Soil organic carbon (SOC) has a significant effect on carbon emissions and climate change. However, the current SOC prediction accuracy of most models is very low. Most evaluation studies indicate that the prediction error mainly comes from parameter uncertainties, which can be improved by parameter calibration. Data assimilation techniques have been successfully employed for the parameter calibration of SOC models. However, data assimilation algorithms, such as the sampling-based Bayesian Markov chain Monte Carlo (MCMC), generally have high computation costs and are not appropriate for complex global land models. This study proposes a new parameter calibration method based on surrogate optimization techniques to improve the prediction accuracy of SOC. Experiments on three types of soil carbon cycle models, including the Community Land Model with the Carnegie–Ames–Stanford Approach biogeochemistry submodel (CLM-CASA') and two microbial models show that the surrogate-based optimization method is effective and efficient in terms of both accuracy and cost. Compared to predictions using the tuned parameter values through Bayesian MCMC, the root mean squared errors (RMSEs) between the predictions using the calibrated parameter values with surrogate-base optimization and the observations could be reduced by up to 12 % for different SOC models. Meanwhile, the corresponding computational cost is lower than other global optimization algorithms.

Soil organic carbon (SOC) is the largest pool of global land carbon (Todd-Brown et al., 2013; Luo et al., 2015). The emission of

Considering the high similarity in the structures of the 11 ESMs, the difference in the SOC simulations mainly results from parameterizations (Todd-Brown et al., 2013). Thus, parameter calibration is among the top priorities regarding the improvement of the prediction of global land carbon cycle dynamics (Luo et al., 2016). However, parameter calibration with global observations has not been widely applied, owing to the high computational cost. A matrix approach has recently been developed to reorganize the carbon balance equations in the original ESMs into one matrix equation without changing any modeled C (carbon) cycle processes and mechanisms (Luo et al., 2003, 2017; Huang et al., 2018). The matrix land carbon cycle model can be semi-analytically saved to obtain steady-state solutions and save more than 90 % computational time compared to original models (Xia et al., 2012). As a consequence, the matrix approach makes parameter estimation and calibration possible. The matrix approach has been successfully used for parameter calibration to constrain SOC turnover and microbial process using the Bayesian Markov chain Monte Carlo (MCMC) algorithm (Harauk et al., 2014, 2015; Shi et al., 2018).

Bayesian MCMC is a sampling-based approach and usually requires a large number of simulations for building an acceptable parameter chain. For instance, over 500 000 simulations are required during the parameter calibration of soil carbon models (Xu et al., 2006). Even using high-performance computers, complex land models, like the latest version of Community Land Model (CLM5.0), require a very long spin-up time for carbon cycle simulation, leading to several hours or days for one simulation. Although the matrix approach has been developed to enable data assimilation of global land carbon cycle models (Harauk et al., 2014, 2015; Shi et al., 2018), Bayesian MCMC is still very computationally expensive for calibrating global land models. More effective and efficient parameter calibration algorithms are urgently needed.

The parameter calibration of SOC models can be formulated as an optimization problem that aims to minimize the output of a cost function. This cost function evaluates the difference between the outputs of model simulation and the corresponding observations and returns a single value (e.g., RMSE) to represent the model error. Global optimization algorithms are introduced to find the minimum value of the nonlinear, non-convex, and black-box problems (Hapuarachchi et al., 2001; Ma et al., 2006; Rocha, 2008). Unfortunately, the number of required simulations for most global optimizations is very large.

To reduce the number of simulations and decrease the computational cost, we, for the first time, present a surrogate-based optimization method (SBO) for calibrating soil carbon models. Surrogate models serve as computationally inexpensive approximations of expensive simulation models (Booker et al., 1999), such as complex geoscientific models. During the optimization process, the surrogate model can be used to determine a new promising point in the parameter space at which the expensive simulation model originally has to be evaluated. With the help of the surrogate model, many unnecessary simulations with bad parameter values, which lead to high prediction errors, are avoided. SBO has been shown to find the near-optimal parameter values within only a few hundred simulations for different problems (Aleman et al., 2009; Giunta et al., 1997; Regis, 2011; Simpson et al., 2001).

