The GLEAM, PT-JPL, and PM-MOD models, and the inputs required to run them globally at a 0.25∘ spatial resolution are extensively described by and . Only the main differences with respect to the original WACMOS-ET runs are fully detailed here. Note that the original 2005–2007 period is extended here to cover 2002–2007, and that the models are only run at daily time resolutions.

Agreement with the towers' ET is analysed by calculating the Pearson correlation coefficient (R), the mean square difference (MSD), and the root mean square difference (RMSD) according to the following expressions: R=N∑i=1NPiOi-∑i=1NPi∑i=1NOi[N∑i=1NPi2-(∑i=1NPi)2]N∑i=1NOi2-(∑i=1NOi)2, MSD=[1N∑i=1N(Pi-Oi)2]=RMSD2, where P and O are the model-derived and observed (or a second model-derived) variate, and N is the number of cases. The MSD can be decomposed into a random (MSDr) and systematic (MSDs) component following by using the following expressions:

MSDr=1N∑i=1N(Pi^-Oi)2=RMSDr2, MSDs=1N∑i=1N(Pi-Pi^)2=RMSDs2, where Pi^=a+bOi is the linear least squares regression of P onto O, a and b being the regression intercept and slope, respectively. Notice that MSD = MSDr + MSDs.

Statistics are calculated for the complete study period, or separately for the boreal winter (DJF), spring (MAM), summer (JJA), and autumn (SON). For the correlations, statistical significance is tested by calculating 95 % confidence intervals. For the correlation differences, a Fisher Z transformation is applied to the correlations, and a Student t test at a 5 % significance level is used to test the significance of the difference. The autocorrelation of the daily time series is taken into account by reducing the degrees of freedom using an effective sampling size .

The GLEAM, PT-JPL, and PM-MOD required that global inputs remain unchanged with respect to and , apart from the precipitation product, and are applied at the same resolution of 0.25∘. Common inputs to the models are the surface net radiation, coming from the NASA and GEWEX Surface Radiation Budget SRB, Release 3.1, and the near-surface air temperature, sourced from the ERA-Interim atmospheric reanalysis . PT-JPL and PM-MOD also require near-surface air humidity, also derived from ERA-Interim, and the vegetation products discussed in Sects.  and . Regarding GLEAM, it requires precipitation, coming from the Multi-Source Weighted-Ensemble Precipitation (MSWEP) version 1 product , soil moisture and vegetation optical depth from the European Space Agency (ESA) Climate Change Initiative (CCI) Soil Moisture v2.3 product , and information on snow water equivalents, from the ESA GlobSnow product for the Northern Hemisphere , and from the National Snow and Ice Data Center (NSIDC) in snow-covered regions of the Southern Hemisphere .

The FLUXNET 2015 synthesis data set (http://fluxnet.fluxdata.org/, last access: 8 July 2018) is used to obtain point-based measurements of evaporation (referred to as tower ET), and it is processed as in to retain only high-quality data appropriate to evaluate the evaporation estimates. Starting from the original time resolution (generally 30 min or 1 h), the processing involves (1) masking measurements using the originally provided quality flags; (2) masking measurements for rainy intervals, only leaving observations if both the global precipitation product and the local measurements (if available) do not indicate precipitation (as eddy-covariance measurements are less reliable during precipitation events); and (3) aggregating to daily values if more than 75 % of remaining sub-hourly data exists for a given day. This quality check yielded 97 stations. This sample was further reduced to 84 by visually inspecting aerial pictures of the tower surroundings and removing stations close to water bodies, or not representative of the overall land cover within the 0.25∘ cells of the gridded ET estimates. The geographical locations of the 84 stations, and their location in an air temperature and precipitation space, are plotted in Fig. , with the station names, land covers (based on the International Geosphere-Biosphere Programme (IGBP) classification), and reference or principal investigator listed in Table . Note that nearly all stations are in Europe and the US, with only two stations located in the Southern Hemisphere.

