were used in this study (Table 2). The sites cover a latitudinal band from 60 to 67∘ N with annual mean temperatures ranging from 0.8 to 7.1 ∘C, and precipitation from ∼550 to ∼850 mm. Leaf area index (LAI) at each site was treated as one lumped LAI, i.e. all the canopy layers were included in one unique layer. The total (all-sided) LAI varies between ∼3.8 and ∼12 m2m-2 offering a good possibility to address both climatic and LAI controls on forest GPP and ET. A brief summary of the sites is provided in Table 2, and complete descriptions can be found in the respective references.

The NEE and ET were measured above the forest canopies by the eddy-covariance method and the 1/2 h fluxes computed according to common practices (Aubinet et al., 2012). Gaps in data caused by instrumental failures or methodological issues, such as insufficient turbulent mixing, were gap-filled, and NEE was partitioned into component fluxes before the 1/2 h data was aggregated into daily averages or sums. The gap-filling of NEE was done using a combination of look-up tables and mean diurnal variability according to Reichstein et al. (2005). The gaps in meteorological data were filled either by linear interpolation or by the mean diurnal variability determined in a 14 day moving window.

The GPP was separated from the measured NEE as GPP =-NEE+Re, where the ecosystem respiration Re is (Kolari et al., 2009; Reichstein et al., 2005) Re=R10Q10[(T-10)/10] The R10 (µmolm-2s-1) represents the temporally varying base respiration rate at 10 ∘C temperature (T) and Q10 (unitless) represents the short-term temperature sensitivity, which is assumed constant in time but can vary among the sites. The Re model parameters were determined for each site by a non-linear least squares fit of Eq. (22) to nighttime NEE measured in turbulent conditions (friction velocity μ∗ exceeding an empirically defined site-specific threshold) using measured soil or air temperature as an independent variable. The Q10 was first computed by pooling all available growing season data, defined here as May–September (June–August in Sodankylä due to northern location). Secondly, Q10 was fixed and the temporal variability of R10 was determined by fitting Eq. (22) to data in four-day non-overlapping windows, and linearly interpolating between the window centres. Finally, Re was computed by extrapolating the obtained regression model to daytime temperatures, allowing the GPP to be approximated for each 1/2 h period.

After gap-filling, the fluxes and meteorological variables were aggregated at daily time step. A quality flag (F) varying between 0 and 1 was assigned to each day to represent the fraction of gap-filled data used to compute the daily value, and used in later analysis to weight the observations error (see “Model calibration and model comparison” section).

Bayesian calibration (BC) and Bayesian model comparison (BMC) were used to quantify the uncertainty in model parameters and model structure. For a comprehensive understanding of model behaviour, the Bayesian analyses were combined with a model-data mismatch analysis and a global sensitivity analysis (i.e., Morris method, Morris, 1991) following the framework proposed by van Oijen et al. (2011) and improved by Minunno et al. (2013a).

The work consisted of three analyses where we compared multi-site (M-S) and site-specific (S-S) calibrations. M-S has the advantage that the data involved in the calibration cover a wider variability in terms of climate and forest structure since they come from different sites, including measurement and other errors which may or may not partially cancel out when all data are used in parameter inference. In contrast, S-S could provide good correspondence to local data, but may not be spatially generalizable, firstly because the processes may not be generic, and secondly because the risk of bias increases with less measurements. More specifically, we conducted the following comparative analyses:

Peltoniemi et al. (2015a) found that variation of output sensitivity to parameters is regulated by soil moisture status. Here we take these analyses further and quantify sensitivities for all sites, thus taking a sample of site conditions and weather inputs that can affect the sensitivities. We also calculate sensitivities with a global sensitivity analysis method (Morris, 1991). It allows us to determine which parameters have linear or additive effects on model output and which ones have non-linear effects and interact with other parameters.