Most studies on both global and surrogate optimizations focus on the mathematical function benchmarks like “Comparing Continuous Optimizers”, abbreviated as COCO (Hansen et al., 2010; Wang et al., 2014). However, the optimization of the mathematical functions may be extremely different from the parameter calibration of complex real-world models. In this paper, we explore the state-of-the-art surrogate optimization method for parameter calibration of three SOC models: CLM coupled with Carnegie–Ames–Stanford Approach biogeochemistry submodel (CLM-CASA'), and two microbial models such as those used in studies by Hararuk et al. (2014, 2015). Although the three models are computationally attainable for parameter calibration, we compare the performance of surrogate-based optimization to advanced global optimization algorithms and the data assimilation method to examine the potential of SBO. The SBO method may be extended to other complex global land models.

In this paper, we present the structure and parameters of three SOC models in Sect. 2. Section 3 introduces the algorithm design of SBO. The parameter calibration results and the analysis of different parameter calibration algorithms are presented in Sect. 4. Section 5 discusses the calibrated results using SBO. Finally, we draw conclusions in Sect. 6.

Earth system models (ESMs) are a fundamental tool for simulating climate impacts on the carbon cycle at the global scale. There are many common properties among structures of different global land carbon modules of ESMs (Luo et al., 2016). Almost all models have multiple carbon pools and carbon is transferred among these pools (Weng and Luo, 2011). In this study, we selected three SOC models, which have previously been calibrated for their parameters with the Bayesian MCMC algorithm (Hararuk et al., 2015). The first model is the soil carbon component of CLM coupled with the Carnegie–Ames–Stanford Approach biogeochemistry submodel (CLM-CASA') (Oleson et al., 2004, 2008). The CLM is the land model for the Community Earth System Model (CESM). The other two SOC models are microbial models, which consider the nonlinear regulation of SOC dynamics with microbial biomass.

The CLM-CASA' is embedded in CLM3.5. The latter includes biogeophysics and
biogeochemistry submodels. CLM-CASA' inputs carbon through net primary
productivity (NPP), which is partitioned to three live biomass pools (wood,
leaves and fine roots) (Fig. 1a). Dead plant materials become litter and are
transferred separately to four litter pools. Litter decomposition results in
part of the carbon being released to the atmosphere as heterotrophic respiration and
part of the carbon being stabilized into soil carbon pools. Organic carbon in
the soil pools is partially decomposed to be released as

Schematic representations of

The CLM-CASA' model simulates soil carbon decomposition as a first-order
decay process (Oleson et al., 2004, 2008). Based on theoretical analysis,
the carbon cycle of most ESMs can be summarized with one matrix equation
(Luo et al., 2001; Luo and Weng, 2011; Xia et al., 2013)
as follow:

Parameter description of CLM-CASA' C-only model.

Microorganisms catalyze various processes of the land carbon cycle, such as the decomposition and stabilization of SOC (Kuzyakov et al., 2000; Luo et al., 2001; Peng et al., 2009). However, most conventional SOC models, such as CLM-CASA', do not explicitly represent microbial processes. Microbially explicit models usually represent SOC decomposition by considering extracellular enzyme activities rather than simple decay constants as in the CLM-CASA' and other traditional SOC models (Schimel and Weintraub, 2003). In this study, we focus on two enzyme-driven decomposition models; one has two pools (Fig. 1b) introduced by German et al. (2012) and the other has four pools (Fig. 1c) introduced by Allison et al. (2010). We call these two models the two-pool microbial model and the four-pool microbial model, respectively. C inputs for the two models are NPP and the outputs are the carbon content of each pool at a steady state.

The two-pool microbial model is described using the following equations:
(Hararuk et al., 2015).

