CARAIB is a physically based dynamic vegetation model that was developed for the simulation of the carbon cycle on the global scale . It calculates the carbon fluxes through the soil–vegetation–atmosphere continuum by simulating ecophysiological processes: photosynthesis, carbon allocation to plant pools, and autotrophic and heterotrophic respiration. The CARAIB model has been used in numerous paleoclimatology, vegetation and crop modelling studies. The reader is referred to the aforementioned references for a full model description.

For C3 plants, photosynthesis is computed according to the model of . The stomatal conductance governing the flux of CO2 through the stomata is described on the leaf scale with the Ball–Berry approach , using the model of with further adaptations from to account for soil water stress affecting the stomatal conductance. Photosynthesis and respiration processes are computed at 2-hour time steps on a half-day basis, and the model assumes a symmetry with respect to solar noon time; that is, computation of these processes is made for half the day and further aggregated using a daily time step. Other processes, e.g., related to soil hydrology or carbon allocation, are computed on a daily basis.

In this study, a single plant functional type (PFT) is considered (BAG 22 as defined in ) corresponding to the flora that can be encountered in European grasslands, i.e., species of Poaceae and Asteraceae. The model was adapted for simulating the grassland sites by adding management functions for grass cutting and grazing. Grass cutting is modelled by the removal of a part of the plant carbon mass so that the model matches given values of leaf area index after cutting. Grazing is modelled such that a given fraction of the plant carbon mass is removed every day according to the grazing charge. The dates of the grass cutting and the duration of the grazing periods were known and fixed in the simulations. Daily meteorological data recorded at the experimental sites were used in the model, i.e., minimal and maximal temperature, precipitation, solar radiation, relative air humidity, and wind velocity. Although they can affect vegetation modelling , uncertainties in the meteorological data were not considered in this study.

Thirty-three parameters per PFT are set in CARAIB. These parameters govern photosynthesis, plant physiology process (e.g., specific leaf area, carbon-to-nitrogen ratio), allocation of carbon and residence times in the different pools of carbon, including plants and soil pools, land surface–atmosphere interactions (albedo, roughness length), and tolerance to extreme conditions (thresholds and response times). During the model development, parameter values in CARAIB were mainly taken from the literature and further compared with observed values (remote sensing, field data and paleorecords). So far, no model inversions were performed with the CARAIB model.

In this study, 10 model parameters were sampled (Table ). They were chosen according to their presupposed importance – that is, the model sensitivity to these parameters – and because some parameter values were already known in the measured data from the experimental sites. Default values that were defined during the model development and used in previous research are given in Table . These parameters govern the main processes of the model, namely, the photosynthesis, the respiration and carbon transfer between carbon pools:

The slope g1 and the intercept g0 (µmol m-2 s-1) of the stomatal conductance as described in are directly related to the photosynthesis since they govern the stomatal conductance. They are thus related to the gross primary productivity (GPP) and evapotranspiration (ET) with respect to the meteorological conditions. While most of ecological models, including CARAIB, use an empirical approach for stomatal conductance, derived from the Ball–Berry model, recently reconciled the empirical approach with the theoretical background based on the optimal stomatal behaviour , which states that there is a trade-off for stomata between maximizing carbon gain (photosynthesis) and minimizing water loss (transpiration). These new developments in the theoretical understanding of the empirical relationship push forward the necessity to measure or calibrate the stomatal conductance parameters under different environmental conditions. Although single values of these parameters are used for regional or global modelling of C3 plant photosynthesis e.g.,, it is known that stomatal conductance parameters actually vary through time and space according to the environmental conditions and plant species.

To acknowledge that measurements and modelling errors are inevitable, the inverse problem is commonly represented by the stochastic relationship F(z)=d+e, where F is a deterministic, error-free forward model that expresses the relation between the uncertain parameters z and the measurement data d and where the noise term e lumps measurement and model errors.

Inversions were performed within a Bayesian framework, which treats the unknown model parameters z as random variables with the posterior probability density function (pdf) pz|d given by p(z|d)=p(z)p(d|z)p(d)∝p(z)L(z|d), where pz denotes the prior distribution of z and Lz|d≡pd|z signifies the likelihood function of z. The normalization factor pd=∫pzpd|zdz is obtained from numerical integration over the parameter space so that pz|d scales to unity. The quantity pd is generally difficult to estimate in practice but is not required for parameter inference. In the remainder of this study, we will focus on the unnormalized posterior pz|d∝pzLz|d. For numerical stability, it is often preferable to work with the log-likelihood function, ℓz|d, instead of Lz|d. If we assume the error e to be normally distributed, uncorrelated and with an unknown constant variance, σ2, the log-likelihood function can be written as ℓ(z|d)=-N2log⁡(2π)-N2log⁡(σ2)-12σ2∑i=1Ndi-Fi(z)2, where σ can be fixed beforehand or sampled jointly with the other model parameters z.

