The concepts used to assess the effect of overparameterisation and parameter equifinality on model predictions are illustrated using four models simulating soil C and N dynamics without mineral protection of SOC based on the SESAM v3.0 model of, termed rhizosphere models. Next, these concepts are applied to a microbially-driven model simulating mineral protection of SOM and the Δ14C value of the simulated SOC pools, referred to as the SOM model. To assess the identifiability of model parameters, i.e., which parameter combinations can be optimised together without parameters compensating for each other, realistic values for all parameters need to be perturbed by a very small amount. Such values were obtained for all models by performing a frequentist calibration using the Differential Evolution (DE) algorithm , given as many constraints on simulated pools as realistically possible. Next, to assess how different amounts of available calibration data affect model simulations in steady state, every model was calibrated using the Differential Evolution Markov Chain with snooker updater (DEzs) algorithm for parameter sets which were either identifiable (termed the identifiable parameter model; IPM) or non-identifiable (termed the full parameter model; FPM). Last, to show the effect of parameter equifinality on model predictions, steady-state model outcomes were perturbed by doubling OC inputs for a period of 100 years. All simulations and analyses were performed in R .

In this study, artificial SOC data were used to calibrate model parameters. We created “measurements” of the total SOC stock and fractions for one point in time, to mimic the common assumption of an SOC stock in steady state. The intention was not to replicate a soil at a specific location under a specific land use, but to use reasonable values measured across different land uses. The total SOC stock down to 0.2 m depth was calculated assuming an SOC concentration of 2 % and a bulk density of 1.2 gcm-3 , resulting in an SOC stock of 4800 gCm-2 down to 0.2 m. It was assumed that 25 % of SOC is POM and microbes in the rhizosphere i.e., 1200 gCm-2,, which was divided into 96 % POC (1152 gCm-2) and 4 % microbes (48 gCm-2). The remaining SOC was assumed to be mineral-associated organic carbon (MAOC, 3600 gCm-2). The standard deviation of the SOC pools was assumed to be 10 % of their size.

The C:N ratio of plant litter inputs, microbes and enzymes were fixed at 30, 10 and 3, respectively, following . Assuming POM consists of 95 % plant litter and 5 % microbial residues, the C:N ratio of POM was calculated to be 29. Similarly, MAOM was assumed to consist of equal amounts of microbial residues and unprocessed plant-derived organic matter, resulting in a C:N ratio of 20.

The Δ14C values of POC and MAOC were estimated using data for density-fractionated forest soils from the ISRAD database . Selected Δ14C data were limited to samples collected in the top 20 cm of the soil between 2004 and 2009 in the northern hemisphere. This resulted in Δ14C values of 75.2 ‰ for POC, and 17.6 ‰ for MAOC, with the standard deviation for both values being assumed to be 10 % of their size. The average year at which these values were measured was 2007, which was taken to be the final year of the performed simulations.

Two SOM models were used to assess the effect of parameter equifinality on model predictions (Fig. ). The first is a microbially-driven model simulating coupled C and N dynamics without mineral protection of OC based on the SESAM v3.0 model , referred to as the rhizosphere model (RM, Fig. a). For this model, four different sets of mathematical equations were used to simulate depolymerisation of POM and microbial turnover, resulting in four rhizosphere models. The second model, termed the SOM model, is identical to the SESAM model but additionally simulates mineral protection of OC (Fig. b) and keeps track of the size of the DOM pool. The processes are identical to the rhizosphere model, with the addition that DOM can be stabilised by soil minerals. Competition for DOM between microbes and minerals is simulated using the equilibrium chemistry approximation . The SESAM model was selected because it was developed with consideration for trade-offs between model complexity and data availability , the topic of this study.

Simulated inputs of organic matter (C and N) to the soil come from plant litter, with a fixed C:N ratio. In addition, OC enters the soil through rhizodeposits, which are assumed to contain no N, and inputs of mineral N come from atmospheric N deposition. The SESAM model keeps track of the size of four pools: POM consisting of microbial residues and deactivated enzymes (POMres) and plant litter (POMlit), microbes (MIC) and mineral N (Nmin). In addition, in the rhizosphere model dissolved organic matter (DOM) and enzymes mediating the depolymerisation of plant litter (ENZlit) and microbial residues (ENZres) are simulated without explicitly keeping track of their size, assuming they are in quasi-steady-state (i.e., their size determines the rate of depolymerisation, but they are instantly transferred to other pools upon their creation). The SOM model, however, explicitly tracks the size of the DOM pool to simulate competition for DOM between microbes and mineral surfaces. The relative production of these enzymes is calculated in proportion to their revenue . Inputs of C and N from plant litter enter the POMlit pool, while rhizodeposit C enters the DOM pool. Nitrogen leaves the system through leaching, while C leaves the system as CO2 through microbial maintenance respiration, overflow respiration, growth respiration, and CO2 losses upon the turnover of dead microbes through grazing by predators. All model pools contain both C and N, except Nmin, and are referred to as organic matter pools, for example POMlit. When referring to the C content of a pool, they are referred to accordingly, for example POClit. Dead microbes are transferred to the POMres pool, while decayed enzymes are divided between the POMres and DOM pools. A full description of SESAM v3.0 is provided in .

