A simple statistical model is calibrated at each GRACE mascon individually, meaning that model parameters are space-dependent. One model is calibrated for each combination of the two GRACE products (Table 1) with the three precipitation products (Table 2). The meteorological forcing is always spatially averaged over the spatial footprint of the GRACE mascons. Because the model described here does not have any explicit constraint in terms of mass or energy conservation, we refer to it as a statistical model; however its formulation is largely inspired from basic principles of hydrological modeling. Assuming a linear water store model, water outputs are directly proportional to the storage and to the residence time of the water store (e.g., Beven, 2012), so that the temporal evolution of the storage can be approximated as 1TWSt=TWSt-1⋅e-1τ(t)+Pt, where t is a daily time vector, TWS(t) is the storage, P(t) is the precipitation input, and τ(t) is the residence time of the water store.

Small (large) values of the residence time indicate that water inputs tend to leave the reservoir quickly (slowly), through either runoff or evapotranspiration. Here we introduce seasonal changes in residence time (e.g., related to snow accumulation during the cold season or increased evaporative demand during the warm season) using a temperature-dependent relationship. The residence time used in Eq. (1) is formulated as a function of de-trended daily air temperature: 2τt=a+b⋅TZ(t), where a and b are calibrated model parameters with a positive sign and TZ(t) is a transformation of the original de-trended daily air temperature T(t). The purpose of this transformation is to first make τ only sensitive to changes in temperature when temperature is higher than 0 ∘C, 3T0=0,T<0T,T≥0, and to moderate the influence of extreme temperature values by applying a sigmoid transform to the standardized temperature: 4TZ=1-tanh⁡T0-meanT0SDT0. As a result of this transformation, TZ approaches a value of 1 (0) when temperature gets colder (warmer) and thus the residence time increases (decreases) (Eq. 2). Note that different or more complex formulations (e.g., also involving net radiation) were tested but did not yield significant improvement compared to the relatively simple approach presented here. The result of this model is illustrated in Fig. 1a, which depicts the temperature-dependent residence time (red line), the daily precipitation input (blue bars) and the resulting terrestrial water storage time series (blue line).

The initial value of the storage (TWS(t) at t=0) is computed from the analytical solution for the equilibrium state of Eq. (1) given the mean precipitation input and the mean residence time: 5TWS(0)=mean(P)1-meane-1τ(t). The initial value of the storage is thus obtained as the ratio between the mean rate of water input and the mean rate of water loss (also see the full development in the Supplement). Using this solution (Eq. 5) requires the assumption that the storage is close to equilibrium at the start of the reconstruction but avoids the loss of 6 years for model spin-up as was done in previous work (Humphrey et al., 2017). Still, we note that reconstructed TWS anomalies at the very beginning of the time series (typically the first year) should be interpreted with care.

The daily water storage time series (Eq. 1) is averaged to monthly temporal resolution (tm) in order to make it comparable with the monthly GRACE time series. Calibration is conducted at a monthly scale against de-seasonalized and de-trended GRACE TWS observations (Fig. 1b), such that 6anomGRACEtm=β⋅anomTWStm+ε, where β is a calibrated scaling factor, ε corresponds to an error term, and anom() is an operator indicating that the seasonal cycle and the linear trend are removed as mentioned above. The trends are removed during model calibration because many trends in GRACE are caused by anthropogenic activities (Humphrey, 2017; Rodell et al., 2018), which our climate-driven model cannot explain by definition. We note that as a result, the choice of the GIA model used in GRACE processing (Table 1) does not impact the model calibration. Removing the seasonal cycle lets the model focus on capturing the inter-annual variability correctly. The three model parameters (a, b: Eq. 2 and β: Eq. 6) are calibrated at each mascon using a Markov chain Monte Carlo (MCMC) procedure minimizing the sum of squares of the residuals between the predicted and observed monthly TWS anomalies (Haario et al., 2006; Humphrey et al., 2017). The MCMC procedure provides distributions of equally acceptable parameter sets which are later used in the generation of ensemble members (Sect. 2.4).

