A proportionality-based multi-scale catchment water balance model and its global verification

Journal of Hydrology 582 (2020) 124446

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Research papers

A proportionality-based multi-scale catchment water balance model and its global verification

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Shulei Zhanga,b, Yuting Yangb, , Tim R. McVicarc,d, Lu Zhangd, Dawen Yangb, Xiaoyan Lia a

State Key Laboratory of Earth Surface Process and Resource Ecology, School of Natural Resources, Faculty of Geographical Science, Beijing Normal University, Beijing, China b State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing, China c CSIRO Land and Water, Canberra, Australia d Australian Research Council Centre of Excellence for Climate System Science, Sydney, Australia

A R T I C LE I N FO

A B S T R A C T

This manuscript was handled by Marco Borga, Editor-in-Chief, with the assistance of Luca Brocca, Associate Editor

Unifying the description of catchment hydrological partitioning across spatial and temporal scales has long been an important research topic within the hydrological community. As an effort to that, here we develop a conceptual water balance model based on the proportionality hypothesis (denoted PWBM) and for the first time, test the proportionality hypothesis using hydrologic observations across global catchments at varying time scales. The PWBM has a transparent and parsimonious model structure with only two model parameters, and requires precipitation, potential evapotranspiration and leaf area index as model inputs. To test PWBM, we apply the model to simulate streamflow (Q), evapotranspiration (E) and storage change (ΔS) in 22 large basins and 255 small basins globally and find an overall good model performance in reproducing all three water balance components at the daily (only in small basins), monthly, seasonal, annual and mean-annual time scales. Compared with three other conceptual hydrological models developed for specific time scales, the PWBM substantially outperforms the SCS model at the daily scale and the ABCD model at the monthly scale and performs similarly as the long-standing Budyko model at the mean-annual scale. In summary, our study verifies the robustness of PWBM and offers convincing evidence regarding the applicability of the proportionality hypothesis for enhanced hydrological system understanding at multiple spatial and temporal scales.

Keywords: Proportionality hypothesis Catchment water balance Multiscale modelling

1. Introduction Catchment hydrological partitioning—the partitioning of precipitation into streamflow (Q), evapotranspiration (E) and storage change (ΔS)—underlies the complex interactions between climate, hydrology, vegetation, soil, geology and other land surface processes (Rodríguez-Iturbe, 2000; Trancoso et al., 2016; Troch et al., 2013). An accurate quantification of hydrological partitioning is imperative for understanding catchment hydrological behaviors and provides the quantitative basis for water resources management (Oki and Kanae, 2006; Troch et al., 2015). As a result, there has been continuous interest in developing methods to quantify hydrological partitioning across a range of spatial and temporal scales (e.g., L’vovich, 1979; Beven et al., 1980; Zhao, 1992; Liang et al., 1994; Burnash, 1995; Yang et al., 1998; Andréassian et al., 2012; Liu et al., 2016; Xing et al., 2018). Current methods to model hydrological partitioning can be broadly classified into two types: (i) the ‘bottom-up’ (or Newtonian) approach;

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and (ii) the ‘top-down’ (or Darwinian) approach. Compared with the ‘bottom-up’ approach that requires large data inputs and parameterization of individual processes, the ‘top-down’ approach has attracted considerable attention within the hydrological community primarily due to its simplicity and transparency (Sivapalan et al., 2003; Andréassian et al., 2016; Donohue et al., 2012; Liu et al., 2016; Yang et al., 2016a; Zhang et al., 2001). The ‘top-down’ approach analyzes the hydrologic behavior of a system without detailing individual processes, and involves identifying simple and robust spatial and/or temporal patterns in hydrologic behavior from observations and postulating a theory to connect the observed patterns with the processes that created them (Harman and Troch, 2014). As a result, the ‘top-down’ approach often has a parsimonious model structure and parameterization to describe hydrological behaviors at the scale(s) of interest (Blöschl and Sivapalan, 1995; Beven, 2006). Different spatial and/or temporal patterns can emerge from hydrological behaviors at varying scales, which can make the model

Corresponding author. E-mail address: [email protected] (Y. Yang).

https://doi.org/10.1016/j.jhydrol.2019.124446 Received 16 July 2019; Received in revised form 11 October 2019; Accepted 9 December 2019 Available online 12 December 2019 0022-1694/ © 2019 Elsevier B.V. All rights reserved.

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developed a unified water balance model accordingly. Unfortunately, this theoretical framework has not been applied in real hydrological modelling nor tested against hydrological observations. This is partly because, for example, six parameters need to be quantified with only three equations in Zhao et al.’s (2016) model, potentially leading to the issue of over-parameterization (Beven, 1993). It is still unclear if a unified pattern exists at different scales that can be identified and modelled using the proportionality hypothesis, and it is imperative to resolve this before assessing testing its underlying optimality principle. Thus, our objectives were to: (i) develop a multi-scale catchment water balance model based on the proportionality hypothesis (termed as PWBM); (ii) calibrate and validate PWBM with catchment observations globally at the daily, monthly, seasonal, annual and mean-annual time scales; (iii) compare the model performance of PWBM with other conceptual hydrological models developed for specific time scales; and (iv) explore the sensitivities and characteristics of model parameters across temporal scales.