To evaluate the sensitivity and identifiability of model parameters, it was necessary to have a set of parameter values that leads to realistic model results with the model output being sensitive to local changes in parameter values. Therefore, a deterministic calibration was performed using the differential evolution (DE) algorithm as implemented in the DEoptim package in R . For the rhizosphere models, between 3 and 5 parameters were optimised (depending on model variant, Table S5 in the Supplement), while for the SOM model, 9 parameters were optimised (Table S6 in the Supplement). The parameter values were optimised by minimising the sum of squared relative errors (SSRE), formulated as: 11SSRE=∑i=1nmeasi-modimeasi2 Where n is the number of measurements, and measi and modi are the measured and modelled pool sizes, respectively.

For the rhizosphere models, artificial measurements of POC, MIC and the C:N ratio of total SOM were used to optimise model parameters with a population size of 180 parameter sets for 500 iterations. To make sure that the rate modifiers of the Michaelis–Menten equations for depolymerisation of microbial residues and litter were sensitive to changes in the size of model pools, their artificial measured value was assumed to be 0.5, and the SSRE calculated accordingly. For the rhizosphere model with first-order microbial mortality, the decay constant (kmic.fo) was kept constant at a value of 0.01 d-1, to simulate a turnover rate of the microbial biomass of 100 d. For the rhizosphere models with density-dependent microbial turnover, the turnover rate of microbes was calibrated to be as close as possible to 100 d. The optimised parameter values for the rhizosphere models are shown in Table S7 in the Supplement.

For the SOM model, artificial measurements of POC, MAOC, total OC, the C:N values of POM and MAOM, and the Δ14C values of POC and MAOC were used to optimise model parameters with a population size of 180 parameter sets for 500 iterations. Similar to the rhizosphere model, the rate modifiers for the Michaelis–Menten equations simulating depolymerisation of litter and microbial residues were optimised to be as close to 0.5 as possible. The turnover rate of microbes was calibrated to be as close as possible to 100 d. The optimised parameter values for the SOM model are shown in Table S8.

To assess which parameter combinations were identifiable, given different available data sets for parameter optimisation, the methods developed by were applied using the FME package in R . For the rhizosphere models, three scenarios of data availability were tested: (1) only data on total SOC, (2) data on total SOC and total N, and (3) data on POC, MIC and the C:N ratio of total SOC. The first scenario is one with a minimum data availability, while the second scenario is considered the most common (as C and N are often measured together). The third scenario assumes that in addition to total C and N, also microbial biomass C and POC were measured. Also for the SOM model, three scenarios of data availability were tested: (1) data on total SOC and N, (2) data on POC, MAOC and their C:N ratios, and (3) data on POC, MAOC, their C:N ratios, and the Δ14C values of POC and MAOC. The choice of these scenarios was based on common practices in SOM modelling, although data availability may be more extensive in other studies. As the identifiability analysis needs to be performed with a parameter set that leads to an acceptable model output , the identifiability analysis was performed using the optimal parameter set obtained from the deterministic DE calibration (see Sect. ).

To perform the identifiability analysis, the local sensitivity of selected model state variables to variations in parameter values was quantified using a normalised, dimensionless sensitivity index : 12sij=∂yi∂ΘjwΘjwyi Where yi is the ith state variable, Θj is the jth model parameter, wΘj is a weighting factor for Θj,and wyi is a weighting factor for yi. The obtained sensitivity matrix (s) thus quantifies the rate of change in the value of model output yi for a small change in the value of parameter Θj. The weighting factor for the parameter value (wΘj) was set to the optimal parameter value obtained by the DE optimisation, while the weighting factor for the state variable (wyi) is the steady-state size of this pool obtained using the optimal parameter values from the DE optimisation. The weights were thus constant for every assessed combination of state variable and parameter.