The four-pool microbial model from Allison et al. (2010) is described as
follows:

Fifteen parameters of the four-pool microbial model are also described in Table 2.

Parameter and description of the four-pool microbial models.

Microbial models and CLM-CASA' C-only models divide the world into

IBGP-DIS soil carbon distribution. Soil carbon varies from
0 kg m

The goal of parameter calibration is to improve the SOC predictions to
better fit the observations. Therefore, we use the root mean squared errors
(RMSEs) between the model SOC predictions and the observations at all grid
cells as the cost function. This cost function can be described by the
following formula:

The parameters of most soil carbon models and land models have traditionally been tuned manually (Luo et al., 2001, 2016). The manual tuning method might be effective for simple models but still highly depends on expert experience. Complex models may consist of various components from different disciplines and have hundreds or thousands of parameters. In these cases manual tuning consequently becomes impractical.

Different parameter calibration algorithms are available, which have been developed based on optimization theory. Gradient search algorithms like the quasi-Newton method are introduced to search for a set of parameters with better performance in the parameter domain. These algorithms are usually efficient and fast. However, gradient search algorithms are designed for finding the local optimum. They cannot be used to solve multimodal problems derived from complex Earth system models. In addition, they are based on the gradient information which is unavailable for most soil carbon and land models, as these models are too complex to obtain such details. Thus, parameter calibration usually becomes a black-box optimization problem. Global optimization algorithms, such as genetic algorithms and particle swarm optimization algorithms, are based on parameter generation and selection strategies. They are basically still gradient independent but can be easily used for parameter calibration of complex Earth system models. Global optimization algorithms are designed to find the global minimum. However, the number of samples (model runs) might still be too large to be applicable to complex models with large numbers of parameters (Jones et al., 1998). Moreover, complex Earth system models, for example CLM, require several hours over hundreds of cores for just one sample run and pose a special challenge regarding the feasibility of automatic parameter calibration.

SBO is an efficient and effective automatic parameter calibration framework. It fits a surrogate model (or response surface) based on the previous samples and uses this surrogate model to emulate the output behaviors of original models with an acceptable level of accuracy. The main advantage of SBO is to save computational costs during the global optimization by using the surrogate model instead of the original model. Furthermore the surrogate model can be continuously improved by exploiting new sample runs with the original model. With the surrogate model, the algorithm can make full use of previous samples' information and reduce the sample size, the time-to-solution, as well as the computation cost. SBO has been successfully used to solve the parameter calibration of computationally expensive black-box problems (Vu et al., 2016).

A flowchart describing the SBO is presented in Fig. 3.

A flowchart of the surrogate-based optimization.

First, initial sets of parameter values are generated using a sampling method. These sets are then used as inputs to run the real simulation model. Second, a surrogate model is constructed by fitting the outputs of these sample runs. The surrogate model serves as a computationally inexpensive approximation of the expensive simulation model (Booker et al., 1999). Then in each iteration, new sample points simulated by the real model are generated according to a specific strategy. This strategy can make use of the information gained from the surrogate model and only exploits the avoidable real model runs to meet the accuracy requirement. The new sample points and their simulation outputs are used to update the surrogate model at the same time. Finally, when some stop criteria (typically the maximum number of simulations allowed) are met, the algorithm returns the optimized parameter values. During the SBO process, quite a few sample runs are generated based on the evaluation of the surrogate model and most meaningless simulations with bad parameter values are avoided. As a result, the computationally expensive model is only simulated at a few selected promising parameter points, and the surrogate model replaces the real model during the calibration process. Thus, the computation cost is substantially reduced.

Different surrogate-based optimization algorithms may have different choices
with respect to the following:

The sampling method to generate the initial set

The surrogate model, which predicts the output

How to decide the new points at which to run the real model in each iteration.

