Random forest (RF) is a machine-learning algorithm that can be used for classification or regression analyses (Breiman, 2001). In this study the RF models consist of a large ensemble of regression trees. Each individual regression tree is built by training it with a random subset of training data and the trees are trained independently of each other. The RF model output is then the average of all the predictions made by individual regression trees in the forest. Hence, the RF algorithm applies the bootstrap aggregation (bagging) algorithm and takes full advantage of the fact that ensemble averaging decreases the noise of the prediction. In addition to random selection of training data, the predictor variables used in split nodes are also selected from a random sample of all predictors, which minimizes the possible correlation between trees in the forest (Breiman, 2001) and decreases the possibility of overfitting. The predictor variables can be either categorical or continuous. The variables are then used in the split nodes to divide the data into two (e.g. categorical variable true or false or a continuous variable, such as temperature above or below 5 ∘C).

Performance of RF algorithms to predict CO2 and energy fluxes across FLUXNET sites have been compared against other machine-learning algorithms, such as artificial neural networks and multivariate regression splines, by Tramontana et al. (2016), who showed that differences between methods were negligible. We anticipate a similarly negligible effect of machine-learning algorithm choice for CH4 fluxes. For a thorough description of the RF algorithm for flux upscaling purposes, the reader is referred to Bodesheim et al. (2018) (and references therein).

In this study, the RF models were developed using the MATLAB 9.4.0 (R2018a) TreeBagger function with default values similar to those of Bodesheim et al. (2018). These settings included a minimum of five samples in a leaf node and used mean squared error (MSE) as a metric for deciding the split criterion in split nodes. Each trained forest consisted of 300 randomized regression trees.

The RF model performance was evaluated against independent validation data using a set of statistical metrics, which were related to different aspects of model performance. During the RF model training MSE was optimized as follows: 1MSE=o-p2‾, where o and p are vectors containing the observed and predicted values, respectively, and the overbar denotes averaging.

The Nash–Sutcliffe model efficiency (NSE; Nash and Sutcliffe, 1970) was used to evaluate how well the model was able to predict validation data when compared against a reference (typically the mean of the validation data): 2NSE=1-∑i=1noi-pi2∑i=1noi-o‾2, where i is index running over all the n values in the o and p vectors. When NSE is equal to 1, there is a perfect match between prediction and observations. Values above 0 imply that the model predicts the observations better than the mean of observations and values below 0 indicate that the predictive capacity of the model is worse than the mean of validation data. Note that NSE calculated with Eq. (2) above is equivalent to the coefficient of determination calculated using residual sum of squares and total sum of squares. However, following the approach used in previous upscaling studies (e.g. Bodesheim et al., 2018; Tramontana et al., 2016), we opted to call this metric NSE. Instead, the coefficient of determination (R2) was estimated as the squared Pearson correlation coefficient. Note that R2 and NSE are equal when there is no bias between o and p and the residuals follow Gaussian distribution. Pearson correlation coefficients obtained with different model runs are compared using Fisher's r to z transformation.

The standard deviation (σ) of the model residuals was used to evaluate the spread of model residual values (RE): 3RE=σo-p, whereas biases between model predictions and validation data were used to estimate the systematic uncertainty in the upscaled fluxes (BE): 4BE=o-p‾. Note that RE equals RMSE when there is no systematic difference between the model predictions and observations (i.e. when BE equals zero).

For upscaling CH4 fluxes using the developed RF model, the LST data were acquired from the aggregated product MOD11C3 (Wan et al., 2015b) and snow cover data from MOD10CM (Hall and Riggs, 2018). Distribution of permafrost in the northern latitudes were estimated using the circum-Arctic map of permafrost derived by National Snow and Ice Data Center (Brown et al., 2002). The resolution of the gridded data was adjusted to match the resolution of the wetland maps using bilinear interpolation if needed. Additionally, land and ocean masks (Jet Propulsion Laboratory, 2013) were utilized when processing the gridded datasets.

