We examine two simple models – the baseline and the abcd model (both lumped and distributed scheme) in order to identify a suitable one for serving as a HE.

In terms of the abcd model, we evaluate both the lumped and distributed model schemes, although most previous applications of the model are conducted in a lumped scheme (Bai et al., 2015; Fernandez et al., 2000; Martinez and Gupta, 2010; Sankarasubramanian and Vogel, 2002, 2003; Vandewiele and Xu, 1992; Vogel and Sankarasubramanian, 2003). In the lumped scheme, each of the 235 river basins is lumped as a single unit, and each of the data input (see Sect. 2.3.1) are the lumped average across the entire basin, and thus all the model outputs are lumped as well. In terms of the distributed scheme, however, each 0.5∘ grid cell has its own data inputs, and likewise, the model outputs are simulated at the grid level. Although the two schemes differ in the spatial resolution of their inputs and outputs, their within-basin parameters are uniform. We use basin-uniform rather than grid-specific parameters for the distributed scheme for two reasons: (1) to enhance computational efficiency and (2) to avoid drastically different parameters for neighboring grid cells that may be unrealistic. Note that lateral flows between grid cells and basins are not included at this stage for the abcd model. For the baseline model, as documented in Sect. 2.1.1, every 0.5∘ grid cell of each basin has its own monthly climatology runoff estimates for each of the 12 calendar months.

Typically, most applications of the abcd model utilize single-objective optimization for total runoff (or streamflow) during the calibration process to minimize the difference between measured and simulated streamflow (Bai et al., 2015; Martinez and Gupta, 2010; Sankarasubramanian and Vogel, 2002). While this may lead to a good fit for simulated total runoff, however, it may result in inappropriate partition of total runoff between direct runoff and baseflow. To improve the accuracy of the simulated total runoff and the partition between direct runoff and baseflow, we introduce the baseflow index (BFI) into the objective function.

Unlike the baseline model, the abcd model requires a calibration step for reasonable parameterization so as to enable good prediction. As mentioned above, we incorporate BFI into the objective function during the calibration process. On one side, we maximize the Kling–Gupta efficiency (KGE) (Gupta et al., 2009), which is used as a metric to measure the accuracy of the simulated total runoff relative to the VIC benchmark runoff. The KGE is defined as the difference of unity and the Euclidian distance (ED) from the ideal point; thus, we maximize KGE through minimizing the ED. The KGE and ED are calculated as follows (Gupta et al., 2009): KGE=1-ED,ED=(r-1)2+(α-1)2+(β-1)2,r=Covsoσs⋅σs,α=σs/σo,β=μs/μo, where r, α, β, and Covso are relative variability, bias, correlation coefficient, and covariance between the simulated and observed values (here we treat the VIC runoff as the observed), respectively; μ and σ represent the mean and SD (subscripts “s” and “o” stand for simulated and observed values). On the other side, we also nudge the simulated BFI towards the benchmark BFI (here we treat the benchmark BFI as the observed) – the mean BFI of the four products from (Beck et al., 2013). Then, the objective function is as follows: min⁡ED+absBFIobs-BFIsim, where min⁡ stands for minimizing the value in the parenthesis, abs represents absolute value, ED is the Euclidian distance between the simulated and observed total runoff (Gupta et al., 2009), and BFIobs and BFIsim are the observed and simulated BFI, respectively. Here we treat the benchmark runoff from the VIC and BFI from Beck et al. (2013) as observed values. We then minimize the objective function for parameter optimization by utilizing a genetic algorithm (GA) routine (Deb et al., 2002). Note that for the distributed model scheme, we aggregate the grid-level total runoff estimates to basin level and then nudge it toward the basin-level benchmark total runoff during the calibration process.

To evaluate the predictability and efficiency of the baseline and the abcd model so as to identify a suitable one to serve as a HE, we have conducted a series of simulations. Specifically, for the baseline model, no simulations are needed as it uses the interannual mean value for each month – 12 monthly values – as prediction, so we just replicate the 12 monthly runoff values for 1971–2010 and for each of the global 235 basins and then compare them against the benchmark runoff product. For the abcd model, two sets of model simulations across the global 235 basins are conducted, with one set for calibration and the other one for validation, for both the lumped and distributed model schemes. For the first set, we run the model for each basin for the period of 1971–1990 to get basin-specific parameters by using the GA approach (see Sect. 2.4). For the second set, using the parameters identified in the first set of simulation, we run the model for the period of 1991–2010 to validate the model predictability and also evaluate the computational efficiency. Model inputs and outputs in the distributed scheme are at a spatial resolution of 0.5∘, whereas those in the lumped scheme are all in lumped single units for each basin. All model simulations are conducted at a monthly time step. Note that users can run the identified HE for the global 235 basins, or for as many basins as they want for either scheme, as all the related basin-specific input data and calibrated parameters for both schemes are open-source.

