Sediment fluxes rather than oxic methanogenesis explain diffusive CH4 emissions from lakes and reservoirs

Methane emissions from lakes and reservoirs are a major natural source in the global budget of atmospheric CH4. A large fraction of these emissions are due to diffusive transport of CH4 from surface waters to the atmosphere. It was suggested recently that CH4 production in the oxic surface waters is required to compensate for diffusive CH4 emissions from lakes. In contrast, we demonstrate here that typical diffusive CH4-fluxes from sediments in shallow water zones, Fsed,S, suffice to explain CH4 emissions to the atmosphere. Our analysis is based on the combination of an exceptional data set on surface concentrations of CH4 with a mass balance model of CH4 that is focused on the surface mixed layer and considers CH4-fluxes from sediments, lateral transport, gas exchange with the atmosphere, and includes temperature dependencies of sediment fluxes and gas exchange. Fsed,S not only explains observed surface CH4 concentrations but also concentration differences between shallow and open water zones, and the seasonal variability of emissions and lateral concentration distributions. Hence, our results support the hypothesis that diffusive fluxes from shallow sediments and not oxic methanogenesis are the main source of the CH4 in the surface waters and the CH4 emitted from lakes and reservoirs.

Scientific RepoRts | (2019) 9:243 | DOI: 10.1038/s41598-018-36530-w resuspension events 27 , the presences of plants 3,31,32 , and/or because of the temperature dependence of CH 4 production in sediments and the comparatively high temperatures of littoral sediments 33,34 . Statistical analysis of a large data set on surface CH 4 distributions in several lakes revealed that basin wide average CH 4 concentrations are described better by the ratio of the area of the shallow water zone to the area of the entire surface than by surface area alone 23 . Based on these observations Encinas et al. 23 suggested that diffusive fluxes from sediments in the shallow water zone are a major source of surface water CH 4 and diffusive CH 4 emissions from lakes. However, this hypothesis was not tested by a quantitative analysis comparing sediment fluxes and emissions. Donis et al. 30 employed a full mass balance of CH 4 in Lake Hallwil to quantitatively compare the losses of CH 4 by diffusive emissions at the lake surface with diffusive CH 4 fluxes from sediments, CH 4 oxidation and other sources and sinks of CH 4 . Donis et al. 30 claim, that the amount of CH 4 emitted at the lake surface exceeds the CH 4 provided by diffusive fluxes from sediments by a factor of 26. They conclude that a large additional source of CH 4 is required and hypothesize that the major part of this source is production of CH 4 in the open water providing 22 times more CH 4 than the total diffusive fluxes from the sediments in the surface mixed layer. This argument implies that without the additional source of CH 4 , i.e. without net-production in the mixed surface layer, diffusive fluxes from the sediments in the surface mixed layer, F sed,S , required to compensate the total diffusive emissions of CH 4 from the lake surface would be extremely large (>40 mmol m −2 d −1 ; using the sediment flux of Donis et al. 30 ) and beyond reasonable values expected from measured sediment fluxes. We test this conclusion by estimating F sed,S required to provide sufficient CH 4 to compensate total diffusive CH 4 emissions to the atmosphere, E atm , in several lake basins and reservoirs. We compare these values of required F sed,S with the typical range of measured F sed,S in other system and with the diffusive sediment flux F sed,Hal obtained from the analysis of pore water concentrations measured by Donis et al. 30 in a sediment core collected in the surface mixed layer of Lake Hallwil.
In our analysis we combine a mass balance model for CH 4 in the mixed surface layer with one of the largest data sets on CH 4 distributions within lakes and reservoirs. The CH 4 mass balance model considers as source of CH 4 diffusive fluxes from the sediments, loss of CH 4 due to diffusive emissions from the water surface to the atmosphere, temperature dependence of these sources and losses, and lateral transport of CH 4 by turbulent mixing within the surface mixed layer. We demonstrate that, in contrast to the conclusion from the argument by Donis et al. 30 , F sed,S required to explain E atm is not ~20 times larger but on average smaller than F sed,Hal . This result suggests that net-production of CH 4 in the surface mixed layer is not required to close the mass balance of CH 4 . However, this argument has the weakness that it compares atmospheric emissions with sediment fluxes from a different system. We therefore additionally re-analyzed the data of Donis et al. 30 and confirm that also in Lake Hallwil the measured diffusive sediment flux F sed,Hal provides sufficient CH 4 to compensate the total diffusive losses of CH 4 to the atmosphere from this system. Finally, we dynamically simulate the CH 4 development along a transect in Lake Uberlingen over several seasons. We demonstrate that the simple model, which does not include net-production of CH 4 in the water, is sufficient to adequately describe the seasonal development of CH 4 concentrations and the seasonal changes in the lateral distribution of CH 4 .