structure scale dependent. For example, at the mean-annual time scale, Budyko (1974) proposed a semi-analytical model (also known as the Budyko model) to quantify the relationship between the mean-annual E ratio (i.e., the ratio of E over P) and the mean-annual aridity index (i.e., the ratio of potential evapotranspiration EP over P). Despite variations around the original Budyko model being observed in subsequent studies (e.g., Milly, 1994; Potter et al., 2005; Zhang et al., 2001; Yang et al., 2007, 2009; Donohue et al., 2007; 2010; 2012; Williams et al., 2012), the Budyko model is used to represent the first-order control of climate variability on catchment hydrological partitioning and applies globally (Roderick and Farquhar, 2011; Roderick et al., 2014; Yang et al., 2016a). At the inter-annual or intra-annual scale, the catchment water balance can be equally affected by climate spatio-temporal variability, water storage change and vegetation dynamics (Sivapalan et al., 2003; Zhang et al., 2008; Liu et al., 2016). As a result, the corresponding modelling approach must account for those factors and the interactions among them. Examples of such approaches include the classic two-stage partitioning theory proposed by L’vovich (1979) and the ABCD conceptual hydrological model developed by Thomas (1981). At the event scale, hydrological partitioning is often primarily controlled by other factors in addition to those introduced above, including rainfall intensity (Horton, 1933), soil infiltration (Green and Ampt, 1911), soil storage capacity (Dunne and Black, 1970), slope and terrain (Beven and Kirkby, 1979), among others (Wang et al., 2012). Consequently, various approaches based on different conceptualizations of hydrological processes were proposed to describe the streamflow generation process at the event scale for a specific catchment or region, such as the TOPMODEL originally developed for humid catchments (Beven and Kirkby, 1979), the Xinanjiang Model for humid catchments in southern China (Zhao, 1992) and the SCS model based on small rural catchments in the United States (SCS, 1972). Nevertheless, despite that the controlling factors of hydrological partitioning vary greatly with temporal scales and study regions, commonality exists among the behavior of hydrological partitioning across scales and serves as signatures from the co-evolution of natural systems (Sivapalan, 2005; Newman et al., 2006; Gentine et al., 2012; Harman and Troch, 2014; Trancoso et al, 2016; 2017). Thus, over the last 40 years there has been enduring interest to develop a generic theory to effectively unify the description of hydrological partitioning across spatial and temporal scales (Eagleson, 1982; Schymanski et al., 2008; Zehe et al., 2010). Inspiring progress was made by Wang and Tang (2014) who identified the theoretical commonality between three hydrological models at different temporal scales (i.e., the Budyko model at the mean-annual scale, the ABCD model at the monthly scale and the SCS model at the daily scale) using the proportionality hypothesis. The proportionality hypothesis expresses the competition between different catchment functions and their symmetry identified in empirical functional representations of hydrological partitioning. Further studies were conducted to explore the optimality principle behind the proportionality hypothesis. For example, Wang et al., (2015) and Zhao et al., (2016) claimed that the maximum entropy production (MEP) theory was the optimality principle behind the proportionality hypothesis and

2. Methodology 2.1. The proportionality hypothesis The proportionality hypothesis used here expresses the competition between different catchment functions and their symmetry identified in empirical functional representations of hydrological partitioning (Poncea and Shetty, 1995). Assuming the amount of available water is Z, which is partitioned into X and Y over a specified time interval. X has an upper bound denoted as XP, whereas Y increases with Z in an unbound fashion. Then, the competition between X and Y would follow the proportionality formula:

X Y = XP Z

(1)

If an initial value X0 is allocated to X before the competition between X and Y is calculated, the partitioning is determined by:

X − X0 Y = XP − X0 Z − X0

(2)

This hypothesis has been successfully applied for modelling runoff at the event scale (SCS, 1972), inter-annual runoff variability (Thomas, 1981; Poncea and Shetty, 1995) and mean-annual water balance (Wang and Tang, 2014; Wang et al., 2015). However, in these conceptual hydrological models, different model structures and parameters were developed for specific time scales. Consequently, the applicability of the proportionality hypothesis to unify hydrologic behaviors across temporal scales is still not demonstrated and remains elusive. 2.2. A multi-scale catchment water balance model based on the proportionality hypothesis As illustrated in Fig. 1, we consider the total available water for a catchment during period i as the cumulative precipitation during period i (Pi, L) and water storage at the beginning of period i (Si-1, L). The total

Fig. 1. Schematic diagram illustrating the hydrological partitioning in a catchment during period i. The subscripts i represents the period i; the subscripts 0 represents the initial value; the subscripts P represent the maximum possible value (capacity). 2

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μi = 1 − e−β × LAIi

available water is then partitioned into three components: streamflow (Qi, L), evapotranspiration (Ei, L) and water storage at the end of the period (Si, L), which can be described as:

where β is a model parameter accounting for local effects (e.g., rainfall intensity, catchment slope, and so on) that modifies the μ-LAI relationship. We further verified this relationship spatially and found that using μ estimated by Eq. (8) greatly improved model performance compared with fixed μ (see Supplementary Text S2 for details).

(3)

Pi + Si − 1 = Qi + Ei + Si

In this process, a portion of precipitation is available for direct evaporation (evaporation of water intercepted by vegetation and retained at the soil surface; Wang and Tang, 2014), which is defined as the initial evaporation (E0, i, L). There may also some water retained at the surface ponding (Shi et al., 2009) and is defined as the initial storage (S0,i, L). The remaining available water Pi + Si-1-E0, i-S0, i is then partitioned into continuing evapotranspiration (Ec, i, L), continuing soil storage (Sc, i, L) and runoff (Qi, L). As precipitation increases, continuing evapotranspiration is increasingly limited by atmospheric evaporative demand and asymptotically approaches a constant value of EP,i-E0,i, where EP,i is the potential evapotranspiration during period i; continuing soil storage is limited by continuing catchment storage capacity SP-S0, i, where SP,i is the catchment storage capacity; and runoff is constrained by the available water Pi + Si-1-E0, i-S0, i. Applying the proportionality hypothesis, one can obtain:

Ei − E0,i Si − S0,i Qi = = EP,i − E0,i SP − S0,i Pi + Si − 1 − S0,i − E0,i

2.3. Model verification To test PWBM, we firstly applied the model to 22 global large catchments (see section 3.1 for more information) at a monthly timestep. The two model parameters (i.e., SP* and β) for each catchment were determined by calibrating the model from 1984 to 1995 with the objective function to minimum the root-mean-square error (RMSE) between observed and simulated Q. Then, based on the calibrated parameters SP* and β, we simulated Q, E and ΔS for the validation period (i.e., 1996–2006). Furthermore, to evaluate the model performance across temporal scales, we respectively calibrated and validated PWBM at the seasonal (i.e., January to March as the first season in a year, and so on), annual and mean-annual scale (i.e., sequential threeyear mean; Yang et al., 2016b) with similar procedures. The calibration and validation periods are consistent at multi-time scales. Additionally, as it is entirely plausible for the model performance and parameters to change with spatial scales (Li et al., 2013; Yang et al., 2015; Liu et al., 2016; Trancoso et al., 2016; 2017), we also tested PWBM at 255 global small unimpaired catchments at multiple time scales (i.e., daily, monthly, seasonal, annual and mean-annual) with the former half data used for calibration and the latter half data used for validation (each small catchment contains > 12 years continuous Q observations; see section 3.1 for more information).

(4)

Note: Eq. (4) is consistent with the Budyko model at the mean-annual scale, the ABCD model at the monthly scale and the SCS model at the event scale (see Supplementary Text S1 for details). There are three parameters in Eq. (4), namely, E0,i, SP and S0,i. Defining μi = E0,i/ EP,i, so that a dimensional parameter (E0,i) will turn to be dimensionless (μi) and Eq. (4) can be rewritten as:

Ei − μi EP,i Si − S0,i Qi = = EP,i − μi EP,i SP − S0,i Pi + Si − 1 − S0,i − μi EP,i

(5) 2.4. Comparison of PWBM with three other conceptual hydrological models

As S0,i is often much smaller than SP , i.e., water ponded over the surface is usually much less than that stored in the entire soil zone, it is reasonable to assume that Sp-S0,i is a constant. With this and further definingSP* = SP − S0 , we have:

Ei − μi EPi Si − S0,i Qi = = EPi − μi EPi Pi + Si - 1 − S0,i − μi EPi SP*

Additionally, we compared the performance of PWBM with three other conceptual hydrological models, including the SCS model at the daily scale, the ABCD model at the monthly scale and the Budyko model at the mean-annual scale. The model structures and parameters needed in the three other conceptual hydrological models are provided in Supplementary Text S1. The periods and procedures of calibration and validation for these three models are the same with that described in section 2.3 for PWBM. The ABCD and Budyko models are compared in both 22 global large catchments (Fig. 2b) and 255 global small catchments (Fig. 2b), whereas the SCS model is only compared in 255 global small catchments with daily streamflow observations (Fig. 2b). The performance of these three models and PWBM are assessed by three evaluation metrics (Krause et al., 2005), i.e., the Nash–Sutcliffe Efficiency index (NSE; Nash and Sutcliffe, 1970; McCuen et al., 2006), rootmean-square error (RMSE; Willmott and Matsuura, 2005) and relative root-mean-square error (RRMSE).

(6)

Finally, combining Eq. (6) with the catchment water balance equation (i.e., Eq. (3)) and an additional equation for water storage change (i.e., ΔSi = (Si − S0,i ) − (Si - 1 − S0,i ) ), the simplified proportionality-based catchment water balance model (termed as PWBM hereafter) is expressed as: E −μ E

S −S

Q

i Pi i 0,i ⎧ E − μi E = S * = P + S − Si − μ E i i-1 0,i i Pi ⎪ Pi i Pi P P S Q E S + = + + − i i 1 i i i ⎨ ⎪ ΔSi = (Si − S0,i ) − (Si - 1 − S0,i ) ⎩

(8)

(7)

The above model (i.e., Eq. (7)) contains four equations with three parameters (i.e., μi, SP* and S0,i). Among these three parameters, S0,i can be easily eliminated if we were to estimate ΔSi instead of Si. As a result, there are only two remaining parameters (i.e., μi and SP* contained in the 4 equations), which can be determined by calibrating the model against hydrological observations thus avoiding the equifinality problem (Beven, 1993). Parameter μi (i.e., μi = E0,i/ EP,i) determines the relative importance of EP in the control of E. Additionally, the value of μi should range from 0 to 1 as EP, i sets the upper limit of E0, i. The parameter E0 is related to canopy interception and should consequently increase with vegetation coverage for given rainfall amount and intensity (Wang and Tang, 2014). Using satellite-derived leaf area index (LAI) as a surrogate of vegetation coverage, it follows that: E0 → 0 and μ → 0, when LAI → 0, and E0 → EP and μ → 1, when LAI→∞. The Beer’s Law type of function is then adopted to estimate μi as a function of LAI:

3. Data 3.1. Water balance datasets at global catchments Catchment-average monthly time series of water balance components, including precipitation, evapotranspiration, streamflow, and water storage change, for 32 major (i.e., > 200,000 km2) river catchments across the globe from 1984 through 2006 were provided by Pan et al. (2012). The water balance component dataset of Pan et al. (2012) represents an optimal combination of different data sources (i.e., in situ observations, remote sensing observations, land surface model outputs and reanalysis products), which has been considered as the best-available water budget dataset to-date (Li et al., 2013) and are regarded as observations in the current study. However, as PWBM does not explicitly consider the effect of snow and/or glacial melting on 3

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Fig. 2. Location of (a) the 22 large catchments and (b) the 255 small catchments.

were quantified using the “artificial areas” class of the GlobCover v2.3 map (http://ionia.esrin.esa.int). The location and capacity of dams and reservoirs were determined based on the Global Reservoir and Dam (GRanD) database (v1.01) (Lehner et al., 2011a,b).

