Simultaneous numerical representation of soil microsite production and consumption of carbon dioxide, methane, and nitrous oxide using probability distribution functions

2.1 Site description and data collection

We measured GHG fluxes in a mature boreal-transition forest with a hummock–hollow microtopography, Howland Forest research site (45.20°N, 68.74°W) from central Maine, United States. Mean annual temperature and mean annual precipitation are +5.5°C and 1,000 mm, respectively. Soils of the Howland Forest upland sites are characterized as Skerry fine sandy loam, frigid Aquic Haplorthods. More information on the Howland Forest research site can be found in Fernandez, Rustad, and Lawrence (1993).

High-frequency (sampling frequency: 1 Hz) real-time soil CH₄ and N₂O fluxes were measured using an Aerodyne quantum cascade laser integrated with soil CO₂ flux measurements by LI-COR IRGA assembly. Triplicate chambers were each sampled once every 2 hr. Chamber tops were closed for 5 min and automated fluxes were calculated by fitting a linear regression on the change in headspace GHG concentrations followed by temperature and pressure corrections. We characterized the uncertainty of measurements by the standard deviation estimates for all three GHGs. Soil temperature and soil moisture were measured at each chamber location at 10 cm depth once every hour using a Type-T thermocouple and Campbell Scientific CS616 water content reflectometer probes, respectively, and stored on a Campbell Scientific CR10X data logger (Campbell Scientific). We used daily average values of both drivers (soil temperature and soil moisture) and GHG fluxes for modeling purposes to smooth high measurement noise observed at subdaily timescale. See Savage and Davidson (2003) for more details on our chamber design and automated sampling system. Quality control protocols for soil GHG fluxes can be found in Savage, Davidson, and Richardson (2008) and Savage et al. (2014).

2.2 Modeling scheme

Aerobic and anaerobic processes in soil are linked through heterotrophic dependence on fixed C sources for energy, but with contrasting effects of O₂ as either essential substrate or potential inhibitor (Figure 1; Figure S1). To date, most biogeochemical models use separate model versions for simulating soil organic matter decomposition resulting in CO₂ emissions and the processes affecting CH₄ and N₂O emissions, but here we simultaneously simulate biogeochemically linked multiple GHG emissions using the same biophysical framework.

We used soluble C as a proxy for the reducing power needed for methanogenesis. However, future studies may explicitly represent specific substrates for acetoclastic and hydrogenotrophic methanogenic pathways, if parameterization of that type of model structure can be constrained by the availability of data on concentrations of organic acids (acetate, formate) and hydrogen (H₂), which was not the case for our study. We also assumed that respiration is the dominant pathway of CO₂ production in soil. Thus, we did not account for the minor contribution of acetoclastic methanogens to CO₂ production and hydrogenotrophic methanogens to CO₂ consumption. Likewise, we considered soil respiration is the major sink of O₂ and ignored the otherwise small fraction of O₂ consumed by methanotrophs.

We added a nitrification module to account for the N₂O production during nitrification using the observed seasonal dynamics of ammonium () in our study area (Fernandez, Lawrence, & Son, 1995). The Howland Forest is a strongly nitrogen-limited system, with porewater nitrate () concentration always close to detection limits by inductively coupled plasma-mass spectrometry, ICP-MS (Fernandez et al., 1995). N₂O production during classical denitrification is mechanistically simulated using seasonally averaged porewater data along with microsite PDFs of soil C and soil moisture. Nitrous oxide is reduced to N₂ during classical denitrification following the Hole-in-the-Pipe conceptual model, including possible reduction of atmospheric N₂O that diffuses into the soil (Firestone & Davidson, 1989). See Supporting Information (Section S2) for DAMM-GHG model equations.

2.3 Microsite probability distribution functions of soil C and soil moisture

The microsite PDFs of soil C substrate and soil moisture were generated from a bivariate truncated log-normal distribution. The PDF for soil C was truncated at 0.001 and 0.15 (g/cm³), respectively. The soil C PDF was distributed with mean equaling the observed soil C value (see Figure S3 for more information on the PDF of soil C). Likewise, the soil moisture PDF was distributed with mean equaling the observed soil moisture value, truncations at 70% and 200% of the observed mean soil moisture values. The spatial heterogeneity of soil C and soil moisture were constrained by optimizing the parameters (standard deviation and/or coefficient of variation) that control the skewness of soil C and soil moisture PDFs by enveloping the bounds reported by Stoyan et al. (2000). If the upper truncation limit of the soil moisture PDF exceeded the soil pore volume, we reset it to 95% of the porosity value. We constructed the microsite PDF as the product of two log-normal distributions of soil C and soil moisture. The PDF was evaluated at 10 × 10 equally spaced quantiles for soil C and soil moisture, respectively.