Results
The diffusive fluxes of CH 4 from sediments in shallow waters, F sed,S, estimated for all campaigns in all lakes and reservoirs for which spatially well resolved CH 4 data were available, range between 0.16 and 7.4 mmol m −2 d −1 (Fig. 1a) with an average F sed,S of 2.0 mmol m −2 d −1 and a standard deviation of 1.8 mmol m −2 d −1 . These sediment fluxes were determined assuming steady state conditions and are the sediment fluxes sufficient to compensate the total diffusive emissions of CH 4 from the respective lakes and reservoirs to the atmosphere. F sed,S increases strongly with increasing water temperature (Fig. 1). The temperature dependence can be well described by Boltzmann-Arrhenius law with an apparent activation energy E a = 0.877 eV (R 2 = 0.47, df = 29, p < 0.001). This activation energy agrees well with that determined for CH 4 production in sediments and CH 4 ecosystem emissions 17 . Note that the temperature dependence of F sed,S can be described similarly well using an exponential function with an exponent of 0.122 °C −1 (R 2 = 0.46, df = 29, p < 0.001) which closely agrees with the temperature dependence of CH 4 emissions from surface waters of lakes 18 .
The overall range of the values for F sed,S is about the same as the range of published CH 4 diffusive fluxes from surface sediments (0.03 to ~7 mmol m −2 d −1 ) that were estimated based on measured CH 4 concentration gradients in the pore water of sediment cores 33,35 or on measurements of the CH 4 flux from sediment cores into overlying water 36 . However, the wide range of values is partially due to the temperature dependence of F sed,S . At 20 °C the relation of F sed,S as function of temperature provides on average F sed,S (20 °C) = 2.2 mmol m −2 d −1 (Fig. 1a). This value is ~20% smaller than the diffusive sediment flux F sed,Hal = 2.8 mmol m −2 d −1 determined from the re-analysis (see Supplementary Section S3) of the pore water concentrations measured by Donis et al. 30 in the sediment core collected at 3 m water depth in Lake Hallwil (yellow star in Fig. 1a,b). Thus, in the lakes considered in our study the sediment fluxes required to compensate total diffusive emissions to the atmosphere at 20 °C are smaller than the diffusive sediment flux measured in Lake Hallwil, i.e. on average the required sediment fluxes are smaller than F sed,Hal = 2.8 mmol m −2 d −1 . The largest sediment fluxes required to compensate total diffusive emissions are at most 2.5 times larger than F sed,Hal (i.e. in LLC). These results are in conflict with the CH 4 mass balance of Donis et al. 30 for Lake Hallwil. According to Donis et al. 30 the total diffusive CH 4 emission to the atmosphere is ~26 time larger than the total CH 4 flux from the littoral sediments ( Table 2 in Donis et al. 30 : 5040 mol d −1 versus 196 mol d −1 ) which implies that in Lake Hallwil F sed,S required to compensate total CH 4 emissions to the atmosphere is ~26 times larger than the sediment flux Donis et al. 30 obtained from their pore water measurements. However, our re-analysis of the CH 4 mass balance in the surface mixed layer of Lake Hallwil reveals that the F sed,S required to compensate the total diffusive CH 4 emissions to the atmosphere in Lake Hallwil is only F sed,S = 2.5 to 2.7 mmol m −2 d −1 , which is smaller and not orders of magnitude larger than the measured CH 4 flux from the sediments F sed,Hal = 2.8 mmol m −2 d −1 (a detailed re-evaluation of the data from Lake Hallwil is provided in the Supplementary Section S3). Thus the CH 4 mass balance in Lake Hallwil implies that F sed,Hal is Scientific RepoRts | (2019) 9:243 | DOI:10.1038/s41598-018-36530-w sufficient to compensate the total diffusive emissions at the surface of Lake