In 2015 we established a transect of sites in the northern boreal forests of Canada (Fig. ). All of these sites are located near Eagle Plains, Yukon (66∘220′ N, 136∘430′ W), and Tsiigehtchic, Northwest Territories (67∘260′ N, 133∘450′ W). The mean annual air temperature at these sites is -8.8 ∘C. The sites are evergreen needle forests dominated by Picea mariana (Mill.) BSP and Picea glauca (Moench) Voss species. Site selection and physical characteristics of the sites are also described in and .

Chronosequence sites were selected from the time since the last fire (in 1968, 1990, and 2012) that burned all aboveground vegetation. We also included a control site, where the last fire was more than 100 years ago. The date and boundaries of the fires were determined from geographic data from the Canadian Wildfire Information System . We visually corroborated the geographic location of our sites with reported fire boundaries. Previous studies with these data classified the 1968 site as 1969, which we attribute to this site being classified by fire season rather than the year of burn. For this paper we will refer to a site as a categorical variable by the year it was burned (2012, 1990, 1968) or the control site, as “control”. sites will be ordered by the fire year (2012, 1990, 1968, or control).

At each site we measured soil temperature, fluxes of CO2, microbial biomass assays, soil carbon, tree biomass (foliage, branches, and stems), and other auxiliary measurements by establishing three different lines at each site and, within each line, three replicate plots . Additionally, at each plot, soil samples were collected for further analysis in soil temperature incubation experiments. Roots were excluded from incubation soils; we assume the measured respiration from these samples is RH. The soil samples were incubated at 1, 7, 13, and 19 ∘C for 24 h, and the respiration was measured from syringe samples taken at the end of each 24 h period. The method is described in more detail in .

The field data measured total soil carbon in the top 30 cm, whereas the incubation data included measurements of soil carbon to a given depth (which extended to 50 cm). To determine the total soil carbon to a given depth in the field data we applied a multistep process. This process assumes that the soil carbon profiles in the incubation and field data are similar. First, for the soil carbon in each of the incubation samples (for each replicate line and plot described above) we computed the cumulative proportion of soil carbon (g C m-2) to 50 cm (dots in Fig. ). We acknowledge that soil carbon is present in deeper layers (estimated to be 59 100 g C m-2 in the top 100 cm at our sites; see , and https://bolin.su.se/data/ncscd/, last access: 26 September 2021). However, the objective of this process is a representative empirical estimate of soil carbon for the field data. Second, at each incubation sample we fit a saturating function to the cumulative proportion of soil carbon. The function we fit had the form yi=1-e-kDi, where yi is the cumulative proportion of soil carbon at depth Di in incubation sample i. Third, we computed the median ensemble average and 95 % confidence interval from the saturating functions grouped by chronosequence site (2012, 1990, 1968, and the control sites, Fig. ). The median ensemble average allowed estimation of the proportion of soil carbon up to a given depth (5 or 10 cm) at each field site (Table ). These proportions were then used for determining the amount of soil carbon at 5 or 10 cm for the field data.

The incubation data included measurements of the available soil organic carbon extracted from incubation soils, denoted here as CA, as described in . Briefly, soil dissolved organic C content was measured using a total organic C analyzer (Shimadzu TOC-V CPH, Shimadzu Corp., Kyoto, Japan) from soil extracts extracted with 0.5 M K2SO4. Microbial carbon used in the FireResp model was extracted using the chloroform fumigation extraction method . Briefly, 3 g dry weight equivalent of soil was fumigated at 25 ∘C with ethanol-free chloroform for 24 h and extracted with 0.5 M K2SO4. The conversion factor, also known as the extraction efficiency, for estimating the microbial carbon is 0.45 . For the field data, we approximated CA as linearly associated with total soil carbon CS at a given depth, extrapolated from linear regression in the incubation data (results not shown).

For the field samples an estimate of root carbon CR was assumed to be proportional to total tree biomass collected at each plot . A summary of all input variables is reported in Table .

