Methanogens prefer recently assimilated fresh carbon as their energy source, for instance, the root exudates of vascular plants . A connection between ecosystem productivity and CH4 emission has been observed in several wetland studies . However, anoxic decomposition of litter and older peat also produces CH4 . Many models form CH4 substrates by extracting directly a fraction of the net primary production , and some rely on heterotrophic peat respiration only . In sqHIMMELI, both primary production and anaerobic peat decomposition were included.

The modified sqHIMMELI model contains an exudate pool description, from which it produces methane (Eqs.  and ). The exudate pool itself is described by Eq. (), detailing how the modeled NPP turns into root exudates. Effectively, a fraction of NPP determined by the parameter ζexu (–) produces root exudates, which are then distributed as anaerobic respiration according to the root distribution into the peat column at the rate determined by the model parameter τexu (s). The part ending up under the water table produces CH4 and CO2, depending on the oxygen content of the water, and above the water table the exudates are respired into CO2.

The second source of anaerobic respiration, the anaerobic peat decomposition, is modeled in sqHIMMELI with a simple Q10 model adopted from . The peat under the water table is prescribed a turnover time, based on which anaerobic respiration and CH4 are produced according to Eqs. () and ().

The sqHIMMELI model differs from HIMMELI in the details regarding the root distribution model. Compared to measurement data of root distributions of aerenchymatous sedges from , the original root distribution π(z), adopted from and described by π(z)∝exp⁡(-z/λroot), does not describe the distribution of roots well. Here, z is depth, and λroot is a parameter describing the steepness of the decaying exponential curve. This formula is replaced with π(z)∝C0exp⁡-(z-z0)2λroot2+C1.

With the Gaussian shape, the new root density decreases faster with depth. Without this change, the optimization process calibrates the model to have very high root masses below 50 cm underground. The other difference between the models is that in the original model there are vanishingly few roots below the depth of 1 m, but according to , sedge roots can reach to as low as 2.3 m under the surface. The term C1 in Eq. () was added to remedy this.

Before starting the optimization, the parameters C0, C1, and z0 were fitted to data from , resulting in values of C0=215, C1=6, and z0=0.105. The different root distributions are shown in Fig. .

Methane is produced from anaerobic peat decomposition at all peat depths in the sqHIMMELI model, and its transport and oxidation affect the modeled CH4 emission. The homogeneous model description of the peat column is highly idealized, as in reality the peat column varies from place to place with respect to CH4 production rate, production depth, and gas transport. We model the peat column to be 3.4 m deep, which is 85 % of the maximum observed depth of the Siikaneva wetland. Small uncertainty in the value of the parameter is acceptable since the parameter τcato, which regulates the rate of peat decomposition into CH4, can partly compensate for this uncertainty.

The parameters for the optimization were chosen to constrain the processes most important for the CH4 emission. Of the optimized parameters, all but ζexu (–) and Q10 (–) are the same for all years. However, ζexu and Q10 change year to year to reflect the changes in the relative CH4 input to the system from peat decomposition and NPP-based production. This will allow to analyze the year-to-year changes in relative importance of the production pathways. The setup is natural; for example, report the Q10 values changing from measurement date to another, even within a single year. As the values reported for minerotrophic lawn in indicate that they may vary quite irregularly within a growing season, the modeling performed here does not take intra-annual variations into account and concentrates on the year-to-year variation. Possible mechanisms for the parameter variations include variations in substrate supply and desiccation stress, and are discussed in, e.g., . Table  shows the parameters that are used in the equations below but not optimized in this work, along with their values and explanations of why they were left out. The list of calibrated parameters along with their physical meanings is presented below.

The version of HIMMELI presented here describes processes for CH4 production and transport. It differs from the version presented in in that the model presented there does not contain the processes for anaerobic respiration but rather takes them as input, the idea being that such input would be available when using HIMMELI as a part of a larger model. Hence, the equations presented in Sect.  are specific to the version used in this study. The other difference between the models is the difference between the root distributions described in Sect. .