Here, we focused on spatial heterogeneity across soil microsites at the mm and sub-mm scale. Within stand heterogeneity at the meter scale, such as variation in bulk density and porosity along topographical gradients, is not account for in this study.

2.4 Parameter optimization and uncertainty analysis

We optimized model parameters within a Bayesian Markov chain Monte Carlo (MCMC) framework (see Section S1 for more details on the optimization algorithm). We implemented the MCMC algorithm using mcmc and doParallel packages (Revolution Analytics & Weston, 2015) in the R (version 3.3.2) statistical programming language (R Core Team, 2018). We applied a posterior predictive procedure to estimate the uncertainty of the optimized parameters. We implemented the posterior predictive analyses using the R-INLA package (Lindgren & Rue, 2015; Rue, Martino, & Chopin, 2009). We divided daily-average soil GHG flux measurements into alternative synoptic-scale periods of 10 days, where we used one half of measured GHG fluxes for model calibration and another half for model validation.

2.5 Sensitivity analysis

We evaluated the sensitivity of model parameters using a global variance-based sensitivity analysis and a collinearity (or parameter identifiability) analysis. We implemented a variance-based sensitivity analysis using the R-multisensi package, where a generalized sensitivity index (ranging between 0 and 1, extracted from the first axis of principle component analysis) was used to determine the sensitivity of multiple GHG fluxes to each model parameter value (Bidot, Lamboni, & Monod, 2018). The global sensitivity analysis quantifies the proportion of variability accounted by each of the parameters on model outputs, where a high GSI value indicates that the simulation results are highly sensitive to that parameter (Lamboni, Makowski, Lehuger, Gabrielle, & Monod, 2009).

We implemented the collinearity analysis using the collin function of R-FME package, where the collinearity index (CI) was used to determine the linear dependence of model parameters to each other (Brun, Reichert, & Künsch, 2001; Soetaert, 2016; Soetaert & Petzoldt, 2010). In general, higher values of CI indicate increased equifinality (or decreased number of identifiable parameters) of model parameters. One can compensate % of the effect of a change in one parameter by modifying the values of other parameters. Hence, CI values can range between 1 (when all terms are orthogonal or all subsets of parameters are identifiable) and infinity (when all terms are linearly dependent, or no single subset of parameters is identifiable). The CI value of 15 is considered as a threshold above which approximate linear dependence of model parameters increases and poor identifiability can be expected (sensu Omlin, Brun, & Reichert, 2001).

3.1 Seasonality of soil greenhouse gas fluxes

Soil CO₂ fluxes followed the typical seasonal trend of soil temperature, where both seasonal average and peak CO₂ fluxes were comparable between 2015 (average [min–max]: 171 [73–297] mg CO₂-C m⁻² hr⁻¹) and 2016 (average [min–max] measurement period: 168 [52–281] mg CO₂-C m⁻² hr⁻¹; Figure 2). This is due, in part, to the comparable seasonal soil temperature ranges between 2015 (average: 14.4°C, ranged between 8.4 and 18.6°C) and 2016 (average: 13.3°C, ranged between 5.6 and 17.9°C) measurement periods (Figures 2 and 3). Soil CO₂ fluxes exponentially increased with soil temperature (R² = .75; Figure 3a). We also observed a typical bell-shaped relationship between CO₂ flux and soil moisture (R² = .16), with the optimum for CO₂ flux at intermediate water contents (Figure 3d). Although the effect of soil moisture on CO₂ fluxes was always secondary to soil temperature (Figure S5), the fit with soil moisture was better when it was more limiting in the dry summer of 2016 than the wet summer of 2015 (R² = .24 and .52 in 2015 and 2016, respectively; Figure S6d).