The FireResp model predicts plot-level soil respiration (RS) and its components: autotrophic respiration (RA), microbial maintenance respiration (RM), and microbial growth respiration (RG). All respiration units are reported as g C m-2 d-1. The FireResp model expresses respiration components with two primary functions; the different combinations of these functions yield different submodels (described in detail below). First, we assume that RA and RM both follow an exponential Q10 relationship (Eq. ) parameterized by soil temperature (Tsoil; ∘C): 1RX=kXCX⋅r(fW)⋅Q10,X(Tsoil-10)/10. Equation () is a commonly applied (empirical) paradigm for respiration, motivated by temperature dependencies of enzymatic reactions . This exponential temperature model is applied for RA and RM, similar to process models for these components at the ecosystem scale . The function r(fW) is an empirical function developed by to represent the response of respiration across a range of soil moisture conditions, where fW represents volumetric soil moisture (%) and r(fW)=3.11fW–2.42fW2. The variable CX represents a soil carbon pool (g C m-2). For RA this CX equals root carbon (CR); for RM this CX equals soil carbon (CS) or microbial carbon (CM) depending on the type of submodel considered (e.g., Null, Microbe, Quality, Microbe-mult, or Quality-mult; all described below). Equation () has two parameters: kX, the base rate of respiration (d-1) for pool CX, and Q10,X, the temperature response of respiration (Q10 value) (no units) for pool X. To aid the representation of model equations, we will write Eq. () as RX=gXCX, where gX=kXCXr(fW)Q10,X(Tsoil-10)/10. As an example, autotrophic respiration RA would be written as RA=gRCR.

Second, we model microbial growth respiration (RG) via Michaelis–Menten kinetics : 2RG=ϵμCXCMkA+CX.

Equation () arises from first-order microbial enzyme kinetics under quasi-steady-state assumptions . In Eq. (), ϵ is the efficiency converting substrate to microbial biomass (no units), μ is the maximum microbial uptake rate (h-1), kA (g C m-2) represents the half-saturation rate, and CX represents the substrate for respiration. Depending on the model variant, CX may be total soil carbon (CS) or available soil organic carbon (CA), which represents more labile carbon for ingestion by microbes.

The FireResp model has five different submodels which arise through different combinations of these functional representations of respiration. These submodels are slightly modified from a similar approach in .

Null submodel. The Null submodel assumes soil carbon consists of a single pool . Here, soil maintenance respiration depends on soil carbon (so RM=gSCS). Microbial carbon is not considered in the Null submodel, so total soil respiration (RS) is the sum of autotrophic and maintenance respiration (Eq. ).3RS=RA+RM=gRCR+gSCS

Table  summarizes the different parameters for each model and their allowed ranges when estimating parameters.

The different submodels (Null, Microbe, Quality, Microbe-mult, and Quality-mult) may be nonlinear with respect to the parameters. For parameter estimation we applied the Levenberg–Marquardt algorithm . The Levenberg–Marquardt algorithm optimizes an objective function, which in this case is the residual sum of squares between measured and modeled soil respiration RS. The algorithm also requires (1) the Jacobian of the model to accelerate convergence to the optimum value, (2) an initial guess for parameters, (3) and bounds for all parameters.

The Levenberg–Marquardt algorithm may converge to a local (rather than global) optimum, or the estimated parameter values may be at the boundaries of the allowed range. To ensure that parameter estimates converged to a global (rather than local) optimum, initial parameter guesses for the method were drawn from a uniform distribution with reasonable bounds on parameters (Table ). The Levenberg–Marquardt algorithm is implemented in R with the package nlsr .

For parameter estimation, we applied a quasi-factorial design with the field and incubation data. This design allowed us to investigate how predictions for autotrophic (RA) and heterotrophic (RH) respiration varied when different data are incorporated into the parameter estimation routine. Four different data combinations were used for parameter estimation.

Field. All model parameters (e.g., Q10,M, kM, kA, μ, ϵ, kS, Q10,R, and kR, depending on the type of model) were estimated with the field data only.

Table  shows the relationship between the different parameter estimation approaches studied.

Table  lists the parameters estimated for each submodel and parameter estimation approach. Data used for parameter estimation consisted of combinations from five different categories of sites (2012, 1990, 1968, control, or all sites together) and three different depths (5 cm, 10 cm, or both depths together). Additionally, with the four different parameter estimation approaches (field, field linear, incubation field, and incubation field linear) and five different submodels (Null, Microbe, Microbe-mult, Quality, and Quality-mult), 300 separate parameter estimations were computed.

When parameters were estimated using (1) the incubation data, (2) field parameter estimation approach, and (3) field linear parameter estimation approach, we applied 1000 iterations of the Levenberg–Marquardt algorithm. Following these iterations we reduced post-processing computational time in two ways. First, duplicated parameter sets were reduced to a single instance. Second, we excluded parameter sets for which the residual sum of squares was outside the 50 % centered confidence interval. For the incubation field and incubation field linear approaches, we used these filtered parameter sets for subsequent estimation of the remaining parameters with field data.

