We employ a simple ODE model to describe CH4 emission from the soil column, accounting for the competing influence of methanogenesis and methanotrophy. We denote by Ms (mgCH4cm-2) the total mass of CH4 in the soil column per unit ground area. Methane is produced in the soil column at rate fp (mgCH4cm-2d-1), oxidised at rate fo (mgCH4cm-2d-1), and emitted via plant transport, ebullition, and diffusion at rate fe (mgCH4cm-2d-1). Hence, the CH4 dynamics in soil can be summarised as 1dMsdt=fp-fo-fe, where the reaction and transport terms are parameterised by the water table depth, soil temperature and availability of carbon substrate due to plant biomass in a similar manner to. Figure 1 illustrates how CH4 production, oxidation, and emission respond to these key environmental drivers.

Wetland methanogenesis is governed by three primary environmental drivers: water table depth, soil temperature, and the availability of carbon substrate . Hence, we assume that the rate of CH4 production in the soil column is given by 2fp=kpfT(p)fb(p)fw(p), where kp (mgCH4cm-2d-1) is a calibrated constant for each wetland, and the terms fT(p), fb(p), and fw(p) are dimensionless scaling functions representing, respectively, the temperature dependence of methanogenesis (see Fig. 1b), the availability of carbon substrate due to plant biomass decay (see Fig. 1c), and the effect of water table depth on methanogenesis (see Fig. 1d).

Methanogenesis occurs in the anoxic layers of the soil and is therefore strongly dependent on water table depth. When the water table lies below the soil surface, higher water table depths increase the anoxic zone, promoting CH4 production. However, waterlogged conditions can reduce CH4 production if the available carbon substrate becomes distributed over an increasingly large water column above the soil surface, effectively reducing substrate availability for methanogenesis depending on the degree of mixing. As illustrated in Fig. 1d, we use the same piecewise water table dependence as . In particular, we have 3fw(p)=zw-zbzs-zb,zw<zszw-zbzs-zb-p1,zw≥zs, where 0≤fw(p)≤1 (see Fig. 1d). The exponent p1≥0 characterises CH4 production in the region below the water table but above the soil column; p1=0 corresponds to no mixing of carbon substrate, whereas p1≫0 corresponds to well-mixed conditions. In other words, CH4 production increases linearly with water table depth as the anoxic zone expands, such that fw(p)=0 when the water table lies at the base of the soil column and fw(p)=1 at the soil surface. Beyond the soil surface, further inundation reduces production, following a negative power-law relationship with fw(p)→0 as zw→∞. Note that, whilst methanogenesis predominantly occurs in anoxic soil layers, small amounts of CH4 may also be produced in oxic layers via micro-anoxic microsites within the soil column . These contributions are not explicitly resolved in our model, which focuses on bulk soil CH4 production as the dominant source.

Three methanogenic pathways contribute to CH4 production in wetlands: hydrogenotrophic, whereby bacteria reduce CO2 with hydrogen to form CH4; acetoclastic, in which bacteria convert acetic acid from decomposing organic matter into both CH4 and CO2; and methylotrophic methanogenesis, in which specialised methanogens convert methylated compounds (e.g. methanol, methylamines, or methyl sulphides) into CH4, and may be particularly important in PPR wetlands . Biomass availability thus plays a crucial role in all three pathways: directly through litter input for acetoclastic methanogenesis, indirectly via CO2 production for hydrogenotrophic methanogenesis, and by supplying methylated compounds for methylotrophic methanogenesis. Despite differences in substrates and microbial mechanisms, we combine these pathways into a single effective methanogenesis term for model simplicity. To represent this dependence, we use the Normalised Difference Vegetation Index (NDVI) as a proxy for available plant biomass. NDVI, which ranges from -1 to 1, quantifies vegetation density and health based on the differential reflectance between near-infrared and red light, typically derived from satellite data. To account for the lag between vegetation growth and availability of decomposable organic matter, we introduce a delay τ days between NDVI and biomass availability see, such that 4fb(p)=1+NDVI(t-τ)2p3, where p3≥0 (see Fig. 1c). This functional form ensures that 0≤fb(p)≤1.