In contrast to CO₂ fluxes, soil CH₄ fluxes mimicked the seasonal trend of soil moisture for both years (Figure 2b,g). Soil moisture and net CH₄ fluxes were positively related (R² = .70), although the slope of the linear regression line was steeper during 2016 than during 2015 (Figure 3e; Figure S6e). We observed relatively smaller net CH₄ oxidation in the spring followed by higher net CH₄ oxidation in summer months, and again lower net CH₄ oxidation in the autumn. Although seasonal average values were generally similar, the range of CH₄ flux values in 2016 (average: −0.07 µg CH₄-C m⁻² hr⁻¹, min–max range: −0.13 to 0.004 µg CH₄-C m⁻² hr⁻¹) was wider than in 2015 (average: −0.05 µg CH₄-C m⁻² hr⁻¹, min–max range: −0.08 to −0.03 µg CH₄-C m⁻² hr⁻¹).

Unlike CO₂ and CH₄, the seasonal trend of soil N₂O fluxes contrasted between 2015 and 2016 (Figures 2c,h and 3f; Figure S6f). The 2015 growing season was significantly wetter than the 2016 growing season. The cumulative precipitation of the summer months (June 1 to September 30) of 2015 and 2016 was 439 mm and 279 mm, respectively (source: https://www.ncdc.noaa.gov/crn/). Consequently, the average soil moisture was generally higher (24.8 v/v, min–max range: 14.8–32.6 v/v) during the 2015 growing season (measured over June 11 to October 17) than the average soil moisture (19.7 v/v, min–max range: 8.9–27 v/v) during the 2016 growing season (measured over May 3 to November 6; Figures 2d,i and 3f; Figure S6f).

We observed low rates of net N₂O consumption during the spring of 2015 when soil moisture was highest, which was followed by mostly near zero net emissions and a few positive emissions throughout the 2015 summer (Figure 2c,d). In contrast, we observed net positive N₂O emission during the early spring of 2016 and net N₂O consumption throughout most of the 2016 growing season, when soil moisture was minimal (Figure 2h,i). The temperature effect was weak and inconsistent between 2 years (Figure 3c; Figure S6c). On average, we observed small net N₂O emission during the 2015 growing season (average: 0.06 µg N₂O-N m⁻² hr⁻¹, ranged between −0.77 and 1.73 µg N₂O-N m⁻² hr⁻¹) and net N₂O consumption during the 2016 growing season (average: −0.18 µg N₂O-N m⁻² hr⁻¹, ranged between −1.10 and 1.68 µg N₂O-N m⁻² hr⁻¹).

3.2 Performance of the DAMM-GHG model

Overall, the model reproduced the seasonal dynamics of soil greenhouse gas fluxes (Figures 2 and 4). The model explained 72% of the variation in soil CO₂ fluxes (Figure 4a). Likewise, the 1:1 relation between the observed and simulated soil CH₄ fluxes was remarkable (R² = .78; Figure 4b). The model marginally overestimated CO₂ fluxes and underestimated CH₄ fluxes during early spring of 2016 (Figure 2f,g).

Soil N₂O fluxes were relatively noisier as compared to CO₂ and CH₄ fluxes with a few outliers in both years (red triangles and squares in Figures 2 and 4, respectively). The model generally explained the dynamics of N₂O fluxes (R² = .36 and .52 with and without outliers, respectively; Figure 4c). The model did not capture the few very low net N₂O fluxes observed in the peak season of 2016. Most importantly, the model captured the instances when net atmospheric consumption of CH₄ (i.e., net CH₄ oxidation) co-occurred with net atmospheric consumption of N₂O (i.e., net N₂O reduction) within the same soil chamber and at two extremes of the measured soil moisture, during the early wet spring of 2015 and during the driest period of the 2016 growing season (see Figure 2c,h and red circles in Figure 3f).

In general, there was little bias in the relations between the observed and simulated GHG fluxes (slope ranged between 0.98 and 1.10; Figure 4). The 95% CI of the simulated GHGs for all, CO₂, CH₄, and N₂O, were narrow and the model parameter values were generally well constrained. The interquartile ranges in the posterior distributions of all parameters were less than half of their respective prior interquartile ranges (Table 1). The prior interquartile ranges represent the approximate upper and lower bounds of the measured values from the relevant literature.