The empirical residuals (ε) in Eq. (6) correspond to the difference between observed and predicted water storage anomalies. They include measurement and leakage errors from GRACE, structural model errors, and errors introduced by the imperfect meteorological forcing. In this section, we aim to quantify and communicate the magnitude of these errors to end users in a practical way. A classical approach is to provide the standard error σε for every mascon m1,…,i,…,n (Fig. 2a): 7σεmi=varianceεmi. Because it can be shown in our case that the residuals are normally distributed (Fig. 2b), it is relatively safe to use the standard error to estimate the predictive uncertainty (and any confidence interval) over a given mascon. However, in many applications, predictions from individual mascons need to be aggregated, for instance to compute basin-scale averages or global means. In this case, obtaining an error estimate for the aggregated value is not trivial because the spatial covariance of the errors needs to be taken into account during the error propagation (Bevington and Robinson, 2003). Because errors are spatially and temporally correlated, any averaging operation (in the time or space domain) potentially requires that error covariance is taken into account.

To provide a practical solution to this problem, we generate ensemble members which incorporate the spatial and temporal covariance structure of the residuals. These ensembles can be easily averaged over any larger area, and once averaged they provide a predictive spread that is representative of the aggregated error. In order to generate these ensembles, we present hereafter a spatial autoregressive (SAR) noise model (Cressie and Wikle, 2011), which aims at reproducing the spatial and temporal autocorrelation structure found in the empirical residuals (ε). The SAR model is used to generate random realizations of these residuals (hereafter denoted ε^) which have a spatial and temporal autocorrelation structure that is comparable to that of the empirical residuals (ε). De-seasonalized ensemble members (GRACEREC) are obtained by combining the monthly water storage predictions (from Eq. 6) with the randomly generated residuals ε^. 8GRACERECtm=β⋅deseasTWStm+ε^tm

In the SAR model (Cressie and Wikle, 2011), residuals (ε^(tm), hereafter noted ε^tm) at a given monthly time step are represented as the sum of (1) the product of the residual of the antecedent month (ε^tm-1) with a local (mascon-specific) autoregressive parameter (φ) and (2) spatially autocorrelated innovations (η) that are randomly generated from a multivariate Gaussian with zero mean and covariance matrix Qn: 9ε^tmm1⋮ε^tmmn=φm1⋅ε^tm-1m1⋮φmn⋅ε^tm-1mnηm1⋮ηmn, where m1,…,n corresponds to the mascon index and squared brackets indicate a n×1 vector. An equivalent vector notation yields 10ε^tm=φ∘ε^tm-1+η,η∼iid Gau0,Qn, where ε^tm, ε^tm-1, φ and η are n×1 vectors, Qn is a n×n spatial covariance matrix and ∘ denotes the Hadamard product (i.e., pair-wise multiplication).

The local autoregressive parameters φm1,…,mn are estimated at each mascon from the lag-1 temporal autocorrelation of the empirical residuals (ε) (φ illustrated in Fig. 2c) (Wilks, 2011). To estimate the spatial covariance matrix of the innovations (Qn), we employ the following procedure. First, an isotropic exponential decay autocorrelation function (Eq. 11) is fitted at each individual mascon (Fig. 3a, b) to represent the spatial autocorrelation (AC) of the empirical residuals, such that 11ACd=e-dk, where d is the distance and k is the parameter to fit. Locations with high (low) values of k (Fig. 3c) indicate regions where the residuals have a strong (weak) spatial autocorrelation. The calibrated AC functions are then used to construct the spatial autocorrelation matrix Pn, which approximates the structure of the spatial autocorrelation matrix of the empirical residuals. From this, the covariance matrix for the innovations is obtained by definition as 12Qn=diagσηPndiagση, where ση is a n×1 vector containing the standard deviation of the innovations at each mascon estimated from (Cressie and Wikle, 2011) 13ση=σε∘1-φ2, where σε is the empirical standard error of each mascon (Eq. 7, Fig. 2a). The multiplication with 1-φ2 scales the empirical standard error under the assumption of an autoregressive process of order 1 (Cressie and Wikle, 2011). This accounts for the fact that the variance of an autoregressive process (σε) is larger than that of the driving white noise process (ση). In the special case where the first residual in Eq. (10) (ε^tm at tm=1) is generated and ε^tm-1 does not exist yet, the multiplication with 1-φ2 is not necessary and the following formulations are used instead of Eqs. (10) and (12). 14ε^1=η,η∼iid Gau0,Q′n15Q′n=diagσε⋅Pn⋅diagσε To summarize, a first residual is generated with Eq. (14) and subsequent residuals are generated from Eq. (10).