Upscaled fluxes were initially estimated in flux densities per wetland area, i.e. amount of CH4 per area of wetland per unit of time. To create a gridded product of CH4 emissions from the northern wetlands, these upscaled flux densities were converted into (amount of CH4) per (grid cell area) per (unit of time) using different wetland maps. Wetland mapping is an ongoing field of research and the usage of different wetland maps contributes to the uncertainty of global wetland CH4 emission estimates (e.g. Bloom et al., 2017a; Z. Zhang et al., 2017). Hence, three different wetland maps (PEATMAP, DYPTOP and GLWD) were used in this study to evaluate how much they affect the overall estimates of northern high-latitude wetland CH4 emissions.

The recently developed static wetland map PEATMAP (Xu et al., 2018) combines detailed geospatial information from various sources to produce a global map of wetland extent. Here, the polygons in PEATMAP were converted to fractions of wetland in 0.5∘ grid cells. While PEATMAP is focused on mapping peatlands, marshes and swamps (typically on mineral soil) are included in the product for certain areas in the northern latitudes. However, most of the wetlands in the northern latitudes are peatlands, and thus PEATMAP is suitable for our upscaling purposes. The dynamic wetland map estimated by the DYPTOP model (Stocker et al., 2014) was used by aggregating peat and inundated areas to form one dynamic wetland map with 1∘ resolution. The widely used Global Lakes and Wetlands Database (GLWD, Lehner and Döll, 2004) is a static wetland map with 30 arcsec resolution and since it has been widely used here it provided a point of reference for the other two maps. The map was aggregated to 0.5∘ resolution and lakes, reservoirs and rivers were excluded from the aggregated map.

The upscaled CH4 fluxes were compared against the output from two process models: LPX-Bern (Spahni et al., 2013; Stocker et al., 2013; Zürcher et al., 2013) and the model ensemble WetCHARTs version 1.0 (Bloom et al., 2017a, 2017b). LPX-Bern is a dynamic global vegetation model that models carbon and nitrogen cycling in terrestrial ecosystems. The model has a separate peatland module with peatland-specific plant functional types (for more details, see Spahni et al., 2013). The wetland extent in LPX-Bern was dynamically estimated using the DYPTOP approach with 1∘ resolution (Stocker et al., 2014). WetCHARTs combines several prescribed wetland maps with different gridded products for heterotrophic respiration and temperature sensitivity (Q10) parameterizations for CH4 production to form a model ensemble of wetland CH4 emissions (Bloom et al., 2017b). Here we used the extended ensemble of WetCHARTs.

The predictors in Table 1 were selected one by one using the procedure described in Sect. 2.1.2. The order in which the predictors were selected is shown in Fig. 2. LSTn alone gave NSE =0.29. After including the category permafrost presence and absence, Rpot, SC and biome class increased NSE to 0.47. However, the influence of SC and biome class on the model performance was marginal based on the small increase in NSE. Additional predictors did not increase the model performance further because (1) they were strongly correlated with a predictor already included in the model (e.g. Tair is correlated with LSTn) or (2) the predictors did not contain any information about CH4 flux variability. The model response to predictors other than biome category was physically reasonable (e.g. permafrost and snow cover decrease fluxes, close to exponential dependence on LSTn), whereas the response to biome category was contrary to expectations. The RF model estimated the CH4 flux magnitude from the different biomes to be in the following order: tundra < temperate < boreal. However, in prior studies it has been shown to be in the following order: tundra < boreal < temperate (Knox et al., 2019; Treat et al., 2018; Turetsky et al., 2014). This discrepancy may be due to the limited number of measurement sites and related sampling bias problems. Hence, in order not to upscale an incorrect pattern of decreasing CH4 emissions when moving from boreal to temperate regions, the biome class was omitted from upscaling. In the subsequent analysis and flux upscaling only the four first predictors (LSTn, permafrost category, Rpot and SC) are utilized.

We further tested whether information about wetland type or sedge cover would improve the model performance even though these categorical variables were not available in gridded format and hence were not usable for upscaling. Including the sedge flag increased the NSE to 0.53, although the increase in Pearson correlation was not statistically significant (p>0.05, comparison of correlation coefficients using Fisher's r to z transformation). Also, wetland type did not have a statistically significant influence on the model performance (p>0.05 and NSE =0.49 if type included). Using too many categorical variables in a RF model may be problematic because each site may end up with a unique combination of categorical variables.