Table 1. Parameters used for simultaneous simulation all three greenhouse gases in the DAMM-GHG model

Parameter	Description	Unit	Prior range	Posterior range	Source
For CO2 module
	Base rate for soil respiration	µmol CO2 L−1 hr−1	2 × 1010 (2 × 109–2 × 1011)	5 × 109 (2 × 109–9 × 109)	Abramoff et al. (2017), Davidson et al. (2018), Sihi et al. (2018)
	Temperature sensitivity for soil respiration	kJ/mol	72 (60–80)	66 (64–72)
	Half-saturation constant of C for soil respiration	µmol C/L	1 (0.1–100)	0.9 (0.7–1.6)
	Half-saturation constant of O2 for soil respiration	µmol O2/L	100 (3–300)	16 (4–38)
Depth	Effective depth	cm	15 (5–30)	7 (6–11)	This study
Total microsite	Total number of microsites	unitless	104 (103–105)	7 × 103 (2 × 103–9 × 103)	This study
Soil CSD	Coefficient that determines skewness of soil carbon PDF	unitless	0.5 (0.1–0.9)	0.5 (0.1–0.9)	Stoyan et al. (2000)
Soil moistureSD	Coefficient that determines skewness of soil moisture PDF	%	20 (5–30)	7 (6–12)	Stoyan et al. (2000)
For CH4 module
	Base rate for CH4 production	µmol CH4 L−1 hr−1	3 × 105 (3 × 104–3 × 106)	3 × 105 (2 × 105–6 × 105)	This study
	Temperature sensitivity for CH4 production	kJ/mol	100 (50–150)	74 (68–78)	Nedwell and Watson (1995), Westermann (1993)
	Half-saturation constant of C for CH4 production	µmol C/L	1 (0.1–100)	1.2 (0.9–2.1)	This study
	Inhibition coefficient of O2 for CH4 production	µmol O2/L	3 (0.3–4.3)	0.43 (0.4–0.9)	Arah and Stephen (1998)
	Base rate for CH4 oxidation	µmol CH4 L−1 hr−1	0.07 (0.007–7)	0.1 (0.08–2)	Davidson et al. (2001)
	Temperature sensitivity for CH4 oxidation	kJ/mol	30 (10–50)	34 (32–37)	Crill, Martikainen, Nykanen, and Silvola (1994)
	Half-saturation constant of CH4 for CH4 oxidation	µmol CH4/L	10–2 (10–3 to 10–1)	0.005 (0.002–0.006)	Davidson et al. (2001)
	Half-saturation constant of O2 for CH4 oxidation	µmol O2/L	43 (3–300)	24 (13–33)	Davidson et al. (2001)
For N2O module
	Base rate for nitrification	µmol N2O/L	102 (101–103)	282 (207–722)	This study
	Temperature sensitivity for N2O production during nitrification	kJ/mol	60 (45–75)	62 (57–67)	Stark (1996), Stark and Firestone (1996)
	Half-saturation constant of for N2O production	µmol /L	15 (8–22)	9 (8–10)	Stark and Firestone (1996), Zarnetske, Matsuo, Yamazaki, and Wada (1987)
	Half-saturation constant of O2 for N2O production	µmol O2/L	100 (9–163)	36 (15–44)	Bodelier, Libochant, Blom, and Laanbroek (1996), Verstraete and Focht (1977), Zarnetske et al. (1987)
	Base rate for N2O production	µmol N2O/L	102 (101–103)	520 (438–958)	This study
	Temperature sensitivity for N2O production during denitrification	kJ/mol	60 (45–75)	67 (65–71)	Canion et al. (2014), Holtan-Hartwig, Dörsch, and Bakken (2002), Vieten (2008)
	Half-saturation constant of for N2O production	µmol /L	26 (15–57)	17 (16–19)	Zarnetske et al. (1987)
	Inhibition coefficient of O2 for N2O production	µmol O2/L	14.3 (4.3–43)	41 (40–42)	Körner and Zumft (1989), Zarnetske et al. (1987)
	Half-saturation constant of C for N2O production and reduction during denitrification	µmol C/L	1 (0.1–100)	0.6 (0.3–1)	This study
	Maximum velocity for N2O reduction	µmol N2O/L	103 (102–104)	3,413 (1,585–9,583)	This study
	Temperature sensitivity for N2O reduction	kJ/mol	50 (45–75)	47 (46–47)	Canion et al. (2014), Holtan-Hartwig et al. (2002), Vieten (2008)
	Half-saturation constant of N2O for N2O reduction	µmol N2O/L	0.16 (0.05–0.27)	0.19 (0.13–0.26)	Holtan-Hartwig et al. (2002), Vieten (2008)
	Inhibition coefficient of O2 for N2O reduction	µmol O2/L	7.5 (4.3–20.1)	19.5 (18.6–19.7)	Körner and Zumft (1989), Vieten (2008)

Note

• Prior range represents initial (min–max) of prior interquartile range. Posterior range represents median (95% CI) of posterior interquartile range. Units for all base rates, that is, α(s), are in µmol concentration dissolved in water.
• Abbreviation: PDF, probability distribution function.