Soil temperature strongly regulates both major CH4 production pathways through temperature-dependent enzyme kinetics . This influence is two-fold: higher temperatures enhance fermentation processes such as increased root exudation, and directly stimulate methanogen activity. The temperature dependence of methanogenesis typically follows an Arrhenius-type (i.e. exponential) relationship . Following previous models e.g., we represent this effect using a Q10 formulation: 5fT(p)=0,T≤0°CQp110(T-30),0°C<T<30°C1,T≥30°C, where T (°C) is the soil temperature, Qp≥1 is a fitted parameter, and 0≤fT(p)≤1 (see Fig. 1b). This form reflects enzymatic saturation above an optimal temperature, which we set at 30 °C due to the mid-latitude wetland focus, where temperatures rarely exceed this threshold. Under future climate scenarios with more frequent extreme heat, CH4 production may continue to increase above 30 °C, in which case our current formulation would be conservative. Alternatively, enzymatic and microbial activity may decline above their thermal optimum due to heat stress, potentially suppressing production. The net effect will likely vary among wetlands with different thermal regimes and microbial communities, and warrants further modelling investigation.

Methane produced in the soil column may be oxidised before reaching the surface, thereby preventing its emission into the atmosphere. This oxidation occurs predominantly under aerobic conditions in the unsaturated soil zone and is therefore strongly modulated by the position of the water table. Additionally, soil temperature regulates the activity of methanotrophic bacteria .

We refine the model of by assuming that CH4 oxidation follows first-order reaction kinetics that depend on water-table depth (cf. CH4 production, which is treated as a zero-order process, reflecting that methanogenesis is primarily controlled by substrate availability rather than the instantaneous CH4 concentration within the soil column). In our formulation, oxidation is further modified by temperature through a multiplicative factor, such that 6fo=kofT(o)fw(o)Ms, where ko (d-1) is the maximum oxidation rate under atmospheric oxygen availability throughout the soil column; fT(o) and fw(o) are dimensionless functions describing the dependence of methanotrophy on soil temperature and water table depth, respectively. For simplicity, oxygen transport via plant roots, which could enhance methanotrophy, is neglected here. Moreover, our model assumes that CH4 oxidation occurs predominantly under aerobic conditions in the unsaturated soil zone. Anaerobic oxidation of CH4 in saturated layers, which requires alternative electron acceptors e.g. sulphate or nitrate, see, is not explicitly resolved and may slightly reduce net CH4 flux in some wetlands.

In the current model, we only resolve aerobic oxidation and neglect these secondary anaerobic pathways for simplicity. Following , we model the water table dependence of oxidation as an exponential decay with increasing water table depth: 7fw(o)=e-p2(zw-zb), where p2>0 (cm-1) is a control parameter (see Fig. 1e). Thus, oxidation is maximal when the water table is at the base of the soil column (zw=zb) and decreases sharply as the water table rises. To capture temperature dependence, we employ a Q10 relationship analogous to that used for CH4 production e.g.: 8fT(o)=0,T≤0°CQo110(T-30),0°C<T<30°C1,T≥30°C, where Qo≥1 is a constant.

Methane produced in the soil column is emitted into the atmosphere by three distinct mechanisms: diffusion, plant transport, and ebullition. We assume emission by diffusion occurs at rate kD (d-1), emission due to plant transport at rate kPT (d-1), and emission due to ebullition at rate kEB (d-1). We employ a simple functional form (see Fig. 1f) for CH4 emission: 9fe=keMs,ke=kD+kEB+kPT. Emission is treated as first-order, dependent on the total soil CH4 pool. Such a simple form neglects the vertical motion of CH4 e.g. detailed in models by but has been demonstrated to successfully describe key aspects of CH4 emission .

Diffusive release involves CH4 movement down gradients from regions of high CH4 concentration in the anaerobic soil to the atmosphere. This transport is slow and subject to methanotrophic oxidation, but typically dominates when soils are moist or water columns are deep and oxygenated. The effectiveness of diffusion decreases with water table depth, since the diffusive path lengthens and so the rate of diffusive release scales inversely. Hence, the rate of diffusive transport of CH4 is given by 10kD=Dzw-zbp4, where D is the diffusion coefficient of CH4 and p4=2 for Fickian diffusion. Soil temperature can provide an additional layer of complexity, by affecting water viscosity and gas diffusion, though we neglect this.

