We use a modern-day reference data set of observed FW, the term observed being used to distinguish this from our later model results, with corresponding environmental data to develop an algorithm for the prediction of FW in the past, i.e. we assume that there exists a relationship between FW and the environmental variables compiled in the reference data and then apply that relationship to predicting FW in the past. We use the recently developed SWAMPS-GLWD (Poulter et al., 2017), which improves on the Surface Water Microwave Product Series (SWAMPS; Schroeder et al., 2015) using the static inventory of wetland area from the Global Lakes and Wetlands Database (GLWD; Lehner and Doll, 2004), correcting the SWAMPS data set in regions where this satellite-derived data set fails to detect water beneath closed canopies. We calculated the average monthly FW at each 0.5∘ × 0.5∘ grid cell for the years 2000 to 2012 on a monthly time step to give a modern-day FW (FWobs; annual max shown in Fig. 1). Corresponding climate data on the same spatial and temporal resolution were obtained from CRU-NCEP v4.0 (Wei et al., 2014) and averaged to give monthly values for a mean modern-day year over the same time interval. The climate data for this mean year were then used to drive two DGVMs: the Sheffield Dynamic Global Vegetation Model (SDGVM; Woodward et al., 1995; Beerling and Woodward, 2001) and the Lund-Postdam-Jenna model (LPJ; Wania et al., 2009) to produce corresponding vegetation data. The combination of these yielded a reference data set of FW, climate (temperature and precipitation) and vegetation (leaf area index, net primary productivity, transpiration, evapotranspiration, soil water content and surface runoff) variables (either SDGVM or LPJ) for a set of 0.5∘ × 0.5∘ spatial and monthly temporal resolution sites for a single modern-day average year. Some variables, such as transpiration and evapotranspiration, are available from both climate and vegetation models. In such cases we use those from the vegetation model as they will be calculated from a more advanced vegetation scheme. To ensure that wetlands in areas dominated by agriculture or areas where one of our vegetation models, SDGVM, predicts bare land did not bias our FW predictions, such grid cells were removed from the reference data. For the latter, this was done simply by removing those grid cells that SDGVM predicted to be bare land. For the former, we removed those that were 50 % or more, by cover, classed as cultivated and managed or mosaic cropland (Global Land Cover 2000 database, 2003).

Many of the methods that can be used to analyse the reference data and predict FW require that the data are scaled so that each variable covers a similar range of values. Therefore, we scaled the values of each environmental variable, X, using their global mean, μx, and global standard deviation, σx, i.e. for a given grid cell, J, each variable was scaled as 1X′J=XJ-μxσx. This scales all variables such that they have a global mean of 0 and standard deviation of 1.

A modern-day test data set was made by interpolating the reference climate data to 2.5∘ × 3.75∘, the spatial resolution often used for paleoclimate models. The DGVM simulations were driven by this interpolated data to yield the vegetation outputs. All climate and vegetation variables were scaled in the same way as the reference data, using the global means and standard deviations of the reference data. The paleoclimatic assessment of our model was performed using an early Eocene test data set made using a single year of output, on a monthly time step, from a three-dimensional fully dynamic coupled ocean–atmosphere global climate model HadCM3BL-M2.2 (Valdes et al., 2017), on a 2.5∘ latitude by 3.75∘ longitude grid. To simulate the early Eocene a Ypresian paleogeography and high CO2 concentration (4 times modern; 1120 ppm; Anagnostou et al., 2016) was used. SDGVM and LPJ were both run with these model-simulated climate data to produce the vegetation variables required, as was done for the reference data set, whereas temperature and precipitation were derived directly from the climate model. All variables were again scaled using the means and standard deviations of the reference data. Therefore, for each climate, modern day and early Eocene, we have two test data sets for a mean year on a monthly time step at 2.5∘ × 3.75∘ spatial resolution and both with the same climate data, one with SDGVM vegetation data and one with LPJ vegetation data. Predictions for each test data set were made with the corresponding vegetation model's reference data set. The reference and test data sets are summarised in Table 1.

Before discussing the model we employed to predict paleoclimate FW, it is useful to describe briefly other strategies that we attempted but did not yield robust predictions when evaluated against modern-day data. The first of these was to examine FW vs. individual environmental variables graphically from the reference data to ascertain if we could define ranges for those variables that corresponded to predominantly low or high FW; this is similar to the approach of Shindell et al. (2004), who proposed threshold values of standard deviation of topography, ground temperature, ground wetness and downward shortwave flux for wetland development. However, this proved unsuccessful, revealing only the rather obvious relationship that wetlands do not usually occur when mean monthly temperature is below 0 ∘C. Although we expected to identify relationships for FW with other environmental variables (i.e. ground wetness), none were found. This is due to the combined effects of wetland occurrence being the function of multiple factors and the fact that most grid cells have FW≈0 for all months of the year and the number of grid cells with significantly non-zero FW is quite small. Therefore, environmental variables associated with high values of FW also tend to be associated with FW≈0. Poor correlation of FW with environmental variables is also due to the important control exerted by the topography; regardless of climate, wetlands cannot form in landscapes where excess water flows away rather than remaining in situ. Collectively, these factors caused significant overlap in the range of environmental variables associated with both low and high FW.