We applied two different approaches to evaluate the reasonableness of a model–data fit. The first approach relied on Taylor diagrams , which facilitates intercomparison between models when compared to measured values (in this case RS). The Taylor diagram is structured as a polar coordinate plot; here, the radius ν is the normalized ratio between modeled and measured standard deviation σmodel/σmeasured and the angle θ corresponding to the correlation coefficient r for measured and modeled RS. Two comparisons can be visually inferred from the Taylor diagram. First, the point located at (ν,θ)=(1,0) represents a set of modeled values of RS that perfectly match measured RS. Values of ν less than unity indicate that modeled RS has less variability. Second, the distance from a point on the diagram to (ν,θ)=(1,0) is the centered-pattern root mean square distance. Concentric circles from the point (ν,θ)=(1,0) help assess the centered-pattern root mean square distance for modeled results.

A second approach relies on Akaike's information criterion (AIC) . The AIC is defined as -2⋅LL+2⋅p, where LL is the log-likelihood and p the number of parameters in the model. The submodel with the lowest AIC is defined as the best approximating model for the data. We apply the AIC to compare across submodels for a parameter estimation approach to control for sample size effects in the AIC.

We had two categories of datasets for this study: the type of data (incubation or field data) or the depth at which measurements were made (5 cm, 10 cm, or both depths together). This controlled experimental design is also represented in the Taylor diagrams (Fig. ), which comparatively shows a centered-pattern root mean square distance (distance between a point on the Taylor diagram and (ν,θ)=(1,0)) ranging from 0.25–1 and r ranging 0.7–0.9. For the field data (Fig. ), the centered-pattern root mean square distance ranged from 0.5–1 and r 0.3–0.9. We attributed the differences between Figs.  and to the soil temperatures from the incubation experiments spanning 1–19 ∘C, allowing for a wider temperature range to characterize any exponential temperature profile. In contrast, field measurements ranged from 4–9 ∘C (Table ). For both Figs.  and , the 5 cm depth had higher values for r and a smaller centered-pattern root mean square distance compared to the 10 cm depth.

We did not find any noticeable site differences in submodel outputs depending on the depth of the soil used for data assimilation (5 cm, 10 cm, or both depths together; Figs. , , ). While soil model parameters (such as Q10) are expected to vary with soil depth we did not observe any significant depth-dependent differences in parameter estimates (see the figures in the Supplement). The primary reason for this result is that the inter-site variability is larger than the variability by depth at a given site (Table  and Fig. ). We also did not find any improvements in our results when all data from sites were pooled together (Figs.  and ). From these conclusions, we will limit the discussion to evaluating model results generated from data at the 5 cm depth.

We cannot eliminate a parameter estimation approach (field, field linear, incubation field, or incubation field linear) simply by the magnitude of the estimated fluxes RA (Fig. ). Measured autotrophic respiration in actively growing high-latitude boreal forests or inferred from synthesis studies can range from 0.5–4 g C m-2 d-1. Most of the modeled values of RA for all the parameter estimation approaches are within that range. The incubation field and field parameter estimation approaches predicted higher RS values outside this range at the 1968 site.

While there is no universal pattern to RH following forest fire disturbances , we have reason to believe that the near-zero modeled values for RH for the 1968 site in Fig.  may be an underestimate. For our sites we expect modest, and perhaps decreasing (but not zero), changes in RH from the time of disturbance for three reasons. First, factors influencing recovery of RH are burn severity or intensity and decomposition of pyrogenic litter . The fires at our sites combusted a significant amount of soil organic matter resistant to decomposition , thereby minimizing any increases in RH from the decomposition of labile litter. Additionally, from this chronosequence, reported increased temperature sensitivity (Q10,M) in recently burned sites, but this was tempered by decreases in soil organic matter quality . Second, as succession occurs, the increase in aboveground vegetation insulates the soil, decreasing the active layer and thereby decreasing RH . Third, at the same chronosequence sites found constant C:N:P and fungal-to-bacterial ratios for microbes, indicating homeostatic regulation of the microbial community. The cumulative effect of these confounding factors may translate into RH remaining constant across the chronosequence.

Our models implicitly assumed an increasing exponential relationship between temperature and respiration. The temperature sensitivity of respiration (Q10) across ecosystems can vary (usually around 2-5) and is generally expected to be greater than 1, but the Q10 value may decrease as soils warm . Some degree of additional variability is expected when considering the biochemical or thermodynamic foundations of respiration , the methodological approach used to measure soil respiration , or variation in the soil organic matter supply .

However, an increasing exponential relationship between temperature and respiration may not be robustly supported by observed data at the chronosequence sites. The forest fires at each site burned a large portion of soil organic matter and killed the roots. Immediately following a fire, RS will be lower even if there are higher soil temperatures. In late-successional forests, the soil is colder and the active layer depth is shallower, even though there may be more soil respiration due to higher quantities of roots and soil organic matter; we observed such patterns across the chronosequence. The 2012 and 1990 sites had the highest values of Tsoil (Table ) but the lowest overall respiration (Fig. ). Across the chronosequence, scatterplots of respiration with temperature had a null or negative relationship (results not shown). Empirically the negative association of respiration with temperature would imply a Q10 value less than unity. As a result, to compensate for these opposing tendencies the RH parameters tend to be edge-hitting (Fig.  and the Supplement).