Meanwhile, plant transport and ebullition provide more efficient emission routes. In the case of plant transport, CH4 travels along gas conduits (e.g. roots, stems) in aerenchymous plants or tree stems by means of convective flow, molecular diffusion, or pressure-driven transport . This allows CH4 to bypass much of the (aerobic) oxidation zone, preventing methanotrophy . The effectiveness of this bypass increases with higher water tables, particularly when zw>zs, as plant-mediated transport can transport CH4 directly from saturated soil to the atmosphere, thereby avoiding oxidation in near-surface oxic zones . On the other hand, ebullition provides a sporadic emission mechanism due to bubbling of supersaturated CH4. These emission pulses bypass methanotrophy almost entirely and are triggered when gas pressure overcomes surface tension, typically in response to fluctuations in hydrostatic or atmospheric pressure, or changes in temperature . We assume, in an identical manner to , that plant transport and ebullition occur at a constant rate, which for the remainder of this paper we couple into one term, kEP=kEB+kPT. However, future model refinements could link plant transport to NDVI and water table depth, and ebullition to changes in weather conditions and water table depth.

We solve the single ODE (Eq. 1) in Matlab using the inbuilt stiff solver ode15s and employ default error tolerances. The results are well converged, even for long simulation times. Model parameters are calibrated to individual wetlands by minimising the normalised root mean square error (nRMSE) between observed (oFi) and predicted (pFi) CH4 flux at each data point i=1,2,…,N, defined as 11nRMSE=RMSE/σ,RMSE=∑i=1NpFi-oFi2N, where σ is the standard deviation of observed fluxes. This normalisation facilitates comparison of model performance across wetlands with differing flux magnitudes .

To identify the set of optimised parameters for a particular wetland, we use the inbuilt Matlab genetic algorithm (ga), a population-based, stochastic global optimisation method. This algorithm is particularly suited for solving non-linear and discontinuous optimisation problems where gradient-based methods struggle to converge or become trapped in local minima. In particular, we minimise the nRMSE between model output and observations by calling ga, which searches the global parameter space until certain convergence criteria are met. Following , the effective diffusivity of CH4 in the soil is fixed at D=1.3cm2d-1 in all simulations. All other parameters are adjusted during calibration and fitted individually for each wetland to capture site-specific environmental conditions. This approach allows the model to account for variation in soil properties, hydrology, and vegetation among wetlands, thereby reflecting wetland-specific biogeochemical and hydrological dynamics. The ranges of possible parameter values are detailed in Table A2.

We fit model parameters for each wetland using the dataset provided by , which includes data at approximately fortnightly (occasionally coarser) intervals for NDVI, soil temperature, water depth, water-filled pore space (WFPS) at the soil surface and CH4 flux throughout the growing season (April–November). To reduce bias from extreme outliers in CH4 flux measurements, likely caused by ebullition or anthropogenic disturbance, we exclude data points where CH4 fluxes exceed 20 times the interquartile range for each wetland. The possible parameter ranges for each wetland are given in Table A2 in Appendix A, based on an extensive review of the literature. Given our focus on the wetland centre, we use flux measurements from a single chamber located near the centre of each wetland (Chamber 1 in the dataset). Measurements from other wetland or upland chambers are not included in the analysis.

When the water table lies above the soil surface, direct measurements of water depth are available. For periods when the water table falls below the surface, observed water depths are truncated at 0 cm, so subsurface positions must be estimated. For simplicity, to estimate zw when the water table is below the surface, we assume a linear relationship with WFPS. Zero water depth measurements are replaced with these estimates and capped at zw=-49.9cm to remain within the model domain. This approach neglects lateral flow and potential time lags between water table changes and CH4 emissions, but provides a first-order seasonal estimate adequate for budget calculations, which are usually dominated by periods when the water table is at or above the soil surface.

