This study considers eight evolving data models starting from a data model that assumes independent, homoscedastic, and Gaussian residuals to a data model that relaxes all three assumptions. The eight data models are based on the generic normalized residual, 1at=εtσtat∼X, where εt=dt-Yt is the residual (the difference between data dt and its corresponding model simulation Yt) at time or location t, σt is the standard deviation of the residual, and X is the probability density function (PDF) of at. The eight data models are formulated with different forms of εt, σt, and X. The standard least square (SLS) data model is 2at=εtσ0at∼N(0,1), where σt=σ0 is a constant for all of the data (i.e., homoscedasticity), and X is the standard normal distribution, N(0,1). The unknown parameter σ0 is estimated along with the unknown physical model parameters. If σt is not a constant (i.e., heteroscedastic), SLS becomes the weighted least squared (WLS) data model. While heteroscedasticity can be accounted for via residuals transformation (e.g., Thiemann et al., 200; T. Smith et al., 2010) or other similar approaches (Gragne et al., 2015), a linear heteroscedastic model σt=σ0+σ1Yt is assumed here following the studies of Thyer et al. (2009), Schoups and Vrugt (2010), and Evin et al. (2013, 2014). With the linear model, there is no need to estimate σt for each piece of data. Instead, σt is calculated by estimating only two parameters, σ0 and σ1. The WSL data model is written as 3at=εtσ0+σ1Ytat∼N(0,1). The two unknown parameters σ0 and σ1 are estimated along with the unknown physical model parameters. The linear model assigns smaller weights to data with larger simulation values, Yt. If the simulation value is small and σ0≫σ1Yt, the weight becomes constant for all data. Both SLS and WLS assume that at is independently and identically distributed.

It is not uncommon that residuals are correlated in space and time, due to the propagation of measurement errors (Tiedeman and Green, 2013) and model structure errors (Evin et al., 2014; Kavetski et al., 2003; Lu et al., 2013). The temporal correlation that occurs in the numerical example of this study can be accounted for by using a p order autoregressive model. This leads to the standard least square data model with autocorrelation (SLS-AC): 4at=εt-∑i=1pϕiεt-iσ0at∼N(0,1), where p is the order of autocorrelation, and ϕi is an autocorrelation coefficient. The unknown ϕi and σ0 are estimated along with the unknown model parameters. Extending the concept of correlated residuals to WLS leads to the weighted least squared with autocorrelation (WLS-AC): 5at=εt-∑i=1pϕiεt-1σ0+σ1Ytat∼N(0,1). The unknown parameters of σ0, σ1, and ϕi are estimated along with the physical model parameters. Equations (2)–(5) assume that the residuals are Gaussian.

The next four data models are similar to the previous four models except that the standard normal distribution of at is replaced by the skew exponential power distribution, SEP(0,1,ξ,β), with a zero mean and unit standard deviation (Schoups and Vrugt, 2010): 6pat|ξ,β=2σξξ+ξ-1ωβexp⁡-cβaξ,t2/(1+β), where ξ is skewness, β is kurtosis, aξ,t=μξ+σξatξsign(μξ+σξat), μξ=Mξ-ξ-1, ωβ=Γ1/231+β/21+βΓ3/21+β/2, σξ=1-M2ζ2+ζ-2+2M2-1, M=Γ1+βΓ1/231+β/2Γ1/21+β/2, and cβ=Γ31+β/2Γ1+β/21/1+β are derived variables of β and ξ, and Γ[.] is the gamma function. The kurtosis parameter β∈R:-1≤β≤1 determines the peakedness of the PDF such that the β values of -1, 0, and 1 give uniform, Gaussian, and Laplace distributions, respectively. The skewness parameter ξ∈R:0.1≤ξ≤10 determines the skewness of the PDF such that the ξ values of 0.1, 1, and 10 give positively skewed, symmetric, and negatively skewed distributions, respectively. Setting β=0 and ξ=1 leads to μξ=0, σξ=1, ωβ=12π, cβ=12, and aξ,t=at, and the skew exponential power distribution SEP(0,1,ξ=1,β=0) becomes the standard normal distribution, 7pat|ξ=1,β=0=12πexp⁡-12(at)2, which is the SLS data model in Eq. (2).

