## Abstract

Evaluation criteria play a key role in assessing the performances of hydrological models. Most previous criteria are based on the standard least square method, which assumes model residuals to be homoscedastic and is, therefore, not suitable for assessing cases with heteroscedastic residuals. Here, we compared a heteroscedastic and symmetric efficiency (*HSE*) criterion with the Nash–Sutcliffe efficiency (*NSE*) and the heteroscedastic maximum-likelihood estimator (*HMLE*) by running a monthly water balance model with four parameters (i.e., the *abcd* model) in 138 basins located in the continental United States derived from the Model Parameter Estimation Experiment dataset. The results show that compared to the *NSE*, the *HSE* and *HMLE* are both more effective for stabilizing variance and producing more uniform performances with flow magnitude, and the latter is slightly more effective than the former on stabilizing the residual heteroscedasticity, with the aid of an additional parameter.

## INTRODUCTION

*et al.*2005; Ewen 2011). Quantitative measures for model performance assessments are also necessary during model calibrations, as they are objective functions in most hydrological modeling applications (Gupta

*et al.*1998). A number of assessment criteria have been developed in the past several decades for hydrological studies (Legates & McCabe 1999; Krause

*et al.*2005; Moriasi

*et al.*2007; Ewen 2011; Pushpalatha

*et al.*2012; Bennett

*et al.*2013; Harmel

*et al.*2014), among which the Nash–Sutcliffe efficiency (

*NSE*) (Nash & Sutcliffe 1970) has been mostly adopted (Legates & McCabe 1999; Ritter & Munoz-Carpena 2013). The NSE is mathematically equivalent to the standard least square (SLS) method (i.e., the Euclidean distance) for parameter calibration (Thyer

*et al.*2009) and is, thus, referred to here as an SLS-based performance measure. SLS-based criteria assume that the model residuals follow a Gaussian distribution with a zero mean (unbiasedness) and a constant variance (homoscedasticity) (Gupta

*et al.*1998; Beven 2006; Thyer

*et al.*2009; Schoups & Vrugt 2010). The model residual is defined as follows: where

*t*represents time,

*q*

_{sim}(

*t*) represents the simulated flow,

*q*

_{obs}(

*t*) represents the observed flow, and represents the constant standard deviation. However, heteroscedasticity (i.e., inhomogeneous variance) has been increasingly recognized to be more common in hydrological modeling (Sorooshian & Dracup 1980; Gupta

*et al.*1998; Beven 2006; Schoups & Vrugt 2010; Pushpalatha

*et al.*2012). In addition to hydrological model residuals, Di Baldassarre & Montanari (2009) found that standard deviations in discharge measurements varied with observed discharge, indicating that heteroscedasticity might be a natural phenomenon existing in hydrological systems. Therefore, given that the assumption of homoscedasticity may not hold under certain conditions, the use of SLS-based criteria for evaluating model performance and calibrating model parameters might be problematic (Nourali

*et al.*2016). Moreover, some theoretical works were based on the assumptions of unbiasedness and homoscedasticity. For instance, Cheng

*et al.*(2014) showed that the

*NSE*was equivalent to a maximum-likelihood method with an identical Gaussian error distribution.

To resolve the aforementioned problems, it is essential to acquire quantitative knowledge of the heteroscedasticity of model residuals (Beven & Smith 2015) and develop variance-stabilizing techniques. It is a common practice to use transformations, such as logarithmic or square root transformations in hydrology, to adjust the variances on different flow levels (Oudin *et al.* 2006). The other method is the weighted least squares method, whose major problem is the weight calculation. For instance, Sorooshian & Dracup (1980) developed a heteroscedastic maximum-likelihood estimator (*HMLE*) method by assuming that the standard deviation in the model residuals followed a power function of the predicted flows *q*_{sim}(*t*). In a later study, Sorooshian & Gupta (1983) modified *q*_{sim}(*t*) to be replaced by the observed flows *q*_{obs}(*t*) in the *HMLE* method (Yapo *et al.* 1998; Nourali *et al.* 2016). There have also been attempts to use a linear function to represent the heteroscedasticity of model residuals (e.g., Schoups & Vrugt 2010). The above methods can effectively stabilize the variance in model residuals; however, their formulas contain unknown parameters that must be inferred from the model and data via complicated iterations (Duan 1991; Sorooshian *et al.* 1993). Thus, the objective function cannot be fully determined before calibration. Meanwhile, the unknown parameters vary with different applications, leading to certain difficulties when comparing the performances of different case studies (e.g., multi-models or multi-basins). The primary objective of this study was to construct an efficiency criterion which was suitable for heteroscedastic cases without any additional parameters, and it can be computed based on only the observed and simulated flows.