The most important predictor for the model was temperature, similar to numerous studies showing that wetland CH4 emissions are strongly correlated to soil temperature (Christensen et al., 2003; Helbig et al., 2017; Jackowicz-Korczyński et al., 2010; Rinne et al., 2018; Yvon-Durocher et al., 2014; Knox et al., 2019). Selection of LSTn as the primary driver instead of the other temperature variables was likely an outcome of the available data and the algorithm used to select the drivers. With slightly different dataset (more sites) other temperature variables (e.g. Tair) might have been more important drivers for the CH4 flux variability. Estimating apparent Q10 from the RF model LSTn dependence yielded a value of 1.90±0.03 and for validation data it was slightly higher (1.97±0.06) (Fig. 3). These values are comparable to the ones reported in Turetsky et al. (2014) for CH4 chamber measurements at bog and fen sites. The temperature dependence of CH4 production is modelled in many process models with the parameter Q10 value close to 2 (Xu et al., 2016b), which agrees with the CH4 emission temperature dependence shown here. However, one should note that CH4 oxidation also depends on temperature and the derived apparent Q10 value describes the temperature dependence of surface CH4 emission, which is always a combination of CH4 production and oxidation.

The overall systematic bias (BE) between the RF predictions and validation data was negligible (Fig. 4), whereas the spread of the data (RE) was more pronounced (Fig. 4). Following Moffat et al. (2010), RE was analysed further by binning the data based on CH4 flux magnitude and calculating RE for each bin. RE was clearly correlated with flux magnitude (RE=(0.52±0.06)FCH4+(3.3±2.0) nmol m-2 s-1, where FCH4 denotes CH4 flux), indicating that the relative random error of the RF model prediction was nearly constant and approximately 50 % for high fluxes. The systematic error BE did not show a clear dependence on flux magnitude. The RF model performance was worse on a site mean level than with monthly data. When comparing site means, NSE and R2 were both 0.25 and RE and BE were 27.0 and 1.5 nmol m-2 s-1, respectively. Possible drivers causing the remaining CH4 flux variability not captured by the RF model (i.e. the scatter in Fig. 4) are discussed in Sect. 4.2.1.

When considering the model performance for each site separately, the agreement shows different characteristics (see Fig. 5 for four examples). For individual sites the magnitude of BE is typically somewhat higher (median of absolute value of BE approximately 11 nmol m-2 s-1), whereas RE is lower than for the overall agreement (median RE approximately 10 nmol m-2 s-1). These results indicate that the upscaled CH4 fluxes have, in general, relatively low bias and high random error, whereas individual pixels in the upscaled CH4 map may have higher bias but lower random error.

The mean annual cycle of CH4 emission predicted by the RF model agrees well with the mean annual cycle calculated from the validation data (not shown). During the non-growing season the RF model slightly overestimates the fluxes (15 % overestimation) but such differences were negligible during the rest of the year (<1 %). However, for individual sites CH4 emission seasonality agrees less. For instance, at US-Los the modelled CH4 emissions start to increase 1 month earlier in the spring (Fig. 5b). The non-growing season fluxes are overestimated at four example sites (FI-Sii, US-Los, US-Atq and RU-Ch2; Fig. 5). The mean flux magnitude is modelled well at FI-Sii (Fig. 5a), whereas at US-Los (Fig. 5b) and US-Atq (Fig. 5c) the RF model overestimates and at RU-Ch2 (Fig. 5d) underestimates the CH4 emissions. The flux bias had a relatively large impact on site-specific NSE. For example, for US-Atq NSE was -1.85, meaning that the observation mean would be a better predictor for this site than the RF model (see the NSE definition in Sect. 2.2). The RF model is not able to replicate the between-year differences in CH4 emissions at the example sites. Capturing interannual variability has also been difficult in previous upscaling studies of CO2 and energy fluxes (e.g. Tramontana et al., 2016).

In general, the RF model performance was better for permafrost-free sites than for sites with permafrost (r=0.66 and r=0.51, respectively; p<0.05), which is likely related to the fact that at sites with permafrost the MODIS LSTn is not as directly related to the soil temperature than at sites without permafrost. Hence, LSTn is not as good proxy for the temperature that is controlling both CH4 production and consumption and this results in a worse performance than at sites without permafrost.