The homoscedasticity (i.e., constant variance) assumption for e may be excessively strong in many cases. Considering the residual errors, e, to be heteroscedastic, Eq. () becomes ℓ(z|d)=-N2log⁡(2π)-∑i=1Nlog⁡(σi)-12∑i=1Ndi-Fi(z)2σi2, where the σi represents the individual residual error standard deviations that can be gathered into a vector σ. Here also, σ can either be fixed beforehand or sampled along with z (see Sect. 2.3.4).

In this work, we chose three types of eddy covariance data for the calibration: d1 (GPP), d2 (RECO) and d3 (ET). We further assume that the corresponding residual errors, e1, e2 and e3, are uncorrelated, leading to the following multi-objective log-likelihood function: ℓ(z|d1,2,3)=ℓ(z|d1)+ℓ(z|d2)+ℓ(z|d3). The weighting between the three components of ℓz|d1,2,3 is an important issue. The constant (σ) and non-constant (σi) standard deviations in Eqs. () and (), respectively, basically weight the respective influences of e1, e2 and e3 on the log likelihood defined by Eq. (). Distinct homoscedastic or heteroscedastic residual error models must be specified for e1, e2 and e3. This was done for both the homoscedastic and heteroscedastic cases either by specifying the residual error standard deviations beforehand or by jointly inferring these standard deviations along with the model parameters.

Based on prior knowledge of the measurement errors, the homoscedasticity assumption simply reduces to assigning values to σ1, σ2 and σ3 in Eqs. () and (). These values were fixed to 3 g C m-2 day-1 for the GPP measurements, 1.5 g C m-2 day-1 for the RECO measurements and 1 mm day-1 for the ET measurements. As stated earlier, measurement errors associated with eddy covariance fluxes are, however, typically found to be heteroscedastic, with a variance that is assumed to be linearly related to the magnitude of the measured data : σd,i=12σ0,ddid‾+1, where the variable d denotes either GPP, RECO, or ET measurements, i=1,⋯,N are measurement times and σ0,d is equivalent to σ1, σ2 or σ3 in the homoscedastic case. We refer to the inversions based on these homoscedastic and heteroscedastic error models as HO1 and HE1, respectively. It is worth noting that by fixing the standard deviations to known measurement errors, one implicitly assumes that the model is able to describe the observed system up to the observation errors. This might not be realistic in environmental modelling, where models are always fairly simplified descriptions of a much more complex reality.

Still under the Gaussianity assumption, a more advanced treatment of the residual error models considers the simultaneous inference of the standard deviations with the model parameters, i.e., it considers the standard deviation of the residual errors as unknowns. Doing so assumes that residual errors are expected to be a mixture of both model (equations and inputs) and observational errors. For the homoscedastic case, this simply consists of jointly sampling σ1, σ2 and σ3 along with the model parameters, z.

The heteroscedastic error model then becomes σd,i=adi+b, where the a and b coefficients are to be jointly inferred with z from the measurement data. Using Eq. () thus leads to the addition of six variables to the sampling problem: a1, a2, a3, b1, b2 and b3. We refer to the joint inversions of these homoscedastic and heteroscedastic error models as HO2 and HE2, respectively. In these inversions, a total predictive uncertainty around the model values can be computed by adding to the modelled data a random noise drawn from a normal distribution with mean 0 and standard deviation σ sampled from its posterior distribution (HO2) or computed by Eq. (; HE2).