The analysis consists of many individually randomized one-factor-at-a-time experiments (OAT), i.e. one parameter at a time is changed in turn to evaluate the effect on model output and expressed through an incremental ratio called elementary effect (EE). The parameters are normalized ranging between 0 and 1, and the experimental region (i.e., the parameter space, Ω) is divided into p levels; therefore Ω is a k dimensional p level grid, where k is the number of parameters. Starting from a randomly selected parameter vector (X), a sequence of (k+1) sampling points, called trajectory, is created varying one parameter at time by a constant quantity Δ usually set as multiple of 1/(p-1) (Campolongo et al., 2007). Model output (y) is calculated for each parameter vector of the trajectory; so the elementary effect for parameter i, ei, can be calculated as: ei(X)=y(x1,…,xi-1,xi+Δ,xi+1,…,xk)-y(X)Δ The OAT experiments are designed in order to uniformly cover the whole parameter space; r trajectories are generated, with each trajectory having a different starting point randomly selected. So, r EEs are computed; the mean (μ∗) (Campolongo et al., 2007) and the standard deviation (σ), from the distributions of the absolute values of the EEs represent the sensitivity measures. μ∗ gives the overall importance of a parameter, while σ describes non-linear effects and interactions between parameters. A complete description of the method can be found in Morris (1991) and Campolongo (2007). Morris sensitivity analysis has been used for several process-based forest models (Minunno et al., 2013a; Song et al., 2013, 2012; van Oijen et al., 2011).

Sensitivity analyses of PRELES were carried out for the parameter space defined by the minimum and maximum values (Table 1), corresponding to the parameter space of the prior distribution of the Bayesian calibrations. The prior parameter space was chosen because it helps us to understand model behaviour in relation to the dataset used in the calibration process, thus helping us to interpret the results of the Bayesian calibrations. Morris method was applied to each site in order to test if the sensitivity results change with the climatic conditions in Fennoscandia.

The fraction of absorbed PPFD (faPPFD) is the only stand structural variable used to drive PRELES. In this work, faPPFD of each site was estimated using LAI (including both the main canopy and understory), by means of Beer's law (Eq. 1). In order to quantify model output response to variations in LAI we conducted a sensitivity analysis using the Hyytiälä dataset. Average annual GPP and ET were calculated running PRELES with different LAI values ranging from 0 to 16. Considering that total (all-sided) stand LAI at Hyytiälä was about 8, for this sensitivity analysis LAI was varied ±100 %. Model runs were conducted using the parameter estimates achieved by the multi-site calibration (i.e., all data from all sites were included in the calibration; details are provided in the next section).

Bayesian calibration (BC) provides an updated joint probability distribution of the parameters (posterior distribution) combining existing parameter knowledge (prior distribution) and new information enclosed in the data (likelihood).

The 12 most influential parameters on PRELES outputs (Peltoniemi et al., 2015a) were included in the BC (Table 1); the priors were uniformly distributed between the minimum and maximum values reported (Table 1).

A variable number of GPP and ET data points were available for the BC at each site (Table 2). The data were considered to be normally distributed so the likelihood was a Gaussian distribution and the standard deviation (si) of each data point i was assumed to be proportional to the number of gap-filled data in a day (see the quality flag F defined above) and was calculated using the following equation: si=aj+bjFi where Fi is a the quality flag and the parameters aj and bj were specific for each data type j (i.e., GPP and E) and were included in the BCs. We only used data with a quality flag lower than 0.7 in the calibrations.

The Bayesian approach jointly uses the prior and all the data to calculate the posterior distribution, but it provides little information about the strengths and weaknesses of a model. On the contrary, the more classical analyses of the mismatch between the simulated and the observed data give useful insights about model behaviour. In particular the decomposition of the mean squared error (MSE) provides indications about the accuracy and the precision of the predictions (Minunno et al., 2013a; van Oijen et al., 2011).

MSE can be decomposed into three components: the bias error, the variance error and the correlation error (Kobayashi and Salam, 2000) (Eq. 24). MSE=(S-O)2‾=(S‾-O‾)2+(σS-σO)2+2(σSσO)(1-r) where O are the observations, S the model predictions, σO and σS are the standard deviation of the observed and simulated data respectively and r is the correlation between the O and S.