In order to force our model at the daily scale, we linearly interpolate between observations for all model drivers, i.e. NDVI, soil temperature, water depth and WFPS a similar approach is employed by. We also introduce “ghost” data points two weeks before and after the observed growing season for each year, assuming T=0°C and F=0 mgCH4cm-2d-1 (i.e. no CH4 flux) to reflect dormancy in the winter, whilst holding NDVI and water table depth constant in this two week period. As discussed in Sect. 2, NDVI is introduced as a calibrated time-lagged forcing term, using values from τ days prior to account for the delay between photosynthetic activity and substrate availability due to plant growth and litter decomposition. Note that observed total CH4 emissions are estimated by linearly interpolating daily flux measurements and numerically integrating them over each seasonal period.

Full details on the measurements used to force the model are provided by , but we summarise briefly here. Methane fluxes were measured in the wetland zone of undrained PPR wetlands using static chambers deployed on sediment or floats along a single transect per wetland, with five wetland-zone sampling locations spaced from the edge to the centre. Measurements were collected approximately every two weeks during the growing season (May–November), between 10:00 and 14:00 LT, and included water depth, soil temperature (or water temperature when the water table exceeded 10 cm), air temperature, and soil volumetric water content (with WFPS calculated by dividing by soil porosity). NDVI corresponding to each sampling event was obtained from Landsat imagery to represent vegetation availability. These variables provide the spatially and temporally resolved forcing used in the chamber-scale model. As measurements are limited to the top 5–10 cm of soil (or water surface), daytime sampling, and relatively coarse NDVI (30 m), short-term heterogeneity may be smoothed or under-represented.

For clarity, we briefly reiterate the key structural assumptions implicit in our model formulation. First, subsurface hydrology is represented through a simplified relationship between water-table depth and WFPS, implicitly assuming monotonic pore saturation with rising water tables and neglecting hysteresis, preferential flow paths, and fine-scale soil heterogeneity. Second, CH4 production combines hydrogenotrophic, acetoclastic, and methylotrophic pathways into a single effective term rather than treating each microbial mechanism explicitly; small contributions from CH4 production in oxic microsites are neglected. Third, CH4 emission is treated as a first-order process, with diffusion represented explicitly and plant-mediated transport and ebullition combined into a single term, kEP. In reality, ebullition is episodic and sensitive to temperature and pressure fluctuations, whilst plant-mediated transport depends strongly on vegetation phenology and water-table depth. This simplification neglects detailed vertical transport dynamics and the pulsed nature of ebullition, but is expected to primarily influence short-term emission variability rather than seasonal totals. Fourth, CH4 oxidation is assumed to occur predominantly under aerobic conditions in the unsaturated soil zone, whilst anaerobic oxidation in saturated layers, which requires alternative electron acceptors (e.g. nitrate or sulphate), is not explicitly resolved. Fifth, the model resolves dynamics within a vertically homogeneous soil column at the wetland centre, neglecting lateral transport and edge effects; this assumption is likely less appropriate near the wetland boundary. Finally, to reduce bias from extreme CH4 flux outliers (e.g. large ebullition events or disturbance), we exclude observations exceeding 20 times the interquartile range for each wetland, whilst retaining background ebullitive fluxes.

Together, these assumptions favour interpretability and parameter identifiability, capturing dominant seasonal controls on CH4 flux whilst smoothing short-term variability and episodic pulses. They may, however, bias short-term fluxes under extreme hydrological conditions; for example, linear WFPS assumptions may overestimate production in partially saturated soils, pathway lumping smooths short-term variability, and neglecting oxygen transport may slightly overestimate net emissions. Near wetland edges, lateral flows, variable soils, and heterogeneous vegetation may further alter methane dynamics, meaning central-column assumptions likely break down within a few meters of the boundary.

To assess how well the model captures observed emission patterns, Figs. 3 and 4 show baseline simulations using optimised parameters derived using the method described in Sect. 2.5 (see Table A1 for details). In particular, Fig. 3 compares modelled CH4 fluxes (blue line) with observations (red crosses) for the 2009–2016 growing seasons note thatprovide data for wetland T5 only through 2015. The corresponding cumulative emissions are presented in Fig. 4, where predicted (blue) and observed (red) totals show close agreement. Panels (a)–(f) correspond to wetlands P1, P3, P8, T5, T6, and T9, respectively, spanning permanent and temporary types and representing a range of hydrological settings : three recharge wetlands (P3, T5, T6), one discharge (P1), one flow-through (P8), and one apparently hydrologically isolated site (T9). The full response to modelled environmental drivers are detailed in Sect. S1 in the Supplement.