Hallwil. The total source of CH 4 in the surface mixed layer due fluxes from the sediments S sed,total = 1990 mol d −1 is slightly larger than the total loss of CH 4 due to emissions to the atmosphere at 20 °C (between 1800 and 1900 mol d −1 , see Supplementary Section S3). Apparently, the mass balance in Lake Hallwil does not require an unknown process producing substantial amounts of CH 4 in the open water (for details see Supplementary Section S3). F sed,S required to compensate total diffusive CH 4 emissions at the lake surface were also estimated using surface water CH 4 concentrations measured at the center of several lakes (Fig. 1b) assuming that these CH 4 concentrations are representative for the lake-wide average surface concentration. The temperature dependence of F sed,S derived from the CH 4 concentrations in the center of the lakes (Fig. 1b) and from the average surface concentration ( Fig. 1a) is essentially the same. However, the F sed,S in Fig. 1b are generally smaller than those in Fig. 1a. This difference may be explained by the fact that the surface concentration at the center of a lake is typically smaller than the lake-wide average surface concentration of CH 4 23 . However, the combination of data from different lakes and reservoirs may also contribute to the difference between the values of F sed,S in Fig. 1a,b F sedS is the flux per unit time and unit sediment area from sediments in the shallow water zone which provides sufficient CH 4 that the overall flux from the sediments in the shallow water zone (F sedS · A s ) compensates the overall flux from the lake surface into the atmosphere ( F atm ·A Surf ). The calculation of F sedS assumes that the CH 4 in the surface mixed layer originates only from sediments of the shallow water zone and has no source in the open water. In this case CH 4 concentrations should be larger in the shallow than in the open water zone. If, however, all methane is produced by oxic methanogenesis within the water column of the surface mixed layer one would expect the opposite, i.e. larger CH 4 concentrations in the open water than in the shallow water zones, because CH 4 production per unit surface area is larger in the open water than in the near shore zones where the water depth is smaller than the depth of the surface mixed layer.
To test whether the estimated sediment fluxes F sed,S explain the observed horizontal distribution of CH 4 within the surface mixed layer of the basins, the model was applied to simulate the concentration distribution of CH 4 at steady state for each campaign assuming radial symmetry of the basins and by using the F sed,S of the respective basin and time depicted in Fig. 1a. The difference between the average CH 4 concentration in the shallow and the average CH 4 concentration in the open water, ∆CH 4,av , was determined from the model results and from  F sed,S measured in Lake Hallwil Figure 1. Diffusive CH 4 fluxes from the sediments of the shallow water zone, F sed,S , required to compensate total diffusive CH 4 emissions to the atmosphere. F sed,S were calculated based on spatially averaged CH 4 concentrations in the surface water utilizing spatially highly resolved distributions of CH 4 available from numerous campaigns on several lakes and reservoirs (a), and on time series of CH 4 concentrations measured in the surface water at the center of several lakes (b). The diffusive sediment flux derived from pore-water measurements in Lake Hallwil, F sed,Hal , is shown by a yellow star. The temperature dependence of F sed,S can be well described by the Boltzmann-Arrhenius law (black regression lines) or an exponential law (red regression lines). The degree of freedom df is 29 and 48 in (a) and (b), respectively.