The simultaneous inference of model parameters with homoscedastic or heteroscedastic error model parameters requires the definition of their prior probability distributions. Based on the available prior information, uniform (flat) priors are used for the 10 model parameters contained in z (see Table ). We follow two guidelines for specifying the prior densities of the error model parameters. First, we would like to obtain posterior standard deviations that are as small as possible within the range permitted by the model and measurement data errors in order to get the lowest possible data misfits. Second, the magnitudes of the different prior distributions should reflect the desired weights of the different data types within the multi-objective inference. These weights translate the modeller's relative preferences among the three modelling objectives in Eq. (). We therefore use normal distributions with mean 0 truncated at 0 to avoid negative values. The prescribed weights then correspond to the different standard deviations of these normal distributions: pX=1σXBϕX-μXσX∝ϕX-μXσX, where the X variable is either σj, aj or bj for j=1,2,3; where the value of σX expresses the modeller's preference for objective j compared to the other objectives (the smaller σX, the larger the relative weight of objective j); where ϕ⋅ signifies the probability density function of the standard normal distribution; where μX is set to 0 for maximizing the prior density of X towards small values; and where the constant B depends on the lower (v) and upper (w) limits of the truncation interval B=Φw-μXσX-Φv-μXσX, in which Φ⋅ denotes the cumulative distribution function of the standard normal distribution.

This treatment of multi-objective Bayesian inference is in line with the work of , who further considered different statistical models for model and observation errors. Overall, this resulted in four different ways of treating the eddy covariance data uncertainties: fixed homoscedastic (HO1) and heteroscedastic (HE1) error models and jointly inferred homoscedastic (HO2) and heteroscedastic (HE2) error models. Using the HO1 and HE1 models led to a total of 10 inferred parameters, whereas using the HO2 and HE2 models resulted into a total of 13 and 16 inferred parameters, respectively. Table  lists the marginal prior distributions used for all sampled parameters. The upper and lower bounds of these distributions were either set to their maximal and minimal physically possible values or determined on the basis of expert knowledge.

The goal of the inference is to estimate the posterior distribution pz|d where the 10-, 13- or 16-dimensional z vector contains all sampled parameters and d signifies the conditioning data: d=d1,2,3 herein. As an exact analytical solution of pz|d is not available, we resort to Markov chain Monte Carlo (MCMC) simulation to generate samples from this distribution. The basis of this technique is a Markov chain that generates a random walk through the search space and iteratively finds parameter sets with stable frequencies stemming from the posterior pdf of the model parameters see, e.g.,for a comprehensive overview of MCMC simulation.

The MCMC sampling efficiency strongly depends on the assumed proposal distribution used to generate transitions in the Markov chain. In this work, the state-of-the-art DREAM(ZS) (DiffeRential Evolution Adaptive Metropolis) algorithm is used to generate posterior samples. A detailed description of this sampling scheme including convergence proof can be found in the literature cited and is thus not reproduced herein.

Convergence of the MCMC sampling to the posterior distribution is monitored by means of the potential scale reduction factor of , R^. For each parameter of interest, this statistic compares the average within-chain variance to the variance of all the chains mixed together. The smaller the difference between these two variances, the closer to 1 the value of the R^ diagnostic. Values of R^ smaller than 1.2 are commonly deemed to indicate convergence to a stationary distribution. In this study, posterior distributions of the parameters were drawn from the point where all parameters achieved R^<1.2. This is more conservative than the conventional practice of stopping the inference when R^<1.2 for every parameter. The mean acceptance rate of the proposed samples, AR (%), is an important sampling property and is thus also reported. An excessively small fraction of accepted candidate points indicates poor mixing of the chains due to too wide a proposal distribution. In contrast, a very large acceptance rate signals too narrow a proposal distribution, causing the chains to remain in the close vicinity of their current locations. The optimal value for AR depends on the proposal and target distributions, but a range of 10–30 % generally indicates good performance of DREAM(ZS).

The DREAM(ZS) algorithm was run with four parallel chains, initialized by sampling the prior parameter distribution (Table ). As an example, Fig.  shows sampling trajectories of DREAM(ZS) parametrized with four chains for the SLA parameter and inversion HO1 at the Oensingen site. The R^ convergence statistic becomes < 1.2 for each parameter after about 20 000 forward model runs, and the AR over the last 50 % model evaluations is about 18 %. Overall, convergence was achieved for all MCMC trials after some 15 000–30 000 forward runs with AR values in the range of 10–30 %, except for the inversions associated with the Laqueuille site that showed AR values as low as 5 %.