The data reveal a clear positive correlation between CH4 fluxes and both soil temperature and NDVI, which overall the model reproduces well. Emissions typically peak in mid- to late-growing season (July–September), when temperatures are highest and plant deterioration begins, demonstrating the importance of including these environmental drivers. The model captures the major flux dynamics, responding realistically to both soil temperature and vegetation activity, and also reproduces the near-zero CH4 emissions observed in early spring and late autumn, when low temperature and limited substrate constrain methanogenesis. Meanwhile, the relationship between emissions and water-table depth is more complex (see Sect. 5.3 below). Variations in water level influence CH4 production, oxidation, and transport not only directly, but also indirectly by changing factors such as wetland salinity or the concentration of alternative electron acceptors. These effects contribute to the significant heterogeneity in peak fluxes between wetlands. In particular, field observations indicate temporary wetlands generally exhibit higher emissions .

Focusing first on the permanent wetlands, model performance is strong for wetland P1 (Fig. 3a), particularly during the 2009, 2010, 2015, and 2016 growing seasons. In 2015 the model captures two distinct seasonal emission peaks that align with observed soil-temperature maxima, reflecting its sensitivity to temperature forcing. Wetland P3 likewise demonstrates the need for the additional environmental drivers: during the 2010, 2011, and 2015 seasons, water depth remains nearly constant early in the year whilst emissions increase in tandem with lagged NDVI and temperature. Moreover, rare anomalously high CH4 fluxes outside this window often coincide with elevated lagged NDVI or temperature, such as the early 2012 peak (Fig. 3b), driven by a large carbon substrate pool (see Fig. S1f in the Supplement). Performance for wetland P8 is more mixed. The model generally reproduces the overall pattern of low spring and autumn fluxes and higher summer emissions, together with the early 2012 peak, but performs poorly in 2009 and 2011. During both years emissions remain unexpectedly low despite favourable temperatures, water depths, and typical vegetation growth, leading the model to overestimate total seasonal fluxes (Fig. 4c). This discrepancy may reflect additional hydrological complexities: as a flow-through wetland, P8 may experience transient salinity or redox dynamics not represented in the model.

Overall, the model also performs well for the temporary wetlands, particularly for wetland T5 (Fig. 3d), where it correctly captures the trend in each season and reproduces the pronounced peak in CH4 emissions during the 2011 growing season (in response to elevated soil temperatures, see Fig. S4e in the Supplement). Moreover, the model replicates muted emissions in 2012, when the water table remained below the soil surface (Fig. S4d). This clearly demonstrates the modulating influence of water table depth; for example, unlike in wetland P3, emissions in T5 do not appreciably rise early in 2012 despite anomalously early plant growth (Fig. S4f). Results are more mixed for wetland T6 (Fig. 3e) but relatively strong for wetland T9 (Fig. 3f). In 2012 both wetlands also experienced unusually early plant growth; in T6, the model correctly captures the peak emissions in April–May, followed by a decline. In both T6 and T9, the model captures the high summer 2011 fluxes, driven by elevated soil temperatures (Figs. S5e and S6e in the Supplement). Meanwhile, for T9 the model reproduces negligible 2016 emissions, attributable to low water table, cooler soil, and limited substrate, but it misses a sharp late-season spike in 2015. Excluding this outlier substantially reduces the nRMSE, making T9 one of the best-described sites overall. The 2015 spike coincides with a rapid water-table drop and is likely due to ebullition or enhanced diffusion, suggesting the need for more detailed representation of these processes in future model iterations.

In Fig. 3d we also compare our full model with the reduced formulation, demonstrating the benefit of including vegetation and temperature drivers. Focusing on wetland T5, the framework of capably describes steady-state behaviour, but fails to reproduce the strong seasonal variability in CH4 fluxes. Incorporating a Q10 temperature dependence for methanogenesis and oxidation substantially improves the timing and magnitude of predicted emissions, highlighting the dominant role of temperature under inundated conditions. Yet temperature alone cannot explain all observed fluctuations. Including NDVI as an independent proxy for substrate availability is crucial, particularly in reproducing anomalous early-season peaks such as those in various wetlands in 2012. Given the central roles of temperature and substrate availability, we also tested whether including water-table depth improves model performance (results not shown). For permanent wetlands, which typically remained inundated, the effect is minor, consistent with the optimised parameters in Table A1 that indicate a weak exponential decay of oxidation and a near-zero depth dependence of methanogenesis. In contrast, the influence of water-table depth is much stronger in temporary wetlands, where CH4 emissions drop sharply once the water table falls below the soil surface.