The RF model developed in this study was used together with the gridded input datasets (Sect. 2.4) and wetland distribution maps (Sect. 2.5) to estimate CH4 emissions from northern wetlands in 2013 and 2014. The mean CH4 emissions of the 2 years from the RF model are plotted in Fig. 6 together with CH4 wetland emission maps from the process model LPX-Bern and model ensemble WetCHARTs. Differences between the process model estimations and upscaled fluxes are shown in Fig. 7. In general, the spatial patterns are similar among emission maps, which is not surprising given that the spatial variability is largely controlled by the underlying wetland distributions. One noteworthy difference is that WetCHARTs, RF-PEATMAP (i.e. RF modelling with PEATMAP) and RF-GLWD show higher emissions from western Canada than LPX-Bern or the upscaled fluxes using the wetland map from that process model (RF-DYPTOP). The other difference is that RF-GLWD show negligible emissions from Fennoscandia (Fig. 6c). These differences are related to differences in the underlying wetland maps. While the wetland maps differ, there is no consensus on which is more accurate, so comparisons indicate the uncertainty in upscaling emanating from uncertainties in wetland distribution.

Three statistical metrics (NSE, R2 and RE) were calculated between RF-DYPTOP and LPX-Bern for each grid cell (Fig. 8). The figure illustrates how well the temporal variability of CH4 emissions estimated by RF-DYPTOP and LPX-Bern agree in each grid cell. NSE values are low in areas where the systematic difference between RF-DYPTOP and LPX-Bern was high (compare Figs. 8a and 7a) since the bias strongly penalizes NSE. The R2 values are high throughout the study domain, likely due to the fact that the seasonal cycle of CH4 emissions dominated the temporal variability in most of the grid cells and the seasonal cycles were in phase between RF-DYPTOP and LPX-Bern. RE values calculated between RF-DYPTOP and LPX-Bern were high in areas where the emissions estimated by RF-DYPTOP were also high (compare Figs. 8c and 6a). This is likely due to the fact that, even though the seasonal cycles were in phase, their amplitudes were different which increased the variability between LPX-Bern and RF-DYPTOP (i.e. increase in RE).

The uncertainties of the upscaled fluxes were estimated from the spread of predictions made with the ensemble of 200 RF models (Fig. 9). The uncertainty mostly scales with the flux magnitude (compare Fig. 6a–c with Fig. 9a–c), meaning that grid cells with high fluxes tend to also have high uncertainties. However, the relative flux uncertainty does have some geographical variation (Fig. 9d–f). The highest relative uncertainties are typically at the highest and lowest latitudes of the study domain. In these locations the dependencies between the predictors and the CH4 flux are not as well defined as in the locations with lower uncertainties leading to larger spread in the ensemble of RF model prediction. For instance, at low latitudes LSTn may go beyond the range of LSTn values in the training data (see the range in Fig. 3), and hence the RF model predictions are not well constrained in these situations. On the other hand, lower relative uncertainties are typically obtained for locations close to the measurement sites incorporated in this study (compare Figs. 1 and 9), since the dependencies between the predictors and the CH4 flux are better defined.

The seasonalities of the upscaled fluxes and CH4 fluxes from process models are similar with the highest CH4 emissions in July–August and the lowest in February. This seasonal pattern is consistent throughout the study domain (Fig. 10). Warwick et al. (2016) and Thonat et al. (2017) showed that the northern wetland CH4 emissions should peak in August–September in order to correctly explain the seasonality of atmospheric CH4 mixing ratios and isotopes measured across the Arctic. Hence, the wetland CH4 emissions presented here are peaking approximately 1 month too early to perfectly match with their findings. CH4 flux magnitude agrees well between WetCHARTs and the upscaled flux during spring and midsummer (April–July), whereas LPX-Bern estimates lower fluxes (0 % and 26 % difference, respectively). During late summer and autumn (August–October) both process models estimate slightly lower fluxes than the upscaled estimate (17 % and 19 % difference, respectively). The upscaled fluxes also show somewhat higher emissions during the non-growing season (November–March) than the two process models (27 % and 35 % difference; see Table 2), and the upscaled estimates of non-growing season emissions are relatively close to a recent model estimate (Treat et al., 2018). This result promotes the recent notion that process models might be underestimating non-growing season fluxes at high latitudes (e.g. Treat et al., 2018; Xu et al., 2016a; Zona et al., 2016).