Eddy-covariance measurements are subject to errors, both random and systematic, and any merging technique using them as reference is likely to be impacted by those errors. Systematic errors can arise from instrumental calibration and unmet assumptions about the meteorological conditions, while random errors are typically related to turbulence sampling errors, the assumptions of a constant footprint area, and instrumental limitations . Estimating these errors is far from simple and typically requires dedicated experiments . As such, reporting them is not a widespread practice and error statistics for the individual sites are not commonly available.

The propagation of systematic errors typically results in the lack of energy balance closure observed at many eddy-covariance sites . Methods to correct the energy unbalance exist, with the Bowen ratio approach and the energy balance residual approach being the most frequently adopted. Corrected fluxes are typically preferred over the original uncorrected observations, but these corrections imply the need for surface radiation and soil heat flux measurements, which are not routinely measured at all stations. At the sites where they are available, the FLUXNET 2015 data set offers a test product containing a corrected version of the heat fluxes based on the Bowen ratio approach, i.e. assuming that the measured Bowen ratio is correct. For the 84 stations selected here, 26 do not have Bowen ratio corrected (BRC) fluxes. For the remaining 58 stations, the relative mean difference between the original and BRC latent heat fluxes averaged over all stations is 6.1 %, with a maximum value of 16.5 %. If the correlation coefficient between original and BRC fluxes is calculated at each station and then averaged over all stations, we obtain 0.96, showing that the original and BRC ET correlate well in time. Also, if the weights of Eq. () are calculated with the original and BRC fluxes, they display a 0.91 average correlation over all stations and models, with an average RMSD of 0.035. These numbers do not suggest strong differences between the two, and thus the original (uncorrected) fluxes for all stations are retained for our analyses in order to maximize the number of sites.

Moreover, not all stations completely cover the 2002–2007 period, with 6, 14, 24, 9, and 31 stations reporting 2, 3, 4, 5, and 6 years of data within the period, respectively. At stations where inter-annual variability is large, the weights may not be representative of the overall climate conditions at the tower if only a relatively short number of years exist. Limiting the study to stations with a relatively large number of years could minimize this drawback, but it would severely reduce the number of towers, so this filtering has not been applied. For instance, if we only derive weights for towers with at least 4 years of data, half of the towers would have been removed. Notice also that due to the masking of the tower data the 61 consecutive daily estimates required to estimate our temporally varying weights (see Sect. ) are generally not all available. Therefore, in the case of the tower data we set a minimum threshold of 15 daily values within the 61-day running window for the error to be estimated. Most stations have weights for nearly all days, but in a few stations there are recurrent gaps. A clear example is the tropical BR-Sa3 station, where the frequent rainy episodes complicate the derivation of the weights.

Because the substantial mismatch between the size of the model grid cells and the tower footprint is likely to result in representativeness errors, ancillary data sets are required to characterize the spatial homogeneity of the grid cells where the stations are located. Two data sets are considered: the MODIS Land Cover Type product MCD12Q1 at a native resolution of 500 m, and the Terra MODIS Vegetation Continuous Fields product MOD44B, available at a spatial resolution of 250 m. A homogeneity index (Ih) is constructed as follows: Ih=12FgtIGBP+12(1-∣Fgbare-Ftbare∣-∣Fgherb-Ftherb∣-∣Fgforest-Ftforest∣), where FgtIGBP is the fraction of MCD12Q1 500 m cells included in the 25 km model grid cell containing the tower and having the same IGBP land cover than the model cell, Ftbare, Ftherb, and Ftforest are, respectively, the bare, herbaceous, and forest fractions of the MOD44B 250 m cell containing the tower, and Fgbare, Fgherb, and Fgforest are the same fractions but calculated for the entire 25 km model grid cell where the tower is situated. The first term is the mismatch between the land cover at the tower and at the grid cell level, and the remaining terms are the net mismatch in land cover types across the two resolutions. Ih takes values in the range [0,1], the larger the value the more representative the grid cell is of the landscape of the tower footprint. Finally, to evaluate the merged products, we use river run-off from a compilation of monthly data using different sources, as described in , and annual precipitation estimates from WorldClim and MSWEP .