As mentioned in Sect. 2.3, the Markov chain Monte Carlo (MCMC) procedure for model parameter estimation additionally provides a distribution of equally acceptable model parameters (a, b, and β). Each parameter set provides one ensemble member for which the entire procedure described here is repeated. Thus, ensemble members combine (1) a model parameter uncertainty arising from the distribution of calibrated model parameters and (2) an estimate of the predictive uncertainty. Here, we provide 100 randomly sampled ensemble members. This number was chosen as a compromise between the size of the final dataset and the minimum number of ensemble members required to derive a reasonable estimate of the 90 % confidence interval.

The result of the above-described procedure is briefly illustrated and evaluated in Fig. 4. For illustration, Fig. 4a shows the empirical residuals ε for the month of April 2002 and Fig. 4b shows one instance of the randomly generated residuals ε^. As expected, both the empirical and the randomly generated residuals exhibit spatial autocorrelation. The generated residuals also have approximately the same variance (Fig. 4c) and lag-1 temporal autocorrelation (Fig. 4d) as that of the empirical residuals. The confidence intervals derived at a regional- or basin-scale level reliably cover the actual GRACE-based regional average, which was the initial motivation for the presented approach (illustrated for the Mississippi basin in Fig. 4e). We evaluate the overall reliability of the ensemble hindcast for regional averages over 90 large (>500000 km2) river basins using a rank histogram (or Talagrand diagram) (Fig. 4f). In the ideal case (perfect reliability), the observed TWS ranks lower than the pth percentile of the reconstruction only p percent of the time (for instance, GRACE observations should be lower than the fifth percentile of the reconstruction only 5 % of the time). According to this first-order metric (see, e.g., Hamill, 2001, for a discussion), we conclude that regional averages of the ensemble members provide reliable forecasts (Fig. 4f), with only a minor tendency to miss extreme positive TWS anomalies.

The presented method represents one amongst many possible approaches to the generation of ensemble members. This method has the advantage of reflecting the uncertainty of the reconstruction (compared to GRACE measurements) and mimics the empirical spatiotemporal autocorrelation structure of the errors while only requiring a minimal degree of model complexity and parameterization. We note that while the SAR model also represents errors coming from the GRACE solution itself, it does not include any anisotropic error structure (e.g., due to striping) due to the isotropic nature of Eq. (11). The uncertainty related to the choice of the input precipitation or training GRACE dataset can be explored independently by comparing the six different versions of GRACE-REC (see Table 3).

Finally, we note that our modeling approach could in principle be evaluated with a cross-validation experiment, using only a subset of the data to calibrate the model parameters and then evaluate the performance against the other unused data (as done in Humphrey et al., 2017). However, this would go beyond the scope and objective of this paper, which is to document the generation of the GRACE-REC product. We prefer to evaluate the ability of the final product to extrapolate beyond the model calibration period in later sections by comparing the model predictions with fully independent datasets (Sect. 4.3 to 4.5).