We recommend either the incubation field or incubation field linear parameter estimation approach for two reasons. First, values of the proportion of the respiration that is autotrophic (pA=RA/(RA+RH), Fig. ) for the field or field linear approaches are unexpectedly and unrealistically large, attributed to the variation in RH (Fig. ). As a baseline, reported values of RA/(RA+RH) to be approximately 0.50, which has also been supported in meta-analyses (Soil Respiration Database, ). Second, the incubation field and incubation field linear approaches in Fig.  show a temporal pattern in pA similar to patterns reported in and the predicted pA inferred from . The modeled values of pA are larger at late-successional sites (0.75–1), which may be an effect of the timing of field collection (August) when RA is at a seasonal peak .

Our first hypothesis concerned the dependence of RA on tree biomass. We developed this hypothesis from our previous studies, which concluded that tree biomass was a key factor explaining patterns of soil respiration across the chronosequence . For all submodels and the field linear or incubation field linear parameter estimation approaches, RA is proportional to CR, which is proportional to tree biomass. Values of CR increase across the chronosequence (Table ). However, even with this proportional association, the results in Fig.  indicate less support for our first hypothesis for two reasons. First, some modeled values RA at the 1990 site are higher than expected, especially given the association with RA to CR. Since CR is still comparatively low at this site, we might expect RA (and by association pA) to be near zero as well. Additionally, the near-zero values of RA are not a consequence of parameters relating to RA (kR, Q10,R, or gR) being estimated as zero. (Otherwise, the values for these aforementioned parameters in Fig.  or the Supplement for all the different models and approaches would be edge-hitting and indicated with blue-colored dots.) Second, and perhaps more importantly, all parameter estimation approaches in Fig.  predict RA to decrease between the 1968 and control sites. The modeled decreases in RA are a result of observed decreases in RS (Fig. ) as CS increases. To compensate, estimated parameters kR or gR decrease across the chronosequence sites (Fig.  or the Supplement). The patterns of kR or gR may be due to the parameter estimation routine compensating for the confounding effects of increasing CR with decreasing RS. In summary, even though there is evidence for association between RA and tree biomass in earlier chronosequence sites (2012 and 1990 sites), additional work is needed to explain the reasons for the decline in RA for later chronosequence sites (1968 and control sites). Future work could quantify field estimates of root mass, production, and turnover to corroborate the values of CR used here and with the estimated decreases in kR across the chronosequence.

Our second hypothesis concerns the structural representation of soil respiration for soil models. Our submodels are arranged on a continuum of complexity (Null, Microbe, Quality, Microbe-mult, or Quality-mult). When parameterizing more complex models, parameters may be non-informative and/or edge-hitting . Reducing parameter dimensionality is a key consideration for model–data assimilation in the carbon cycle . Considering the incubation field linear approach only, across the range of submodels the Microbe submodel had the smallest percentage of edge-hitting parameters (10 %), ranging from 30 %–50 % for the other models.

While the AIC suggests a preference towards the Null submodel, we do not believe it is a sufficient criterion to choose it over the Microbe and Quality submodels. There was no noticeable improvement with the Null submodel in the Taylor diagrams for the field data (in both the values of r and the centered-pattern root mean square difference; Fig. ) or with the adjusted R2 or AIC values (Fig. ). While all models could not account for a majority of the variance in observed soil respiration (the adjusted R2 values in Fig.  ranged from 0.25–0.61), no submodel significantly improved the adjusted R2 or AIC. In other words, the model statistics indicated that the parameter estimation approaches all performed similarly. This model result similarity conforms to a study by , which synthesized a range of experimental data with different types of process-based models to predict long-term soil organic carbon storage.

A design constraint was to construct models with the greatest potential to be fully parameterized from the collected data. For the Quality-mult and Microbe-mult submodels, kA was estimated at the lower end of its range (Fig. ), essentially reducing these models to the Quality and Microbe submodels, respectively. Even though we cannot definitively conclude which of the two submodels (Quality or Microbe) is the better approximating model, we recommend that some consideration of microbial growth and maintenance respiration be considered using Michaelis–Menten kinetics as a starting point . Several frameworks already exist for incorporating Michaelis–Menten kinetics or substrate quality degradation . Continuous (daily or sub-daily) soil respiration measurements could better support more complex soil models . Each of the models could be incorporated into a dynamic model of ecosystem carbon cycling that also includes temporal changes in permafrost active layer depth .