The effective depth in the DAMM-GHG model was optimized to a median value of 7 cm (min–max range: 6–11 cm; Table 1), indicating that most of the important processes affecting the net GHG fluxes that we measured with chambers were occurring within the topsoil horizons at this site. The number of microsites within the 0.07 m² chamber footprint was optimized to a median value of 7,000 (min–max range: 2,000–9,000; Table 1), indicating that the simulated microsites were about 3.6 mm in diameter, which could include macroaggregates and clusters of fine roots and pockets of organic debris.

3.3 Sensitivity analysis

Soil moisture primarily (and soil temperature secondarily) controlled the microsite PDFs of production, consumption, and diffusion processes of CO₂, CH₄, and N₂O. The net flux is the net effect of production, consumption, and diffusion of individual gases (Figure 5; Figure S10). Of all parameters, the most sensitive ones were those that control the V_max terms in the Arrhenius equation (E_a and α) for production and consumption of individual GHGs, followed by the half-saturation constants (kMs) and O₂ inhibition coefficients (kIs) for each process (Figure 6). The linear dependence of the DAMM-GHG model parameters was generally low and was usually below the threshold of 15 (with a few exceptions) identified for potential equifinality issues (Figure 7).

4.1 Microsite representation captures co-occurrence of methane oxidation and nitrous oxide reduction

Consumption of atmospheric N₂O via classical denitrification should occur only under reducing conditions. Yet, we have observed net uptake of atmospheric CH₄ (oxidation) and uptake of atmospheric N₂O (reduction) simultaneously in well-drained soils of Howland forest under both low and high soil moisture levels. With the advent of high frequency and high sensitivity flux measurement technology, we can be confident that these modest uptake rates of both CH₄ and N₂O are significantly different from zero and are not measurement errors or artifacts (Figures 2 and 4; also see Figure S4). These seemingly contradictory observations have been qualitatively explained by describing diffusional constraints of gas transport as follows: both CH₄ and N₂O can diffuse into well-drained soils; the CH₄ is oxidized at microsites where O₂ is abundant; while N₂O is reduced at other microsites where N₂O is present and Rh is sufficiently rapid to consume O₂. Here, we demonstrate that this qualitative explanation can be expressed in a mathematically consistent biophysical process model that is numerically consistent with simultaneously measured fluxes of these gases.

The area under a soil chamber was partitioned according to a bivariate log-normal PDF of soil C and moisture across a range of microsites, which leads to a PDF of CO₂ production and O₂ consumption among microsites. The resulting broad range of microsite O₂ concentrations determines the PDF of microsites that produce or consume CH₄ and N₂O according to Michaelis–Menten and Arrhenius functions for each process (Figure 1; Figure S1). Concentrations of below ambient N₂O (hot spots of N₂O reduction) occur in microsites with simulated high C and high moisture. Net consumption and production of CH₄ and N₂O are simulated within a chamber as the average of all soil microsite simulations. To demonstrate that it is numerically feasible for microsites of N₂O reduction and CH₄ oxidation to co-occur under a single chamber, we discuss three different scenarios where mean soil moisture levels cover the envelope of observed soil moisture in our study area (Figure 5).

