We perform optimisations by relying on a Bayesian framework to include prior knowledge about the parameters (xb). Assuming that the errors associated with data observation, model output and parameters follow Gaussian distributions , we seek to obtain a posterior optimal parameter set xopt which corresponds to the minimum of the cost function J(x): 1J(x)=(M(x)-y)TR-1(M(x)-y)+(x-xb)TB-1(x-xb). For a given parameter set x, J(x) measures the mismatch between observations y and the corresponding model outputs M(x) as well as the mismatch between the prior, or background, parameter set xb and x. Each of these terms is weighted by its error covariance matrices, R and B, for the observations and parameters, respectively . In this study, we set both matrices to be diagonal. For B, we define the prior distribution of each parameter to be 40 % of the prior range. For R, we define the observation error (variance) as the mean-squared difference between the observations and the prior model simulation so that this variance reflects not only the measurement errors, but also the model errors. Furthermore, since we do not consider error covariances, R is diagonal, and therefore we can decompose the first term of Eq. () into different terms for each assimilated data stream: 2J(x)=KFlx(MFlx(x)−yFlx)T(σFlx-1)(MFlx(x)-yFlx)+KFACE(MFACE(x)−yFACE)T(σFACE-1)(MFACE(x)-yFACE)+(x-xb)TB-1(x-xb). The Flx and FACE subscripts are used to denote the FLUXNET and FACE parts of the equation; Ki denotes the weighting using for each data stream, σi denotes the observational error, and Mi and yi denote modelled and observed data streams.

There exist many different approaches we can use to find the set of parameters which minimise J(x). These range from simple manual tuning, which are very computationally demanding and inefficient, to more complex algorithms either based on deterministic gradient descent methods or stochastic random search methods. Using “ORCHIDAS”, the ORCHIDEE data assimilation tool developed at the Laboratoire des Sciences du Climat et de l'Environnement , we performed a couple of preliminary experiments to determine which algorithm to use. We tested a gradient descent method based on the L-BFGS-B algorithm (limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm with bound constraints BFGS; ) and a random search method based on the genetic algorithm (GA; ). We found that the GA method outperformed the gradient method in reducing the cost function. These initial results are coherent with the study, which optimised the GPP and latent heat fluxes of a former version of ORCHIDEE against a number of FLUXNET site measurements and also found that the GA algorithm outperformed the other methods, notably by allowing a full exploration of all the parameter space.

The genetic algorithm consists in applying the laws of evolution to our set of parameters by considering the set of parameters as a chromosome, with each parameter as a gene. At each iteration, the algorithm fills h chromosomes with parameter values. The first pool of chromosomes is created by randomly perturbing the value of the parameter. For the following iterations, the chromosomes are created from the previous iterations' chromosomes. Two processes come into play: (a) a crossover process, where we have an exchange of genes between two chromosomes; and (b) a mutation process, where random genes are perturbed. To ensure that the best chromosomes get the most descendants, each chromosome of each iteration is ranked as a function of the cost associated with the parameter's value in the chromosome.

In this study, we consider two sites from the FACE network in nitrogen-limited temperate forest ecosystems: Oak Ridge ORNL;, a site dominated by broadleaf deciduous forests (TeBS, for temperate broadleaf summergreen forests); and Duke DUKE;, a site dominated by needleleaf evergreen forests (TeNE, for temperate needleleaf evergreen forests). The data for these sites come from the FACE Model Data Synthesis project https://facedata.ornl.gov/facemds/, last access: 15 February 2024. For each site, we use the data from two experimental plots (with their associated error bars), one with unaffected atmospheric CO2, i.e. ambient (AMB), and one with elevated atmospheric CO2 (ELE). Although the DUKE experiment also has ammonium nitrate treatments at half of its plots from 2005 onwards (Feng et al., 2010), we only consider the data from the plots without nitrogen fertilisation.