Replacing at∼N(0,1) with at∼SEP0,1,ξ,β in Eqs. (2)–(4) leads to the SEP, WSEP, SEP-AC, and WSEP-AC data models as follows: 8at=εtσ0at∼SEP0,1,ξ,β9at=εtσ0+σ1Ytat∼SEP0,1,ξ,β.10at=εt-∑i=1pϕiεt-1σ0at∼SEP0,1,ξ,β11at=εt-∑i=1pϕiεt-1σ0+σ1Ytat∼SEP0,1,ξ,β In comparison with the Gaussian data models, the SEP-based data models have two more parameters (ξ and β) that are estimated along with the physical model parameters. The WSEP-AC data model, which is known as the generalized likelihood function, is the most commonly used SEP-based data model (e.g., Vrugt and Ter Braak, 2011; Hublart et al., 2016; Scholz et al., 2018). A summary table of the eight data models showing the corresponding parameters is provided in the Supplement.

Consider a Bayesian inference problem for a nonlinear model, f, used to simulate state variables (e.g., CO2 efflux), d=Y(θ)+ε, where d is a vector of data, θ is a vector of model parameters, and ε is a vector of residuals that may include errors in data, model parameters, and model structures. The goal of Bayesian inference is to estimate the posterior distributions, p(θ|d), of model parameters, θ, given data, d, using Bayes' theorem (Box and Tiao, 1992): 12pθ|d=pd|θpθ∫pd|θpθdθ, where p(θ) is the prior distribution, and p(d|θ) is the likelihood function to measure goodness-of-fit between model simulations, Y(θ), and data, d. The prior distribution can be obtained using data from previous studies (e.g., Elshall and Tsai, 2014) or expert judgment. When prior information is lacking, a common practice is to assume uniform distributions with relatively large parameter ranges so that the prior distributions do not affect the estimation of posterior distributions.

The data models above can be used to construct the likelihood functions. For the Gaussian data models given in Eqs. (2)–(5), the corresponding Gaussian likelihood functions are straightforward (see Eq. 7 for an example). For the SEP data models, the corresponding likelihood, which is called generalized likelihood function, is (Schoups and Vrugt, 2010) pd|θ=pεt|θ13=∏t=1nσt-12σξξ+ξ-1ωβexp⁡-cβaξ,t2/(1+β), where n is the dimension of d. The Gaussian likelihood functions are special cases of the generalized likelihood functions. For example, by setting β=0, ξ=1, ϕi=0, σt=σ0, σξ=1, μξ=0, ωβ=12π, cβ=12, and aξ,t=at, Eq. (13) becomes the likelihood function corresponding to the SLS data model. Replacing σt=σ0 by σt=σ0+σ1Et, Eq. (13) becomes the likelihood function of the WLS data model.

In this study, the posterior distributions of the data model parameters are estimated along with the soil respiration model parameters using the MT-DREAM(ZS) code (Laloy and Vrugt, 2012). MT-DREAM(ZS) implements a Markov chain Monte Carlo (MCMC) algorithm by running multiple Markov chains in parallel with adaptive proposal distribution, multiple-try sampling, and sampling from an archive of past states. These state-of-the-art features assist in overcoming common challenges in the sampling space such as multimodality, ill-conditioning, and high dimensionality, and thus allow for accurate exploration of the targeted distributions.

Zhang et al. (2014) studied the Birch effect (the peak soil microbial respiration pulses in response to episodic rainfall pulses), and developed five models, evolving from an existing four-carbon-pool model to models with additional carbon pools and/or explicit representations of soil moisture controls on carbon degradation and microbial uptake rates. Three of the five models are used in this study, and they are denoted as 4C, 5C, and 6C. Note that model 4C is model 4C_NOSM from Zhang et al. (2014), not their model 4C. Figure 1 is the diagram of model 6C, the most complex of the five models. The simplest model, model 4C, has four carbon pools, i.e., soil organic carbon (SOC), dissolved organic carbon (DOC), microbial biomass (MIC), and enzymes (ENZ), and does not consider the soil moisture control on carbon degradation and microbial uptake rates. Models 5C and 6C have an explicit representation of soil moisture controls on the rates. Based on the dual Arrhenius and Michaelis–Menten kinetics model, the original SOC degradation rate, Vdecom, is (Davidson et al., 2011; Davidson and Janssens, 2006) 14Vdecom=Vmax⁡CENZCSOCKm+CSOC, where Vmax⁡ (s-1) is the maximum SOC degradation rate per unit enzyme when the substrates is not limiting, CENZ (g C m-3) is enzyme pool size, CSOC (g C m-3) is SOC pool size, and Km is the half-saturation for SOC. The original microbial uptake rate, Vuptake, is (Davidson et al., 2011; Davidson and Janssens, 2006) 15Vuptake=Vmax_upCMICCDOCKm_up+CDOCCO2Km_upO2+CO2, where Vmax⁡_up (s-1) is the maximum DOC uptake rate when the substrates is not limiting, CMIC (g C m-3) is the MIC pool size, CDOC (g C m-3) is the DOC pool size, CO2 (m3 m-3) is the gas concentration of O2 in the soil pore, and Km_up (g C m-3) and Km_upO2 (m3 m-3) are the corresponding half-saturation constants for DOC and O2, respectively. With the explicit representation of soil moisture control, the two rates become (Zhang et al., 2014) 16Vdecom=Vmax⁡CENZCSOCKm+CSOCθθsVuptake=Vmax_upCMIC17CDOCKm_up+CDOCCO2Km_upO2+CO2θθs, where θ (–) is the volumetric soil moisture, and θs (–) is the porosity.