Treat et al. (2018) adjusted WetCHARTs model output so that it matches with their estimates of non-growing season CH4 emissions and then estimated annual wetland CH4 emissions north of 40∘ N to be 37±7 Tg(CH4) yr-1 using this adjusted model output. The estimates derived here for the annual emissions using the three wetland maps are similar (see Table 2), especially when considering our slightly smaller study domain (above 45∘ N). The two process models included in this study estimated slightly lower mean annual emissions than the upscaled fluxes (11 % and 26 % difference between the mean upscaled estimate and WetCHARTs and LPX-Bern, respectively; see also Table 2). However, given the uncertainties in upscaling as well as in process models, this can be regarded as relatively good agreement. Different process models may be driven with different climate forcing data and they may have discrepancies in the underlying wetland distributions, in addition to the different parameterizations and descriptions of the processes behind the CH4 emissions. These sources of uncertainty should be recognized when models are compared against each other or against upscaling products.

In order to further evaluate the agreement between the upscaled fluxes and process models we focused on two specific regions: Hudson Bay lowlands (HBL) and western Siberian lowlands (WSL) (see locations in Fig. 1). The upscaled fluxes indicate higher annual emissions for both subdomains compared to the two process models or previously published estimate (Table 2). For WSL the upscaled estimates are within the range of variability observed between process models and inversion modelling in WETCHIMP-WSL (Bohn et al., 2015) and close to Thompson et al. (2017). The upscaled estimates by Glagolev et al. (2011) might underestimate CH4 emissions from the WSL area (Bohn et al., 2015). Furthermore, the process models in Bohn et al. (2015) are likely underestimating the non-growing season CH4 emissions which might partly explain the discrepancy to the upscaled estimates in this study. Hence, the upscaled CH4 emission estimates for the WSL area, while large, are still in a reasonable range.

For HBL, the discrepancy between upscaled emission estimates and the estimates based on process models or previous studies is larger (Table 2). The upscaling results agree with Zhang et al. (2016) and Melton et al. (2013) but show emissions that are twice as large for the HBL than the other estimates (Table 2). This cannot be explained by wetland mapping since the difference also holds when the DYPTOP wetland map is used in upscaling. There are only few long-term EC flux studies conducted in the HBL area and the only one found (Hanis et al., 2013) showed on average 6.9 g(CH4) m-2 annual emissions at a subarctic fen located in the HBL. If the upscaled CH4 emissions are downscaled back to ecosystem level in the HBL area with wetland maps, we get on average 11.0 g(CH4) m-2 annual CH4 emission for the HBL area based on the RF model output, which is 1.6 times larger than the estimate by Hanis et al. (2013). While Hanis et al. (2013) studied only one wetland during different years than those used here (years 2008–2011 in Hanis et al., 2013, here 2013–2014), it is still noteworthy that the relative difference between Hanis et al. (2013) and this study is similar to the discrepancy between this study and the inversion estimates (Pickett-Heaps et al., 2011; Thompson et al., 2017) at the whole HBL scale. Pickett-Heaps et al. (2011) and Thompson et al. (2017) show near zero CH4 emissions during October–April and onset of CH4 emissions in mid-May or even June, largely dependent on when the ground was free of snow and unfrozen. This is somewhat surprising given the fact that only 32 % of wetlands in the area are underlain by permafrost (based on amalgam of PEATMAP and the permafrost map), and hence the soils are likely not completely frozen and some non-growing season CH4 emissions are likely to occur in such conditions (e.g. Treat et al., 2018). The upscaled non-growing season CH4 emissions show on average 1.1 Tg(CH4) yr-1 emissions for the HBL area. This partly, but not completely, explains the discrepancy between the CH4 emission estimates for the HBL area. All these results suggest that the upscaled product likely overestimates CH4 emissions from the HBL area.