A summary of daily weight statistics over all the sites belonging to a given land cover group is given in Fig. . These weights have been derived based on the differences between the ET product anomalies and the tower ET anomalies as explained in Sect. . As expected, the simple average product (SA-merger) equally weights all products with a value of one-third and is added here as reference. Notice that the weights can take negative values, although the sum of the weights is still one. This happens when the full error covariance matrix has large off-diagonal values reflecting the correlation between the different product errors e.g.. This correlation is expected given that the products share some common inputs and model formulations, and it is specially noticeable for GLEAM and PT-JPL. On average, GLEAM has the largest weights and contributes more to the weighted anomalies, but the relative weight of each model is not uniform per season or land cover. For instance, for the forest class PT-JPL is more weighted than GLEAM in winter, while the reverse is true in autumn.

An example of the temporal variability of the weights at three towers is given in Fig. . At the FR-Pue site, a Mediterranean forest located in France , GLEAM starts to be clearly more weighted for the second part of the year. The correlation between the GLEAM and PT-JPL anomalies is visible in the anti-correlation displayed by the weights. At the US-SRM site, a semi-arid grassland site in the south-west of the US , PM-MOD is typically more weighted than GLEAM and PT-JPL in spring, and all weights depart less from the 0–1 range, suggesting more independent errors at this particular station. The last site, the US-Ne1 cropland station situated in North America , is an example of closer weights for all models for some periods of the year. This happens during the first half of the year. For the second part of the year, the weights change more, with PT-JPL being the most weighted product during some months.

Figure  shows – for the same three towers in Fig.  – time series of ET from the three products, SA-merger and WA-merger, and the in situ measurement for 2006. At the FR-Pue site, for this specific year all products disagree with the tower ET for a large part of the year, with PT-JPL and PM-MOD having much larger absolute values overall. Differences between SA-merger and WA-merger are mainly visible in spring and summer, where GLEAM is weighted more strongly, making WA-merger follow the GLEAM estimates more closely. The US-SRM site shows a relatively large ET seasonal variability, with the ET tightly linked to the precipitation and associated increases in soil moisture . GLEAM and PT-JPL capture this variability, especially the sudden increase in ET values at the beginning of summer, which is related to the rainfall coming from the North American monsoon. For the first half of summer there are sometimes large differences between SA-merger and WA-merger, with WA-merger correlating better with the tower ET. For the second half, all products fail to replicate the ET increase measured by the tower, and WA-merger and SA-merger are closer to each other as the models' anomalies cannot provide information to guide the merging. The US-Ne1 is an irrigated maize–soybean site, where the seasonal cycle of ET is expected to be more pronounced, and accompanied by higher absolute values resulting from irrigation . The original products have more similar values, not capturing well the ET rise associated with start of the growing season. This may have to do with irrigation not being well captured by any of the models. The closer weights shown in Fig.  result in closer SA-merger and WA-merger ET.

The performance of the individual and merged products across the different stations is summarized in Fig.  by plotting seasonal averaged correlations and RMSDs for the three land cover classes. The statistics are presented for the ET anomalies, and for the absolute values. For both, in 10 out of the 12 cases presented (4 seasons × 3 land covers) the correlation of WA-merger is higher than for SA-merger, indicating that an appropriate characterization of the errors – and derived weights – results in a better fit to the tower ET. The relative increases in correlation between SA-merger and WA-merger are larger for the ET anomalies, but still occur for the ET absolute values. This highlights that when the weighted anomalies are added to the multi-product climatology, the resulting product combination still overcomes the simple average. Note that the lowest correlations occur in wintertime, reflecting the low values and low intra-seasonal variability in this period, while the largest correlations are observed in spring and autumn where vegetation greening and browning typically results in larger ET variability. Note also that correlations are not significant for some stations and periods; non-significant correlations are typically found in wintertime.