This local sensitivity analysis as implemented in the FME package in R evaluates the sensitivity for a time series of model outputs and respective measurements. As we assumed to only have steady-state measurements at one point in time, the sensitivity of the model output could only be evaluated for this one time point. To do so, the sensitivity analysis from the FME package was run multiple times, each iteration changing the parameter values with a different relative amount (from 1×10-4-5×10-4 in steps of 5×10-5). These results were then combined to construct the sensitivity matrix (Eq. ), with rows containing the sensitivity indexes (sij) quantifying how varying parameters by a different relative amount affected the selected model output, and columns showing this for different parameters. One alteration made to the sensitivity analysis from the FME package (in the function sensFun) is that the tested parameter values were not restricted to be larger than the relative deviation as was implemented by, to make sure the proposed range in parameter values was tested and not replaced by the relative deviation.

The sensitivity matrices were subsequently used to assess the parameter identifiability through the analysis of collinearity between all possible combinations of columns (i.e., parameters). As every column quantifies how every evaluated model state variable changes when a parameter value is altered, columns having a high collinearity indicate parameter combinations that are not identifiable, as a change in one parameter can be compensated by a change in one or more other model parameters. Before the collinearity was assessed, every column in the sensitivity matrix was normalised using the Euclidean norm (i.e., the square root of the sum of the squared values), following : 13sj^=sj‖sj‖ Where sj^ is the normalised column for the jth parameter in sij, sj is the original column, and ‖sj‖ is the Euclidean norm of column sj. Combining these columns for all parameters results in the normalised sensitivity matrix S^. For every combination of columns (i.e., parameters) in S^, a collinearity index was calculated : 14γ=1min⁡(EV[S^TS^]) Where γ is the collinearity index, S^TS^ is the vector product of S^ and the transposed vector S^, and min⁡(EV) is the smallest eigenvalue of this product. The interpretation of γ is that a change in one parameter can be compensated by 1-1/γ when one or more parameters are appropriately changed. To label a parameter set as being identifiable, we used a threshold of γ of 10, meaning that a parameter change can be undone for 90 % by changes in other parameter values. It is noted that various studies have used different values for this threshold, generally in the range 5<γ<20 . A detailed description of the methods presented in this section is provided in , and examples of its application are shown in, among others, , and .

Parameter optimisation was performed to obtain as many parameter combinations as possible that produce behavioural models, i.e., model outputs that cannot readily be rejected given available data. These are defined here as predictions within one standard deviation of the measurements. To account for measurement uncertainty, a Bayesian calibration was performed using the Differential Evolution Markov Chain with snooker updater (DEzs) algorithm , as implemented in the BayesianTools package in R . As no prior information on the distribution of parameter values was known, uniform priors were used within specified bounds (see Table S6) and the log-likelihood was calculated as : 15L(x|Ỹ,σ^2)=-n2log⁡(2π)-∑i=1nlog⁡(σi^)-12∑i=1nyĩ-yi(x)σi^2 Where n is the number of observations, σi^ is the standard deviation of the ith observation, yĩ is the ith observation, and yi(x) is the model prediction of this observation.

For the rhizosphere models, the DE_zs algorithm was run using 5 internal chains, higher than the three internal chains which have been shown to be sufficient to explore a high dimensional parameter space . This was done 3 times (i.e., with three independent chains), to ensure an optimal exploration of the parameter space. Each internal chain was run for 20 000 iterations. The Bayesian calibrations of the SOM model were run with a number of internal chains equal to twice the number of optimised parameters (Table ) with 20 000 iterations each, repeated 3 times to obtain 3 independent chains. For these calibrations, the chance of a snooker jump was increased to 0.2 (from the default 0.1), to enhance exploration of the parameter space and avoid the algorithm getting stuck in a local maximum.

The four rhizosphere models were used to illustrate the concept of parameter identifiability and the consequences of equifinality on model predictions. To do so, each of the rhizosphere models was calibrated twice. In a first scenario (termed the full parameter model; FPM) all parameters for which no realistic estimates could be made or found in the literature were optimised (either 5 or 6 parameters, depending on the model; Table S9 in the Supplement). These parameter sets were non-identifiable. In a second scenario (termed the identifiable parameter model; IPM), only identifiable parameters for the assumed data were optimised: Vmax,lit and Vmax,res (see Tables S11, S13, S15 and S17 in the Supplement). These parameters were identified using the parameter identifiability analysis for each model separately (Sect. ). In addition to these parameter values, also the rate modifiers for the depolymerisation of POMlit and POMres (see Eq.  and ) were optimised to be as close as possible to 0.5. For both optimisation scenarios, it was assumed that only measurements of total SOC and the C:N ratio were available, as these data are most commonly available. The values of the parameters that were not optimised were fixed at the values obtained through the deterministic calibration (Sect. ). This means there is an additional hidden uncertainty, as the values of the fixed parameters could not be confidently determined, while choosing different values would likely have led to other calibrated values for the optimised identifiable parameters. This uncertainty is, however, not assessed in the present study.