In order to be able to assess the annual parameter and CH4 transport pathway changes, a hierarchical description for two of the parameters was used. These parameters were Q10 (–) controlling the temperature dependence of the peat decomposition rate, and ζexu (–) regulating the production of root exudates from NPP.

The “hyperparameters” are the means and variances defining the Gaussian priors of the hierarchical parameters Q10 (–) and ζexu (–). They were updated using fixed Gaussian “hyperpriors” with Gibbs sampling. The sampling distribution depends on the current values of the hyperparameters. The role of the hyperprior is to constrain the distribution from which the hyperparameters are sampled.

Technically, a “Metropolis-within-Gibbs” method for sampling the hierarchical parameters, non-hierarchical parameters, and the hyperparameters was used, presented briefly in Appendix . The model parameters (i.e., everything except the hyperparameters) were sampled with the adaptive Metropolis (AM) MCMC algorithm , which uses a Gaussian proposal distribution, whose covariance matrix is adapted as the chain evolves, and over time the acceptance rate gets closer to an optimal value, which is 0.23 for Gaussian targets in large dimensions . If the algorithm proposes values outside the hard parameter limits listed in Table , the model will not be evaluated and the value is rejected.

Our empirical data for the hierarchical model were the 9 years from 2006 to 2014, meaning that for each of these years there were corresponding ζexu (–) and Q10 (–) parameters in the optimization. The model was spun up for each annual flux estimation in order to have a realistic column of gas concentrations available. For this reason, the previous year was always also simulated, and for the likelihood only the residuals from the latter year were included in the calculations. Therefore, the year 2005 did not contribute directly to the values of the objective function. The different years were run in parallel to save execution time.

As in many practical uncertainty quantification applications, a major part of the parameter estimation problem is the proper definition of the objective function. For MCMC, it is defined here based on a priori information about the measurement uncertainties, based on information from the model residuals, and based on additional prior information. For the importance resampling, we modify the error model for the CO2 and CH4 residual components of the objective function based on an analysis of the MCMC results.

The parameter values used in the analyses are shown in Table . The MAP and posterior mean estimates agree on the value of the water diffusion rate coefficient fD,w (–), and the posteriors shown in Fig. k show that the estimates are close to the middle of the marginal distribution and slightly above the prior value. In tests with a shallower peat column, smaller values of this variable were obtained (not shown).

Contrary to this, the air diffusion rate coefficient, fD,a (–), finds its best values lower, and the variability of the parameter is larger than that for the diffusion rate coefficient in water-filled peat.

The root distribution parameter, λroot, is optimized larger than expected, and again the MAP estimate is close to the posterior mean. This implies that the model optimizes best when the CH4 produced from the photosynthesis-induced exudate production goes relatively far below the surface: with a value of 0.3, 49% of the roots are deeper than 25 cm, 15 % of the roots are deeper than 50 cm, and just 2.5 % are deeper than 75 cm; see Fig. . In relation to these numbers, the water table depth is most of the time above the depth of -20 cm. Additionally, a larger λroot will facilitate the emission of the CH4 produced by peat decomposition in the catotelm.

The values of the exudate pool turnover time τexu are close to the default value of 2 weeks, with the MAP estimate at a little under 14 days and the posterior mean at 2.5 days more. The results from the importance resampling show that the spread is around 3 days around this posterior mean value. However, the value of ζexu‾ controlling amount of exudates produced from photosynthesis is smaller than the default value at roughly 0.15–0.45, with the MAP and posterior mean estimates at 0.343 and 0.292, respectively. In contrast to this, and balancing the effect of a relatively low ζexu‾, the parameter fCH4exu (–), controlling how much methane is produced from anaerobic decomposition of exudates, has a skewed posterior marginal distribution with most of the mass above the value of 0.7, as can be seen in Fig. .