We further assess model performance in Fig. 5, which also compares predicted and observed CH4 emissions for each wetland during the 2009–2016 growing seasons (using the same optimised parameters detailed in Table A1). In particular, we present a scatter plot comparing total seasonal emissions in each wetland, perfect model agreement would place data points on the 1:1 grey reference line (i.e. the predicted and observed emissions are equal). At the daily scale, model performance is generally reasonable but shows a consistent bias toward overestimating low emissions, a pattern most evident in temporary wetlands where the water table frequently drops below the surface. However, because these low-flux periods contribute little to the seasonal CH4 budget (except for brief spikes as the water table falls), integrated emissions remain generally well captured. Indeed, seasonal totals are reproduced reasonably for both permanent and temporary wetlands, including high-emission years such as T9 in 2011, though the model occasionally underestimates emissions when large, infrequent flux events occur and overestimates in years with persistently low observed fluxes (Fig. 5). Performance is notably stronger in permanent wetlands, where water levels typically remain above the surface and fluxes are more stable.

Overall, it is clear from Figs. 3–5 that the model captures the dominant seasonal and inter-annual trends in CH4 fluxes driven by vegetation, hydrology, and temperature. As expected, a model of this simplicity cannot reproduce every short-term fluctuation. Temperature and hydrology alone explain only about 75 %–90 % of the observed variability in emission dynamics , and additional processes (e.g. rainfall-driven dilution of electron acceptors) may further modulate methanogenesis (represented here through the parameter kp). However, because these higher-frequency fluctuations tend to average out, the model still predicts total seasonal emissions well (Figs. 4 and 5), even in years when individual daily fluxes are less accurately captured (e.g. 2013–2014 for wetland P3). This reflects a broader modelling trade-off: whilst low-dimensional ODE models are tractable, interpretable, and suitable for large-scale projections, they are inherently limited in representing abrupt or non-linear dynamics that govern short-term CH4 variability.

We note that no simple monotonic relationship is evident between fitted parameter values (Table A1) and wetland area or classification (e.g. permanent versus temporary). This is not unexpected, as the calibrated parameters represent effective process rates that integrate multiple interacting controls, including vegetation structure, sediment organic matter quality, hydrological regime, and redox conditions. Consequently, parameter variability primarily reflects site-specific biogeochemical and hydrological heterogeneity.

Our model also highlights how the role of oxidation differs between temporary and permanent wetlands. Figure 6 shows, for each wetland – (a) P1, (b) P3, (c) P8, (d) T5, (e) T6, and (f) T9 – the fraction of CH4 produced over a season that is emitted, oxidised, or retained in the soil. As expected, oxidation is more significant in temporary wetlands, where water tables frequently fall below the soil surface, allowing CH4 to be rapidly oxidised before it escapes. In contrast, in permanent wetlands, oxidation is weaker, and in some years, a large fraction of CH4 produced is emitted rather than consumed. This difference reflects how hydrology controls oxygen availability in the soil: lower water tables increase aeration and microbial access, enhancing oxidation, whereas consistently inundated soils remain largely anoxic. These results emphasise that the balance between production and oxidation, which determines net emissions, is highly sensitive to seasonal water-table dynamics, with implications for wetland management and greenhouse-gas mitigation, including the potential role of repeated drainage in oxidising CH4 trapped in the soil column. Moreover, the fraction of CH4 oxidised in the soil column relative to that emitted generally decreases with increasing soil temperature. In most wetlands considered, soil water temperatures peaked in 2011, coinciding with the largest fraction of CH4 emitted. This pattern arises from the weaker Q10-dependence of oxidation compared with production: as temperatures rise, CH4 production increases not only in absolute terms but also relative to oxidation, thereby enhancing net emissions.