Scientific RepoRts | (2019) 9:243 | DOI:10.1038/s41598-018-36530-w the observations. Observed and simulated ∆CH 4,av agree well and both indicate that CH 4 concentrations in the surface mixed layer are typically larger in shallow near shore than in the open water zones (Fig. 2). The regression line has a slope of 0.97 and differs significantly from zero but not from 1 (p < 0.001, p 1 = 0.8), and the intercept is −0.01 μmol L −1 and does not significantly differ from 0 (p = 0.9). The results from the regression analysis support the conclusion that the concentration differences between shallow and open water can be explained by the assumption that the source of CH 4 in the surface waters of lakes and reservoirs is the CH 4 flux from sediments in shallow waters. The statistical analysis of measured and simulated differences between shallow and open water CH 4 concentrations combining several lakes and reservoirs could be affected by differences between these systems in trophic state or other properties influencing CH 4 .We therefore have included an analysis which focuses on data from a single large lake, Lake Uberlingen, and applied the model to dynamically simulate the temporal development of the CH 4 distribution over two years. Measured and simulated seasonal changes in the CH 4 concentrations agree well (Fig. 3a). Furthermore, nearshore CH 4 concentrations are typically larger than the concentrations at larger distance from shore, and simulated and measured concentrations at the same distance from shore agree well with each other (Fig. 3a). Note that the simulation results are derived from a time continuous model that required only three time constant parameters. Inverse fitting of the model to the 56 data points provides as best fit parameters the dispersion coefficient K h,disp = 1.4 m 2 s −1 , and E a = 0.823 eV and C = 33.6 of the Boltzmann-Arrhenius law. The diffusive CH 4 flux from the shallow water sediments obtained from these parameters increases with temperature and is F sed,S = 2.8 mmol m −2 d −1 at 20 °C. This value of F sed,S , which is the sediment flux sufficient to compensate CH 4 emissions from Lake Uberlingen to the atmosphere, is the same as the sediment flux F sed,Hal = 2.8 mmol m −2 d −1 obtained from the pore-water measurements 30 in Lake Hallwil. The dispersion coefficient obtained by the inverse fitting, K h,disp = 1.4 m 2 s −1 , is slightly larger than the dispersion coefficient provided by the empirical equation of Lawrence et al. 37 Fig. 3b), but at a smaller rate than the sediment fluxes. Figure 3c directly compares data and model results at different distances from shore illustrating that the highest simulated and measured concentrations occur closest to shore and agree well with each other. According to linear regression of measured versus simulated concentrations the model explains 66% of the variance (R 2 = 0.66, df = 54) and the regression line has a slope of 0.94 ± 0.09 that does not differ significantly from 1 (p 1 = 0.50) and an intercept of 0.02 ± 0.03 that does not differ significantly from 0 (p = 0.47).

Discussion
In spite of the simplifications in the model, the simulated CH 4 concentrations agree well with field data. Because we assume that net-production of CH 4 is zero the F sed,S determined from the model are the diffusive sediment fluxes that are sufficient to compensate diffusive emissions to the atmosphere. F sed,S are within the range of measured diffusive sediment fluxes 33,35,36 and are at 20 °C on average smaller than the sediment flux obtained from the pore-water measurements by Donis et al. 30 in Lake Hallwil. This implies that in the systems investigated by us, diffusive sediment fluxes on the same order as the sediment flux in Lake Hallwil, F sed,Hal = 2.8 mmol m −2 d −1 , result in a total flux of CH 4 from the sediments in the surface mixed layer that is sufficient to compensate the total . The intercept does not differ from zero (p = 0.9) whereas the slope significantly differs from zero but not from one (p < 0.001, p 1 = 0.8).
Scientific RepoRts | (2019) 9:243 | DOI:10.1038/s41598-018-36530-w diffusive flux of CH 4 from the lake surface to the atmosphere at 20 °C. Hence, these sediment fluxes suffice to close the mass balance of CH 4 without requirement of substantial net-production. This conclusion is in conflict with the central argument of Donis et al. 30 who claimed that the total diffusive emissions from the surface of Lake Hallwil are 26 times larger than the total source of CH 4 due to the flux from sediments in the surface mixed layer and that therefore substantial net-production in open water is required to close the mass balance. The temperature dependence of F sed,S obtained from our analysis agrees well with the temperature dependence of CH 4 production in sediments and emissions from ecosystems and lakes 17,18 . F sed,S increases with temperature at a larger specific rate than the CH 4 concentration in the surface water (Fig. 3b). Because the gas transfer velocity increases with temperature, the CH 4 emissions to the atmosphere must increase with temperature at a larger specific rate than the CH 4 concentrations in the surface water. Closure of the CH 4 mass balance requires that the temperature dependence of F sed,S and of the emissions are similar and thus both must have a larger specific rate of increase with T than that of CH 4 concentrations, which is consistent with the results of our model.
At same temperature F sed,S differs between lakes and reservoirs which may be explained by different trophic states, properties of the sediments, e.g. porosity or grain sizes, exposure of the sediments to currents or biotic factors such as biofilms, macrophytes or reed belts. For example at ~20 °C F sed,S in Schwarzenbach reservoir is smaller than in all other lakes and reservoirs investigated which may be explained by the comparatively low alkalinity and low pH of the water in this reservoir and by the comparatively thin sediment layer that was dry in 1997 when the reservoir was emptied. These factors may have negative effects on CH 4 production. The sediment fluxes F sed,S are largest in the three basins of LLC possibly because these basins have particularly large reed belts in the shallow water zone 38,39 .