In addition to using the new rate equations, models 5C and 6C have more carbon pools. In model 5C, DOC is split into two sub-pools for the wet zone and the dry zone of soil pores, and only the wet DOC is used by MIC, as shown in Fig. 1. The moisture-controlled microbial uptake rate becomes Vuptake=Vmax⁡_upCMIC18CDOC_WKm_up+CDOC_WCO2Km_upO2+CO2θθs, where CDOC_W (g C m-3) is the DOC pool size in the wet soil pores. Model 6C is more complex in that ENZ is further split into two sub-pools for wet and dry pores, and both the wet and dry ENZ are subject to degradation, as shown in Fig. 1. The moisture-controlled SOC degradation rate becomes 19Vdecom=Vmax⁡CENZ_WCSOCKm+CSOCθθs for the wet ENZ and 20Vdecom=Vmax⁡CENZ_DCSOCKm+CSOC1-θθsεD for the dry ENZ, where CENZ_W (g C m-3) is the wet soil pores enzyme pool size, CENZ_D (g C m-3) is the enzyme pool size in the dry soil pores, and εD is the catalysis efficiency of the dry zone enzyme.

Due to considering the moisture control and adding more soil pools, model 5C is expected to be significantly better than model 4C for simulating the Birch effect. As the accumulated ENZ in dry soil is secondary, model 6C is expected to be slightly better than model 5C. In terms of model structural error, model 4C has the largest model structure error, model 5C has significantly less model structure error, and model 6C has the smallest model structural error. In other words, model 6C has the highest model fidelity (i.e., lowest model discrepancy) among the three models. As shown below, the degree of model structural error is reflected in the process of Bayesian inference and verified by the cross-validation.

Figure 2 plots the time series of 17 016 observations of soil moisture and CO2 efflux used in this study. The observations were obtained during the entire year of 2007, covering a long period of dry season prior to the monsoon and episodic rainfall events during the monsoon. The first two-thirds of this dataset are used for the Bayesian inference, and the last third is used for cross-validation. The inference and cross-validation periods have both dry and wet periods, as shown in Fig. 2. The observation site is located within the Santa Rita Experimental Range (SRER; 31.8214∘ N, 110.8661∘ W, elevation 1116 m) outside of Tucson, Arizona (Barron-Gafford et al., 2011; Scott et al., 2009). This savanna site was covered by 22 % perennial grass, forbs, and subshrubs and 35 % mesquite. The soils are uniformly comprised of Comoro loamy sand (77.6 % sand, 11.0 % clay, and 11.4 % silt). The half-hourly atmospheric forcing data were collected from measurements via an eddy covariance tower (Scott et al., 2009). This includes downward shortwave radiation, longwave radiation, precipitation, wind, air temperature, humidity, and pressure. The volumetric CO2 concentration was measured at a half-hourly intervals using compact probes. The CO2 efflux was estimated from the gradient of the CO2 concentration measured at two depths (2 and 10 cm) using Fick's first law of diffusion, and the estimates were validated against measurements from a portable CO2 gas analyzer.

The parameters estimated in this study include the parameters of the soil respiration models (4C–6C) and the parameters of the data models described in Sect. 2.1. The estimated parameters of models 4C and 5C include the microbial carbon use efficiency (CUE) (g g-1), enzyme production rate, ke (g m-3 s-1), microbial turnover rate, τm (1 s-1), and enzyme turnover rate τe (1 s-1). Uniform distributions are used as the prior in the Bayesian inference, and the ranges of the four parameters are 0.2–1.00, 1×10-12–1×10-7, 1×10-12–1×10-5, and 1×10-11–1×10-6, respectively. The values of other parameters are fixed at the values used in Allison et al. (2010). Model 6C has two more parameters, and they are the catalysis efficiency εD (–) and the turnover rate of the dry-zone enzymes τen (1 s-1). The priors of the two parameters are uniform distributions with the ranges of 0.2–0.8 and 1×10-12–1×10-8, respectively.