Of the two growing seasons, we measured highest bulk soil moisture (32.6 v/v) on June 29 (DOY 180), 2015. Consequently, microsite PDFs of soil moisture ranged from 23.7 v/v to as high as 47.7 v/v (solid line in Figure 5b). Microsites with high soil moisture limited diffusion of gaseous O₂ through air-filled pore space. Simultaneously, soils had warmed up enough during the late spring of 2015 for soil respiration to exceed 100 mg CO₂-C m⁻² hr⁻¹, which created significant O₂ demand in microsites with high soil C. Relatively high soil respiration along with limited O₂ diffusion resulted in a large fraction of total soil microsites with low O₂ concentrations (solid line in Figure 5d). Production of N₂O was high in microsites with high soil C. However, reducing environments favored N₂O reduction more than N₂O production in microsites with low O₂ concentrations. Together, the PDFs of soil microsites resulted in a modest net negative mean flux of N₂O (solid line in Figure 5l). Classical theories of biological denitrification processes fit with our observations of net negative fluxes of N₂O under conditions of high soil moisture (and soil C) when enzymatic reduction of N₂O to N₂ under reducing environment outcompete N₂O production rates, especially in nitrogen-limited systems like our field site (Davidson, Keller, Erickson, Verchot, & Veldkamp, 2000; Davidson et al., 1993; Firestone & Davidson, 1989). Although N₂O reduction slightly exceeded N₂O production under these conditions, net uptake rates of atmospheric N₂O were low, because diffusion of atmospheric N₂O into the soil, while occurring and thus supporting some uptake, was also limited by high water-filled pore space.

The diffusion of atmospheric N₂O into the soil increased during the drier 2016 summer, thus enabling somewhat larger net uptake of atmospheric N₂O. The observed soil moisture content of 18.2 v/v on July 30 (DOY 212), 2016, approximately represents the first quantile of observed soil moisture across 2015 and 2016 growing seasons. Rates of N₂O production during nitrification and denitrification were low in a large majority of microsites due to the oxygen inhibition effects as well as the diffusional constraints of soil C, ammonium, and nitrate substrates at low soil moisture. However, greater diffusion of atmospheric N₂O to soil microsites also increased the microsite concentrations of N₂O (dotted line in Figure 5j), including a small fraction of microsites with sufficiently low O₂ concentrations to not fully inhibit N₂O reduction (i.e., the simulated O₂ was near the kI for N₂O reduction () in approximately 10% of total microsites, resulting in net negative flux of N₂O (dotted line in Figure 5d; Table 1). Our observation of a net soil sink of atmospheric N₂O during summer drying events match with reports from other natural and managed ecosystems, ranging from tropical to temperate climates, where net uptake of atmospheric N₂O has been measured when mean bulk soil moisture was drier than would normally be expected for N₂O reduction (Donoso, Santana, & Sanhueza, 1993; Flechard, Neftel, Jocher, Ammann, & Fuhrer, 2005; Goldberg & Gebauer, 2009; Verchot et al., 2000; Yamulki, Goulding, Webster, & Harrison, 1995). In this case, however, we can demonstrate quantitatively that reducing conditions in only about 10% of the microsites was sufficient to enable net uptake of atmospheric N₂O.

In contrast to the above two examples, intermediate soil moisture conditions are favorable for production to exceed consumption, resulting in modest net emissions of N₂O from soil to the atmosphere. Within this context, observed soil moisture content of 25.4 v/v on July 10, 2016 (DOY 192; dashed lines in Figure 5), approximately represents the third quantile of observed soil moisture across 2015 and 2016 growing seasons (Figure 5b). As expected, N₂O production was greatest at this intermediate soil moisture (Figure 5j), especially in microsites with high soil C. Because N₂O production was much higher than the very low N₂O reduction rates (Figure 5k) in a sufficiently large number of total soil microsites, a net positive mean N₂O flux resulted (Figure 5l).

Previous studies indicated that N₂O production during biological nitrification and denitrification often peak at 50%–80% of water-filled pore space (Davidson, 1991; Metivier, Pattey, & Grant, 2009), when soil moisture may be sufficiently high such that nitrate and nitrite are more available than O₂ as the alternate elector acceptor in many microsites, but the soil O₂ content is still high enough to mostly inhibit the reduction of N₂O to N₂. This is the basis of the soil moisture function of the conceptual hole-in-the-pipe model (Firestone & Davidson, 1989). This relationship between soil moisture and N₂O production is often represented in models as an empirical statistical algorithm, such as a polynomial or similar function (e.g., Del Grosso et al., 2001; Potter, Matson, Vitousek, & Davidson, 1996). In contrast, the responses to soil moisture in the DAMM-GHG model produce the response pattern across simulated microsites (Figure S7) predicted by the conceptual hole-in-the-pipe model as an emergent property of the role of O₂ as substrate (for nitrification, see Figure S1) or inhibitor, as represented by different kI values ( and ) used in DAMM mechanistic functions (Table 1).