The version of ORCHIDEE we use in this study has yet to be optimised against FLUXNET data using a Bayesian framework, as was done with previous nitrogen-free versions of the model e.g.. Therefore, we also consider TeBS and TeNE sites from the FLUXNET2015 dataset . This dataset provides gap-filled half-hourly meteorological data measured at each site (air temperature, humidity, pressure, wind speed, rainfall and snowfall rates, short-wave and long-wave incoming radiation; see ). It also provides net carbon flux measurements, such as net ecosystem exchange (NEE) further split into GPP and total ecosystem respiration (TER) following a classical night-time vs. day-time flux partition . For each of the two vegetation types, sites with over 60 % vegetation coverage are kept. We exclude sites with overly large discrepancies with the prior model output, such as with no apparent seasonal cycle, large data gaps or only 1 year of data. The list of in situ sites used can be seen in Table  partitioned by vegetation type.

Before performing the optimisations, for each of the sites in this study, a two-step spin-up is performed. The first step helps to put the prognostic variables, including vegetation state, soil carbon pools and soil moisture content, at equilibrium. The available meteorological forcing is cycled over several millennia (with pre-industrial CO2 concentrations) to ensure that the long-term net carbon flux is close to zero. After reaching equilibrium, a second simulation is performed (transient) from the year 1860 to 1 year before the first forcing year while increasing the CO2 concentration at each simulation year following global historical observations.

Before performing the optimisations, we also conduct a sensitivity analysis of the parameters (as described in Sect.  and shown in Table ). A sensitivity analysis tests how differently the model outputs change with respect to different parameters. This is done to ensure that only parameters showing some sensitivity to the model outputs are used in the optimisation, thereby minimising the risk of using parameters that are weakly constrained by the fluxes. This is an important step since we want to avoid constraining parameters that will have a small impact on the optimisation but that have the potential to significantly degrade the model–data fit against processes not included in the calibration.

Once spun up and with the list-sensitive parameters, we perform two main sets of optimisations, always starting from this spin-up. The first is over the FLUXNET sites only, while the second also includes data from the FACE sites. Due to the CO2 fumigation over the FACE sites, NEE is not measured at these sites, and therefore GPP and TER estimates cannot be derived. Instead, for the FACE sites, we have annual NPP and daily LAI data. Throughout this study, we perform multi-site (MS) optimisations, i.e. optimisations executed over multiple sites of the same PFT simultaneously in order to find one common set of optimised parameters. Each optimisation is run for 20 iterations, which we found to be sufficient for the system to converge. For each iteration, 32 chromosomes are used, i.e. 32 different combinations of parameter values. We leave the last year of each FLUXNET site out of the optimisation to have independent data for the validation step of the analysis.

The first set of optimisations tests two different combinations of gross and net carbon fluxes.

FlxGR: two MS optimisations against daily GPP and TER, one for all of the TeBS sites and one for all of the TeNE sites

The nitrogen cycle version of the ORCHIDEE model used in this study has not yet been optimised against FLUXNET data, although it has been extensively tuned manually. Therefore, the first step is to see whether the fluxes over the TeBS and TeNE sites are well represented in the model and whether they can be optimised using observed data and the sensitive parameters identified in the Morris experiment (Table ). For the FLUXNET optimisation, two tests are conducted. The best-performing optimisation will serve as the starting point for the optimisations including data from the FACE sites. In this section, we present the results from both MS FLUXNET optimisations: FlxGR and FlxGN. Figure  shows the mean seasonal cycle across all the FLUXNET sites for each PFT considered. We show the modelled GPP, TER and NEE fluxes against the observed time series.