The non-hierarchically optimized parameter, VO0 (mol m-3 s-1), controlling the amount of CH4 oxidation taking place is close to the minimum allowed value at one-fifth of the default value. This is also true for the parameter controlling heterotrophic respiration, VR0 (mol m-3 s-1), all of whose optimized estimates reside close to its minimum value, reducing the amount of heterotrophic respiration taking place. The posteriors are very narrow. In contrast to these narrow posteriors, the parameters ΔEoxid (J mol-1) and ΔER (J mol-1), which are present in the same equations as the VO0 and VR0 parameters, have slightly wider posterior distributions, with the former slightly under and the latter slightly above the default values.

Table  shows that the hierarchically optimized parameter Q10 (–), controlling the temperature dependence of the CH4 production from peat decomposition, has slightly different values for the MAP and posterior mean estimates, with the Gibbs-sampled mean value (mean of those values in the case of the posterior mean) at 5.72 and 4.43, respectively.

The parameter τcato (y), also controlling the peat decomposition rate in the catotelm, compensates for the differences of Q10‾ between the MAP and posterior mean estimates by having a faster turnover time for the posterior mean than the MAP estimate. That parameter has a wide posterior, ranging from around 10 000 to 30 000, which was the value used by and the upper limit of the parameters in our work. Our posterior density goes to zero towards the higher limit, and the posterior mean is found at the value of 22 690 years.

The interannual variability of Q10 (–) is mostly similar for both MAP and posterior mean estimates. For instance, the years of the smallest values are 2007 and 2008 in both cases, and the values of the years 2006, 2011, and 2014 are the largest in both cases. For the other hierarchically calibrated parameter, ζexu (–), these similarities do not exist.

Table  lists the cost function values for the MAP and posterior mean estimates, and the annual errors for the MAP, posterior mean, and non-hierarchical posterior mean estimates and default parameter values are shown for each parameter set in Fig. . The cost function value is unsurprisingly lower for the MAP estimate than for the posterior mean estimate, indicating a better fit in terms of the error model. In Fig. b, the non-hierarchical posterior estimate shows a large variance of the annual errors, with early years having a positive bias, and later years having a negative bias. Incidentally, the average discrepancy from observations over the whole period for the non-hierarchical posterior mean is small for both methane and carbon dioxide, as Fig.  indicates. However, the variation for methane is the largest, implying that the annual variation is not reflected well. The model estimates of the annual fluxes are good in that the variance of the errors is small for both MAP and posterior mean experiments, especially, even though the estimates show a negative bias of 25 %. Compared to the default parameters, which strongly underestimate methane emissions (and even more overestimate the carbon dioxide emissions), the flux estimates are much improved. This is to be expected as the results shown are not for an independent validation dataset. Rather, the motivation with the MAP and posterior mean estimates is to see what the model fit looks like for optimized parameters and how the features differ from the unoptimized ones. It is, however, worth noting that the target objective function did not aim at minimizing annual discrepancies but daily residuals that were considered correlated.

A cross validation of the regression modeling in terms of the annual errors is shown in Figs. b and . While the annual estimates are not on average better than the ones from the simulation with the non-hierarchically obtained posterior mean, the spread of the errors are acceptable, particularly if the strong negative bias in 2007, which is mostly due to lack of observations during the season, is disregarded. Additionally, the overall biases are surprisingly slightly better than with the optimized parameters, due to effects of the prior, different data resolution in the cost function, and the non-trivial error model used. The cross validation is described in Sect. .

The positive bias in the CO2 may partly be due to the assumption that 70 % of the NPP comes from the aerenchymatous plants, and this affected the data that the sqHIMMELI model results were matched with.