From Eq. (12), fe* clearly depends on the water table depth, soil temperature, and on the wetland-specific parameters. To identify the water depth zp that maximises emissions (for a given set of parameters, temperature, and substrate availability), we solve for the critical point where dfe*/dzw=0. In particular, we have 14kpfb(p)fT(p)kofT(o)fw(o)+ke2{fw(p)dkedzwkofT(o)fw(o)+ke+kedfw(p)dzwkofT(o)fw(o)+ke-fw(p)kofT(o)dfw(o)dzw+dkedzw}=0, which simplifies to 15fw(p)dkedzwkofT(o)fw(o)+ke+kedfw(p)dzwkofT(o)fw(o)+ke-fw(p)kofT(o)dfw(o)dzw+dkedzw=0. Hence, whilst the dependence of methanogenesis on temperature and substrate govern the overall emission magnitude (Eq. 12), they do not determine the peak location (though this would change if vertical temperature gradients were incorporated).

Given the functional forms of fw(p), fw(o), and ke (see Sect. 2), Eq. (15) involves a mixture of exponential and polynomial terms and must be solved numerically for zw=zp. For zw<zs, we have dfe*/dzw>0 for typical parameter ranges and Eq. (15) cannot be satisfied, so peak emissions must always occur at or above the soil surface. For zw>zs Eq. (15) yields 16-2D(zw-zb)3(kofT(o)e-p2(zw-zb)+kEP+D(zw-zb)2)+kep2kofT(o)e-p2(zw-zb)-kep1zw-zb(kofT(o)e-p2(zw-zb)+kEP+D(zw-zb)2)+2Dke(zw-zb)3=0, or, alternatively: 17kofT(o)[kep2-p1zw-zb-2D(zw-zb)3]e-p2(zw-zb)-kEPzw-zb[p1ke+2D(zw-zb)2]+D(zw-zb)3[(2-p1)ke-2D(zw-zb)2]=0.

Whilst one could in principle solve Eq. (17) directly to obtain the depth of peak emissions, the specific functional form of ke given in Eq. (9) (see also Fig. 1f) simplifies the analysis. For zw>zs, the derivative satisfies dke/dzw≈0, so the first term on the left-hand side of Eq. (15) can be neglected. A practical approximation is therefore obtained by solving the more interpretable equation 18kofT(o)p2-p1zw-zbe-p2(zw-zb)-p1kEPzw-zb+(2-p1)D(zw-zb)3=0. The accuracy of this reduced formulation is illustrated in Fig. 7 (red crosses), which shows excellent agreement with solutions to Eq. (17) (black lines).

The existence of a positive root requires that the sum of the positive terms outweighs the negative ones for zw>zs; otherwise, provided p1≠0, emissions are maximised at the soil surface (i.e. zp=0). These dependencies may be able to explain conflicting conclusions for optimal water depth for emissions. For example, report exponential decay in CH4 emission fluxes with water depth above the surface, which contrasts with the data synthesis performed by ; this may result from stronger oxidation or weaker plant transport in the wetlands studied by . Similarly, report that water depth for peak emissions depends on the wetland type (e.g. bog vs. fen).

Equation (18) reflects a balance of competing effects: the exponential decline in oxidation against the hyperbolic decay in production with depth (first term), plant-mediated transport (second term), and diffusion (third term). Since the diffusion term depends strongly on depth, its relative influence diminishes when zw-zb is large. Furthermore, if p1>2 and p1>(zw-zb)p2, no positive solution exists and emissions peak at the surface, driven by the rapid decline in production with depth. Other parameter combinations can also preclude a solution, depending on how oxidation is modulated by temperature and other environmental factors (see discussion in Sect. 5.3).

We explore these dependencies in Fig. 7, which shows, for fixed baseline parameters, the influence of (a, b) the oxidation decay rate, (c, d) plant transport and ebullition, and (e, f) production decay on peak emission depth. The chosen parameters correspond to those identified as influential by means of a Morris sensitivity analysis (Appendix B). Figure 7a, c, and e display representative CH4 flux curves for different parameter families, whilst panels (b), (d), and (f) plot the associated peak depths (insets show the corresponding maximum emission rate).