Generally, we find that the baseline model performs worse than the abcd model (Fig. 2). The baseline model exhibits a lower global mean KGE value (0.61) than the lumped and distributed schemes of the abcd model (0.75 and 0.79, respectively). In addition, our analysis indicates that the incorporation of BFI into the objective function leads to a significant improvement in the partition of total runoff between direct runoff and baseflow (Figs. 3 and S4), without compromising predictability for total runoff, i.e., the global mean KGE values for modeled total runoff with or without the incorporation of BFI are almost the same (0.75 vs. 0.76). Specifically, for the case of involving both the total runoff and BFI in the objective function, the correlation efficiencies (r) between the long-term annual benchmark and modeled direct runoff and between benchmark and modeled baseflow from the lumped scheme across global basins are both 0.98 (Fig. 3) and are much higher than those of 0.86 and 0.72 in the case of only involving the total runoff in the objective function (Fig. S4). Given the superiority of the abcd model over the baseline model, we focus in the following sections on evaluating the predictability and computational efficiency of the abcd model and its potential to serve as a HE.

In terms of total runoff, we find that the lumped and distributed schemes are comparably capable of simulating long-term mean annual quantity, temporal variations, and spatial patterns for the vast majority of river basins globally (Figs. 3–5). Estimates of long-term mean annual total runoff from both the lumped and distributed schemes match very well with that of VIC total runoff across the 235 basins, with a correlation coefficient (r) higher than 0.96, for both the calibration and validation period (Fig. 3). Similarly, the basin-level estimates of long-term mean annual direct runoff and baseflow also match well with those of the VIC across the globe, for both schemes and both periods (Fig. 3). This suggests that both schemes possess the capability of partitioning total runoff.

Furthermore, both schemes display good capability of capturing the seasonal variations in the total runoff for both the calibration and validation period (Figs. 4 and S5). Meanwhile, although the spatial patterns of annual total runoff from the lumped scheme present a general match with that of the VIC, this does not reflect the spatial variations inside a basin, which are, however, captured by the distributed scheme (Fig. 5). Likewise, overall much lower percentage differences between the modeled runoff from the distributed scheme and the VIC runoff product than those between the VIC and the lumped scheme further corroborate the significantly better performance of the distributed scheme (Fig. S6). Both schemes still show large percentage differences in some dry (e.g., North Africa) or cold regions (e.g., Tibetan Plateau). This is because the runoff there is at a low magnitude, and thus small changes in runoff will lead to large percentage differences. Therefore, the distributed scheme provides overall slightly higher KGE (Fig. 6), with a global mean KGE value of 0.79 as compared to 0.75 for the lumped scheme (Fig. 2).

To ensure good model predictability for the major water fluxes, we also evaluate the modeled ET estimates. The modeled ET compares reasonably well with the VIC ET product as well as with the mean synthesis of the LandFlux-EVAL ET product (Mueller et al., 2013), displaying similar spatial variations (Fig. S7). Likewise, the distributed abcd scheme tends to have better capability in presenting spatial heterogeneity than the lumped one. In addition, the percentage differences between our modeled ET and the VIC ET product further confirm that the distributed scheme significantly outperforms the lumped one (Fig. S8), with much lower differences from the VIC ET product, although discrepancies still exist in some extremely cold (e.g., Greenland) or dry regions (e.g., North Africa), which is because small differences in ET will lead to a large percentage difference in those regions with low ET. Further, given that the changes in basin-scale monthly soil moisture are relatively small, precipitation should approximate to the sum of ET and runoff according to the water mass balance; the good predictability of seasonality in runoff as illustrated in Fig. 4 also reflects similar performance for ET.

The distributed scheme appears to outperform the lumped scheme in terms of goodness of fit, especially in some cold (e.g., Arctic, northern European, interior Tibet) and in some dry (e.g., North Africa) regions (Fig. 6). This is possibly because distributed inputs can reflect basin-level heterogeneity, and thus better capture the characteristic of the hydrological conditions in those regions. However, neither scheme performs well at the southern end of the Andes Mountains (Fig. 6). This may be attributed to the complex land surface characteristics in that mountainous area, which cannot be resolved due to the coarse spatial resolution. Moreover, the distributed scheme seems not to perform very well in some cold regions (Fig. 6), which is possibly due to a lack of representation of permafrost in the model.