streamflow, we excluded catchments located north of 60° N. As a result, only 22 of Pan et al.’s (2012) 32 catchments of were used herein (Fig. 2a). Additionally, daily streamflow observations at 255 small (1,480 – 13,662 km2, Fig. 2b) unimpaired catchments from 1982 to 2010 across the globe (south of 60° N) were collected from four sources: (i) Global River Discharge Centre (GRDC; http://www.bafg.de/GRDC); (ii) Geospatial Attributes of Gages for Evaluating Streamflow (GAGES)-II database; (iii) Water Information Research and Development Alliance (WIRADA) between CSIRO and Australian Bureau of Meteorology; (iv) Model Parameter Estimation Experiment (MOPEX). The selected catchments have at least 12 years’ continuous streamflow observations, with the former half data (i.e., longer than 6 years) being used for model calibration and latter half for model validation. To ensure the selected catchments are minimally affected by human activities, the selected catchments must satisfy the following criteria: (i) no significant forest gain or loss (less than2% of the total catchment area); or (ii) irrigated areas less than 2%; or (iii) urban areas less than 2% or (iv) the absence of large dams (defined as total reservoir capacity > 10% of the mean-annual catchment observed Q. Areas of forest change were determined from Hansen et al. (2013) based on 30-meter Landsat imagery from 2000 through 2013; noting there is currently no other reliable, high-resolution, global-scale dataset on vegetation cover change from 1981 onwards. Irrigation areas were derived from the Global Map of Irrigation Areas-GMIA (Siebert et al., 2007). Urban areas

3.2. Global climate, vegetation and land surface properties datasets Global daily meteorological data, including air temperature, wind speed, surface pressure and specific humidity, at a spatial resolution of 0.5° from 1982 to 2010 were obtained from the WFDEI (WATCH Forcing Data methodology applied to ERA-Interim data; Weedon et al., 2014). Global monthly net radiation from 1982 to 2010 at a spatial resolution of 0.5°×0.5° was provided by the Multiscale Synthesis and Terrestrial Model Intercomparison Project (Wei et al., 2014). Global three-hourly precipitation data at 0.25° spatial resolution from 1982 thought 2010 was derived from the Multi-Source Weighted-Ensemble Precipitation (MSWEP) version 1 dataset, which represents an optimal combination of the highest quality P data sources available as a function of time scale and location (Beck et al., 2017). Monthly LAI at a spatial resolution of 1/12° (approximately 9 km) from 1982 to 2010 was derived from Zhu et al. (2013) based on AVHRR GIMMS-3 g NDVI data (Pinzon and Tucker, 2014), which has been shown to be the optimal AVHRR NDVI dataset for tracking dynamics (Beck et al., 2011). Soil properties, including saturated hydraulic conductivity and root-zone storage capacity at 0.5° spatial resolution were 4

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and 16.4% for modeled ΔS. Note that we use the ratio of RMSEΔS over mean monthly P to calculate RRMSEΔS instead of the ratio of RMSEΔS over mean monthly ΔS, as mean monthly ΔS over a long-term would approach zero. For the validation period, the NSE, RMSE and RRMSE values for simulated Q, E and ΔS are all very close to those found for the calibration period, suggesting the robustness of the model structure and stability of the model parameters. To further illustrate the model performance from a time series prospective, Fig. 4 shows the comparison of observed and simulated monthly time series of Q, E and ΔS in the Yangzi River Basin, Nile River Basin and Senegal River Basin, respectively and these catchments represent humid, sub-humid/semi-arid and arid climates. It shows that the temporal dynamics of the three hydrological variables are generally well captured by the model, except for that the model tends to slightly overestimate peak flows and underestimate E at the lower end for the Nile and Senegal river basins. With the increase of time scale (i.e., from monthly to the meanannual scale) all three evaluation metrics for simulated Q, E and ΔS gradually decrease (Fig. 3). Lower RMSE and RRMSE suggest reduced overall simulation error as the time scale increases, whereas a smaller NSE value indicates a weakened model capacity to capture the temporal variations of the water balance components. Nevertheless, the lower NSE may also be caused by a lower temporal variability of observed hydrological variables at longer time scales as NSE reflects more the aspect of temporal variability of a time series and is sensitive to extreme values (Nash and Sutcliffe, 1970). Fig. 5 shows the values of NSE, RMSE and RRMSE of simulated water balance components for individual catchments at the monthly scale. Similar spatial patterns of model performance were found for the seasonal and annual time scales (Supplementary Figure S2). For simulated Q, lower NSE and high RMSE/RRMSE values are concentrated in high latitude regions (i.e., the Don, Ural, Danube, Volga and Dnieper Basin), whereas lower NSE values for simulated E are found in three tropical rainforest catchments (i.e. the Amazonas, Congo and Mekong Basin) (Fig. 5). Nevertheless, the mean RRMSE of monthly simulated E for the three tropical rainforest catchments is only 11.2%, which is even lower than the mean RRMSE of 18.4% over all 22 large catchments. For simulated ΔS, it does not show any notable spatial pattern in all three metrics. 4.1.2. At global small catchments Using the small catchments distributed across the globe dataset, we firstly test the model performance in simulating daily Q at the 255 catchments with daily Q observations (their locations are provided in Fig. 2b). As shown in Fig. 6a, there are 193 catchments (~75.7%) showing an NSE value larger than zero during the calibration period with an averaged NSE of 0.40. Averaged RMSE for daily Q estimates during the calibration period are 34.8 mm month−1 (Fig. 6b), with 158 catchments (~62.0%) showing a RRMSE lower than 100%. Similar with the results of the 22 large catchments (Fig. 3), the model performance during the validation period shows only slight difference compared with that during the calibration period. At the monthly scale, 227 (~89.0%) catchments show a positive NSE value with 135 of them (~50%) showing an NSE value higher than 0.5 for simulated Q during the validation period (Fig. 6d). The mean positive NSE of simulated Q during the validation period is 0.49. Averaged over all 255 small catchments, the RMSE and RRMSE for monthly Q are 18.9 mm month−1 and 47.0%, respectively (Fig. 6e and 6f), during the validation period, with 91.4% catchments showing monthly Q RRMSE lower than 100%. The PWBM performs, again, better at longer time scales in the small catchments in terms of RMSE and RRMSE. Averaged over all small catchments, the mean RMSE and RRMSE decreases to 14.1 mm month−1 and 38.2% at the seasonal scale, 6.2 mm month−1 and 19.0% at the annual scale and 5.0 mm month−1 and 16.5% at the mean-annual scale, respectively (Fig. 6). Although we also find notable reductions in Q NSE at longer time scales (Fig. 6g and 6j), this, as