The bias error quantifies the distance of the predictions from the data; the variance error expresses if the model is able to catch data variability; the correlation error indicates if the model is able to reproduce the pattern of data fluctuations. The latter component expresses the lack of positive correlation between the observed and simulated data and is weighted with standard deviations (Eq. 3), therefore there is an overlap between the variance and the correlation error (Kobayashi and Salam, 2000). In practice this MSE component seems to capture all types of error, such as random errors, left after accounting for the bias and the differences in the variances. MSEs were calculated for both GPP and ET and for each site.

The Morris sensitivity metrics, μ and σ, for GPP indicate that X0 (temperature threshold), χ (non-canopy evapotranspiration), α (canopy evapotranspiration), γ (light saturation, and β (potential GPP) were the most influential parameters (highest μ) with respect to the modelled photosynthetic activity in Hyytiälä (Fig. 2a). The results for Hyytiälä are representative of all sites except Norunda where the sensitivity of GPP to χ was much lower (Fig. 2b). Moreover, GPP response to the more influential parameters was non-linear, because they had the highest σ values. The τ (delay of temperature effect) was the parameter with the lowest impact to GPP, while the rest of the parameters had a medium-low effect on GPP.

Three of the five most important parameters for GPP (i.e., χ, X0, and α) were also the most influential on ET that was no-linearly related to these parameters. At Norunda X0 was the parameter to which ET was most sensitive (Fig. 3b); for the other sites χ was the most important parameter (Fig. 3a). β, γ, λ (effect of VPD on ET), ρP (soil water threshold for GPP) and Smax (temperature effect) had a medium impact on ET and the remaining parameters, i.e., κ (effect of VPD on GPP), τ, υ (effect of soil moisture on ET), ρE (soil water threshold for ET), were the least influential on evapotranspiration.

The ET and GPP response to changes in LAI followed an exponential curve (Fig. 4). The relative changes of ET and GPP were calculated using PRELES outputs generated with LAI =8. For low values of LAI, a small difference in leaf area causes big changes in GPP. A decrease of 50 % in LAI (LAI =4) causes a reduction of 40 % in GPP and 10 % in ET; a change of +50 % in LAI (LAI =12) increases the GPP of 15 %, while ET increment is small.

The data were highly informative in determining the values of the parameters that were assessed highly influential in the Morris sensitivity analysis, as demonstrated by the constrained posterior distributions (Fig. 5a) of the parameters compared with their priors (Table 1). In the multi-site calibration even the less important parameters were well constrained in the posterior distributions, however, a lot of uncertainty remained in some of these parameters in the site-specific calibrations (Fig. 5b). These include υ for Alkkia, Kalevansuo, CAge12yr, CAge75yr and Skyttorp site-specific calibrations; ρE for Alkkia, CAge12yr, CAge75yr, Skyttorp and Flakaliden; ρP for CAge75yr and Skyttorp (Fig. 5b). Parameter estimates across the different calibrations (i.e., S-S and M-S) were consistent for the most influential parameters, in particular for α, γ and χ (Fig. 5a), whereas differences occurred in estimates of the parameters at which model outputs are less sensitive (Fig. 5b). In the CAge12yr site-specific calibration γ, X0 and β marginal posterior distributions were quite different from the rest of the calibrations.

Bayesian calibration provides a joint posterior distribution of model parameters, i.e. BC also considers the interactions between parameters. This kind of information is not derivable from the marginal posterior distributions (Fig. 5), but it can be expressed through the correlations (r) between parameters calculated from the posterior sampled by the MCMC. To simplify the result presentation we briefly summarize the results below. In the M-S calibration the highest correlations were between β and γ (r=0.83) and between X0 and Smax (r=-0.78). Significant correlations were also found between α and χ (r=-0.67), and between α and λ(r=-0.64). For the Kalevansuo, Flakaliden and Norunda S-S calibrations parameter correlations were similar to the M-S calibration; while for the other sites some differences in the parameter correlations of the posterior distributions were found. For instance β was positively correlated to Smax in the calibrations for Knottåsen (r=0.84), CAge75yr (r=0.71), Hyytiälä (r=0.66), CAge12yr (r=0.63) and Alkkia (r=0.63).