Figure  presents marginal posterior histograms of the 10 model parameters for all experimental sites, considering the inferred homoscedastic error model (inversion HO2). In the remainder of this document, results are mainly detailed for this inversion scenario, since it generally led to the lowest data misfit statistics in calibration. For some parameters (e.g., SLA and C / N1), the marginal posterior distributions are narrow compared to the prior parameter range. This indicates a large sensitivity of the model to the considered parameter. In contrast, some other parameters such as γ2 are poorly resolved, demonstrating a relative insensitivity. Asymmetric edge-hitting distributions are also observed such as for C / N1 and C / N2 in Monte Bondone. In a Bayesian inversion of eddy covariance data obtained from a forest site, found that 7 out of 26 marginal parameter distributions were edge-hitting. Extending the prior parameter ranges would lead to unphysical or implausible parameter values. Edge-hitting distributions reveal model inadequacies and/or large systematic measurements errors. For some parameters, posterior distributions were fairly distinct from the default values that were used in previous studies (Table ), such as high g1 values. Values of the characteristic mortality time τ also generally increased compared to the default value.

Table  shows the most likely parameter values for the four experimental sites; these parameter values resulted in the highest values of the log-likelihood function. Some of the parameters present contrasting values between inversion scenarios and/or experimental sites, which may be related to the different ecological characteristics of the sites as discussed in section . Depending on the width of the posterior distributions, the most likely parameter values are well resolved or largely uncertain. As a result, a comparison between the experimental sites must account for the posterior distributions of the parameters.

As the parameter sampling resulted in posterior distributions of the parameters instead of single values, ensembles of posterior modelled signals can be represented as a graph. In Fig. , measured and modelled eddy covariance data are depicted for the experimental site of Monte Bondone for inversions with the inferred homoscedastic error model (inversion HO2). The posterior ranges of the modelled signals are represented by the dark grey shaded areas for the prediction uncertainty due to parameter uncertainties and by the light grey shaded areas for the total predictive uncertainty (at 95 % confidence level). This total prediction uncertainty is computed using the standard deviation of the residual errors σ as sampled by the inversions and, therefore, cannot be computed for the NEE. The site of Monte Bondone was chosen here since there is one single cut a year (indicated by the vertical arrows in Fig. 3) that is clearly identifiable, which facilitates the interpretation of the fluxes. The dates of cutting corresponded to a sudden drop in the GPP in the middle of the year, which was followed by a gradual increase. They were also observed in the NEE graphs, with a sudden increase in the NEE.

There was overall good agreement between measured and modelled signals. It is worth noting that the posterior ranges of modelled data were not constant over time and were not related to the magnitude of the signals. The ranges due to parameter uncertainties were relatively small and did not encompass the measured data. Overall, it could be observed that measured eddy covariance data have stronger dynamics than the modelled signals, meaning that the CARAIB model cannot follow the fast fluctuations of the GPP (and other signals) over time. In particular, the model could not simulate the highest peaks in GPP well.

Considering the other three experimental sites (Fig. ), there was a similar agreement between measured and modelled signals, although the sites displayed different behaviour in terms of GPP as their management varies: there are several cuts per year in Grillenburg and Oensingen, while Laqueuille is a grazed meadow. In general, the peaks in GPP cannot be simulated well by the model. The modelled GPP seemed averaged out when compared to the measured signals, as observed before in Monte Bondone (Fig. a).

All the graphical comparisons between measured and modelled signals could not be shown but are summarized in Table  for the homoscedastic and heteroscedastic cases, and with the fixed and inferred error models, using the root mean square error (RMSE), the R2 and the model efficiency criterion (E) between measured and modelled signals. The latter criterion takes values from -∞ to 1. A value of 1 means a perfect match between measurements and model simulations, a value of 0 indicates that the mean of the observed data is as accurate as the modelled values, and an efficiency less than 0 occurs when the mean of the observed data reproduces the observations better than the modelled values. The maximum log-likelihood value “ml” that was obtained by the algorithm is also indicated. Note that performance criteria were also computed for the NEE, although these data were not used in the model inversions. Overall, the best agreement was found for the Monte Bondone site and the worst for the Laqueuille site. The lowest model efficiencies E were found for the NEE, which is not surprising since these data were not accounted for in the model inversions. While the ml values were generally the highest for the heteroscedastic inversions HE2, RMSE appeared larger for these inversions.