The DREAM-based MCMC simulation is conducted for a total of 24 cases, the combinations of eight data models and three soil respiration models. For each case, the parameter distributions are obtained after drawing a total of 5×105 samples using five Markov chains. The Gelman and Rubin (1992) R statistic is used for the convergence diagnostic, and it approaches 1 in less than 40 000 samples. The initial 50 % of the samples are discarded during the burn-in period.

Three criteria are used to evaluate the predictive performance of the soil respiration models and data models: the central mean tendency, the dispersion, and the reliability. Each criterion is measured by a single metric. In addition, a newly defined metric by Elshall et al. (2018) is also used to simultaneously measure the three criteria.

The central mean tendency is measured in this study using the Nash–Sutcliffe model efficiency (NSME) coefficient (Nash and Sutcliffe, 1970), 21NSME=1-∑i=1ndi-Yi‾2∑i=1ndi-d‾2, where n is the number of cross-validation data, di is the ith data, d‾ is the mean of the data, and Yi‾ is the mean of the prediction ensemble, Yi, for di. The NSME ranges from -∞ to 1, with NSME =1 corresponding to a perfect match between data and mean prediction, i.e., the ensemble is centered on the data. NSME =0 indicates that the model predictions are only as accurate as the mean of the data, whereas an efficiency NSME <1 indicates that the mean of data is a better prediction than the mean prediction.

In addition to the central mean tendency, it is also desirable that the ensemble is precise, with small dispersion, and reliable to cover all of the data. This study uses a nonparametric metric for dispersion, which is the sharpness of a prediction interval (e.g., M. W. Smith et al., 2010): 22Sharpness=1n∑i=1nMaxYi-MinYi, where Yi is the prediction ensemble within the 95 % prediction interval, the Bayesian credible interval, not the confidence interval used in nonlinear regression (Lu et al., 2013). Smaller sharpness values indicate better prediction precision. Reliability is measured using predictive coverage (e.g., Hoeting et al., 1999), which is the percentage of data contained in the prediction interval. Larger predictive coverage values are preferred.

To account for the trade-off between the three metrics, Elshall et al. (2018) defined relative model score (RMS) that simultaneously measures all three criteria. Scoring rules are commonly used in hydrology to assess predictive performance (e.g., Weijs et al., 2010; Westerberg et al., 2011). The RMS is used in this study to measure the relative predictive performance of the combinations of soil respiration models and data models. For combination Mj, RMS is defined as 23RMSMj=∑i=1npdi|Yij,Mj∑j=1mpdi|Yij,Mj×100, where m is the number of combinations; the ensemble prediction Yij is similar to Yi above with index i over time and index j specific to the jth combination. The density function pdi|Yij can be evaluated by first obtaining the density function pYij of the ensemble prediction Yij (e.g., by using the kernel density function) and then evaluating pdi|Yij using interpolation methods based on the intersection of Yij and di. More details about evaluating RMS can be found in Elshall et al. (2018). This evaluation is based purely on the model predictions, and does not involve any assumptions on the models, their parameters, or their likelihood functions. Larger RMS values indicate better overall predictive performance. A figure displaying our workflow scheme is presented in the Supplement.

Figure 3 shows residual plots for model 6C based on the SLS and WSEP-AC data models. SLS is the simplest data model with the assumptions of homoscedastic, independent, and Gaussian residuals, and WSEP-AC is the most complex model without the assumptions. Model 6C is the most complex model and also the best model as ranked by Zhang et al. (2014) using Bayesian model selection. The variable at, plotted in Fig. 3a–c and Fig. 3d–f, is defined in Eqs. (2) and (11), respectively. Figure 3a–c show that all three residual assumptions are violated when SLS is used, as (i) the residual variance is not constant, but increases as a function of the simulated CO2 efflux (Fig. 3a); (ii) the autocorrelation function at most lags is beyond the 95 % confidence interval (Fig. 3b); and (iii) the standard normal density function cannot adequately characterize the residuals (Fig. 3c). Figure 3d–f show that, after relaxing the three assumptions, the processed residuals, at, can be well characterized by WSEP-AC. Figure 3d shows that, after normalizing εt with the linear variance (σt=0.034+0.099Et), the variation of the variance of at becomes significantly smaller, although the variance is still not constant. Figure 3e shows that, after removing a first-order autoregressive model from εt, at becomes less correlated, although the correlation is not fully removed. The two coefficients of the autoregressive model are ϕ1=0.989 and ϕ2=4.5×10-6; the small value of ϕ2 indicates that there is no need to attempt an autoregressive model of higher order. Figure 3f shows that at follows the SEP distribution with the estimated skewness coefficient of ξ=0.933 and kurtosis coefficient of β=0.998. As a summary, Fig. 3 shows that it is important to examine the residuals and to determine whether the selected data model is adequate for characterizing the residuals. While WSEP-AC still cannot perfectly characterize εt, it is significantly better than SLS.