In this section, the ensemble mean of GRACE-REC is compared against GRACE observations. Note that this does not constitute an independent evaluation because GRACE-REC is calibrated with GRACE data (comparisons with independent sources are provided in Sect. 4.3 to 4.5). We evaluate model performance with the Pearson correlation coefficient (Fig. 5) and the Nash–Sutcliffe efficiency (Fig. 6). Model performance is highest especially in regions with dense meteorological observing systems (e.g., Europe, western Russia, North America, India, Australia) where we expect precipitation datasets to have the highest accuracy. Over South America and central Africa, the performance of the century-long reconstruction (GSWP3-based products, Figs. 5c, d and 6c, d) is slightly inferior to that of multisource and reanalysis precipitation datasets such as MSWEP and ERA5. Interestingly, there is no clear difference in performance when GRACE-REC is calibrated with the 3∘ JPL mascons (top row) or the 1∘ GSFC mascons (bottom row). We conclude that in terms of model performance, the choice of the GRACE product used to calibrate GRACE-REC is of secondary importance compared to the accuracy of the input precipitation datasets.

We compare these performance metrics with the scores obtained by hydrological models and land surface models of the Water Resources Reanalysis version 2 (WRR2) (Schellekens et al., 2017; Dutra et al., 2017), which were also forced with MSWEP precipitation. Compared to the simple modeling approach used in GRACE-REC, WRR2 models are forced with additional meteorological information (such as radiation and humidity), were calibrated using various data streams, sometimes including GRACE observations (Dutra et al., 2017; Decharme et al., 2011, 2012, 2016; Vergnes et al., 2014; Krinner et al., 2005; de Rosnay et al., 2002; Van Der Knijff et al., 2010; Döll et al., 2009; Sutanudjaja et al., 2011, 2014; van Beek and Bierkens, 2008; van Beek et al., 2011; Wada et al., 2011, 2014; van Dijk et al., 2013, 2014), and are potentially able to resolve more complex processes that are relevant for TWS, such as snow dynamics, the effect of vegetation phenology on evapotranspiration, and runoff routing through the river system. We calculate TWS in WRR2 models by summing over all simulated water reservoirs (this includes soil moisture, snow, groundwater, and surface waters whenever these are represented in the models). It is important to underline that unlike WRR2 models, GRACE-REC is directly calibrated to reproduce GRACE observations. Therefore, GRACE-REC should be interpreted here as a benchmark, indicative of the performance that is at least achievable for a given precipitation dataset. In terms of Nash–Sutcliffe efficiency, GRACE-REC often obtains better scores than the WRR2 models (Fig. 7a). This is because the reconstruction better fits the local amplitude and variance of the observed TWS signal, as already diagnosed in previous work (Humphrey et al., 2017). We note that the reconstructions driven with ERA5 precipitation are most often superior to those driven with the other two precipitation datasets.

Global averages of all GRACE-REC products are illustrated in Fig. 8a. Differences caused by different precipitation forcing datasets are much greater than the differences related to different GRACE training datasets. This is particularly true for long-term (>20 years) trends as we find that, over the overlapping period 1979–2014, the two MSWEP-based products both produce a positive climate-driven TWS trend while GSWP3-based and ERA5-based products yield a negative TWS trend. As mentioned above (see Sect. 3.5), discrepancies in long-term trends in GRACE-REC largely depend on the trends initially present in the driving precipitation data and also do not incorporate effects such as groundwater depletion or potential long-term changes in evaporative demand.

Comparisons with the de-trended GRACE global average are shown in Fig. 8b, c. We find that all GRACE-REC products produce a very similar inter-annual variability at the global scale and compare well against actual global mean GRACE, without applying any global constraint to the locally calibrated statistical model. Correlations between global means of GRACE-REC and global means of GRACE are larger than 0.75 (Fig. 9a) (evaluated over the common period 2003–2014). Compared to global means from the WRR2 models, GRACE-REC is on average better correlated (Fig. 9a) to the observed GRACE global mean and has a lower root-mean-square error (Fig. 9b), regardless of the GRACE dataset used for evaluation.