For the deciduous sites (TeBS; Fig. , left-hand column), we see that both GPP and TER are overestimated by the prior model, and the NEE sink is underestimated. This overestimation is the most severe for TER, where the prior model simulates a maximum of approximately 9 gC m-2 d-1 when the maximum TER observed is half that. In contrast, the overestimation for GPP is very slight when found at and after the peak. Both optimisations improve the model–data fit against GPP by correcting the overestimation found after the peak. FlxGN performs best, with the average seasonal peak now the same as the observations. FlxGR on the other hand reduces the peak below that observed. Similarly, both optimisations now start production later in the year, degrading the fit to the observations in the early months. In ORCHIDEE, deciduous sites lose all their leaves in winter, and therefore no photosynthesis occurs before the leaves start growing back in spring. In contrast, the observations never go to zero, implying that there is undergrowth or evergreen vegetation present that we are not accounting for in the model set-up. When looking at the root-mean-square deviations (RMSDs) of the individual sites, we also see that FlxGN reduces the spread relative to the prior. For TER, both optimisations improve the model–data fit over the whole period, with FlxGR performing slightly better. This is not surprising since this optimisation directly considers the TER component of NEE. The optimisations mainly change the magnitude of the peak and do not correct for its late timing. When looking at the RMSD, we can see that a group of sites is driving the overestimation of TER, with values close to 4.5 – this is corrected for in both optimisations. For NEE, the FlxGN optimisation performs better than FlxGR, especially in fitting the autumn month. However, the overestimation of summer TER with this parameter set means we do not attain the minimum of the NEE trough. We note that, for NEE, the posterior spreads of the RMSDs over all the sites are the same for both optimisations.

For the evergreen sites (TeNE; Fig. , right-hand column), the optimisations improve the GPP underestimation in the prior model by increasing the peak by approximately 2.5 gC m-2 d-1. However, this falls short of correcting the full overestimation, which is closer to 4 gC m-2 d-1. The FlxGN optimisation performs best, with the timing of the average peak closest to the observed value. However, both optimisations degrade the average fit to TER, increasing the overestimation found when using the prior set of parameters. Both optimisations move the summer peak up by between 0.5 and 1 gC m-2 d-1, with FlxGN increasing the most. When considering the fit of the individual sites (right-hard part of each panel), we note that two anomalous sites are driving this behaviour. These sites (IT-Lav and US-Wi4) have respiration rates much lower than the other sites. Since we cannot match the respiration rates, we cannot capture the full NEE dip in summer. However, the improved GPP with both optimisations does mean that the NEE is slightly improved compared to the prior mode.

In these optimisations, we have included a parameter, KSoil, which acts as a multiplicative factor of the initial soil and nitrogen pools (slow, passive and labile). We have used such a parameter in past ORCHIDEE optimisations at the FLUXNET sites e.g. and found that it played a large role in improving the model–data fit against respiration. Therefore, it seems counter-intuitive that we do not improve the fit to respiration as much as expected when including KSoil in the ORCHIDEE v3 optimisations, especially for the TeNE sites. The past ORCHIDEE experiments all used a previous version of ORCHIDEE without the nitrogen cycle, so this factor solely acted on the carbon pools and heterotrophic respiration. Here KSoil multiplies both carbon and nitrogen pools to maintain the carbon/nitrogen ratio. However, by acting on the nitrogen pools, we directly impact the mineralisation rate and thus indirectly the plant N uptake, leaf N content, Vmax and, therefore, GPP. While KSoil used only to impact soil respiration, it now impacts both respiration and GPP, and so the optimisation needs to find a compromise to fit both. To adjust the respiration, KSoil is decreased, reducing the carbon and nitrogen pools in the soils, but, at the same time, GPP is significantly reduced, deteriorating its fit to observations.

Overall, the ORCHIDEE model reasonably represents the TeBS and TeNE carbon fluxes, although respiration at the TeNE sites is high, even after optimisation. The FlxGN optimisation results in the best-simulated production for both types of vegetation.

Given the results from the previous section and our motivation to improve the model performance regarding ecosystem productivity, we will further include the FACE data in the FlxGN optimisation.

To conclude this study, we test how the optimised model parameters impact the model responses to CO2 increase. We especially want to consider the additional information gained from including FACE data in the optimisation. As such, for this last experiment, we use parameters of the FlxGN (FLUXNET-only) and FlxGN-BOTH (FLUXNET and FACE data) optimisations. The FlxGN-BOTH optimisation resulted in the best compromise in simulating NPP under ambient and elevated conditions as well as their ratio while also maintaining a satisfying model–data fit to GPP over the FLUXNET sites.