There are various surrogate models, such as multivariate adaptive regression splines (MARS) (Friedman, 1991), polynomial regression models (Myers et al., 2016), radial basis functions (RBFs) (Gutmann, 2001; Müller and Shoemaker, 2014; Powell, 1992; Regis and Shoemaker, 2007, 2009; Wild and Shoemaker, 2013), and kriging (Davis and Lerapetritou, 2009; Forrester et al., 2008; Jones et al., 1998).

The MARS model is an extension of naïve linear models introduced by
J. H. Friedman (Friedman, 1991). The form of MARS is presented as
follows:

The RBF model is a real-valued function. The prediction at a point

Both the kriging model and the Gaussian process regression model predict the
output using a Gaussian process governed by prior covariance. The

In addition, many machine learning regression models are also introduced, such as support vector regression (Zhang et al., 2009), artificial neural network (Behzadian et al., 2009) and random forest (Breiman, 2001).

The strategies of parameter point generation are iterative algorithms that use data acquired from previous iterations to guide new parameter point generation. Most strategies convert the parameter point generation to optimization problems using an evaluation criterion (Fig. 3). There are many different generation strategies, including “Minimizing an Interpolating Surface” (MIS) (Jones, 2001) and “Maximizing Expected Improvement” (MEI) (Schonlau et al., 1997; Picheny et al., 2012). In MIS, the minimum of the surrogate model response surface is found and treated as the new parameter point to evaluate the real simulation model and then update the surrogate model. MEI introduces the “expected improvement” criterion. This criterion estimates the uncertainty of the surrogate model and balances the exploration and exploitation. Exploration refers to searching in an unfamiliar area of the parameter space to learn about it and avoiding being trapped in the local optimum. Exploitation means fast convergence in an area. Balancing the exploration and exploitation ensures SBO can find real global optimum and does not waste simulations on meaningless parameter sets and areas. Another parameter generation strategy is the candidate point approach (CAND) (Regis and Shoemake, 2007). In the CAND strategy, the criterion for exploitation is MIS and the criterion for exploration is the distance of the candidate point to the set of sampled parameter points from previous iterations. The previously sampled points represent the explored region and we can estimate the uncertainty from the distance to the explored region. A weighted sum of these two criteria is used to determine the new parameter point during the SBO.

Elaborating on the previous introduction of SBO, a detailed account of the SBO procedure follows.

Step 1: Generate an initial sampling set

Step 2: Run the real model and calculate the output error of the parameter points of

Step 3: Build the surrogate model using the parameters and outputs generated in Step 2.

Step 4: Predict the output errors of those points that do not belong to

Step 5: Run the real model again for the new parameter points from Step 4 and calculate the output errors of these selected points.

Step 6: Update the surrogate model with the new data from Step 5.

Step 7: Iterate through Steps 4 to 6 until the end condition has been met.

The SBO scheme mentioned in previous sections is a parameter calibration framework. The key components introduced in Sect. 3.2 must be selected when calibrating the parameters of soil carbon models. The LHS can cover the whole parameter space with a limited number of sample points while Monte Carlo sampling usually requires a much larger number of samples. Therefore, we choose the LHS as the initial sampling strategy.

As mentioned in the previous section, many kinds of surrogate-based models have been introduced and developed. The machine learning regression models do not perform as well as RBF and kriging models, according to an evaluation of similar cases (Wang et al., 2014). In this study, we use the RBF surrogate model (RBF-SBO) as our default choice because it has been shown to perform better than other surrogate models (Müller and Shoemaker, 2014) and is easily implemented. Our algorithm framework also includes other surrogate models, such as kriging and MARS, and can introduce others in the future.

The surrogate model is not accurate enough to represent complex and nonlinear models when the SBO starts. The MIS can be very efficient but is easy to trap into local optima, as the strategy does not consider the uncertainty of the surrogate model and only selects the optimum of the surrogate model. The MEI eliminates the disadvantage of MIS but can only be used for the kriging surrogate model because the calculation of the expected improvement requires the standard error at the parameter point and only the kriging (Gaussian Process) surrogate model can provide this (Jones et al., 1998). Finally, we use the CAND strategy as the parameter generation strategy in our algorithm to balance the exploitation and exploration of the uncertain region.