Although the Gaussian assumption used in SLS is violated for model 6C (Fig. 3c), this is not generally the case for other data models and soil respiration models. This is shown in Fig. 4, which presents the quantile–quantile (Q–Q) plot for the eight data models and the three soil respiration models. For SLS, WLS, SLS-AC, and WLS-AC, the theoretical quantiles are based on the standard normal distribution, N(0,1); for SEP, WSEP, SEP-AC, and WSEP-AC, the theoretical quantiles are based on the standard skew exponential power distribution, SEP(0,1,1,0). If the residuals follow the assumed standard distributions, the Q–Q plots fall on the 1:1 lines, marked as the theoretical lines in Fig. 4. If the residuals are Gaussian or SEP but not standard, the Q–Q plots fall on a straight line but not on the 1:1 line. Figure 4a and e show that, for all of the soil respiration models, the Q–Q plots of SLS and SEP deviate significantly from the theoretical lines and exhibit fat-tail behaviors, which are an indication of outliers (Thyer et al., 2009). The deviation is reduced after accounting for autocorrelation in SLS-AC and SEP-AC, as shown in Fig. 4c and g. It is interesting to observe from the two figures that the Q–Q plots of the three models are visually almost identical. The deviation is almost fully removed after accounting for heteroscedasticity in WLS and WSEP in that their corresponding Q–Q plots fall on the 1:1 lines, especially for models 5C and 6C (as shown in Fig. 4b and f). However, the Q–Q plots start deviating from the 1:1 lines as shown in Fig. 4d and h, after accounting for both heteroscedasticity and autocorrelation in WLS-AC and WSEP-AC. In summary, Fig. 4 shows that, for the numerical example of this study, either the Gaussian or the SEP distribution is valid if heteroscedasticity is accounted for in the data models. However, accounting for autocorrelation in the data models does not help improve the characterization of the residual distributions.

While Figs. 3 and 4 help understand the validity of the three assumptions used in the data models, the impacts of the data models on estimating model parameter distributions must be evaluated separately. This section discusses the impact of the data model selection on parameter estimation with the objective of understanding whether the incorrect specification of the data model necessarily leads to biased parameter estimates. Such assessment is not a trivial task for two main reasons. First, microbial soil respiration models aggregate complex natural processes and spatial details into simpler conceptual representations. As a result, several model parameters are effective values of several complex natural processes that cannot actually be measured in the field, as discussed by Vrugt et al. (2013). Second, even for model parameters that can be measured in the field, as the model structure is imperfect, calibrated parameter values are sometimes beyond their physically reasonable range, as discussed by Pappenberger and Beven (2006). This is often undesirable, if we seek to make the models more mechanistically descriptive.

We focus our discussion on the carbon use efficiency (CUE) for microbial growth due to two reasons: (1) the CUE is a fundamental parameter in microbial soil respiration models, and (2) a physically reasonable range can estimated for the CUE. The concept of microbial CUE (Allison et al., 2010; Bradford et al., 2008; Manzoni et al., 2012; Wieder et al., 2013) has been used to present fundamental microbial processes in recent microbial enzyme models (Allison et al., 2010; German et al., 2011; Schimel and Weintraub, 2003; Wang et al., 2013). The microbial CUE, which is marked between MIC and CO2 in Fig. 1, controls microbial growth, enzyme production, and microbial respiration. A physically reasonable range of the CUE can be estimated from the physical viewpoint (Tang and Riley, 2014). Sinsabaugh et al. (2013) showed that the thermodynamic calculations support a maximum CUE of 0.60 and that previous studies that estimate CUE in terrestrial systems report a mean value of 0.55. Theoretically, there is no lower limit for the CUE as it can approach zero, and CUE <0.1 has been reported for terrestrial ecosystems (e.g., Fernández-Martínez et al., 2014) and used in modeling studies (Li et al., 2014). Note that, for inverse modeling with MCMC sampling, we did not assume a CUE maximum value of 0.6. In other words, for parameter estimation and predictive performance we did not impose the constraint that the CUE is less than 0.6. We merely use this CUE maximum value of 0.6 to evaluate whether the posterior CUE parameter samples obtained using different data models and different soil respiration models are within the physically reasonable range of 0–0.6.