Previous studies investigating the credibility of lumped and distributed hydrological models indicate that, in many cases, lumped models perform comparably to or just as well as distributed models (Asadi, 2013; Brirhet and Benaabidate, 2016; Ghavidelfar et al., 2011; Michaud and Sorooshian, 1994; Obled et al., 1994; Reed et al., 2004; Refsgaard and Knudsen, 1996; Yao et al., 1998). However, distributed models may have advantages for predicting runoff in ungauged watersheds (Reed et al., 2004; Refsgaard and Knudsen, 1996) and for capturing the spatial distribution of runoff due to heterogeneity in rainfall patterns or in land surface (Downer et al., 2002; Paudel et al., 2011; Yao et al., 1998). Our results on the predictability of the lumped and distributed abcd model are in line with previous findings in the literature.

The good agreement between our modeled water fluxes, including total runoff, direct runoff, baseflow, and ET, and the benchmark products provides confidence in the capability of both the lumped and distributed schemes in estimating temporal and spatial variations in major water fluxes across the globe. In addition, to identify a suitable HE, the required computation cost is another key factor as detailed below.

While the performance of model predictability is comparable for the lumped and distributed schemes as elucidated above, great disparities still exist for the runtime of the two schemes and the VIC model (Table S1 in the Supplement). Take the Amazon Basin that covers a total number of 2002 0.5∘ grid cells as an example: it takes 11.05 min for model calibration via the GA method for the distributed scheme but only 0.16 min for the lumped one. A similar disparity is also found for model simulation with calibrated parameters, with a runtime of 0.03 and 3.20 s for a 1000-year simulation of the Amazon Basin for the lumped and distributed schemes, respectively. However, according to the authors' experience, it will take ∼ 1 week for the VIC model to accomplish the same job, which is far more computationally expensive. In general, the computational efficiency of the lumped scheme is 2 orders of magnitude higher than the distributed one, although that of the distributed one is still much higher than the VIC (∼ 5 orders of magnitude) and many other GHMs and LSMs. Note that all of the simulations here are conducted on the Pacific Northwest National Laboratory (PNNL)'s Institutional Computing (PIC) Constance cluster using one core (Intel Xeon 2.3 GHz CPU) with the same configuration.

The good predictability and computational efficiency of both the distributed or lumped schemes as elucidated in Sects. 3.2 and 3.3 suggest its suitability for serving as HEs that can efficiently emulate complex GHMs (e.g., the VIC or others). The source codes, input data, and basin-specific parameters across the globe for both the lumped and distributed schemes are open-source and well-documented, which will make the HE ready to use and facilitate the schemes' wide and easy use with minimal efforts.

The choice of either the distributed or lumped scheme as HE depends on the user's specific needs. There is a tradeoff between the model predictability and computational efficiency. While the distributed scheme tends to better capture the spatial heterogeneity of water fluxes and can produce grid-level outputs that the lumped scheme cannot, it incurs a higher computational cost than the lumped scheme. For applications that aim to strike a balance between predictability and computation cost, such as a practical assessment of water resources or an estimation of water supply for integrated assessment models (IAMs) or a quantification of uncertainty and sensitivity analyses, it would be reasonable to employ the lumped scheme as a HE. The lumped scheme is especially advantageous due to its minimal calibration and computational cost, parsimonious efforts for model implementation, and reasonable fidelity in estimating major water fluxes (e.g., runoff, ET). For users from the IAM community, the lumped scheme might be sufficiently suitable for their needs since (1) the lumped scheme can operate at the same spatial resolution at which IAMs typically balance water demands and supplies (Edmonds et al., 1997; Kim et al., 2006, 2016) and (2) the inherent uncertainty of the lumped scheme is likely comparable to or even overshadowed by the intrinsic uncertainty of IAMs (Kraucunas et al., 2015; O'Neill et al., 2014). Similarly, for users who aim to conduct uncertainty and sensitivity analyses, the high computational efficiency of the lumped scheme allows the users to emulate the hydrological model of interest (e.g., GHMs, LSMs) and then run a large number of simulations to conduct the model's uncertainty and sensitivity analysis (Scott et al., 2016). Therefore, the high computational efficiency makes the lumped scheme more appealing as a HE in these cases. However, if the research questions hinge on the gridded estimates or emphasize the spatial heterogeneity of the water fluxes or pools, it would be more desirable to deploy the distributed scheme as a HE instead. For example, a follow-up work is coupling the distributed scheme of the HE with a widely used IAM, the Global Change Assessment Model (GCAM; Edmonds et al., 1997) and then using the coupled model to investigate the impacts of a variety of land use policies on global water scarcity, where the HE is used to estimate grid-level runoff globally under different land use policies.