Concerning the RMSDs, they are slightly lower for WA-merger for all seasons except for winter months. As SA-merger and WA-merger share their climatology (see Sect. ), large differences between both are not expected. This means that the biases between the merged products and the tower ET are preserved for both mergers, indicating that most of the differences in RMSD is coming from changes that are also reflected in the correlations.

The local weights at the 84 stations have been extrapolated by the NN as described in Sect. . The seasonal averages of the weights are presented in Fig. . Overall, the spatial patterns of the extrapolated weights for each product do not change substantially across the seasons. Some exceptions are Europe and northern Asia for GLEAM and PT-JPL. The PM-MOD weights are mostly positive, apart from forested areas in the tropics and some dry areas in Asia and Australia, and are more confined than GLEAM and PT-JPL to the 0–1 interval, indicating smaller error correlation with the other products. For GLEAM and PT-JPL, the weights are a mixture of positive and negative values, and a clear anti-correlation of the weights is visible, i.e. positive GLEAM weights correspond to negative PT-JPL weights, and vice versa, similar to the pattern observed in the local weights for some periods.

The seasonally averaged ET differences between WA-merger and SA-merger, normalized by the seasonal SA-merger, are plotted in Fig. . The large differences in (semi-)arid areas or the northern latitudes in winter are related to the very low ET absolute values. For the remaining land, most of the relative differences are within the ±25 % range. Overall, there are more negative than positive differences, indicating that the WA-merger results in smaller absolute values. Given that SA-merger and WA-merger have a common climatology, this suggests that the weighting results in an overall reduction in the anomalies at many regions.

Some geographical structures and seasonal changes are visible in some regions. For instance, in the sub-Saharan transition zone the differences are positive in the first half of the year (WA-merger > SA-merger), but negative in the second half. Over India the differences are positive in autumn and winter, but negative in spring and summer. In contrast, some regions do not display large seasonal changes. For instance, in most of Europe WA-merger is smaller than SA-merger over all seasons.

Our inverse error variance weighting is based on the differences between the model and tower ET anomalies. However, it is expected that part of the difference between in situ measurements of ET and model estimates respond to the mismatch in spatial resolution (tower footprint versus model cell). The RMSD of SA-merger against the towers' ET, normalized by the mean annual tower ET, is displayed in Fig.  for all the available stations, together with the station Ih described in Sect. . The towers are sorted from maximum to minimum Ih, i.e. starting with the towers that better represent the grid cells where they fall. Nonetheless, low and high normalized RMSDs can occur at stations with comparable Ih, indicating that spatial heterogeneity is only one of the contributing factors to the ET differences. In fact, if the RMSD is linearly regressed on the Ih, the slope of the fit is close to zero, as shown in Fig. . Also, for the separate products (GLEAM, PT-JPL, and PM-MOD) and WA-merger, no significant correlation between their RMSD against in situ measurements and Ih was found (results not shown). This indicates that for the calculated Ih, and the selected sample of ET products and stations, the error related to the inconsistencies between the tower footprint and the model pixels does not dominate the total error budget.

The objective on an inverse error-variance weighting is to find the estimate that minimizes the variance of the random error . As such, the merging only results in the optimal weights if applied over an ensemble of unbiased estimates. Strictly speaking, this requires removing the bias between the model ensemble and in the situ observations prior to the merging, which is not the case here (see Eqs.  to ). The objective here was to correct the product anomalies towards the tower anomalies, but not to correct the original estimates toward the tower in absolute terms. On the one hand the tower observations have their own systematic errors, as discussed in Sect. . On the other hand, debiasing toward the tower ET would require a global correction of the gridded products towards a global tower climatology. If the ultimate objective is to reproduce the tower fluxes, other approaches like regressing the tower ET on either the ET products or the ET explanatory drivers may appear more straightforward and be possibly more appropriate.