Figure 5 plots the CUE posterior marginal density of the three soil respiration models obtained using the eight data models. The physical range between zero and 0.6 is marked in yellow. Figure 5 shows that the CUE posterior parameter distribution of model 6C (obtained using the data models) that does not account for autocorrelation is within the physically reasonable range. For models 4C and 5C, the posterior parameter samples are outside the range for six data models. For model 4C, the posterior parameters are only within the physical range for the SEP and WSEP data models; for model 5C, the two data models are WLS and WSEP. It is not surprising to find the posterior parameter distribution of models 4C and 5C, which have a certain degree of model structure error, to be outside of the physically plausible range. This can be attributed to two reasons. First, the model solution can be biased toward the missing processes in the model structure such as the additional carbon pool in both 4C and 5C or missing the explicit accounting for soil moisture in 4C. Second, biased parameter estimation can compensate for model structure inadequacy and other sources of discrepancy in both the physical models and the data models.

In addition, it is important to understand how accounting for autocorrelation, heteroscedasticity, and non-Gaussian residuals can affect the parameter estimation. First, it is observed in Fig. 5e–h that biased parameter estimates are outside of the physically reasonable range when autocorrelation is explicitly accounted for. This may suggest again that accounting for heteroscedasticity is desirable but accounting for autocorrelation is not. A possible reason is that filtering autocorrelation may reduce the residual space such that the transformed residual space cannot correspond to the parameter space of the models. In other words, parameter information may be lost due to filtering out autocorrelation. However, it is not fully understood why this does not occur for model 6C under data model SLS-AC (Fig. 5e), and more research is warranted. Second, unlike accounting for autocorrelation, accounting for heteroscedasticity alone (i.e., WLS and WSEP) only amplifies or reduces the variance without affecting the structure of the residual space. Figure 5c and d show that accounting for heteroscedasticity (i.e., WLS and WSEP) tends to improve the parameter estimation in comparison with the homoscedastic data models (i.e., SLS and SEP) shown in Fig. 5a and b. Finally, with respect to non-Gaussian residuals, Schoups and Vrugt (2010) suggested that, compared to Gaussian PDF, the peaked PDF of the SEP with a longer tail is useful for making the parameter inference robust against outliers. To a certain degree, this can be substantiated by the results in Fig. 5a–d, in that SEP and WSEP provide more favorable parameter estimates than SLS and WLS.

Finally, Fig. 5a shows that the posterior parameter distributions of SLS are very narrow for the three soil respiration models. The narrow distributions can be attributed to several reasons. As a SEP distribution can have longer tails than a Gaussian distribution, this can further increase the sample's acceptance ratio from tails resulting in a wider distribution (Fig. 5b). In addition, accounting for heteroscedasticity will result in a wider posterior parameter distribution (Fig. 5c) due to accepting higher variances at peak effluxes. Moreover, filtering correlation (Fig. 5e–h) increases the entropy, and leads to wider distributions.

In this section the ensemble is generated by running the soil respiration models with the posterior samples (obtained from the Bayesian inference) of the physical model parameters. In other words, the ensemble addresses parametric uncertainty of the soil respiration models only. Considering the relative contribution of parametric uncertainty only will provide insights for modeling approaches that attempt to segregate various sources of uncertainty (e.g., Thyer et al., 2009; Tsai and Elshall, 2013). The four statistics above (i.e., NSME, sharpness, coverage, and RMS) are calculated for the three soil respiration models and the eight data models. Taking the SLS and WSEP-AC data models as examples, Fig. 6 plots the data (for the calibration and cross-validation periods separately) along with the mean and 95 % credible intervals of the prediction ensemble for the three models.

Figure 6 shows that the data models affect model simulations for all of the models. The statistics, especially the RMS, indicate that WSEP-AC has better predictive performance than SLS. This is most visually obvious for model 6C during the cross-validation period after 330 d, as the prediction ensemble of SLS (Fig. 6k) cannot cover the observations, whereas the prediction ensemble of WSEP-AC can (Fig. 6l). This conclusion that WSEP-AC outperforms SLS agrees with the conclusion drawn from Figs. 3 and 4.