All years of hierarchically optimized experiments show at least a small negative annual bias in the methane flux when compared to the available observations. This can be due to the high day-to-day variability of the summertime fluxes, which dominate year-round total fluxes, and the fact that the model can not, without data about the fine structure and heterogeneity of the wetland, match the high variability fluxes. The proportional model–data residual error component αyt (Appendix ) allows the model to underestimate the high peaks more than the low flux values. The error model favors the baseline of the lower values during periods when observed variance is very high, for instance, in the peak emission season of 2010. This is also true for periods of increased ebullition, and such fluxes are very difficult to fit into. These periods contribute to both the cost function values and the underestimation of the total methane flux. Any temporal shifts of peaks of seasons are penalized heavily, and the optimized parameter values rather produce less peaks than right size peaks at a slightly wrong time.

Another reason is that the carbon dioxide fluxes are overestimated by the model, leading to need to balance between the two, and as methane production in the wetland also produces carbon dioxide, the optimization algorithm will find a middle ground between the conflicting needs of minimizing carbon dioxide and maximizing methane production.

Additionally, the wintertime methane fluxes are underestimated systematically, and the emissions start slightly late in early summer, which produces a negative bias to the total flux even though visually the fit is good, as can be seen in Fig. . This figure also reveals that the observations for the vast majority fall within the confidence margins suggested by the ARMA model for the residual. The variation from the full posterior is higher because the uncertainty shown in Fig.  does not take the parameter variations into account.

The carbon dioxide time series against flux observations are shown in Fig. . This figure reveals that sqHIMMELI and the error model most of the time are able to explain the carbon dioxide fluxes well, even though some of the largest deviations are not captured. Since in an observational time series outliers can come from an underlying process that is not well explained by these models, having a small number of such deviations is not surprising.

The input data have a role in affecting the model fit to the data, and since NPP is a modeled quantity, there is some additional uncertainty stemming from that modeling involved. For LAI, we note that even though in reality it is not identical every year, in the model, it follows the same pattern (see Appendix ). The parameter calibration must then favor parameters producing a good fit in terms of average model performance.

The sqHIMMELI model produces the CH4 from anaerobic respiration that originates from peat decay and the decay of root exudates. These production components, along with the different output pathways, CH4 oxidation, and model residuals, are plotted as functions of water table depth in Fig.  for the MAP, posterior mean, non-hierarchical posterior mean, and default parameter values. The process correlations and covariances are shown for the year 2012 in Fig. .

In the following, “all ebullition” refers to any ebullition in the peat column regardless of whether the bubbles reach the peat column surface. “Ebullition” refers to the part of all ebullition which reaches the surface. Most of the time, the water table is under the peat surface, and at those times ebullition is zero, although all ebullition can be substantial. In that case, the ebullition flux does not go directly into the atmosphere, but into the first air-filled peat layer above the water table level, and continues from there via other pathways. The reason for this separation comes from implementation details of HIMMELI. In all experiments, ebullition reaching the surface is a minor fraction of the total CH4 emission.

For the posterior mean estimate, the flux components and oxidation are shown as time series in Fig. . Optimizing the model leads to increased production of methane from peat decay, as can be seen in Fig. f. A similar effect is seen also in the plant transport component in Fig. b.

Comparing results from simulations with optimized parameters to results using the default parameter values (shown in Table ) shows that the optimization somewhat decreases the role of the plant transport pathway in favor of the diffusion pathway, especially for the years 2010, 2011, and 2013. Diffusion and all-ebullition fluxes are closely tied to each other, as can be seen in Fig. a, in that in many years (2007–2008, 2012–2014) their values are close to each other for all estimates. This is also visible in the flux component time series in Fig. .

