What is the hydrologically effective area of a catchment?

We define a new metric, the ECI, to describe the deviation of the effective catchment area from the topographic catchment area. It is an evolution of the discharge/recharge ratio introduced by Schaller and Fan (2009) and adds the following advantages: (i) the sign of ECI indicates whether a catchment is gaining water (+) or losing water (-); and (ii) the absolute value of ECI indicates the strength of water gains and losses. Catchments with similar absolute values but different signs of ECI have the same relative influence of water gains or losses. ECI enables us to estimate the effective area of a catchment in contrast to its topographic area and to quantify inter-catchment groundwater flow. It is calculated as follows:

$$\begin{equation}{\text{ECI}} = \log \left( {\frac{Q}{{P - AET}}} \right)\end{equation} \tag{ 1 }$$

where Q [mm/d] is the long-term observed average specific streamflow at the catchment outlet (volumetric flow rate normalized by topographic catchment area). P [mm/d] and AET [mm/d] are long-term average estimates of precipitation and actual evapotranspiration, respectively (see section 2.2). For ECI > 0: catchments gain water, i.e. their effective area is larger than their topographic area; for ECI < 0: catchments lose water, i.e. their effective area is smaller than their topographic area.

Assuming the presence of inter-catchment groundwater flow (±IGF), the continuity equation can be defined by dS/dt = P-AET-Q + IGF, where dS/dt is the change of storage. Assuming further that the long-term water balance has similar starting and ending storage conditions, the influence of storage becomes negligible (dS/dt ≅ 0). We can then quantify the relationship between P-AET and Q using ECI, which quantifies the significance of IGF. The observed mean discharge at the catchment outlet, Q, represents the response of the effective catchment (${A_{eff}}$ [km²]), while the difference of catchment average precipitation (P) and actual evapotranspiration (AET), P-AET, represents the response of the topographic catchment (${A_{topo}}$ [km²]). Therefore, the ratio of the effective to the topographic catchment area (${A_{eff}}/{A_{topo}}$) can be calculated as follows:

$$\begin{equation}\frac{{{A_{eff}}}}{{{A_{topo}}}} = \frac{Q}{{P - AET}}\end{equation} \tag{ 2 }$$

Combing equations (1) and (2) yields:

$$\begin{equation}\frac{{{A_{eff}}}}{{{A_{topo}}}} = {10^{ECI}}\end{equation} \tag{ 3 }$$

where the uncertainty in estimates of this ratio is considered with the uncertainty in the ECI estimates (see section 2.3).

In our study, discharge observations are obtained from several sources (table 1) described in detail by Beck et al (2019a). Based on the best availability of discharge observations, we choose the 10-year time period 2000–2009 to calculate long-term averages of ECI. We further selected precipitation and actual evapotranspiration datasets according to the following quality criteria (i) temporal coverage: the precipitation and actual evapotranspiration datasets should cover the time period 2000–2009 to coincide with good discharge availability; (ii) the spatial resolution should not be coarser than around 0.25° × 0.25°, because forcing with too coarse a resolution may result in unexpected uncertainty for small catchments due to the averaging over a large area. Based on these criteria, three independent precipitation and three independent actual evapotranspiration products were deemed suitable to perform our analysis (table 1).

Topographic areas of our analyzed catchments are in the range of 1 − 50 000 km². The majority (85%) of those are larger than 100 km², and 46% are larger than 625 km² (which is approximately 0.25° × 0.25°). The distribution of catchment topographic areas is provided in figure S1 (available online at stacks.iop.org/ERL/15/104024/mmedia). To further minimize the influence of irrigation on ECI estimates, we only consider catchments (table 1) with irrigation areas < 5% of the total catchment area. By doing so we exclude the influence of pumping to a large extent, even though pumping might still occur for other purposes than irrigation, which would not be captured by our analysis (see figure S3). These considerations result in a final catchment set of 8701 catchments for analysis.

Uncertainty in the ECI calculation originates from uncertainties in the discharge, precipitation, and actual evapotranspiration datasets. Compared to the precipitation and actual evapotranspiration datasets, discharge observations are assumed to contain much smaller uncertainty (Sauer and Meyer 1992, Biemans et al 2009, Khan et al 2018). We therefore focus on the uncertainties introduced through the precipitation and actual evapotranspiration data, by analyzing the range of ECI estimates obtained from the nine possible combinations of the three independent actual evapotranspiration and the three independent precipitation datasets (table 1). We define 'conclusive' catchments as those where at least seven out of nine possible ECI estimates can be calculated (given that some datasets do not cover all catchments) and that these have the same ECI sign (effective catchment areas are therefore either all larger than or all smaller than the corresponding topographic catchment areas). We subsequently use the mean of the conclusive ECI estimates for further analysis. We show in the supplemental material (figure S2) that the variability across the forcing data products is not higher for small catchments than it is for large ones. Thus providing at least some evidence that data bias related to poorer estimates over smaller areas is not causing the ECI results. Results using more stringent criteria are shown in the supplemental material. Adopting those would not change our overall conclusions.