Modelled and measured ∆CH 4,av agree well and indicate that the CH 4 concentrations in the shallow near shore zone are typically larger than the CH 4 concentrations in the open water (Fig. 2). This supports the hypothesis that the main source of the CH 4 in the surface mixed layer are the sediments in the shallow water zone and not production within the water column. The statistical analysis of measured versus simulated ∆CH 4,av is dominated by the results for the basins of LLC because ∆CH 4,av are large due to the high sediment fluxes, the comparatively large surface area and the large ratio of shallow to open water surface area in LLC. In the smaller lakes, e.g., Illmensee and Königseggsee, ∆CH 4,av is small because at small spatial scales concentration differences are more rapidly homogenized by horizontal mixing than at large spatial scales (see Supplementary Section S2). The potential influence of differences in the conditions in the lakes and reservoirs on the conclusion of F sed,S and its temperature dependence has been circumvented in the investigation of the seasonal development of CH 4 along a transect in Lake Uberlingen. Although only three time-constant parameters of the simplified model were fitted, the simulated CH 4 concentrations show a very similar temporal development as the field data over the observation period of more than two years. Furthermore, the results obtained from the transect in Lake Uberlingen agree well with the results from all other lakes and reservoirs studied. In all systems reasonable fluxes from sediments in shallow waters suffice to explain surface concentrations and emissions of CH 4 . Hence, our findings suggest that CH 4 production in oxic surface waters is not required to compensate emissions and may be not the main source of CH 4 in surface waters of lakes and reservoirs as was claimed recently 9,30 . Support for CH 4 production under oxic conditions comes from mesocosm experiments in which CH 4 concentrations remained essentially constant over 28 days 9 . Rates of oxic CH 4 production were estimated by assuming that emissions from the mesocosms and oxidation of CH 4 were compensated by net-production 9 . However, these experiments do not prove that oxic CH 4 production is a large source of CH 4 in the unbounded open water of lakes. CH 4 concentrations in the mesoscosms remained ~ 4-10 times smaller than in the lake water outside the mesocosms suggesting that the mesocosms excluded the major source of CH 4 , e.g. CH 4 fluxes from littoral sediments. Assuming that CH 4 is produced in oxic waters by acetoclastic production Bogard et al. 9 have taken a correlation between Chl-a and CH 4 as evidence for the importance of methanogenesis in oxic surface waters in lakes. However, the data set of Bogard et al. 9 only provides a significant correlation between Chl-a and CH 4 if marine systems and freshwater lakes are combined but not for freshwater lakes alone 23 . Seasonal changes of CH 4 and Chl-a in individual lakes do not support a strong link between CH 4 and Chl-a concentrations 23 .
Considering lateral dispersion of CH 4 and emissions to the atmosphere DelSontro et al. 28 compared steady state CH 4 concentration distributions with the observed decrease of CH 4 from shallow to open water zone. They did not consider fluxes from sediments and therefore could not explain the cause for the increased CH 4 concentrations in the littoral zone. According to DelSontro et al. 28 net-oxidation is required in 30% and net-production in 70% of their lakes to reproduce the observed CH 4 concentration distributions. However, because atmospheric fluxes of CH 4 were calculated using v gas of CO 2 at 20 °C 28 , emissions were underestimated by ~11% in their warmest and overestimated by ~25% in their coldest lakes, respectively, affecting the reliability of the estimated net-production and net-oxidation rates of CH 4 . Furthermore, the conclusions on net-production and net-oxidation are very sensitive to lateral transport. The model of DelSontro et al. 28 underestimates the transport of CH 4 from shallow to open water zones because it uses a horizontal dispersion coefficient that underestimates lateral transport in the near boundary region 37 . Additionally, advective transport may further enhance the CH 4 transport to the open water. Underestimation of CH 4 transport to the open water leads in the model of DelSontro et al. 28 to an underestimation of net-oxidation and overestimation of net-production of CH 4 in the open water.