For the SOM model, Bayesian parameter optimisation was performed for three assumptions on data availability: (1) data on total SOC and N (referred to as OMtot), (2) data on POC, MAOC, particulate N and mineral-associated N (which can be obtained through SOM fractionation, referred to as Fractions), and (3) data on POC, MAOC, particulate N, mineral-associated N and the Δ14C values of POC and MAOC (referred to as Fractions and Δ14C). For each of these data sets, one full parameter model (FPM) was run by optimising eight parameters for which no reasonable estimate could be made (Table ), and one identifiable parameter model (IPM), optimising as many parameters as could be jointly identified: 2 for the OMtot scenario, 3 for the Fractions scenario, and 5 for the Fractions and Δ14C scenario (Table ).

All results obtained by the Bayesian calibration were within one standard deviation from the average respective measurements (behavioural models). This implies that none of these models (and their parameter sets) could readily be rejected given the variability in measurements. To assess how different sets of optimised parameter combinations, i.e., identifiable versus non-identifiable, affect model predictions starting from steady-state pool sizes, OC inputs were doubled for a period of 100 years. Subsequently, the amount of simulated total SOC of the simulated pools was analysed after 100 years. The additional uncertainty caused by parameter equifinality was assessed for the different scenarios by quantifying the absolute value and spread of model predictions upon the doubling of OC inputs.

The simulations with the rhizosphere models showed that the choice of the mathematical formulations has a large impact on predictions. Although all model formulations led to behavioural steady-state models, the use of different equations for the depolymerisation of POM (absolute versus relative Michaelis–Menten kinetics) and microbial turnover (first-order versus density-dependent) led to different predictions in SOM upon a doubling of OC inputs (Fig. ). A similar result was obtained by , who showed that different formulations of the thermal adaptation of soil microbes resulted in large differences in predicted losses of SOC in a soil warming experiment. While many of the recently developed mechanistic SOM models use a range of mathematical formulations , the quantitative evaluation of different equations on predictions made by recently-developed models received little research attention to date. Similarly, microbially-driven models forced with the same input data at the soil pedon scale have been shown to result in divergent predictions after a change in OC inputs or temperature , or to result in different turnover times of SOC, POC and MAOC . Also at the global scale, different microbially-driven models have been shown to lead to different predictions of SOC . As a result, the discussion on the optimal structure of SOM models at the landscape scale is still ongoing, whether on how to improve existing models , or how to drastically change the simulated processes included in these models, and the scale at which these need to be measured . In addition to these discussions, which are vital to direct the field of soil biogeochemical modelling for the coming decades, our results show that the mathematical formulation of simulated processes should not be overlooked, and its effects on predictions evaluated more extensively.

A second aspect of SOM models underlined by our results is that the correct simulation of SOM in steady state is an insufficient criterion to evaluate their predictive capabilities. This was shown by the simulations with the SOM model, for which all calibration scenarios led to the correct simulation of total SOC under steady state, while predictions made with these behavioural models showed a wide variation in responses (Fig. ). Similarly, multiple studies found that while overparameterised SOM models resulted in behavioural models, predictions of changes in SOM for the future diverged widely . This questions predictions made by SOM models, and this uncertainty should be reduced by making better use of available data from field experiments in which environmental forcings are manipulated, independent validation of model predictions with diachronic data or the inclusion of more data on the size and turnover time of internal model pools during parameter optimisation (see Sect. ).

Including additional data in the model calibration process offers several advantages: (1) more parameter values that cannot be measured in the field or from experiments can be optimised, while the value of fewer parameters needs to be fixed (Fig. ), (2) the size and turnover time of different model pools is simulated more accurately (Fig. ) and (3) model predictions are better constrained (Fig. ). Taken together, this increases confidence in model predictions.