Considering homoscedastic or heteroscedastic eddy covariance residual errors resulted in different sampling of parameter posterior distributions and, therefore, different posterior modelled signals. As an example, Fig.  shows the measured and modelled GPP with their posterior ranges for the site of Monte Bondone in 2004, for both homoscedastic (a, c) and heteroscedastic (b, d) cases. For the HO2 and HE2 inversions, the 95 % total predictive uncertainty is depicted using the light grey shaded areas. The measurement uncertainty is depicted only for fixed eddy covariance residual error inversions (a, b) for clarity. The measurement uncertainty is thus constant for the homoscedastic case (namely, ±3 g C m-2 day-1 for HO1), while it varies linearly according to the GPP for the heteroscedastic case (HE1). These two options led to different behaviours of the modelled GPP using the posterior distributions, which better approached the high values of the measured data (in summer) in the homoscedastic cases and better fit the low values (in winter) in the heteroscedastic cases. Overall, in calibration, modelled signals with parameter values from the homoscedastic inversions were in a better agreement with the measured data than with the parameters from the heteroscedastic inversions. The same observation was also made for the other sites (not shown), as can also be observed in Table . However, the total predictive uncertainty range derived from the HE2 inversions was more consistent, as, e.g., it avoids unrealistic negative values of GPP. The standardized residuals, which were computed as the difference between measured and modelled data divided by the standard deviation of the residual error, are depicted in Fig.  to the right of the GPP graphs. Heteroscedasticity of the GPP residual errors was fairly reduced but not fully removed by using the HE1 and HE2 heteroscedastic residual error models. Indeed, the standardized residuals still showed some small but complex heteroscedastic patterns. Partial autocorrelation of the residuals of the GPP was also depicted, and independence of the days of simulation from one another was reached after a few days.

Inversions with the sampling of the standard deviations of the residual errors resulted in posterior distributions of the standard deviation of the residual errors (HO2) and parameters of Eq. (; HE2). Most likely values of these distributions (Table ) depended on the experimental sites, being larger for Laqueuille and Oensingen, which can be related to the poorer agreement between measured and modelled data at these sites. Although the sampled standard deviations of the residual errors were lower than in the fixed inversions, there were no large differences between the inversions with fixed model errors (HO1 & HE1) and inversions with inferred model errors (HO2 & HE2), neither in terms of agreement between measured and modelled signals (see Fig.  and Table ) nor in the parameter posterior distributions (Table ).

Bayesian inversions were conducted considering homoscedasticity and heteroscedasticity in the eddy covariance residual errors. Figure  showed that accounting for heteroscedasticity in eddy covariance residual errors permitted a better simulation of low-magnitude signals (winter), but at the same time, it penalized the modelling of high-magnitude signals (summer). Actually, it is worth remarking that, in carrying out inversions considering heteroscedastic measurement errors, we do not attempt to produce smaller RMSEs between measured and modelled data compared to homoscedastic scenarios since larger errors are considered for high peaks in the signals. However, in validation, the posterior parameter distributions derived from using the heteroscedastic residual error models outperform their counterparts derived from using the homoscedastic residual error models. This important finding reveals that, despite inducing larger RMSE values in calibration, the use of a heteroscedastic residual error model leads to a more robust parameter estimation.

Since eddy covariance data are known to show heteroscedasticity, accounting for a heteroscedastic model of the residual errors in the inversions is more conceptually sound for ensuring unbiased parameter posterior distributions. However, we showed that considering a linear heteroscedastic model of the residual errors only partly removed heteroscedasticity in the standardized residual values (Fig. b and d). Other kinds of heteroscedastic models (i.e., non-linear) might be tested, but the residual distributions did not show any clear trend for all sites.

It is also worth noting that a substantial fraction of the large residual errors is caused by the tendency of the CARAIB model to underestimate the observed GPP summer peaks. As discussed above, this is related to a slower temporal resolution of the model compared to that of the measured data. To overcome this model inadequacy, further model modifications are necessary to increase the time resolution of the model. Another model improvement would be to simulate varying model parameter values as a function of the time of the year, since plant traits actually evolve over the course of the seasons. However, this would come at the cost of a large increase in model complexity.

Sampling the standard deviation of the residual errors, i.e., the inversions HO2 and HE2, resulted in similar parameter samplings and modelling as the inversions HO1 and HE1, respectively. Some performance criteria were better with the sampling of the residual standard deviations, while others were not. As expected, the most likely standard deviations of the residuals errors were close to the RMSE obtained in the inversions HO2. The benefit of these values is that they inform us about the level of the uncertainties in the eddy covariance data with respect to the model used to invert the data, e.g., uncertainties in GPP ranged from 1.79 to 2.29 g C m-2 day-1, in RECO from 1.09 to 1.63 g C m-2 day-1 and in ET from 0.52 to 1.31 mm day-1. They could be used to weight different eddy covariance data in multi-objective inverse modelling.