We evaluated model performances in terms of R2 and the slopes of the simulated vs. observed data, calculated for each calibration and each model output (i.e., GPP and ET) at daily time step (Table 3). The predictions were generated using the maximum a posteriori (MAP, i.e. the modal parameter vector of the posterior distribution) parameter vectors of M-S and S-S (Fig. 6). The variance explained by the model was higher for GPP than for ET, both being in most of the cases higher than 70 % (R2 of Table 3); however the model tended to underestimate carbon and water fluxes (slopes lower than 1) (Table 3). Model fit to the Flakaliden data was generally rather poor. Furthermore, the multi-site calibration significantly underestimated evapotranspiration at Alkkia site (slope =0.62). In general, after BC, model outputs were characterized by low uncertainty (not shown in the plots).

In model predictions of GPP and ET for each site, the differences between the multi-site and site-specific calibrations were small in most of the cases (Fig. 6). The exception is the evapotranspiration at the Alkkia site, where ET by the multi-site calibration was clearly different from the site-specific calibration prediction (Fig. 6a).

Mean squared errors were calculated for GPP and ET and decomposed to the bias, variance and correlation errors for the multi-site and site-specific calibrations (Fig. 7). For the site-specific calibrations the main component of model error was the correlation error, while the other two components were negligible (Fig. 7). Also the MSEs of the multi-site calibration predictions were mainly constituted from the correlation error (Fig. 7); however the other two error components were significant at some sites, varying between 10 and 30 % of MSE. ET predictions at the Alkkia site for the multi-site calibration were characterized by the highest bias error (i.e. 40 % of MSE, Fig. 7b).

Both M-S and S-S calibrations showed robust performances in predicting the photosynthetic activity of boreal forests also at annual time step (Figs. 8a and 9a); while the model was less accurate in reproducing the annual evapotranspiration (Figs. 8b and 9b). Note that to compute the “annual” fluxes, the daily fluxes were summed only if the quality flag was lower than 0.7 (i.e. at maximum 30 % of 1/2 h fluxes were missing and gap-filled for that particular day), and the numbers in Figs. 8 and 9 are not representative for true annual balances. PRELES was able to catch the pattern of GPP inter-annual fluctuations for the sites with the long-term datasets (i.e., Hyytiälä, Sodankylä, Flakaliden, Norunda and Kalevansuo) (Fig. 9a), but at Flakaliden for some years (i.e., 1997, 2002) the relative difference between the observed and modelled GPP was around 50 %. The M-S and S-S annual prediction were really similar (Figs. 8 and 9), apart from the GPP at Flakaliden (Fig. 9a).

M-S and S-S calibrations were evaluated at each site using the validation dataset and considering both output fluxes (i.e., ET and GPP) at the same time. In six sites S-S had 100 % probability of being the best model version, while in the other sites BMCs supported the M-S calibration (Table 4). In general the NRMSEs calculated with the two types of calibration did not differ substantially (Fig. 10). At Hyytiälä and Skyttorp the GPP NRMSE of S-S was 10–20 % higher than the GPP NRMSE of M-S, while at Flakaliden and CAge75yr the M-S calibration error was significantly higher (Fig. 10a). The NRMSEs of the evapotranspiration M-S were always higher than those of the S-S calibration, except for Skyttorp (Fig. 10b). At Alkkia the M-S had a NRMSE of about 60, 20 % higher than the error of the S-S calibration; on the contrary, at Skyttorp the NRMSE of S-S was 20 % higher than the M-S NRMSE. For the rest of the sites the ET NRMSEs of the two calibrations differed less than 5 %.

According to the BMC, the Hyytiälä S-S calibration had 100 % probability of being better than the M-S calibration on every site; on the contrary, the M-S calibrations were always better than the other S-S calibrations (Table 5). The normalised root mean squared errors calculated for Analysis #3 are consistent with the BMC probabilities (Fig. 11, Table 4). The ET and GPP NRMSEs of the M-S calibration were slightly higher than those of Hyytiälä S-S calibration, but the differences are negligible (Fig. 11). The NRMSEs of the other S-S calibrations were always higher than the errors generated from the M-S calibration (Fig. 11).