Fig. 3. Model performance during the calibration (1984–1995) and validation (1996–2005) periods at multi-time scales for 22 global large catchments. Part (a) NSE, RMSE and RRMSE values for monthly, seasonal, annual and meanannual Q during the calibration period and the validation period at 22 global large catchments. The triangle is the averaged value over all catchments and the error bar indicates one standard deviation. Part (b) and (c) are the same as (a) but for simulated E and ΔS, respectively.

acquired from the ORNL DAAC Spatial Data Access Tool (https://doi. org/10.3334/ORNLDAAC/1388). Land-surface slope with 1 km spatial resolution was extracted from HYDRO1K dataset (http://lta.cr.usgs. gov/HYDRO1K). As needed, these gridded data were further aggregated, by averaging, and lumped for individual catchments. 4. Results 4.1. Testing the PWBM at multiple scales 4.1.1. At global large catchments We first applied the model in 22 global large catchments at the monthly, seasonal, annual and mean-annual time scales, respectively. Averaged over the 22 catchments, the model reasonably reproduces the monthly Q series with NSE at 0.42, RMSE at 5.08 mm month−1 and RRMSE at 30.3% on average for validation period (Fig. 3a). For monthly E and ΔS, PWBM performs even slightly better, having an averaged NSE and RRMSE of 0.71 and 18.4% for simulated E and 0.79 5

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Fig. 4. Comparison between monthly time series of observed (black) and simulated (red) streamflow, evapotranspiration and storage change in the Yangzi Basin (ac), Nile Basin (d-f) and Senegal Basin (g-i) during the calibration and validation period.

mentioned above, may also be caused by the relative lower Q temporal variability at longer time scales. 4.2. Comparison of PWBM with three other conceptual models At the global large catchments, we compare the model performance between the PWBM with the ABCD model at the monthly scale and the Budyko model at the mean-annual scale in simulating all three water balance components (Q, E and ΔS). Results show that PWBM performs substantially better than the ABCD model in simulating all three water balance components at the monthly scale, with NSE increased by 0.11, 0.54 and 0.65, RMSE decreased by 2.0 mm month−1, 12.3 mm month−1 and 12.5 mm month−1 and RRMSE decreased by 6.3%, 34.6% and 22.4%, respectively (Fig. 7a-c). At the mean-annual scale, PWBM exhibits a similar performance compared with the Budyko model, with both RMSE and RRMSE of PWBM only slightly higher than those of the Budyko model (Figs. 7d and 9e). At the global small catchments, we compare the performance of PWBM in simulating Q with the SCS model at the daily scale, the ABCD model at the monthly scale and the Budyko model at the mean-annual scale. As shown in Fig. 8a–c, PWBM performs much better than the SCS model at the daily scale with a higher average NSE (0.40 vs 0.23) and lower average RMSE (34.8 mm month−1 vs 47.6 mm month−1) and RRMSE (98.6% vs 138.1%). Similar improvements are also found in PWBM over the ABCD model (Fig. 8d–f). The average NSE (positive), RMSE and RRMSE for PWBM Q are 0.49, 18.9 mm month−1 and 47.0%, whereas the average NSE (positive), RMSE and RRMSE for ABCD Q are 0.35, 22.9 mm month−1 and 58.9%, respectively. In addition, the number of catchments showing negative NSE values reduces by 26.2% and 19.1% in PWBM, compared with that in SCS and ABCD. This suggests a much better performance of PWBM in capturing the observed Q variability than the SCS and ABCD models. At the mean-annual scale,

Fig. 5. Monthly model performance during the 1996–2006 validation period for each of the 22 global large catchments. Part (a) NSE values during the validation period for monthly Q, E and ΔS. Part (b) and (c) are the same as (a) but for RMSE and RRMSE values, respectively. The NSE values less than zero are labeled as −0.1 for simplification.

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Fig. 6. Model performance at multi-time scales during the calibration (the former half data) and validation period (the latter half data) in 255 global small catchments. Part (a) NSE values for daily Q during the calibration and validation period in 255 global small catchments. Part (b) and (c) are the same as (a) but for RMSE and RRMSE values, respectively. Parts (d)-(f) are the same as parts (a)-(c), respectively, but for monthly Q. Parts (g)-(i) are the same as parts (a)-(c), respectively, but for seasonal Q. Parts (j)-(l) are the same as parts (a)-(c), respectively, but for annual Q. Parts (m)-(n) are the same as parts (b)-(c), respectively, but for mean-annual Q.

parameter uncertainties may be largely canceled each other out and consequently leads to an insensitive ΔS modelling to both parameters. Fig. 9 shows the calibrated parameters SP* and β for each catchment at the four time scales. It is found that not only the median value but also the spatial variability of parameters SP* show an evident decreasing trend with increasing time scales. The parameters SP* has a median value of 1400 mm with an upper quartile of 4086 mm and a lower quartile of 620 mm at the monthly scale. These three values respectively decrease to 660 mm, 1476 mm and 410 mm at the meanannual scale. Similar with SP*, parameters β also reduces as the time scale increases, with the median of β being 0.32, 0.26, 0.23 and 0.19 at the monthly, seasonal, annual and mean-annual scale, respectively.

again, PWBM shows a comparable performance with the long-standing Budyko model (Fig. 8g and h).