The sensitivity to lateral transport in the assessment of net-production of CH 4 and of boundary effects prevalent in mesocosm experiments can be avoided by using a mass balance approach considering entire lake basins. According to Donis et al. 30 substantial methanogenesis in oxic waters is required to close the mass balance of CH 4 in the 5 m thick mixed surface layer of Lake Hallwil. They estimated that oxic CH 4 production contributes ~91% of the total emissions and produces 22 time more CH 4 than is supplied by the diffusive flux from the sediments in the mixed surface layer. However, in their mass balance Donis et al. 30 underestimated the total flux of CH 4 from littoral sediments by more than an order of magnitude and also overestimated the CH 4 emissions to the atmosphere. Donis et al. 30 apparently used 0.1225 km 2 as value for the area of the shallow water zone in the surface mixed layer, which is ~6 times smaller than the 0.711 km 2 suggested by the published hypsography of Lake Hallwil 40 . Furthermore, the pore-water concentrations in the top 3 cm of the sediment measured by Donis et al. 30 suggest a concentration gradient of 3.4 10 4 mmol m −4 , which is ~1.7 times larger than the gradient used by Donis et al. 30 (Supplementary Fig. S8 in Supplementary Section S3). As consequence, Donis et al. 30 underestimated the diffusive sediment flux by a factor of 1.7. Re-analysis of sediment flux and emissions from Lake Hallwil (Supplementary Section S3) reveals, that the total source of CH 4 due to diffusive sediment fluxes is slightly larger than the total loss of CH 4 due to diffusive emissions to the atmosphere confirming that diffusive sediment fluxes are sufficient to compensate emissions to the atmosphere. This disproves that a large additional source of CH 4 is required to close the mass balance in Lake Hallwil, i.e. the central argument of Donis et al. 30 for substantial CH 4 production in the open water. Interestingly, the 13 C isotopic composition of CH 4 in the open water and in the pore water at the surface of the sediments in the shallow water zone were essentially the same 30 , suggesting that the CH 4 in the pore water near the sediment surface is the source of the CH 4 in the open water rather than an "unknown production process(es)" 30 generating substantial amounts of CH 4 in oxic waters.
In the mass balance calculation uncertainty arises from the estimated loss of CH 4 due to diffusive emissions to the atmosphere. The larger the CH 4 emissions to the atmosphere the larger the required source of CH 4 , i.e. in our model the diffusive flux from the sediments in the shallow water. The calculations of the atmospheric CH 4 emissions require estimates of the gas transfer velocity. Several empirical equations have been proposed to relate v gas to wind speed 20,[41][42][43] . For wind speeds typical for the systems studied here (~2 m s −1 ) the different equations provide smaller 41 , similar 42 and, depending on the surface buoyancy flux, similar and larger values 43 of v gas than the equation of Cole and Caraco 20 that was used here. We therefore have performed a sensitivity analysis on the implication of choosing different models for the gas transfer velocity on the results on F sed,S (see Supplementary Section S4). Independent of the model chosen F sed,S required to compensate total emissions to the atmosphere at 20 °C is on average smaller than the observed sediment flux F sed,Hal in the mixed layer of Lake Hallwil.
In summary, our results indicate that the CH 4 mass balances in many lakes and reservoirs do not support the conclusion that oxic methanogenesis is required to compensate CH 4 emissions to the atmosphere. In contrast, field data and modelling results suggest that reasonable CH 4 fluxes from sediments in shallow waters are sufficient to explain diffusive CH 4 emissions from lakes and reservoirs, and also explain the seasonal changes in CH 4 concentrations and CH 4 distributions in their surface waters.

Methods
Data. The data set employed in this study includes numerous well resolved spatial distributions of CH 4 measured at several times during a season in different years and in 10 different lake basins and reservoirs of different morphometry. The data set also includes seasonally resolved time series of CH 4 measured at the central station in several of these lakes. Additionally, a seasonally resolved data set consisting of 14 transects collected during two consecutive years is available for one of the lakes (Lake Uberlingen). In total, the data set is based on 1346 individual measurements of CH 4 concentrations in surface waters. Surface water temperatures are available for all measurements and in several lakes also continuously for several years. Wind speeds were determined from the COSMO-2 wind field 44 available continuously for several years for all lakes. In case of the reservoirs, wind data from nearby weather stations were used. Parts of the data are discussed in 23 , detailed information on all data and systems studied is provided in Supplementary Section S1.