The priors of the hierarchical CH4 production-related parameters Q10 (–) and ζexu (–) in Fig. b and d are constrained by the data, as are the hierarchical parameters themselves, shown in Fig. . The priors of these distributions are wider than their posteriors, which is also the case for the other production-related parameters τexu (s) and τcato (y). Both process descriptions for obtaining the anaerobic respiration are clearly needed for a good model fit, because the parameter posteriors do not have remarkable mass in the regions minimizing either of these processes (hierarchical parameters at the lower bounds or turnover rate parameters τexu and τcato at the upper bound). The covariances in Figs.  and  show that the two production processes covary slightly, with correlation coefficient -0.32, and hence they are to that extent interchangeable. Reasonable identifiability of the Q10 parameters is not obvious; for example, optimized a corresponding parameter to end up with the parameter at the lower bound of their prescribed range. However, half of the mass of the production terms in the process correlation plot, Fig. , lies within a region that for production from exudates is roughly 10 % of the total production and for the production from peat decay of the order of 35 %, and hence the production processes can be said to be well constrained.

The posterior distributions of VR0 (mol m-3 s-1) show that sqHIMMELI performs better when the heterotrophic respiration is close to being minimized, which is also aided by a posterior mean value of ΔER (J mol-1) that is lower than the prior mean. For the oxidation parameters VO0 (mol m-3 s-1) and ΔEoxid, the situation is different: the former has the tendency of being very small, but the temperature response has the tendency of being stronger with posterior mean and MAP values above the prior mean.

Whereas the fraction of plant transport is stable and high, but still constrained, not all the parameters affecting root conductivity are constrained by the data, as the root tortuosity posterior distribution follows very closely the prior form. The root-ending cross-sectional area, however, is constrained to its lower side despite there being mass also above the prior mean value. For this parameter, the importance resampling resulted in a changed posterior in that there is a lot more mass at the higher end of the distribution, as can be seen in Fig. h. In addition to this difference, the effects of the resampling were mostly minor. Still, the resampling informed that the roots should reside slightly higher in the peat column than suggested by the MCMC, and that the fCH4exu is constrained to a higher value by the data than suggested by the initial MCMC run.

The transport pathways are well identified, as can be seen in the ranges of variation in the transport characteristics in Fig. . Notably, the transport processes do not strongly anticorrelate implying that they are not obviously interchangeable with each other. The correlation between oxidation and plant transport suggests that uncertainty in oxidation is a major part of the uncertainty in the plant transport portion. On the other hand, there is uncertainty in the absolute magnitude of the total flux (in terms of the posterior uncertainty) and this is reflected in the strong positive correlation between plant transport and the total flux. Similar but weaker positive correlations exist between the total flux and diffusion and ebullition, which is to be expected. The variation of oxidation is around 10 % of the total flux.

The calibrated sqHIMMELI model is able to describe the CH4 flux correctly in times of low water table, which is not obvious, as other studies have indicated the challenges in parameterizations of emission models with respect to the water table depth e.g.,. Figure  shows how the model processes are described under water stress. In times of a very low water table, the plant transport component and methane production from root exudates are decreased somewhat, as is methane oxidation. This results directly from how the model is constructed, as exudate deposition to the peat column is allocated depth-wise according to the root density profile. The fact that the model continues to perform well during these years implies that this method of regulating methane emissions during dry seasons is realistic. The residuals in Fig. h further show that there is a only a slight positive emission bias at the times of the lowest water table levels.

Modeled CH4 flux estimates may have large errors, as was shown in Fig.  with the default parameter set. The negative biases in the calibration phase that were found with the maximum a posteriori and posterior mean estimates are reasonable since the quality of the modeled input data from, e.g., a land surface scheme will also contribute to the uncertainty in the model predictions. Additionally, a known constant bias can be relatively easily accounted for if the interannual variability is correctly modeled.

Compared to the estimate with the optimized annual variations of the methane-production-related parameters, the non-hierarchical posterior mean estimate produces reasonable flux estimates over the assessment period, with twice the variability in fluxes compared to the posterior mean estimate, even though the average of the errors is closer to zero. The variability is seen in Fig. . The hierarchical posterior mean, on the other hand, does produce very steady estimates of the CH4 flux compared with observations even though there is a downward bias of 23 %, and the smaller interannual variance implies better predictive skill. The same is true to a lesser extent also for the maximum a posteriori estimate.