In this study, we select the Bayesian MCMC approach and four advanced global optimization algorithms for comparison with our proposed SBO method. Our SBO algorithm is implemented based on the “Surrogate Model Optimization Toolbox” toolkit (Müller, 2014). Three SOC models and their cost functions are introduced in Sect. 2. The target of parameter calibration is to find the optimal values of parameters to achieve the minimum value of the cost function (average RMSE). Moreover, we repeat the parameter calibration process of each algorithm 50 times and use the average results for algorithm evaluation. We compare the performance of algorithms in terms of both effectiveness and efficiency. The effectiveness refers to the accuracy of the calibrated results and the efficiency can be evaluated by the required number of simulations of the original SOC models.

The Bayesian MCMC approach and the four following advanced global optimization algorithms: differential evolution (DE), particle swarm optimization (PSO), shuffled complex evolution (SCE-UA) and the covariance matrix adaption evolution strategy (CMA-ES), are compared with our RBF-SBO.

DE (Storn and Price, 1997) and PSO (Kennedy, 2011; Shi and Eberhart, 2009) are the representative algorithms of the evolution strategy and swarm intelligence, respectively. They both have the ability to converge quickly and outperform many genetic algorithms and simulated annealing algorithms (Price et al., 2006; Shi and Eberhart, 2009). SCE-UA is designed for the parameter calibration of hydrologic models and has been successfully applied to various hydrology models such as the TOPMODEL, the Xinanjiang watershed model and short-term load forecasting (Hapuarachchi et al., 2001; Ma et al., 2006; Li et al., 2007). SCE-UA tries to keep both effectiveness and efficiency by combining the local (the simplex method) and global optimization methods. Despite the difference in algorithm details, DE, PSO and SCE-UA all generate new parameter points according to some simple mathematical formulas. Unlike these three algorithms, CMA-ES creates new parameter points based on a multivariate normal distribution (Hansen and Ostermeier, 2001; Hansen and Kern, 2004). The dependencies between parameters are represented by the covariance matrix of a normal distribution. CMA-ES has been shown to be the best global optimization algorithm in the BBOB-2009 comparison study (Hansen, 2009).

The Bayesian MCMC approach is typically designed to obtain the posterior
distributions of model parameters but it can also be used to calibrate
parameters to reduce the prediction error. The Bayesian MCMC consists of two
main parts: sampling and parameter estimation. During the sampling part, the
adaptive Metropolis (AM) algorithm, a Markov chain Monte Carlo method, is
used to conduct sampling from the prior parameter distributions and generate a
parameter chain (Haario et al., 2001). The AM algorithm has two steps: the
proposing step and the moving step. A new parameter set

Calibration results of the Bayesian MCMC and our surrogate-based optimization.

The RMSEs of the different optimization algorithms:

Figure 4 presents the calibrated results (RMSE) of the different algorithms we applied. For each algorithm, we only perform 100 simulations to compare the effectiveness if the number of simulations is limited. As the Bayesian MCMC approach requires a large number of samples to reach a stable distribution, over 500 000 simulations were conducted for the algorithm evaluation.

Clearly, the average RMSE of the RBF-SBO is the lowest (0.6 kg m

The results of RBF-SBO also indicate less variation among the 50 repeated experiments than the global optimization algorithms for the three models. For the same reason previously mentioned, the Bayesian MCMC approach has less variation than our RBF-SBO algorithm for the two microbial models. For the CLM-CASA' model, our RBF-SBO is still promising to get stable results. Among the global optimization algorithms, CMA-ES shows a very significant fluctuation (Fig. 4b, c), indicating that it is unreliable when the number of simulations is as small as 100. This is because the CMA-ES requires quite a few simulations on the exploration of the parameter domain and the construction of the parameter covariance matrix. Therefore, RBF-SBO is the most effective and stable algorithm when the number of simulations is limited.