Another approach was a multiple linear regression using the reference data in order to derive an equation for FW in terms of linear functions of multiple environmental variables. However, this yielded equations that predicted a widespread occurrence of very low FW, including those areas where FWobs is very high either seasonally or throughout the year. Similarly, poor predictive models were obtained whether derived for all sites or just those restricted to specific plant functional types. These outcomes likely occur because linear regression optimises a function by minimising the error between predicted and observed values. As most grid cells have FW≈0 (Fig. 1), the “best” regression equation is one that predicts FW to be very low almost everywhere, since in the majority of cases this is quite accurate. Efforts were made to use other optimisation criteria with customised functions that attempted to put more weight on predicting high FW correctly at the expense of larger errors where FW is low. However, these simply over-predicted FW. Therefore, we were unable to find any satisfactory solution based on linear regression. The fact that we did not find a satisfactory regression equation for FW on the reference data suggests that any relationship between FW and the environmental variables must be complex and therefore another approach is required if we are to be able to predict FW.

Given that we were unable to find simple mathematical formula with which to predict FW, we must consider another approach. Nearest-neighbour searches can be used to predict a property for a query by comparing data for that query to similar such data from a reference data set. We find the entry in the reference data set that is most similar to, i.e. the nearest neighbour of, the query, and predict the query has the same value in the property of interest as its nearest neighbour. The reference data set of FW and environmental variable sites, on a 0.5∘ grid at a monthly time step, can be viewed as a set of data points yielding FW at many different locations in a multi-dimensional space. The eight dimensions of that space are the two climate and six vegetation variables; temperature, precipitation, leaf area index, net primary productivity, transpiration, evapotranspiration, soil water content and surface runoff. If we have the same environmental variables for a site of unknown FW, we can search the reference data set for its nearest neighbour and then predict it would have the same FW as that nearest neighbour, as illustrated below.

The set of N environmental variables, suitably scaled, X1, X2...XN defines an N-dimensional space.

The aim of this study was to derive an algorithm for predicting wetland fraction that can then be used to calculate methane emissions. For the latter, we use the empirical method described by Cao et al. (1996), where methane production, mp, and methane oxidation, mo, rates for a specific grid cell and month (both in units of g CH4 m-2 month-1) are given by 3mp=Rhft,4mo=mp0.6+0.3GPPGPPmax, where Rh is absolute soil respiration and absolute GPP is gross primary productivity (both in units of g C m-2 month-1 and obtained from the respective vegetation model). GPPmax is the maximum value of GPP for that grid cell for any month of the year. ft is a function that scales for air temperature, TMP, in degrees Celcius. 5ft=exp(0.04055TMP)3.375 This is capped at a maximum value of 1. In principle there would also be a scaling function for water table depth, but this is defined as 1 for inundated wetlands and we are only modelling inundated wetland fraction, as that is how the SWAMPS-GLWD FW data set is defined.

Methane emission rate, me, is then the difference between methane produced and methane oxidised, scaled by the wetland fraction for that grid cell and month: 6me=mp-moFW.

The modern-day test set explained in Sect. 2.2 was used as a first, simple, test of the nearest-neighbour algorithm for predicting FW described in Sect. 2.4. Since the modern-day test set is simply the reference climate data interpolated from 0.5∘ to the courser HadCM3BL-M2.2 model grid of 2.5∘ by 3.75∘ (with vegetation from the DGVMs), we expect the NN algorithm to yield predicted FW reasonably consistent with a similar downscaling of the SWAMPS-GLWD observed FW. If the NN predicted FW does not achieve this, then that would indicate that the NN algorithm has failed to predict FW sufficiently accurately. Therefore this test is primarily designed to indicate that a nearest-neighbour algorithm either does or does not have the potential to be applied to paleoclimates.

Figure 2 shows maps of seasonal, June–July–August and December–January–February, average FW from the observed SWAMPS-GLWD data interpolated to 2.5∘ × 3.75∘ along with the predicted FW using either SDGVM or LPJ vegetation data test sets. For both vegetation models, the predicted FW maps are similar to the interpolated, observed data. Sparse patches of high FW occur in the tropics, especially the Amazon, throughout the year, and large areas of seasonal summer wetlands occur in Alaska, Canada, and Siberia. The monthly variation in FW north and south of 30∘ N, i.e. essentially comparing boreal and tropical wetlands is shown in Fig. 3. We split the global values into these two zones because there are virtually no Southern Hemisphere boreal wetlands, and any division based purely on latitude is arbitrary. The nearest-neighbour algorithm generates the correct seasonal FW pattern in boreal regions and, as expected, a relatively constant monthly FW in the tropics. However, SDGVM consistently underestimates the amount of tropical wetland, whilst LPJ agrees reasonably well with observations: mean monthly values are 2.11×106, 1.47×106 and 1.90×106 km2 for the observed, SDGVM and LPJ data, respectively. This is due to the fact that SDGVM classes some grid cells as bare land, assumed to have FW = 0 in our algorithm, even though some of these have non-zero FW in the SWAMPS-GLWD database. LPJ does not classify these grid cells as bare land but instead treats them as very low amounts of vegetation, therefore yielding higher global FW that is more consistent with observations. If we exclude those grid cells SDGVM predicts as bare land from the observed data, then the SDGVM prediction matches better the observed data and LPJ predictions (Table 2). These results give confidence to the fact that a nearest-neighbour algorithm is able to reproduce acceptable FW based on these specific climate and vegetation variables.