We use a random forest analysis to identify and rank the most relevant factors influencing the variability of ECI (Breiman 2001). Random forest is an tree-based ensemble learning method for classification and regression (Denisko and Hoffman 2018), which has been widely applied to identify the most 'important' factors that organize a dataset (Tyralis et al 2019). Building on the idea of hydrologic landscapes with respect to expected dominant controls on the water balance (Winter 2001), we use physiographic factors (Beck et al 2015) that characterize a catchment's climate, topography, and geology, while adding aspects of catchment location: aridity index (AI), distance to the coast (D), topographic catchment area (A), mean slope (S), permeability (µ), and mean elevation (H). In addition, the fraction of lakes and reservoirs (f_w) is considered. We apply a 10-fold cross-validation and use different numbers of ensembles (50, 100, and 200) to control the quality of the regression trees. We identify the most relevant factors using importance estimates derived from permutations of the out-of-bag predictor observations of the random forest. To avoid false conclusions due to the pre-selection of conclusive catchments (see section 2.3), we also apply the random forest analysis to the 2760 conclusive catchments as well as to all 8701 catchments. Random forest analyses using different numbers of ensembles demonstrate similar results of the importance ranking for all influencing factors. Therefore, we only show the importance estimates derived from the random forest analysis using 200 ensembles.

We evaluate our ECI estimates for the conclusive catchments in two independent ways. First, we assess the consistency of water gains or losses (the direction of inter-catchment groundwater flow) indicated by our ECI values and those indicated by an independently simulated hydraulic gradient. We use hydraulic head simulations of a steady-state global groundwater model with a spatial resolution of around 9 × 9 km² at the equator (Reinecke et al 2019b). We define buffer areas inside and outside of the catchment boundary to estimate the average value of all head estimates that fall inside these buffer areas. Then, comparing the mean hydraulic head of the two buffer areas that are inside (h_in) and outside (h_out) of the catchment boundary, we obtain the head difference across the catchment boundary Δh_sim = h_in-h_out. This head difference can be used to indicate the direction of inter-catchment groundwater flow (positive or negative ECI). However, estimating actual groundwater fluxes at the catchment boundaries is complicated and requires good knowledge of subsurface properties, which we do not have for all these catchments. Therefore, we only evaluate the direction of water flux using the head simulations. We make the simple assumption that a positive head difference (Δh_sim > 0) indicates a losing catchment (ECI < 0), while a negative head difference (Δh_sim < 0) indicates a gaining catchment (ECI > 0). Due to the spatial resolution of the available head simulations, the calculation of head differences for smaller catchments may not be possible when an insufficient number of head simulations fall within the buffer areas. To not exclude too many catchments in the evaluation, we test different buffer areas with 20%, 30% and 40% of the topographic catchment area. With these three fractions, the mean hydraulic head of the buffer areas can be derived for 70%, 78% and 81% of the conclusive catchments, respectively. We use the buffer area of 30% of the topographic catchment area for the ECI evaluation below. Evaluations using the other two areas show similar results (see section 7 of the supplemental material).

Second, we identify outliers from the expected range of the Budyko curve (Budyko et al 1974). These outliers may be caused by problems with the water balance (Le Moine et al 2007, Bouaziz et al 2018). We therefore evaluate the consistency of our ECI estimates with the outliers in the Budyko framework. We plot the relationship between the long-term dryness index PET/P and the long-term evaporative index (P-Q)/P (figure 4). Using P-Q instead of AET is beneficial to detect the unrealistic water balance values. PET is obtained from the GLEAM dataset (Miralles et al 2011, Martens et al 2017). P is the mean of the three precipitation products (table 1). We adopt the widely used Fu (1981) curve (equation (4)) to represent the Budyko curve, which is identical to the Fu curve with w = 2.6 (Beck et al 2019a). We show a feasible range around the Budyko curve using two other Fu curves with w = 1.8 and w = 3.8, which indicate the 90% confidence interval of the w parameter derived from a set of 411 MOPEX catchments (Greve et al 2015).

$$\begin{equation}\frac{{AET}}{P} = 1 + \frac{{PET}}{P} - {\left[ {1 + {{\left( {\frac{{PET}}{P}} \right)}^w}} \right]^{1/w}}\end{equation} \tag{ 4 }$$

where w [-] is an empirical parameter that represents catchment characteristics.