The average RMSEs with the increase of number of simulations and different optimization
algorithms:

Figure 5 shows the results in terms of average validation RMSE. We don't compare the efficiency of Bayesian MCMC since it is in nature a sampling algorithm, not an optimization algorithm. The average validation RMSE of RBF-SBO is lower than the four global optimization algorithms before the number of simulations increases to 600 for the two microbial models and to 200 for the CLM-CASA' model, respectively. Our RBF-SBO requires fewer simulations than the global optimization algorithms when they reach the same RMSE value and accuracy range. Thus, our RBF surrogate optimization is the most efficient algorithm, requires the minimum number of simulations and has the lowest computational cost. Compared to the global optimization algorithms, SBO has two main advantages. Firstly, SBO samples an initial parameter set to build a surrogate model; this building process is a learning process which can better understand the parameter space and thus help conduct better optimization. Secondly, SBO can avoid some bad parameter points (“bad” means high prediction error), which are not supposed to be evaluated with the help of the surrogate model.

Another important observation is that the difference between the results of our RBF-SBO and the global optimization algorithms decreases as the number of simulation increases (Fig. 5). Moreover, the CMA-ES outperforms the RBF-SBO when the number of simulations exceeds 200 for the CLM-CASA' model (Fig. 5a). Our SBO can quickly build the surrogate model with relatively good accuracy, which helps find a near-optimal solution with lower computation cost. However, the surrogate model is only an approximation of the real model and accuracy might be limited due to the strong nonlinearity and high complexity of the real model. After gaining sufficient knowledge of the original model through many simulations, the excellent global optimization algorithms, such as CMA-ES, may achieve similar performance to or even outperform our SBO, which suggests that our SBO is better fitted for use regarding the parameter calibration of cost-intensive models.

Compared to the two-pool and four-pool microbial models, the CLM-CASA' model has 13 carbon pools and 20 parameters. Despite the increased complexity of the CLM-CASA' model, the SBO obtains better results before conducting 200 simulations of the real model (Fig. 5a). Moreover, the SBO is always the best parameter calibration method for the two-pool and four-pool microbial models before conducting 600 simulations (Fig. 5b, c). In addition, only one global optimization algorithm, CMA-ES, has better performance compared to the SBO on the CLM-CASA' model after 200 simulations. Considering the high variance of CMA-ES on the two microbial models (Fig. 4b, c), our SBO is more effective and more reliable on average.

We select the RBF as the surrogate model in the experiments because the RBF is widely adopted in many SBO algorithms (Müller and Shoemaker, 2014). In this section, we also test two other typical surrogate models, kriging and MARS. The MARS model is simple and has almost no requirements for the sample quality. MARS is very quick to train and predict. Kriging, also known as Gaussian process regression, is a method of interpolation for which the interpolated values are modeled by a Gaussian process governed by prior covariance. Kriging provides the best linear unbiased prediction of the intermediate values under suitable assumptions on the priors.

Figure 6 presents the results of kriging, MARS and RBF in terms of average validation RMSE. The performance of the three surrogate models is similar. The three surrogate models all have reasonable performance in the parameter calibration of the three types of SOC models and perform better than global optimization algorithms, indicating that our SBO is robust.

The average RMSEs with the increase of number of simulations and different surrogate models.

The steady-state global SOC simulations (Eq. 2) using CLM-CASA' with the default and calibrated parameter values are presented in Fig. 7a and b, which are also compared to the observed SOC pools provided by the IGBP-DIS dataset. The SOC simulation results, using the calibrated parameter values from the SBO, match the observation better than the simulation results with the default parameter values (Fig. 7c) and display a relatively lower RMSE. By using the calibrated parameter values the SOC simulations are significantly improved in most parts of the world, except for some grid cells in the west of Canada and the east of Russia (Fig. 7a, b and c). As a result, the CLM-CASA' simulation result with the default parameter values can only explain 33 % of variation in the observed soil C, whereas the simulation with the calibrated parameter values can explain an improved ratio (42 %) of variation in the observed soil C. The unexplained variation is partly due to uncertainty in observations. To further improve the model's accuracy, we need to gain a more in-depth understanding of uncertainty sources from the data, model structure, parameters and forcing.