Figure 4 shows the monthly variation in wetland methane emissions for boreal and tropical areas, calculated using the observed or predicted FW, both vegetation model outputs and Eqs. (3) to (6). The annual methane emission totals are summarised in Table 3, along with other recent estimates from model intercomparisons. The annual and monthly zonal methane emissions are broadly similar for a given vegetation model regardless of whether the observed or predicted FW is used. SDGVM gives global emissions in line with the other modelling studies, whereas those from LPJ are somewhat lower. This is mainly due to differences in tropical emissions. SDGVM yields higher tropical emissions than LPJ but slightly lower emissions north of 30∘ N. The main factors influencing the modelled methane emissions (other than FW) are, according to Eqs. (3) to (5), temperature (which is the same for both vegetation models), soil respiration (Rh) and gross primary productivity (GPP), the latter two differing between the two vegetation models. It appears that differences in Rh lead to the different zonal methane totals. South of 30∘ N, SDGVM and LPJ model annual total Rh of 46 000 and 35 000 Tg C yr-1, respectively, and, using the same observed FW, SDGVM and LPJ model annual methane emissions of 123 and 69 Tg CH4 yr-1, respectively. Therefore, in the tropics the differences in the predicted methane emissions seem to be due to differences in calculated Rh. North of 30∘ N both DGVMs have similar Rh, 20 000 and 22 000 Tg C yr-1, respectively, for SDGVM and LPJ, and similar values of methane emissions, 64 and 65 Tg CH4 yr-1, respectively.

We stress that this was a simple test for a nearest-neighbour approach for reasons outlined at the beginning of this section, and the satisfactory results obtained here merely indicate that this is an approach that has potential to be useful in predicting FW for a paleoclimate.

In the previous section we have shown that a NN method can reproduce FW for a modern-day climate, justifying its application to the early Eocene climate described in Sect. 2.2. However, as noted at the end of Sect. 2.4, a NN method can be extended to KNN, whereby we predict FW based on some function of the FW of K nearest neighbours (noting that in Sect. 3.1, NN is simply 1NN, i.e. KNN with K=1). A 1NN algorithm that works well to predict modern-day FW may not work as well for a paleoclimate of many millions of years in the past. The reference data set we use, Sect. 2.1, is very similar to the modern-day test set, the latter's climate data are simply obtained by interpolating the former to a courser spatial grid. Therefore, we expected and observed a high correlation between modern-day FW predicted from the nearest neighbour in the reference data and the actual FW. The early Eocene test data has significant differences to the reference data since the climate of the early Eocene is obviously not the same as the modern day. Therefore, it will be harder for a nearest-neighbour-based method, searching a space described by climate and vegetation data, to find a nearest neighbour in the modern-day reference data with the correct early Eocene FW, whatever that may be. It may be that for a high FW early Eocene grid cell, the nearest neighbour happens to have quite low FW and vice versa. Figure 1 shows that FW can change from very high to almost zero over relatively small distances, for example in the Amazon basin, and therefore that sites with similar climate and vegetation can have very different FW. The greater the degree of difference between the early Eocene and the modern-day reference data sets, the more likely it is that the first nearest neighbour does not have the correct FW.

FW calculated for the early Eocene using the exact same 1NN method as used for the modern-day test set yields a value of global monthly mean wetland area of 4.07×106 km2 using SDGVM. This is around 33 % higher than that for the modern-day value, 3.00×106 km2, from Table 2. However, this includes a contribution of 1.53×106 km2 from areas south of 30∘ S, which have an almost negligible contribution for the modern day, so the tropics and northern boreal regions actually have lower FW for the early Eocene. Given that the early Eocene was significantly warmer and wetter than the modern day (Carmicheal et al., 2017), we expect greater wetland area than the modern day. Beerling et al. (2011) reported global wetland area for an early Eocene climate using SDGVM; employing their method to our early Eocene climate, so as to eliminate differences arising from the specific HadCM3 model climate and spatial resolution, yields a global monthly mean FW area of 16.29×106 km2, 4 times higher than the value we would calculate from a 1NN method. Therefore, based on a comparison with both the modern-day studies and a previous Eocene study, it appears that a 1NN method may be unsuitable for a paleoclimate that is very different to our modern-day reference climate, and we consider KNN with higher values of K.