The physical interpretation follows directly from the functional forms in Fig. 1. Methane production is highest at the soil surface, rising as the water table approaches the surface as more of the soil column is exposed to anoxic conditions favourable to methanogenesis but declining when the soil column becomes deeply inundated because of substrate dilution. In contrast, oxidation decreases exponentially with depth and diffusive release scales approximately as an inverse square of water depth. When the oxidation decay rate p2 increases, the aerobic oxidation zone quickly collapses with water depth and CH4 is consumed less efficiently. As a result, the emission peak shifts upward toward the soil surface and the total flux rises by orders of magnitude (Fig. 7a and b). At the extreme of large p2 the model predicts almost no oxidation, so production nearly completely controls the flux and the peak coincides with the soil surface (Fig. 7a and b); in particular, in this regime deep inundation is not required to limit CH4 consumption.

Meanwhile, increasing plant-mediated transport and ebullition produces a similar but more modest effect, shifting both the location and magnitude of the emission peak. Greater ebullition and plant transport allow CH4 to rapidly bypass the oxic layer, further reducing the role of oxidation. Since these pathways are largely independent of water depth, and diffusion is weak above the surface, the overall flux pattern is increasingly determined by the near-surface production profile: higher transport rates both increase emission magnitude and move the peak closer to the surface (Fig. 7c and d), as transport outcompetes oxidation of CH4 produced in the soil column. In particular, as the dominant emission pathway shifts from diffusion, which decays with water depth, to constant ebullition and plant transport, emissions become more directly governed by production. Wetlands with abundant emergent vegetation therefore tend to exhibit higher fluxes and shallower emission peaks. As these emission routes depend directly on the total CH4 present in the soil column, a reinforcing feedback occurs: higher emissions deplete CH4 available for future release, further shifting the peak toward the soil surface where production is maximal.

The influence of production decay is subtler in magnitude but far more significant for peak location. For instance, when p1=0, peak emissions occur at zw→∞, with emissions effectively plateauing with increasing inundation, consistent with the absence of depth-dependent decline in production and continued loss of CH4 via oxidation (this balance can also be readily observed in Eq. 18, which becomes strictly positive). As p1 increases, deeper water no longer offsets oxidation because production itself is suppressed; the emission maximum therefore migrates toward the soil surface (Fig. 7e and f). This behaviour can also be inferred from the simplified analytical condition (Eq. 18), which becomes strictly positive when p1=0.

These results explain why a deeper wetland can sometimes emit less CH4 than a shallower one when oxidation remains active, whereas wetlands rich in organic carbon, such as agricultural systems, can become largely anoxic and release much more CH4. They also show that increasing plant-mediated transport, ebullition, or substrate availability consistently raises emissions and shifts the peak toward the soil surface. In contrast, extending the oxidation decay length suppresses fluxes by enlarging the aerobic consumption zone.

A notable consequence of Eqs. (17) and (18) is that the peak location is itself temperature-dependent via fT(o), implying that both the magnitude and position of peak emissions shift throughout the growing season in response to a shifting balance between production and oxidation. Figure 8 illustrates this dependence: panel (a) shows CH4 fluxes at fixed temperatures, whilst panel (b) plots the corresponding peak emission depths and flux magnitudes. As temperature increases, both the intensity of emissions and the depth at which they peak typically rise, shifting the emission maximum further above the soil surface. This behaviour reflects the fact that warming generally amplifies CH4 production throughout the inundated column more strongly than it enhances oxidation (see also Fig. 6), so that the net flux maximum tends to occur at progressively higher positions above the soil surface. However, under some parameter choices, in particular when p1>p2(zw-zb), the temperature-driven increase in oxidation can instead shift the peak closer to the soil surface. Since this typically occurs only when zw-zb is small and p1 is large, it is usually a weaker effect.

This temperature sensitivity carries broader implications. As global temperatures rise, wetlands may exhibit not only higher CH4 emissions but also shifts in the vertical structure of those emissions. Such changes could impact the efficacy of interventions aimed at suppressing emissions, particularly those based on manipulating water levels. Optimal water levels for mitigation may vary substantially across seasons or under future climatic conditions, necessitating adaptive, site-specific strategies.