4.3. Multi-scale model parameters Two parameters, SP* and β (see Eqs. (7) and (8)), need to be determined when applying the model. Sensitivity analysis shows that the model results, in particularly the simulation of Q, is much more sensitive to β than SP* (Table 1). Using the global 22 large catchments as an example, a 50% change in β would lead to a change in monthly Q by 40% ~ 70%, whereas the same change in SP* only leads to monthly Q changes by ~1% (Table 1). The model sensitivity to parameter changes also varies with time scales, with Q sensitivity to changes in β decreasing and Q sensitivity to changes in SP* increasing as time scale increases (Table 1). A similar pattern in the model sensitivity to changes of these two parameters across time scales is also seen in the simulation of E (Table 1). As for ΔS, changes in SP* and β both show very limited impacts on the model performance (Table 1), suggesting an important internal model attribute that errors in simulated Q and E caused by

5. Discussion Previously catchment hydrologists have sought to develop organizing principles based on prior theories derived from physical and ecological sciences (Zehe et al., 2010; Eagleson, 1982; Schymanski et al., 2008; Wang et al., 2015; Zhao et al., 2016). Although these 7

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Fig. 7. Comparison of PWBM and the ABCD model at the monthly scale and PWBM and the Budyko model at the mean-annual scale at 22 global large catchments. Part (a) shows NSE for monthly Q, E and ΔS simulated by PWBM and the ABCD model during the 1996–2005 validation period at 22 global large catchments. Part (b) and (c) are the same as (a) but for RMSE and RRMSE, respectively. Part (d) shows RMSE for mean-annual Q, E and ΔS simulated by PWBM and the Budyko model during the validation period at 22 global large catchments. Part (e) is the same as (d) but for RRMSE.

The present study contributes to ‘top-down’ hydrological modelling by developing a parsimonious model capable of reproducing catchment water balance at multi-scales with the same model structure. The ‘topdown’ approach is usually designed to interpret observed input–output relationships in terms of internal characteristics and/or processes occurring at certain scale (Harman and Troch, 2014; Zhao et al., 2016), thus a common consequential problem with previous “top-down” hydrological modelling is the difficulty to generalize due to the absence of global perspective (Sivapalan et al., 2003). Here, by invoking the proportionality hypothesis to generalize catchment hydrological

organizing principles have brought us comprehensive and new perspectives to understand hydrological systems, they often suffer from the absence of verification across global bio-climate using hydrological observations (Eagleson, 1982; Schymanski et al., 2008; Zhao et al., 2016). In this study, by developing a proportionality-based multi-scale catchment water balance model (PWBM) and verifying it with catchment observations globally, at a range of time scales, we provide the first comprehensive test of the proportionality hypothesis as an optimal principle that unifies catchment hydrological behavior for different time scales.

Fig. 8. Comparison of PWBM with the SCS model at the daily scale, the ABCD model at the monthly scale and the Budyko model at the mean-annual scale at global small catchments. Part (a) shows NSE for daily Q by PWBM and the SCS model during the 1996–2005 validation period at 255 global small catchments. Part (b) and (c) are the same as (a) but for RMSE and RRMSE, respectively. Part (d) shows NSE for monthly Q simulated by PWBM and the ABCD model during the validation period at 255 global small catchments. Part (e) and (f) are the same as (d) but for RMSE and RRMSE, respectively. Part (g) shows RMSE for mean-annual Q, E and ΔS simulated by PWBM and the Budyko model during the validation period at 255 global small catchments. Part (h) are the same as (g) but for RRMSE values.

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Table 1 Relative sensitivity analysis of simulated water balance components to parameters SP* and β across the 22 global large catchments for 1984–2006. Note that the sensitivity analysis was only conducted at the 22 large catchments due to the availability of all three water balance components (i.e., Q, E and △S). Time scale

Monthly

Parameter

S P* β

Seasonal

S P* β

Annual

S P* β

Mean-annual

S P* β

Change (%)

+50% −50% +50% −50% +50% −50% +50% −50% +50% −50% +50% −50% +50% −50% +50% −50%

Impacts Q

E

△S

+1.4% ± 1.5% −1.2% ± 2.3% −36.2% ± 7.7% +71.2% ± 28.2% +3.1% ± 1.2% −4.7% ± 2.1% −29.4% ± 7.5% +51.9% ± 21.1% +4.7% ± 1.1% −6.4% ± 1.5% −18.7% ± 6.8% +26.5% ± 12.0% +4.8% ± 1.9% −6.4% ± 3.5% −16.6% ± 5.8% +22.1% ± 8.4%

−1.5% ± 0.8% +2.5% ± 1.3% +18.2% ± 3.2% −27.4% ± 3.5% −2.9% ± 0.9% +4.6% ± 1.5% +13.7% ± 2.6% −19.2% ± 3.1% −3.9% ± 1.3% +5.4% ± 2.2% +6.1% ± 0.9% −7.9% ± 1.4% −3.7% ± 1.4% +4.8% ± 2.0% +6.5% ± 1.9% −8.1% ± 2.3%

+0.54% ± 0.23% −0.52% ± 0.30% −0.25% ± 0.24% +0.39% ± 0.70% +0.42% ± 0.17% −0.42% ± 0.21% −0.08% ± 0.08% +0.08% ± 0.08% +0.48% ± 0.28% −0.47% ± 0.31% −0.03% ± 0.03% +0.03% ± 0.02% +0.54% ± 0.82% -+0.54% ± 0.53% −0.04% ± 0.76% +0.03% ± 0.78%

Fig. 9. Comparison of calibrated parameters (a) SP* and (b) β at the monthly, seasonal, annual and mean-annual scale in all 277 global catchments (22 large and 255 small). For the boxes: red lines are the median values, blue boxes represent the interquartile range, and the dashed black whiskers extend to the values defined as 1.5 times the interquartile range.