For CH₄, the microsite PDFs of production, consumption, and net emission were relatively straightforward. Methane production was high in a relatively few numbers of microsites with high soil moisture (and soil C). In contrast, CH₄ oxidation was high in the majority of microsites, which had low soil moisture, as the diffusion of both substrates (O₂ and CH₄) increased with decreasing soil moisture. Note that CH₄ production was several orders of magnitude lower than CH₄ oxidation across the simulated range of microsite moisture contents in our study, mainly due to significant oxygen inhibition. Therefore, microsite PDFs of CH₄ oxidation primarily dominated the net CH₄ emission, where net CH₄ emission linearly decreased with decreasing soil moisture (Figure 5e–h).

Taken together, these results demonstrate that representing production, consumption, and diffusion processes as a function of soil microsite PDFs can neatly encapsulate the factors affecting emissions of CO₂, CH₄, and N₂O in field studies and their underlying mechanisms. We also did a comparison of the model performance to a conventional framework, where we kept identical soil C and soil moisture values, set to the average values observed for the bulk soil, across all microsites. Although the performance of the model without microsite PDF was comparable to the one with microsite PDF for CO₂, model performance for CH₄ and N₂O was strongly affected by microsite variability (Figures S8 and S9). The model without microsite PDF simulated less uptake of CH₄ overall, had larger than observed peaks and valleys during in wet-up and dry-down events, and had more biased residuals (Figures S8b,g and S9b,e). For N₂O, the model with PDF representation of microsite variation also had an overall better fit to the observations and less biased residuals (Figures S8c,h and S9c,f).

The model–data fusion algorithm we used tends to reduce the overall model–data mismatch for the entire measurement window, and so it is not surprising that there are periods within that window where simulations do not match observations as well, such as the CO₂ and CH₄ fluxes of spring 2016 (Figure 2). This may also be due to other factors not included in the DAMM-GHG model, such as impacts of spring freeze–thaw cycles on C availability or phenology of root exudates. Additionally, the lower fraction of observed variability of N₂O fluxes accounted for by the model, compared to CO₂ and CH₄ fluxes, can be attributed to: (a) the low signal-to-noise ratio inherent to very low N₂O fluxes; and (b) the increased number and complexity of interactions of substrates (and inhibitors) for production (e.g., C, O₂, , and ) and consumption (e.g., C, O₂, and N₂O) of N₂O during nitrification and denitrification that are represented numerically in the model with additional parameters (Figures 2 and 4). We further discussed how these potential interacting processes might have increased the covariation of parameters related to N₂O dynamics than those related to CO₂ and CH₄ dynamics (see Section 4.3).

4.2 Sensitivity analysis

4.3 Collinearity analysis

As expected, the CI increases as the number of variable parameters increases (Figure 7). Given the parsimonious model structure, we had few problems of identifiability of the processes for CO₂ and CH₄ module, and the most parameter combinations remain below the threshold value of 15 (Brun, Kühni, Siegrist, Gujer, & Reichert, 2002; Omlin et al., 2001). However, we do have some parameter combinations with CI > 15 for the N₂O module, due to probable ambiguity of whether N₂O fluxes are affected more by production via nitrification (affected by substrate), production via denitrification (affected by substrates), or consumption of N₂O (affected by N₂O diffusion). Trade-offs of processes within models can account for inflation of CI values (Keenan, Carbone, Reichstein, & Richardson, 2011; Richardson et al., 2010), which is likely the case for the N₂O module relative to the CO₂ and CH₄ modules. Overall, however, collinearity analysis indicates that the chances of having equifinality issues characterized by biologically improbable process representations were generally low in the DAMM-GHG model.

We believe that this success is largely due to the multiple constraints imposed by simultaneously modeling data streams of three different gases, which enabled us to capture the influence of temporally and spatially varying drivers on GHG fluxes (Myrgiotis, Williams, Topp, & Rees, 2018). For example, when simulating only CH₄ flux, a better fit of the model to the data might be achieved by adjusting either the Michaelis–Menten parameters or the diffusion parameters, but it may be impossible to know which the “correct” adjustment is. However, if N₂O and CO₂ are also being simultaneously simulated, then adjusting the diffusion parameters will affect simulations of all three gases, whereas adjusting the Michaelis–Menten parameters for CH₄ oxidation should have little effect on the other two gases. Hence, the additional constraints help identify which model parameterizations are consistent with all three data streams of flux measurements.