Model. The interpretation of the data is supported by a model that allows dynamic simulation of a simplified mass balance of CH 4 within the surface mixed layer of lakes and reservoirs. The model simulates CH 4 concentrations in the surface mixed layer considering diffusive CH 4 fluxes from sediments in the shallow water zone, diffusive gas exchange of CH 4 with the atmosphere and horizontal mixing. Temperature dependence of diffusive gas exchange and sediment fluxes is also included. The surface layer is assumed to be fully mixed in the vertical and the CH 4 concentrations are therefore vertically homogeneous within the surface mixed layer. Vertical transport of CH 4 across the thermocline is neglected. In the horizontal dimension CH 4 concentrations vary because CH 4 is introduced from the sediments of the shallow water zone into the water column and is transported laterally by turbulent mixing.
In the simulations of entire lake basins we assume that the surface mixed layer is radially symmetric in the horizontal. The surface areas of the radially symmetric basins correspond to the true surface area of the respective basin. Because basins, sources and sinks of CH 4 are radially symmetric, the development of the CH 4 concentrations can be described based on the radial distance r from the basin center: The four terms on the right hand side of equ. 1a describe (i) the change of the CH 4 concentration C(r, t) with time due to lateral transport, (ii) the source of CH 4 due to the diffusive flux from the sediments, (iii) the loss of CH 4 due to gas exchange with the atmosphere, and (iv) net-production of CH 4 , respectively. To test our hypothesis that diffusive sediment fluxes are sufficient to compensate emissions to the atmosphere, i.e. that net-production is not required to close the mass balance, we simulate the CH 4 concentrations assuming no net-production, i.e. P(r, t) = 0.
C(r, t) is the concentration of CH 4 as function of r and time t, K h,disp the effective horizontal dispersion coefficient, and H(r) the spatially varying thickness of the surface layer. In the open water H(r) is equal to the mixed layer depth. Within the shallow water zone H(r) decreases linearly with r from the mixed layer depth to zero at the shore, i.e. at the maximum radius r max . F sed (r, t) is the diffusive flux of CH 4 from sediments, which is zero in the open water and F sed,S in the shallow water zone. F sed,S depends on water temperature T(t). F atm (r, t) is the diffusive flux of CH 4 to the atmosphere, v gas the gas transfer velocity that depends on T(t) and wind speed WS(t), and C eq the equilibrium concentration of atmospheric CH 4 at T(t). r max is the maximum radius, π = r A / surf max , and radius r s is the distance from the center of the lake to the boundary of its shallow water zone, . A surf is the total surface area and A S the surface area of the shallow water zone of the different lakes. At the boundaries horizontal fluxes are zero which implies that dC/dr = 0 at r = 0 and at r = r max .
If diffusive fluxes from the sediments are the predominant source of CH 4 in the surface water of lakes one expects higher concentrations in the shallow water than in the open water zone 23 . The difference between surface concentrations in shallow and open water zones depends on the rate at which CH 4 is mixed in the horizontal dimension 28 . The rate of horizontal dispersion increases with increasing length scale L 37,45,46 and is described in the model by K h,disp . Adopting the empirical relation by Lawrence et al. 37 and considering as relevant length scale the radius of the different basins K h,disp = 3.2·10 −4 ·r max 1.10 (m 2 s −1 ), whereby r max is in m. In addition to the radially symmetric model for investigations considering entire basins we employ a model of a vertically mixed rectangular basin for the simulation of the seasonal development of CH 4 concentrations in the surface mixed layer along the transect in Lake Uberlingen. We assume homogeneous conditions in cross-transect direction and in the vertical dimension. Hence, the model can be condensed to a one dimensional mass balance model using coordinate x in along-transect direction (see Supplementary equs S2a,b in Supplementary Section S2). The spatially varying thickness of the surface mixed layer H(x) is given by the minimum of local water depth and H S . The latter is the surface mixed layer depth in the open water. The shallow water zone is defined as the region in which H(x) < H S . In the transect model the horizontal dispersion coefficient K h,disp is not calculated from the empirical relation of Lawrence et al. 37  Analyses utilizing data and model. The models and inverse modelling techniques provide a basis for the determination of the diffusive flux of CH 4 from the sediments, F sed,S , required to compensate the total emission of CH 4 to the atmosphere, E atm , in different lakes and reservoirs during different seasons. F sed,S can be estimated assuming steady state conditions for the respective measuring campaign. Steady state requires that F sed,S , v gas and C eq are constant in time. At steady state the total CH 4 emission to the atmosphere is equal to the total flux of CH 4 from the sediments of the shallow water zone: Lake atm Lake sed Shallow sed S , Assuming that temperature and wind speed are horizontally homogeneous F sed,S = F atm ·A Surf /A S and the spatially averaged flux to the atmosphere = −

( )
F v C C atm g as eq can be calculated from the spatially averaged CH 4 concentration C. Note that this conclusion is valid in general and does not require assumptions on the morphometry of the aquatic system. We have applied this steady state approach to estimate F sed,S except in the simulations of the transect of Lake Uberlingen.