Spatial correspondence of SOC produced by CLM-CASA' to SOC reported
by IGBP-DIS. Map

Frequency distributions of 20 calibrated parameters of the CLM-CASA' model using the Bayesian MCMC approach (Harauk et al., 2014) and surrogate-based optimization (blue line in each graph).

Figure 8 presents the frequency distributions of the 20 calibrated
parameters based on MCMC and the calibrated parameter values using the
proposed SBO (the blue lines in Fig. 8). Narrow posterior distributions
indicate highly sensitive parameters, consistent with the conclusions of
Hararuk et al. (2014). The calibrated parameter
values of SBO are quite similar to the responding parameter values at the
peaks of posterior distributions for most highly sensitive parameters, such
as the temperature sensitivity of heterotrophic respiration (

Some calibrated parameter values are very close to the assigned bounds of
the parameters in Fig. 8, which is usually related to the correlations among
parameters. Further investigation on the covariance among parameters is
necessary to explain this issue. In addition, the unreasonable setting of
those bounds might be another possible reason. For instance, the calibrated
c(12,12)
value (

As listed in Table 1, the calibrated temperature sensitivity (

According to the calibrated RMSE and

Spatial correspondence of the four-pool microbial model produced SOC to the IGBP-DIS reported SOC.

The microbial models achieve better SOC predictions than the
calibrated CLM-CASA' model in terms of their prediction of the C capacity in
the low-temperature regions (Russia, Europe, North America) and in the
regions with small soil C inputs (Figs. 7b and 9). The SOC contents are
determined by two main factors: the soil carbon inputs and the SOC residence
time (Luo et al., 2003). Considering the same soil carbon inputs in the
CLM-CASA' and the microbial models, the improvement is mostly induced by the
differences in SOC residence time. In all three models, the SOC
residence time is essentially controlled by temperature (Xia et al., 2013).
As a result, the temperature sensitivity (

Posterior probability density functions of the four-pool microbial model parameters (generated by Bayesian MCMC). The blue vertical lines are the final calibrated parameter values from our surrogate-based optimization.

Figure 10 presents the posterior distributions of the parameters calculated
by Bayesian MCMC and the parameter values calibrated using our SBO.
According to the posterior distribution,

Parameter calibration is becoming more and more challenging for SOC model
development, especially for the computationally expensive global land models,
owing to the large number of simulations. In this study, we introduce an SBO
algorithm to the parameter calibration of three SOC models. The main
findings are as follows:

Compared to advanced global optimization algorithms, SBO is more effective and more efficient on average. Our RBF-SBO outperforms other parameter calibration algorithms when the number of simulations does not exceed 200.

The parameter optimization based on the RBF surrogate model gains more accurate calibration results than those of the Bayesian MCMC approach in the three soil carbon models.

The SBO scheme is robust. Various types of surrogate models have similar performance in the parameter calibration tasks of SOC models.

Although SBO is only guided by a single cost function, it can still result in better parameter values than the default algorithms. We carefully analyze the spatial SOC distributions produced by the models with the calibrated parameters using SBO, which indicate that SBO truly improves the model's prediction and simulation capability.

The code and data from the three models and the related algorithm implementations can be found in the Supplement. If you have any problem using the code and repeating the experiments, please feel free to contact the corresponding author of this paper: Wei Xue (xuewei@tsinghua.edu.cn).

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

We would like to thank all the reviewers for their insightful comments and suggestions. This work is partially supported by the National Key R&D Program of China (grant no. 2016YFA0602100 and 2017YFA0604500) and National Natural Science Foundation of China (grant no. 91530323 and 41776010). Edited by: Christoph Müller Reviewed by: Maarten Braakhekke and one anonymous referee