Parameter SP* (i.e., SP*= SP-S0) is mostly controlled by soil storage capacity (SP) as initial soil storage capacity (S0) is often much smaller than SP. As SP* decreases with the increase of time scale this suggests that the boundary of soil volume contributing to hydrological processes varies across time scales (Seibert et al., 2003; Spence, 2007). At the intra-annual scale, groundwater storage and the groundwater-surface water interaction may exert considerable impacts on the catchment water balance (Wang, 2012), while these impacts on catchment water balance might be subtle as the time scale becomes longer (i.e., annual or mean-annual). This is not explicitly considered in our model but implicitly reflected through parameter SP* (Fig. 9a). Parameter β is related with the initial catchment evapotranspiration (i.e., canopy interception and water retained over the soil surface) and the importance of EP in the control of E, implying that both vegetation characteristics and soil properties can impact on β. To verify this, we further explore the relationship between β and a set of catchment properties and find that β is positively correlated with root-zone storage capacity and negatively correlated with mean catchment slope, rainfall intensity and soil saturated hydraulic conductivity (Supplementary Figure S3). Root-zone storage capacity is jointly determined by effective rooting depth and water holding capacity, both of which are generally larger in more humid regions (loam soil with deeper effective rooting depth) compared to more arid areas (sandy soil with shallower effective rooting depth) (Milly, 1994; Zhang et al., 2001). This explains the positive relationship between root-zone storage capacity and β. By contrast, steeper slope, larger rainfall intensity and higher soil hydraulic conductivity all favor the generation of Q, with less E being produced for a fixed climate forcing (i.e., same rainfall depth and EP) (Beven and Kirkby, 1979; Robinson et al., 1995; Yang et al., 2007). Consequently,

partitioning behavior, we provide a potential solution to this longstanding problem in hydrological modelling. It is noteworthy that the developed model herein shows good performance across both spatial (i.e., large and small catchments) and temporal (i.e., daily, monthly, seasonal, annual and mean-annual) scales. Additionally, although previous studies demonstrated the theoretical consistency between the proportionality hypothesis and other conceptual hydrological models developed for specific time scales (Zhao et al., 2016; Wang et al., 2015; also see Supplementary Text S1), these models still perform quite differently in real hydrological model. Here we found that PWBM performs substantially better than the SCS model at the daily scale, the ABCD model at the monthly scale and shows a comparable performance with the long-standing Budyko model at the mean-annual scale when validated with hydrological observations (Figs. 7-8). This is because that both SCS and ABCD models are essentially PWBM with simplifications, during which some key processes may be overseen. For example, the influence of soil storage before a rainfall event on runoff generation is neglected in the SCS model at the daily scale, and the competing relationship between actual evapotranspiration and water storage is not made explicit in the ABCD model at the monthly scale. The better performance of PWBM implies that these processes are not negligible to understand and to model hydrological partitioning at these time scales. The fact that there are only two model parameters (i.e., SP* and β) also guarantee the transparency and parsimony of PWBM. These two parameters represent the combined effects of catchment surface properties, that affect catchment hydrological partitioning, which are similar to the catchment surface parameter in the Budyko model (e.g., Zhang et al., 2001; Potter et al., 2005; Donohue et al., 2007, 2012). 9