In order to be able to utilize the information regarding the annual variability in the posterior mean estimate for the future prediction of CH4 emissions, the values of the hierarchical parameters need to be estimated for the simulation years. A simple regression analysis of the hierarchical variables with respect to relevant input data was performed in order to find out if such estimation is possible. As the explaining variables, means, minimums, and maximums of NPP, water table depth, and soil temperature at different depths and over different periods of time were looked at. These time periods were June, July, August, and various different amounts of days from the start of the year.

The analysis revealed that the mean soil temperature of the first 10 weeks (70 days) of the year at the depth of 30–40 cm, denoted here by T30-4070‾, is the best single-variable predictor of the Q10 value for that year, and for ζexu, it is the sum of NPP from the first 130 days of the year, denoted by NPP130. This is hardly surprising, since the peat decomposition process regulated by the parameter Q10 is driven by soil temperature, and the anaerobic respiration from exudates controlled by the parameter ζexu is driven by the NPP input. These variables also indicate that the timing of the start of the growing season might play a role in determining the parameters. Possible mechanisms could include, e.g., effects of the start of growing season on development of the microbe populations in the spring. However, further analysis would be needed to confirm this.

The p values summarizing the reliabilities of the regressions and the r2 values, which are the coefficients of determination of the fit, are presented in Table . The r2 values explain what fraction of the variance of the dependent (predicted) variable is explained by the independent (explaining) variables. The p and r2 values uncover that the hierarchical modeling reveals a clear-cut reliable relationship between the early NPP and the optimal ζexu parameter (p=5×10-6, r2=0.957). This provides new insight into future model development and exemplifies why such a hierarchical description of variables is valuable in Bayesian optimization in a geophysical model context.

For the other interannually changing parameter, Q10, the soil temperatures explain only slightly over half of the variation (p=0.0185, r2=0.571). Since the effect of this parameter is very important for the total methane flux, this result leaves lots of room for further analysis. The hierarchical parameters Q10 and ζexu for each year can be estimated with Q10=3.86T30-4070‾+1.76ζexu=-46500NPP130+0.431, where the temperatures are in ∘C, and the units of NPP are mol m-2 s-1.

A leave-one-out cross validation (LOO-CV; see, e.g., ) of the regression modeling was performed by optimizing the hierarchical parameters with respect to the cost function in Eq. () leaving one year at a time out, calculating the estimates for the hierarchical parameters based on the results obtained for other years, and predicting the CH4 emissions for the year that was left out. The results of the cross validation are shown in Figs. b and . The cross-validated results are comparable in terms of annual performance to the non-hierarchical posterior mean. Despite the relatively good performance of the non-hierarchical posterior mean simulation, we note that the cross-validated result should be relied on more for prediction, since the well-predictable ζexu parameters contain useful information that is not available in the non-hierarchical posterior mean estimate. A hybrid between these approaches could be also used, using the regression-modeled values for the ζexu parameters and the mean for Q10, to minimize the risk of major annual biases due to unsuccessful prediction of the Q10 parameters.

As Fig. b shows, much of the error in the cross validation actually comes from challenges estimating the year 2007, which is missing the peak season observations, and therefore the error percentage (in terms of the annual observed flux) is easily high, especially as the start of season is modeled with a delay, which is readily apparent in Fig. , and in this sense the negative bias in Fig.  gives an unnecessarily pessimistic view of the model performance. For the CO2 fluxes, it can be noted that there is a persistent positive bias of some tens of percent, but the observations are very noisy and due to the processing for the use in the cost function, they might have biases. The effect of a small bias on the parameter posterior distribution is, however, minor, since the carbon dioxide observations were given less weight in the cost function than the methane observations. Hence, given their uncertainty, the optimized fit to the measurement data can, also in the cross validation as in the other experiments, be seen as acceptable.