To test the model response to CO2 increase, we run an idealised 100-year long experiment increasing CO2 by 1 % yr-1 over all the FLUXNET sites. This is similar to the experiment done in , where the model was also assessed with GPP, transpiration and WUE. Note that the prior projections shown here are similar to those obtained in . We first note that the optimised models predict lower starting values of GPP, transpiration and WUE for TeBS when compared to the prior model (Table ). This is consistent with Figs.  and , where for TeBS the prior had the highest productivity. Similarly, the behaviour for the TeNE sites mirrors earlier findings – the prior model underestimated productivity. The optimisations lead to higher starting values (except for WUE), and including data from FACE is found to induce a larger increase. When considering the total plant N uptake for both vegetation types, the prior starts with a high value and the optimised runs with much lower values.

In Fig. , we consider the net increase over the 100-year simulation. For both vegetation types, the rate of increase in GPP found by using parameters from the FLUXNET-only optimisation is the highest. For TeNE, the curve is starting to plateau due to a progressive nitrogen limitation. However, in both cases, the rate of increase is much faster compared to the runs with parameters from the FlxGN-BOTH optimisation. The fertilisation rate decreases compared to the FLUXNET-only optimisation. In Fig. , we saw that the FlxGN-BOTH optimisation was starting to model the declining productivity response at ORNL. We may still be overestimating the fertilisation effect over the TeBS sites, and we could expect an even stronger response to nitrogen limitation – further reducing this GPP increase. Although the GPP increase is similar to the prior model for forest types, the larger spread among the sites for FlxGN-BOTH is notable – with a negative GPP change at some sites suggesting a stronger nitrogen limitation under these parameters.

For transpiration rates over both TeBS and TeNE, the FlxGN simulations result in the smallest change by the end of the century. In contrast, the simulations FlxGN-BOTH result in the largest change. For WUE, the differences observed in GPP and transpiration are mostly cancelled out, and therefore the different parameters do not elicit a great variation in responses. For TeBS, the prior and FlxGN simulations give a similar increase, with FlxGN-BOTH suggesting a slightly weaker increase. For TeNE, it is the opposite – FlxGN is slightly weaker than the other two simulations.

For plant N uptake, changes over the 100-year period are small in magnitude but vary between the different optimisations. The prior model shows a range of responses over the different sites (evidenced by the large spread), overall decreasing the rate of N uptake for TeBS and increasing the rate of uptake for TeNE. The FLUXNET-only optimisations lead to a slight increase in plant N uptake by the end of the century for both vegetation types. In contrast, the FACE optimisation leads to a decreased plant N uptake. This is especially notable for TeNE, where the spread of responses over the sites is greatly reduced. In all the cases, changes to the rate of change occur during the first half of the simulations, plateauing to a constant value for the rest of the runs.

Note that our experiment only looks at increasing CO2 while keeping the other drivers constant. It is possible that we would see different responses if we were to include changes in meteorological forcing (to mimic climate change) or changes in nitrogen deposition and fertilisation, which would change nitrogen limitation and responses to water stress. Although we performed a sensitivity analysis to select the parameters, it is possible that additional parameters (e.g. ones controlling water stress or controlling the allocation of carbon in the plant) will give a different response. Furthermore, direct structural changes or addition to the model code could also change the results. Nevertheless, the fact that different parameters give such a varied model response to elevated CO2 for a given model structure highlights the importance of finding a robust set of parameters to have faith in. When performing the FLUXNET optimisations, the optimisation against GPP and NEE gave the best fit to both quantities, resulting in sets of parameters that worked well across each PFT. However, when considering production through NPP and LAI under different atmospheric CO2 conditions, we found that the parameters were unable to capture the differences observed in data under different CO2 regimes. Instead, the parameter sets found by optimising the model against FLUXNET data and both ambient and elevated CO2 conditions were found to be more robust. These parameters resulted in the best fits overall against NPP and LAI under two different CO2 regimes. As such, we have more confidence in these parameters and their ability to simulate terrestrial production under different atmospheric CO2 conditions, leading us to have more faith in the model projections performed with them.