While many studies indicate that basin runoff generation is sensitive to factors such as physical characteristics, spatiotemporal variability in storage distribution, and forcing input, evidence also shows that basin response can be captured using a handful of parameters (Hsu et al., 1995; Young and Parkinson, 2002). In this study, the lumped scheme of the HE ignores the spatiotemporal variability in basin characteristics by averaging the input forcing data; consequently, the associated responses in within-basin runoff or ET variations cannot be captured. In contrast, the distributed scheme presents a better performance in capturing spatiotemporal variability in runoff and ET with the use of the same input data, and without increasing the number of parameters. Thus, the use of the distributed scheme is preferred when the tradeoff in the computational efficiency is not a constraining factor.

Moreover, a combination of a top–down approach (Sivapalan et al., 2003) and a multi-objective approach to model evaluation (Gupta et al., 1998) could be used to explore internal basin behavior, wherein the top–down approach would start from a simple structure and then progressively expand based on its caveats in reproducing overall basin behavior (e.g., Jothityangkoon et al., 2001). In this study we adopt a similar framework, by starting from a baseline model and then expanding to the abcd model with snow representation, also by incorporating the baseflow index into the objective function to exert a multi-objective approach. Our assessment indicates that a baseline model characterized by mean seasonal cycle still holds promise in predicting runoff at basins with small variability in basin characteristics, such as the basins of the Ob', Lena, Yenisey, Siberia and Mackenzie in the Arctic area, where the baseline model yields KGE values of greater than 0.90 from our evaluation. Further, while Martinez and Gupta (2010) indicated that the incorporation of the snow component and an additional snow parameter into the original abcd model has greatly improved model performance in snow-dominated regions, areas without prevailing snow (e.g., the tropical zone) could still utilize the original version of the abcd model to keep the model as parsimonious as possible without compromising model predictability. In addition, although our results reveal that the incorporation of the baseflow index into the objective function generally improves the model performance in the partitioning of runoff between direct runoff and baseflow, simply employing a single-objective approach (i.e., only involving total runoff) also works well for some basins such as northern interior Africa and interior Australia. Thus, the single-objective approach is also acceptable for those basins with the advantage of simplicity without compromise in performance. In short, according to specific basin characteristics and the research needs, suitable model complexity and the number of parameters could be identified by following the abovementioned scenarios, such that either the baseline model or a reduced format of the HE (e.g., without snow representation or single-objective) could be potentially utilized with the merits of simplicity, reasonable predictability, and computational efficiency rather than adopting the full format of the HE. This HE could be used to emulate a wide range of models with different spatial and temporal complexities, and its performance may vary from model to model. Thus, examining and comparing the extent to which the HE could mimic the behaviors of different GHMs and LSMs is of future research interest to us. In addition, future research can extend this work by systematically investigating the role of different levels of inputs and parameters on model performance in different basins across the globe.

Based upon our open-source HE and the validated basin-specific parameters across the globe, researchers can easily investigate the variations in water budgets at the basin, regional, or global scale of interest, with minimum requirements regarding input data, efficient computation performance, and reasonable model fidelity. Likewise, researchers can utilize the framework of the HE with any alternative input data or recalibrate the HE to emulate other complex GHMs or LSMs of interest to meet their own needs.

To demonstrate the capability of the examined abcd model serving as a HE, we use the lumped scheme to conduct parameter-induced uncertainty analysis for the runoff simulation at the world's 16 river basins with top annual flow (Dai et al., 2009). Specifically, for each of the 16 basins, we first apply ±10 % change to each of the five calibrated parameters (a, b, c, d, m) to compose varying ranges; note that we just truncate the range to those ranges listed in Table 1 if the ±10 % change exceeds the valid range. Then we randomly sample the five parameters from corresponding ranges for 100 000 times (i.e., 100 000 combinations of parameters). After that, we run the lumped scheme 100 000 times for each basin with the 100 000 combinations of parameters to examine the parameter-induced uncertainty in total runoff. The uncertainty analysis indicates that most basins are robust to changes in parameters, except for the Tocantins, Congo and La Plata (Fig. 7). In other words, for the basins Congo and La Plata, slight changes in parameters may lead to large changes in runoff estimates. Then the uncertainty in the calibrated parameters for the two basins may lead to large bias in the simulated runoff, which may more or less explain why modeled runoff for the two basins tends to have higher biases than that for other basins (Fig. 4). Notably, the 100 000 simulations only take ∼ 80 s on a Dell Workstation T5810 with one Intel Xeon 3.5 GHz CPU, which demonstrates the extraordinary computational efficiency of the lumped scheme and its advantage for serving as a HE.