Figure 7 plots the four statistics for all of the soil respiration models and data models. Figure 7a and b show the predictive performance with respect to the central mean tendency measured by the NSME for both the calibration and cross-validation periods, respectively. The results indicate that, under all data models, the low-fidelity model 4C over-fits the data and results in biased predictions, in that the NSME values become significantly worse (e.g., from 0.6 to -0.6) from the calibration to the cross-validation period. This is confirmed by the visual inspection of Fig. 6a and g for data model SLS and of Fig. 6b and h for data model WSEP-AC. For models 5C and 6C, the NSME values vary with the data models; the central mean accuracy is the worst for SLS-AC that only considers autocorrelation (Fig. 6b).

With respect to the parametric uncertainty estimation, Fig. 7c and d show that sharpness generally increases when the three assumptions in the data models are gradually relaxed from SLS to WSEP-AC. This is even more obvious during the validation period. Given that the prediction ensemble does not center on the data, the increasing sharpness is desirable as it improves reliability. This is confirmed by the reliability plots in Fig. 7e and f. The exceptions are once again for SLS-AC and SEP-AC that generally have the lowest coverage.

With respect to the overall predictive performance measured by the RMS, the same variation pattern and exception are also observed in the RMS plots in Fig. 7g and h. This is not surprising because the RMS is the metric that can be used to measure all three criteria (central mean tendency, sharpness, and reliability). As the prediction ensemble is not centered on the data, the sharpness and reliability are the decisive factors for evaluating the predictive performance.

In summary, while it is necessary to account for heteroscedasticity in a data model, caution is needed when accounting for autocorrelation in the manner described in Sect. 2.1. In addition, after comparing the RMS values of the residuals using the Gaussian and SEP distributions, the conclusion is that the SEP distribution outperforms the Gaussian distribution with respect to predictive performance. Finally, uncertainty underestimation is evidenced by the very small predictive coverage. The underestimation of uncertainty for all of the physical models with all of the data model is not unexpected because only parametric uncertainty is considered in this study. Considering the overall predictive uncertainty is the subject of the next section.

The simulated output Y(θp) is generally not equal to the observed output d, and we have a residual term ε due to measurement, input, and model structure errors such that d=Y(θp)+ε. Accounting for the error term ε can be undertaken by separating various error terms. For example, in Sect. 4.1 we obtained uncertainty due to the physical model parameters. Accounting for other sources of uncertainty can be carried out using a single model approach (e.g., Thyer et al., 2009) or a multi-model approach (e.g., Tsai and Elshall, 2013). Alternatively, we can quantify the uncertainty based on total residuals that separates out parametric uncertainty, so the residual error includes errors in measurements, model inputs, and model structures (e.g., Thyer et al., 2009; Schoups and Vrugt, 2010). This lumped approach is based on sampling the residuals model ε(θε) with parameters θε. SLS has one fixed parameter, the constant variance, and other data models have two to six parameters. Thus, in this section the prediction ensemble addresses parametric uncertainty of not only the soil respiration models but also the data models. When generating the prediction ensemble in the procedure described by Schoups and Vrugt (2010), an ensemble of residuals is first generated by running the data models with posterior samples of the data model parameters for the positive carbon efflux domain; the residual ensemble is then added to the prediction ensemble generated in Sect. 4.1.

We start by undertaking a visual assessment of the predictive performance. Figure 8 is similar to Fig. 6 with the exception that Fig. 8 considers the overall predictive uncertainty (i.e., parametric and output uncertainty), whereas Fig. 6 only considers the parametric uncertainty. Figure 8 reveals a practical observation about accounting for the overall uncertainty using the lumped approach of sampling the data models. For example, Fig. 8b shows that, despite the wide prediction interval of model 4C, the model with significant model structure error cannot capture the birch pulse around day 180. It indicates that properly using a data model for model residuals cannot compensate for significant model structure error.

Figure 9 plots the four statistics (NSME, sharpness, predictive coverage, and RMS) of the three soil respiration models under the eight data models to assess the predictive performance. With respect to the central mean tendency, the NSME values in Fig. 9a and b are visually the same as those in Fig. 7a and b, indicating that the central mean accuracy under parametric uncertainty is the same as that under predictive uncertainty.

With respect to uncertainty, the values of sharpness and predictive coverage increase substantially (Fig. 9c–f). In particular, Fig. 9e and f show that, except for SLS and SEP, the predictive coverage of the rest of the six data models are close to 100 % for all three soil respiration models, indicating that the prediction intervals cover almost all of the data. This is demonstrated in Fig. 6 for WSEP-AC. Similar to Figs. 7c and d, Figs. 9c and d also show a general pattern where the sharpness increases when the three assumptions in the data models are gradually relaxed from SLS to WSEP-AC. The data models that account for autocorrelation are still the exceptions.