4.4 Microsite probability distribution functions permit model parsimony

Simulating a 3-D array of soil aggregates or pore networks (Arah & Vinten, 1995; Ebrahimi & Or, 2014; Yan et al., 2016) is another approach for representing spatial heterogeneity of soil matrix in biogeochemical models. However, explicit representation of soil spatial variability requires detailed information on soil structure, involving high-throughput instrumentations such as X-ray CT scan (see Carducci, Zinn, Rossoni, Heck, & Oliveira, 2017) and may require greater computational power than our PDF approach. Because of the limited measurements on the spatial variability of soil aggregates (or pores) at a plot scale, let alone at larger scales, and because of the increased computational complexity in spatially explicit model structures, scaling up of 3-D soil aggregate (or pore network) models to the ecosystem models and ESMs is still challenging.

Statistically representing microsite variation as PDFs in the DAMM-GHG model offers a relatively computationally efficient, yet mechanistically consistent, alternative way of simulating soil heterogeneity and maintaining model parsimony, as in the original DAMM model (Davidson et al., 2012, 2014; Sihi et al., 2018). We believe that our framework could be used to simulate fluxes of GHGs from other natural and managed systems as well as be scaled up to ecosystem models and ESMs.

4.5 Opportunities for future improvement of the DAMM-GHG model

We represented all microsite-scale processes by optimizing an equivalent depth for Rh, where most of the biological reactions appear to happen in the soil of this study site (posterior range: 6–11 cm) and fixed that depth for simulating processes related to production and consumption of CH₄ and N₂O. This simplification allowed us to use unitless diffusion constants for gaseous and dissolved substrates and to avoid needing to know exact diffusion path lengths (see Davidson et al., 2012 for details). Exploring heterogeneity of diffusivity within and among soil horizons could be an appropriate next step.

Including more complex models of gas diffusion that includes variable diffusivity between intra-aggregate and inter-aggregate pore spaces may be more appropriate for aggregated media like soil (Millington & Shearer, 1971; Resurreccion et al., 2010). Representing soil microsite PDFs in more than one vertically stratified soil horizon by differentiating between the organic and various mineral horizons may be needed for application to other sites (e.g., wetland with a seasonally variable depth to the water table), such that transport of gases between soil horizons and across soil–air boundary can be estimated using Fick's law. Measured vertical concentration profiles of soil gases could serve as additional data constraints for soil gas concentration profiles that become emergent simulated properties of this modeling approach. It would also be useful to have a data stream of heterogeneity of O₂ or redox potentials across microsites, but that would require new generations of microprobes.

Additionally, techniques that disentangle gross production and gross consumption rates of CH₄ and N₂O under field conditions could increase the predictive power of the dynamics of soil GHGs fluxes. For example, Chanton, Powelson, Abichou, and Hater (2008) reported that measuring stable carbon isotope of emitted CH₄ (¹³CH₄) is a feasible way to quantify gross CH₄ oxidation in situ. Likewise, Wen et al. (2016) demonstrated that ¹⁵N₂O pool dilution method can be effective to measure atmospheric N₂O uptake in soil under field conditions. In situ quantification of microbial activities pertaining to gross production and consumption of CH₄ and N₂O can also be pursued following the gas push-pull method (Urmann, Gonzalez-Gil, Schroth, Hofer, & Zeyer, 2005). Quantifying gross nitrogen transformations using stable isotope tracing could help constrain the sources of N₂O emissions (Morse & Bernhardt, 2013; Müller, Rütting, Kattge, Laughlin, & Stevens, 2007; Myrold & Tiedje, 1986).

In addition to nitrification and biological denitrification, other bacterial (Jensen & Burris, 1986; Yamazaki, Yoshida, Wada, & Matsuo, 1987) and fungal (Hayatsu, Tago, & Saito, 2008; Shoun, Kim, Uchiyama, & Sugiyama, 1992) contributions, as well as abiotic (Davidson, Chorover, & Dail, 2003; Vieten, 2008) sinks of N₂O in soil could be explored if warranted. Adding other controlling factors, such as pH effects, temporal dynamics of enzyme synthesis, and root exudation, could improve model performance for some sites (Butterbach-Bahl et al., 2013; Zheng & Doskey, 2015). In all of these cases, however, the potential additional explanatory value of more parameters and model complexity must be balanced with the availability of data to constrain them and with the advantages of model structure parsimony.