When the parameters of a SOM model with internal pools are optimised without sufficient data to constrain the size of all pools, several combinations of internal pool sizes can lead to behavioural models that accurately simulate the combined size of all pools (i.e., the total amount of SOM). This manifestation of equifinality has been shown to occur in various SOM models . In addition, this may lead to behavioural models with an incorrect simulation of the turnover time of the internal model pools, and therefore total SOM . To alleviate these issues, several studies have used additional data besides data on the total amount of SOM in the parameter optimisation process, such as data on stable carbon isotopes δ13C; e.g.,, radiocarbon isotopes Δ14C; e.g.,, a combination of both isotopes e.g.,, and data on the size of internal model pools e.g.,. Similar to the results presented here, studies devoted to the topic of parameter optimisation under different data constraints consistently found that including more data during the calibration process led to (1) parameter ranges that were better constrained and (2) the distribution of simulated organic matter among model pools better matching measurements . When this exercise was done in combination with a parameter identifiable analysis, similar conclusions were drawn, combined with the observation that although additional data were used, not all parameters could be optimised .

Our results and previous research thus show that model parameter values can be better constrained when including more data in the parameter optimisation process. However, this has to be accompanied by an identifiability analysis to confidently determine how many and which parameters can be optimised together. For our SOM model, only three parameters could be optimised when data on total SOM, POM and MAOM were present. This increased to five parameters when this was amended with Δ14C values of MAOC and POC. Similarly, found that no more than 4 parameters of linear-pool SOC models were jointly identifiable. As a consequence, even when data on SOM fractions and Δ14C were used for our SOM model, three parameters needed to be fixed, while this increased to five parameters when only data on POM and MAOM were available, and six parameters when only data on total SOM was present. As the values of these parameters are generally not based on data, but rather estimates, this implies that there is a hidden uncertainty in the quality of these predictions. This can be quantified by assessing how the values of fixed parameters affect predictions .

Only when data on the size and Δ14C values of POC and MAOC were used during parameter optimisation were the sizes of these pools, and their turnover times, correctly simulated (Figs. g, h and e, f). Based on these results, it is recommended to use data on the Δ14C value of internal model pools in the calibration process, in line with . When these data are not available, it is worthwhile to report the Δ14C value or turnover time of simulated model pools, as this may serve as an indication of whether the simulated turnover times are in line with observations . Whether Δ14C data on total SOC is sufficient to achieve similar results as when data on the Δ14C value of internal model pools is present is a topic for future research. The recommendations mentioned above are equally relevant for the validation of SOM models, which has been shown to be often applied insufficiently and inadequately .

The importance of practical identifiability and equifinality has long been recognised across environmental disciplines that use simulation models, as outlined in the introduction. Only more recently have these concepts received attention in the field of SOM modelling e.g.,, and are they mentioned in review articles . In line with these studies, our results show that optimising identifiable parameters of a SOM model leads to better constrained predictions of SOM upon a doubling of C inputs (note that only the precision of predictions was evaluated, as we did not have data to assess the accuracy). From this, it is clear that knowledge about which parameters can be jointly identified, given the quantity and type of data available for calibration, is necessary to reliably apply a model. However, results from an identifiability analysis are rarely reported in articles describing SOM models. A possible explanation is that this concept has been neglected so far, given its historical underrepresentation in the SOM modelling literature. Other reasons are related to the fact that “there is still a great need for development of methods, software and training to ensure all modellers are able to assess and react appropriately to non-identifiability” p. 428. A last reason can be that although many techniques for parameter identifiability have been developed, their description in the literature can be very technical and difficult to implement by non-expert modellers. One way to solve this issue would be to develop accessible software packages that non-expert modellers can use to perform an identifiability analysis on their model of choice. The potential for developing such tools to make identifiability analysis more accessible is evidenced by the incorporation of the sensitivity-based identifiability analysis presented by into the FME package in R , which has been used by most studies assessing the identifiability of parameters in SOM models e.g.,. In addition, developing such tools will make it possible to apply and compare the results of different methods of parameter identifiability analyses. While an overview of the different methods available to perform structural and practical identifiability analyses is beyond the scope of this discussion, the interested reader is referred to , , , , , , and .

Based on our results, it is argued here that parameter identifiability analysis should be an integral part of the model calibration and validation process, together with previously described and equally important aspects to be taken into account e.g.,. We therefore concur with , who recommended that “any modeling study should document whether a model is non-identifiably, the source of potential non-identifiability and how this affects intended project outcomes”. Based on the results presented in this study, we suggest the following:

When developing a novel SOM model, information on the identifiability of model parameters for different representative scenarios on data availability should be provided, in order for model users to know which parameters can be jointly calibrated, given available data.