A catchment with a closed water balance cannot discharge or evaporate more water than it receives as precipitation (over longer time periods), and actual evapotranspiration cannot be larger than potential evapotranspiration due to the energy limit (Bouaziz et al 2018). Therefore, assuming negligible observational error, this suggests that a catchment with Q > P (area below x-axis in figure 4) gains water (ECI > 0), and that a catchment with P-Q > PET (areas above the energy limit in figure 4) loses water (ECI < 0) through the subsurface. Bouaziz et al (2018) further postulate that catchments that plot further away from the theoretical relationship between dryness and evaporative index are more likely impacted by groundwater gains or losses. We would therefore expect that ECI values should grow (either negatively or positively) with increasing distance from the theoretical relationship.

Through the analysis of the 2760 conclusive catchments (figure 1), we find that substantial deviations between topographic and effective catchment areas are abundant across the globe (circles in dark red and dark blue in figure 1). We define substantial as: the effective areas are either larger than double or smaller than half of their corresponding topographic catchment areas. About 36% of the conclusive catchments show such a substantial deviation. Consequently, blue water availability in these catchments will likely be affected by management activities outside their topographic boundaries, or, activities within topographic boundaries will affect water resources outside.

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In North America, our analysis indicates that 2117 of 6172 (∼34%) catchments are conclusive. Among those, 691 catchments show substantial deviations between effective and topographic catchment areas. In the US, our analysis demonstrates that catchments located in the Upper Mississippi River Basin gain water from neighboring areas resulting in larger effective catchment areas (positive ECI, blue circles). While catchments in the Colorado River Basin show smaller effective catchment areas (negative ECI, red circles), indicating losing water conditions. These results agree well with a previous regional study on groundwater exporters and importers in the US (Schaller and Fan 2009). Additionally, many costal catchments in Southern California show smaller effective catchment areas (negative ECI), which suggests that these catchments export water to the North Pacific Ocean via submarine groundwater flow (Luijendijk et al 2020). In Europe, 212 of 629 (∼34%) catchments are identified as conclusive catchments, 43 of which have substantial differences between effective and topographic catchment areas. The result is consistent with a modeling study of a large sample of catchments in France (Le Moine et al 2007), which showed that explicitly representing inter-catchment groundwater transfers is preferable to address the water balance issues in rainfall-runoff models. In Australia, we find that 82 of the 89 conclusive catchments are costal catchments with losing water conditions (negative ECI). 80% of these coastal losing catchments are also in arid regions (PET/P > 1), which suggests the importance of inter-catchment groundwater flow in the coastal and arid areas of Australia.

Repeating the analysis with a requirement that at least eight out of nine or all nine ECI estimates are consistent (to test the robustness of our results), produces 30% (2573) and 28% (2442) conclusive catchments, respectively. There is therefore no significant impact on our conclusions using more stringent criteria (figures S4 and S5). However, our analysis, similar to others done previously (Bouaziz et al 2018), assumes that observational errors are negligible (as long as the independent datasets are consistent). We could in contrast assume that subsurface groundwater interactions are negligible, and that precipitation bias is the only relevant factor—the opposite extreme. Under this assumption, Beck et al (2019a) showed that precipitation in mountain ranges may be underestimated, which would be consistent with widely made assumptions about the quality of precipitation measurements in mountainous regions. The truth likely lies in-between the two assumptions of no-groundwater interactions and perfect precipitation measurements—both are unlikely to be true in most places and it is rather a question of dominance. Further insights will have to come from additional local analyses where additional information on data quality, geological settings etc is available (Le Moine et al 2007, Muñoz et al 2016).

The random forest prediction of ECI values shows consistent results concerning the relative importance of all the influencing factors for all 8701 catchments and for the 2760 conclusive catchments (figure 2). By quantifying the permutation importance estimates of all seven factors that represent a catchment's climate, topography, geology, and location (figure 2), we find that aridity index is by far the strongest influencing factor, followed by mean slope, distance to coast, topographic catchment area, permeability, fraction of lakes and reservoirs, and mean elevation. We, therefore, analyze the relationship between ECI and the most relevant influencing factors in more detail (see section 3.3), i.e. aridity index, mean slope, distance to coast, and topographic catchment area.