Nevertheless, even if optimality in the sense of minimizing the error variance of the WA-merger cannot be assured, weighting the anomalies should result in a decrease in the random error. This is shown in Fig. , where box plots of the random (MSDr) and systematic (MSDs) components of the difference between the products and the tower observations are displayed (see Eqs.  and ). From the original products, GLEAM and PT-JPL have comparative error components, while PM-MOD is more distinctive, having smaller MSDr and larger MSDs. The latter likely relates to the tendency of the PM-MOD to underestimate ET and its variance . Comparing WA-merger to SA-merger, the reduction in the MSDr for WA-merger is indicative of the merging being effective in this regard. There is also a slight reduction in the MSDs, with WA-merger having the smallest median error of all products.

The number of stations used in this merging exercise is certainly limited in terms of covering different biomes and climatic conditions. Hence, the ability to represent the full distribution of ET across time, space, and biomes is questionable. This is verified here by out-sampling the NN training data set in two different ways. In the first test all stations are included in the tower data set, i.e. the standard configuration used to produce the global WA-merger. Before training the NN, 15 % of the days at each station are randomly masked from the training data set, and the prediction statistics are derived over this independent subset. In the second test, the station where the prediction will be checked is entirely removed from the training data set, i.e, the weights for that station are derived using a NN that did not include that station in the training phase (i.e. leave-one out cross-validation).

A box plot summarizing the correlation and RMSD between the station weights and the weights predicted by the NN for these two tests is presented in Fig. . The results clearly show that the correlation and RMSDs between the predicted and the original weights at the stations degrades notably when stations are fully removed from the training data set. This implies that the global extrapolation of the weights will be quite uncertain for conditions not sampled in the available tower data set. For some stations, the out-sampling from the training data set does not have a large effect, because the mapping between the predictors and ET can still be approximated from the relationship presented by other stations. This is for instance the case for the Canadian forest stations CA-NS1-7 (results not shown). However, for other stations, the statistics are good when predicted with the standard data set, but poor with the one-station-removed data set, indicating that the particular conditions of those stations are not well represented in the out-sampled data set. This happens for stations such as US-Wi4 (forest with a snowy winter and warm humid summer) and CN-Dan (grasslands with a polar tundra climate). Finally, there are also stations where statistics are rather poor in both tests, indicating that a link between the model inputs and the related output error could not be established. This is the case for stations such as IT-Col (deciduous broadleaf forest with temperate climate) or MY-Pso (tropical forest). This guarantees that the extrapolation of weights to areas with similar conditions will be very uncertain, even if those conditions were represented in the tower data set.

An additional test to check the representativeness of the tower data set is conducted by globally extrapolating the weights with each of the previous 84 NNs trained without one station, and then checking the variability of the predicted weights. For the conditions well represented in the training data set, it is expected that removing one station will only result in slight changes in the extrapolated weights. However, for regions that are poorly represented, a slightly different data set is likely to result in substantially different weights. This is illustrated in Fig. , where a weight variability index is displayed. The index is calculated by (1) estimating for each global cell the annual standard deviation of the GLEAM, PT-JPL, and PM-MOD weights, normalized by the sum of the absolute annual model weights, and (2) averaging this standard deviation over the three models. To facilitate its display in Fig. , it has been scaled to span the range 0–1. Overall the variability is larger in the Southern Hemisphere than in the Northern Hemisphere, which is expected given that all stations but two are situated in the Northern Hemisphere. The smallest variability in the weights coincides with the regions where the database is more representative, namely the US, central Europe, and some parts of Asia, suggesting a bias in the tower data set linked to the specific location of the towers selected. The variability in tropical regions, where only three stations are part of the database, is in general larger than for the previous regions. The largest variability occurs over the very dry regions, a regime poorly represented in the tower data set as shown in Fig. . While a poor extrapolation of weights is not critical over very dry regions, given their low ET values, uncertain weights over the very humid regions is more of a concern due to their typically large ET values and their significance for the global mean ET.