The temperature dependence of F sed,S was analyzed using linear regression assuming Boltzmann-Arrhenius law: and C is a constant, E a the apparent activation energy, k B the Boltzmann constant, and T a the absolute temperature. In addition, we tested an exponential temperature dependence of F sed,S . The model was applied to simulate steady state distributions of CH 4 in the simulations considering entire basins. Utilizing equ. 1 with the estimated sediment fluxes, concentration differences between shallow and open water zones in different basins and times of the year, e.g. at different water temperatures, were simulated and compared to observations.
The capabilities of the model approach with respect to predicting seasonal changes in the CH 4 concentrations and seasonal differences between CH 4 concentrations in shallow and open water zones is demonstrated by the dynamic simulation of the temporal development of the CH 4 concentrations along the cross-shore transect in Lake Uberlingen. As model domain a rectangular basin extending from shore to shore along the measured transect was used. Model results are evaluated at the times and the locations along the transect for which measurements exist. Three time-constant parameters, i.e. the activation energy E a and the exponent of the pre-scaling factor of the Boltzmann-Arrhenius law describing F sed,S (equ. 3), and K h,disp , were determined by inverse modelling of the data The comparison of model results and data is based on averaged concentrations in four distance ranges from shore (D1: <100 m, D2: 100-300 m, D3: 300-1000 m, D4: 1000-1850 m). Data are available from 14 dates during two seasons providing 56 data points for the fitting of the 3 parameters.
Statistics. Linear regression analysis was performed using the routine "fitlm" of Matlab. In case of the assessment of temperature dependences, the logarithms of F sed or of CH 4 concentrations were used as dependent variables. Model performance was tested by regression of observed versus simulated values. The explained variance is denoted by R 2 and the degrees of freedom by df. Two-tailed t-tests are employed to provide p-values testing whether slope and intercept of the regression line differ from zero (p) and whether the slope differs from 1 (p 1 ).

Re-analysis of data from Donis et al.
We re-analyzed the data of Donis et al. 30 with respect to the diffusive flux from the sediments and the atmospheric emissions of CH 4 in Lake Hallwil. The diffusive flux from the sediments in Lake Hallwil, F sed,Hal , was determined assuming molecular diffusion of CH 4 within the sediment and by using data on pore-water concentrations of CH 4 measured in the sediment core collected on 29 th September 2016 from 3 m water depth in Lake Hallwil (see Fig. 5a in 30 and Supplementary Section S3 Fig. S8). We used the same approach and parameterization as Donis et al. 30 but estimated the near-surface gradient of CH 4 in the pore water from linear regression (see Supplementary Fig. S8 in Supplementary Section S3). Pore-water concentrations were available from the sediment surface down to 3 cm depth and from depths of 7 cm and larger. Linear regression was applied to the uppermost three measurements of the pore water concentration (0, 2, and 3 cm depth). The data are very well represented by the regression line (see Supplementary α. S8 in Supplementary Section S3), suggesting that the slope of the regression line is a good estimator of the pore-water concentration-gradient near the sediment surface.
The diffusive flux to the atmosphere was calculated using several models for the gas transfer velocity 41-43,47,48 assuming a surface water CH 4 concentration of 0.3 mmol m −3 (June 2016 30 and average concentration April to August 2016 30 ), a water temperature of 20 °C (June 2016 30 ), and hourly wind speeds available from station Mosen (MeteoSwiss) located at ~0.5 km distance from the shore of Lake Hallwil.
Published hypsographic data of Lake Hallwil 40 were used to calculate the total source of CH 4 in the surface mixed layer due to the diffusive flux from sediments and the total loss from the lake surface due to diffusive emissions to the atmosphere.
For further details see Supplementary Section S3.