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these three variables show a negative correlation with β. In addition, the value of parameter β shows a decreasing trend with the increase of time scales (Fig. 9b), suggesting that interception accounts for a higher proportion of total evapotranspiration at shorter time scales. Despite the overall good performance of PWBM across multiple spatial and temporal scales, the model behavior does exhibit some spatial heterogeneity. As shown in Fig. 5, a lower PWBM performance in simulating Q was reported in four high-latitude catchments (i.e., the Don, Ural, Danube, Volga and Dnieper Basin) where mean-annual air temperature is less than 5 ℃. A similar pattern of decreased model performance with decreasing air temperature is also found in global small catchments (Supplementary Figure S4). In these high-latitude catchments, we find that the month with maximum Q precedes the month with maximum P (see examples provided in Supplementary Figure S5). This suggests that errors in estimated Q are very likely due to ignoring the extra water input and energy consumption resulted from snow and/or glacial melting in the model which would has considerable influence on the generation of Q in such high-latitude catchments (Barnett et al., 2005; Berghuijs et al., 2014). However, we also find this influence is reduced for long time scales (Supplementary Figure S4). At the mean-annual scale, the effect of snow and/or glacial melting on runoff is negligible when compared with other factors, i.e., the amount of total precipitation. Whereas, snow and/or glacial melt may be the main water input for runoff generation for specific months or season (Immerzeel et al., 2009; Suzuki et al., 2006). Incorporating the effects of snow and/or glacial melt into the current model remains a challenge for future research. Finally, it is worth mentioning that the proportionality hypothesis is a mathematical expression of the competitive relationship between water balance components. However, the reason why the competition between hydrological processes at different time scales follow such a form as presented in Eq. (7) and (8) still remains unrevealed. Several theoretical explorations (Kleidon and Schymanski, 2008; Paik and Kumar, 2010; Westhoff et al., 2014) regard the maximum entropy production (MEP) theory as the foundation of the proportionality hypothesis (PH), as both (i.e., MEP and PH) identify the commonality existing in the hydrological system (Wang et al., 2015; Zhao et al., 2016). In this regard, future studies are needed to explore the linkages between MEP theory and the proportionality hypothesis to better understand the physical basis of the proportionality hypothesis.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work is financially supported by the Qinghai Department of Science and Techenology (Grant No. 2019-SF-A4) and the National Natural Science Foundation of China (Grant No. 41890821). Tim R. McVicar and Lu Zhang acknowledge support from CSIRO Land and Water. All data for this paper are properly cited and referred to in the reference list. A worked example, including MATLAB code, supporting documentation, input data and model output is available at https:// github.com/zslthu/PWBM. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.jhydrol.2019.124446. References Andréassian, V., Lerat, J., Le Moine, N., Perrin, C., 2012. Neighbors: Nature’s own hydrological models. J. Hydrol. 414, 49–58. Andréassian, V., Mander, Ü., Pae, T., 2016. The Budyko hypothesis before Budyko: the hydrological legacy of Evald Oldekop. J. Hydrol. 535, 386–391. Barnett, T.P., Adam, J.C., Lettenmaier, D.P., 2005. Potential impacts of a warming climate on water availability in snow-dominated regions. Nature 438 (7066), 303–309. Beck, H.E., et al., 2011. Global evaluation of four AVHRR–NDVI data sets: Intercomparison and assessment against Landsat imagery. Remote Sens. Environ. 115 (10), 2547–2563. Beck, H.E., et al., 2017. MSWEP: 3-hourly 0.25 global gridded precipitation (1979–2015) by merging gauge, satellite, and reanalysis data. Hydrol. Earth Syst. Sci. 21 (1), 589. Berghuijs, W.R., Woods, R.A., Hrachowitz, M., 2014. A precipitation shift from snow towards rain leads to a decrease in streamflow. Nat. Clim. Change 4 (7), 583–586. Beven, K., 1993. Prophecy, reality and uncertainty in distributed hydrological modelling. Adv. Water Resour. 16 (1), 41–51. Beven, K.J., Kirkby, M.J., 1979. A physically based, variable contributing area model of basin hydrology/Un modèle à base physique de zone d'appel variable de l'hydrologie du bassin versant. Hydrol. Sci. J. 24 (1), 43–69. Beven, K., Warren, R., Zaoui, J., 1980. SHE: towards a methodology for physically-based distributed forecasting in hydrology. IAHS Publ 129, 133–137. Beven, K., 2006. On undermining the science? Hydrol. Process. 20 (14), 3141–3146. Blöschl, G., Sivapalan, M., 1995. Scale issues in hydrological modelling: a review. Hydrol. Process. 9 (3–4), 251–290. Budyko, M., 1974. Climate and Life, 508 pp, edited. Academic Press, New York. Burnash, R., 1995. The NWS river forecast system-catchment modeling. Computer models of watershed hydrology 188, 311–366. Donohue, R.J., Roderick, M.L., McVicar, T.R., 2007. On the importance of including vegetation dynamics in Budyko's hydrological model. Hydrol. Earth Syst. Sci. 11 (2), 983–995. https://doi.org/10.5194/hess-11-983-2007. Donohue, R.J., McVicar, T.R., Roderick, M.L., 2010. Assessing the ability of potential evaporation formulations to capture the dynamics in evaporative demand within a changing climate. J. Hydrol. 386 (1–4), 186–197. https://doi.org/10.1016/j.jhydrol. 2010.03.020. Donohue, R.J., Roderick, M.L., McVicar, T.R., 2012. Roots, storms and soil pores: Incorporating key ecohydrological processes into Budyko’s hydrological model. J. Hydrol. 436, 35–50. https://doi.org/10.1016/j.jhydrol.2012.02.033. Dunne, T., Black, R.D., 1970. Partial area contributions to storm runoff in a small New England watershed. Water Resour. Res. 6 (5), 1296–1311. Eagleson, P.S., 1982. Ecological optimality in water-limited natural soil-vegetation systems: 1. Theory and hypothesis. Water Resour. Res. 18 (2), 325–340. Gentine, P., D'Odorico, P., Lintner, B.R., Sivandran, G., Salvucci, G., 2012. Interdependence of climate, soil, and vegetation as constrained by the Budyko curve. Geophys. Res. Lett. 39 (19). https://doi.org/10.1029/2012GL053492. Green, W.H., Ampt, G.A., 1911. Studies on Soil Physics. J. Agric. Sci. 4 (1), 1–24. https:// doi.org/10.1017/S0021859600001441. Hansen, M.C., et al., 2013. High-resolution global maps of 21st-century forest cover change. Science 342 (6160), 850–853. Harman, C., Troch, P.A., 2014. What makes Darwinian hydrology“ Darwinian”? Asking a different kind of question about landscapes. Hydrol. Earth Syst. Sci. 18 (2), 417–433. Horton, R.E., 1933. The role of infiltration in the hydrologic cycle. EOS, Transactions American Geophysical Union 14 (1), 446–460. Immerzeel, W.W., Droogers, P., De Jong, S., Bierkens, M., 2009. Large-scale monitoring of snow cover and runoff simulation in Himalayan river basins using remote sensing. Remote Sens. Environ. 113 (1), 40–49.

6. Conclusion We developed a conceptual water balance model based on the proportionality hypothesis (PWBM), and successfully simulated the catchment water balance at multiple time scales with the same model structure. With only two model parameters (i.e., SP* and β) needed in PWBM, this guarantees its transparency and parsimony. PWBM was tested in 22 large basins and 255 small basins globally at multi-time scales (i.e., daily, monthly, seasonal, annual and mean-annual) with overall good agreement between simulated and observed water balance components at these different time scales. Additionally, we found that the model performance improved in terms of the root-mean-square error but degraded in regard to capturing the temporal hydrological variability (as assessed with the Nash–Sutcliffe Efficiency index) when the time scale increased. Nevertheless, in terms of both RMSE and NSE, PWBM performs substantially better than the SCS model at the daily scale, the ABCD model at the monthly scale and shows a comparable performance with the long-standing Budyko model at the mean-annual scale. This study contributes to “top-down” hydrological modelling by offering evidence on the applicability of the proportionality hypothesis at multiple spatial and temporal scales. However, future efforts are still needed to explore the physical foundation of the proportionality hypothesis.

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