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For the 2760 conclusive catchments, 17% of ECI values indicate that the effective catchment area is smaller than half of their topographic area, while 19% show that the effective catchment area is larger than double of that suggested by topography (figure 3(a)). Analyzing the relationship between ECI and the four most important influencing factors for the 2760 conclusive catchments, we find that more arid catchments tend to lose more water, which results in smaller effective catchment areas (figure 3(b)). In wetter regions, the other influencing factors, beyond AI, gain more importance. This is in line with the numerical model simulations in a drier climate by Schaller and Fan (2009), who showed that in arid regions the regional groundwater table often falls below stream beds and much of the recharge through local precipitation enters the regional groundwater flow system rather than the river. A decreasing trend in ECI is found with decreasing distance to the coast (figure 3(c)), suggesting that near-coast catchments tend to lose water through subsurface drainage to the sea. This result is consistent with previous studies on submarine groundwater discharge in coastal catchments, which showed that submarine groundwater discharge can be as important as surface river discharge entering the Atlantic Ocean (Moore et al 2008). ECI variability also decreases with increasing slope (figure 3(d)) and increasing catchment area (figure 3(e)), indicating a higher variability of effective catchment area in flatter regions and for smaller catchments. Consequently, it shows smaller deviations for large basins, which one would expect since their areas should be more consistent with regional groundwater systems. There is some statistical dependency between the four influencing factors (figure S6), which can be explained by physiography. For instance, aridity index is positively correlated with distance to the coast since inland regions in general are more likely to be arid than coastal regions (Makarieva et al 2009, Trenberth et al 2011). The relationships between ECI and the influencing factors do not change substantially when using at least eight out of nine or all nine ECI estimates to define conclusive catchments (figures S7 and S8).

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We compare the direction of inter-catchment groundwater flow (water gains or losses) based on our ECI estimates with that derived from the simulated head differences to assess the consistency between the two approaches. For gaining catchments, we find an increasing agreement with simulated head differences for increasing ECI values (figure S9(a)). When ECI > 0.3, more than half of the conclusive catchments show a consistent direction of water gains or losses compared to the groundwater model. With increasing ECI values, the percentage of the consistent catchments between our results and the groundwater model can become greater than 70%. We find a similar pattern for losing catchments (figure S9(b)), with an increasing percentage of consistency when ECI values become more negative. The percentage of consistent catchments can reach up to 80% for strongly negative ECI values. Results are robust when we use different buffer areas to calculate the head differences (figure S9). We would not expect a full agreement between our ECI estimates and the groundwater model due to reasons such as uncertainties in the forcing data that are used to derive ECI and in the groundwater flow simulations, and the steady-state assumption of the groundwater model. However, we expect the high agreement between the two approaches for catchments with significant water gains or losses (those with large positive or negative ECI values), which is indeed showed with the comparison between ECI estimates and the groundwater model. The impacts of uncertainties in both forcing data and head simulations (Reinecke et al 2019b) are likely more pronounced for catchments with smaller absolute ECI values. Furthermore, subsurface heterogeneity, subsurface conductivity and resolution of the groundwater model used will affect our evaluation (Reinecke et al 2019a, 2020). More detailed subsurface conceptualizations and subsequent higher resolution groundwater modeling would be required to resolve these issues.

Inter-catchment groundwater flow leads to a violation of the assumption of a closed water balance for topographic catchments. Catchments with significant inter-catchment flow are therefore expected to deviate from the theoretical relationship defined by Budyko and discussed in section 2.5 (Budyko, 1974; Bouaziz et al 2018). We therefore expect larger deviations from the Budyko curve for catchments with larger absolute values of ECI. Indeed, we find that conclusive catchments with higher water gains or losses (dark blue and dark red circles) deviate more from the Budyko curve (figure 4). This general similar pattern is still largely preserved for the inconclusive catchments (figure S10). We thus find a consistency between the Budyko framework and the water gains/losses identified by ECI. We further divide figure 4 into three zones to discuss the outliers present. (i) An area that is above the energy limit (Zone 1). In this zone, the calculated AET, derived using P-Q, is larger than PET. Due to the limited energy available, AET cannot be larger than PET for catchment with a closed water balance. However, for losing catchments we may wrongly attribute water loss to AET under the assumption of a closed water balance. Indeed, we find a large portion of losing catchments (red circles) identified by ECI are located within this zone. (ii) An area with (P-Q)/P < 0 (below the lower blue dashed line) represents a second water limit (Zone 2). The area above the water limit with Q = 0 (upper blue dashed line) shows that discharge cannot become negative. Whereas, the zone below the water limit with Q = P (lower blue dashed line) shows that specific discharge Q can be larger than P for the topographic catchment. Since Q cannot be larger than P considering a closed water balance, catchments within this zone can be assumed to gain water from outside the topographic catchment boundary. We see that many gaining catchments (blue circles) identified by ECI fall within this zone. (iii) An area that is between the energy limit and the two water limits (Zone 3). In this zone, we find that many losing catchments (red circles) are above the Budyko range, while the gaining catchments (blue circles) are mostly located below the Budyko range. Therefore, inter-catchment groundwater flow could be one likely reason for catchments that are located outside this feasible Budyko range. However, the deviation of catchments from the Budyko curve may also be affected by other factors, such as P and PET seasonality, P phase, vegetation cover, soil moisture capacity, and topography (Beck et al 2019a). Generally, the analysis of Budyko framework helps us to explain a large proportion of gaining and losing catchments identified by ECI.