With respect to the overall predictive performance, the RMS values are largely determined by the mean accuracy and sharpness as the predictive coverage is similar for different data models. Figure 9g and h of RMS show that the predictive performance of the four data models that account for autocorrelation is worse than that of the other four data models. This suggests again that one needs to be cautious when building autocorrelation into a data model. This is consistent with the finding of Evin et al. (2013, 2014) that accounting for autocorrelation before accounting for heteroscedasticity or jointly accounting for autocorrelation and heteroscedasticity can result in poor predictive performance. In summary, Fig. 9g and h show that accounting for heteroscedasticity in WLS and WSEP for both the calibration and prediction periods gives the best overall predictive performance, and accounting for autocorrelation without heteroscedasticity in SLS-AC and SEP-AC gives the worst overall predictive performance. Finally, for the three soil respiration models, RMS shows that model 4C has the worst predictive performance for both the calibration and cross-validation data. Generally speaking, the high-fidelity model 6C outperforms model 5C for both the calibration and cross-validation data, which justifies the complexity of model 6C.

To demonstrate the impacts of the data models on the predictive performance of the soil respiration models, Fig. 10 plots the model simulations and predictions given by model 6C during the calibration and cross-validation periods using all the eight data models. Figure 10 is used to investigate predictive performance characteristics of the different data models. By examining the predictive performance of model 6C, specific predictive performance patterns can be identified. Figure 10a–d show that SLS and SEP have similar predictive performance with SEP generally having better predictive performance especially during the validation period. Not accounting for heteroscedasticity will underestimate the predication uncertainty (Fig. 10b, d). This is mainly because the variance of the efflux residuals increases with the magnitude of the carbon effluxes (Fig. 3a); thus, assuming constant variance is not representative. Accordingly, accounting for heteroscedasticity using WLS (Fig. 10e) or WSEP (Fig. 10h) will make the predictions more sensitive to peak carbon effluxes. This will generally improve the predictive coverage on the expense of sharpness and the central mean tendency. While WLS and WSEP have similar predictive performance, WSEP has better central mean tendency and overall predictive performance than WLS. Figure 10i–l show that accounting for autocorrelation using SLS-AC and SEP-AC results in wider uncertainty bands and insensitivity to peak carbon effluxes compared with SLS and SEP (Fig. 10a–d), which may be due to the reduction in the information content of the residuals. This results in the deterioration of the sharpness, the central mean tendency, and the capturing of peak carbon fluxes, especially during the validation period. Figure 10m–p show that accounting for both heteroscedasticity and autocorrelation using WLS-AC and WSEP-AC makes the inference robust against peak carbon effluxes. However, due to the loss of information content, the uncertainty bands are still wider, and the uncertainty becomes overestimated especially during validation period compared with WLS and WSEP (Fig. 10e–h). The results of models 4C and 5C, which are not shown here, also display the same prediction patterns with respect to non-Gaussian residuals, heteroscedasticity, and autocorrelation.

Finally, we observe in Fig. 10 that the data models that have good overall predictive performance as measured by RMS during the calibration period will maintain this good predictive performance during the validation period. For model 6C, the RMS values for the calibration and validation periods are very well correlated with a correlation coefficient of 0.92. However, we note that for models 4C and 5C the overall predictive performances during the calibration and validation periods are not as well correlated as for 6C, with correlation coefficients of 0.52 for model 4C and 0.61 for model 5C. This suggests that model 6C is more robust than 4C and 5C for forecasting and hindcasting.

Accounting for autocorrelation can lead to biased parameter estimation (Fig. 5) and poor predictive performance (Fig. 10). Autocorrelated residuals may be attributed to model discrepancy, as shown in Lu et al. (2013). The most obvious solution to handle the autocorrelation is to reduce the autocorrelation by improving the soil respiration model. If model improvement is difficult for practical reasons, we can improve the data model to better characterize the autocorrelation. Addressing autocorrelation in a data model is challenging, as it involves several interlinked factors as follows:

Non-stationarity could be a reason for this problem. By drawing on similarity from surface hydrology, the study of Ammann et al. (2018) suggests that autocorrelated residuals might be attributed to non-stationarity due to wet–dry periods with half-hourly data. Accounting for non-stationarity due to wet–dry periods could better address the problem of autocorrelated residuals (Ammann et al., 2018; T. Smith et al., 2010).