The RF model performance was worse when compared against independent validation data than what has been achieved in previous upscaling studies for GPP and energy fluxes (R2>0.7) and ecosystem respiration (Reco; R2>0.6) (e.g. Jung et al., 2010; Tramontana et al., 2016). However, the RF model performance for monthly CH4 emissions was comparable to net ecosystem exchange of CO2 (NEE) (R2<0.5) (e.g. Jung et al., 2010; Tramontana et al., 2016). Likely reasons for this finding include, for instance, that for other fluxes there is simply more data available from several sites spanning the globe. For example, the La Thuile synthesis dataset used by Jung et al. (2010) and Tramontana et al. (2016) consists of 965 site years of data from over 252 EC stations. Here we have data from 25 sites with CH4 fluxes. Furthermore, the drivers (or proxies for the drivers) of, for example, GPP and energy fluxes are more easily available from remote-sensing (e.g. MODIS) and weather forecasting re-analysis datasets (e.g. WFDEI). In contrast, CH4 emissions are more related to below-ground processes, thus drivers for these processes are more difficult to measure remotely. Also, there are temporal lags between changes in drivers (e.g. LSTn) and CH4 fluxes in response to these changes. Consequently, training a machine-learning model such as RF on such data is difficult since the RF model assumes a instantaneous relationship between the change and response. However, one should also note that GPP or Reco are never directly measured with the EC technique, they are always at least partly derived products (Lasslop et al., 2009; Reichstein et al., 2005). Hence, direct functional relationships between GPP and Reco and their environmental drivers are inherently included in these flux estimates, whereas NEE and CH4 emissions are directly measured without additional modelling. Also, both NEE and CH4 emissions are differences between component fluxes (NEE: GPP and Reco; CH4 flux: production and oxidation). Therefore, GPP and Reco upscaling algorithms show better correspondence with validation data than for NEE or CH4 emissions and the results for NEE would be the correct point of reference for the RF model performance presented here.

While the RF model performance in this study was inferior to previous upscaling studies for other fluxes when evaluated using different statistical metrics, it was still comparable to what has been shown before for several process models for CH4 emissions (McNorton et al., 2016; Wania et al., 2010; Zürcher et al., 2013; Zhu et al., 2014; Xu et al., 2016a). For instance, McNorton et al. (2016) validated the land surface model JULES against CH4 flux data from 13 sites and found R2=0.10 between the validation data and the model. Wania et al. (2010) found on average RMSE =29 nmol m-2 s-1 and RMSE =42 nmol m-2 s-1 with and without tuning their model LPJ-WHyME against CH4 flux data from seven sites, respectively. Zürcher et al. (2013) found the time-integrated CH4 flux to be well represented by LPX-Bern model across different sites. A tight correlation (R2=0.92) is found between simulated and measured cumulative site emissions after calibrating the model against the measurements. While Xu et al. (2016a) did not explicitly show any statistical metrics, their model (CLM4.5) comparison against site level CH4 flux data seemed to be somewhat better than in Wania et al. (2010) or McNorton et al. (2016). Xu et al. (2016a) emphasize the importance of non-growing season emissions and the fact that their model was clearly underestimating these emissions. Zhu et al. (2014) calibrated their model (TRIPLEX-GHG) for each measurement site by changing, e.g. the Q10 for CH4 production and CH4 to CO2 release ratio to be site-specific and found on average R2=0.64 when comparing the calibrated model against measurements at 17 CH4 flux measurement sites. However, their findings are not directly comparable to the RF model agreement with validation data shown here due to their model calibration against data before comparison. Nevertheless, their results show that, even after calibration, the process models are not fully able to capture the CH4 flux variability in measurements. Miller et al. (2014) argued that the structure of some of the process models is so complex that the required forcing variables may not be reliable at larger spatial scales. All of these five models (JULES, LPJ-WHyME, LPX-Bern, CLM4.5 and TRIPLEX-GHG) are contributing to the global CH4 budget estimation within the Global Methane Project (Saunois et al., 2016), highlighting that these results summarize the agreement between state-of-the-art process models and field measurements.