*et al.*1993). This question might be answered by a more explicit and practical definition of efficiency criteria based on the metric space theory rather than the traditional vague definition (i.e., closeness). As pointed out by Guinot

*et al.*(2011), the error component of an efficiency criterion should satisfy certain mathematical requirements (i.e., nonnegativity, symmetry, and triangle inequality; (O'Searcoid 2006)), which can be given as follows: where

*x*,

*y*,

*z*∈

*and*

**X***d*is a so-called metric on

**.**

*X*The metric-based distance or similarity measures are widely used in many scientific fields, such as bioinformatics (Sippl 2008), transport geography (L'Hostis 2015), and computer science (Chen *et al.* 2009). However, the benefits of the metric-based definition of efficiency criteria have not been adequately discussed in the field of hydrology. For instance, (1) a linear combination of metrics is also a metric (see Appendix A, available with the online version of this paper) that provides a theoretical foundation for aggregated multi-objective calibrations (e.g., (Zhang *et al.* 2016)) and (2) the equivalence of metrics has been thoroughly studied with the metric space theory (O'Searcoid 2006), which can be applied to hydrological studies to distinguish equivalent efficiency criteria (e.g., the debate between Willmott *et al.* (2012) and Legates & McCabe (2013)). There are also other properties, such as convergence and continuity, which are likely helpful for efficiency criteria and need additional consideration. Due to the aforementioned advantages, heteroscedasticity and the metric-based definition were simultaneously taken into account in the design of the new proposed criterion.

The proposed criterion was compared to the *NSE* and *HMLE* in terms of the effectiveness of variance stabilization using a heteroscedasticity test (i.e., the White test) (White 1980). Further analysis of the residuals of the three objective functions was also conducted to investigate the advantages and disadvantages of the proposed criterion. The hydrological model was run in large samples of catchment basins to arrive at more convincing and general conclusions of the criterion comparisons with a single simulation (Gupta *et al.* 2014).

## METHODS

### Nash–Sutcliffe efficiency

*NSE*(Nash & Sutcliffe 1970) is the most popular criterion used to evaluate the performance of hydrological models and is defined as follows: where represents the mean of the observed flows and

*t*represents time. The

*NSE*varies from −∞ to 1 and generally falls within the range of 0–1 for calibrated simulations (Gupta

*et al.*2009). In Equation (4), the dimensional error term (i.e., the root mean square error (

*RMSE*)) is normalized by dividing the standard deviation of the observed flows. Therefore, a value of

*NSE*= 0.8, for example, means that the model explains 80% of the total variance in the observations (Ritter & Munoz-Carpena 2013). Specifically,

*NSE*= 1 implies that the simulated flows perfectly match the observed flows, whereas

*NSE*= 0 indicates that the performance of the simulated flows is equal to that of the mean observed flows (in other words, the overall results are reliable, but errors in process simulation are considerable).

*NSE*was designed to mimic the coefficient of determination (

*R*

^{2}) by replacing the fitted line of

*R*

^{2}to the 1:1 line for the

*NSE*. The numerator of the

*NSE*is the

*RMSE*, and the denominator is the variance in the observed flows; thus, the

*NSE*represents the signal-to-noise ratio of a hydrograph. During model calibrations, since is kept constant, the

*NSE*is equivalent to the SLS method, which minimizes the mean square error (

*MSE*) (Schaefli & Gupta 2007; Gupta

*et al.*2009; Thyer

*et al.*2009):

SLS-based criteria, including the *NSE*, generally show their high sensitivity to peak flows in model calibrations, which can be largely attributed to the use of the squared differences in formulating these indices (Legates & McCabe 1999; Krause *et al.* 2005; Moriasi *et al.* 2007; Pushpalatha *et al.* 2012). It should also be emphasized here that both the *NSE* and SLS are prone to larger model residuals, which are more likely to occur in high flows because of heteroscedasticity (Sorooshian & Dracup 1980; Gupta *et al.* 1998; Schoups & Vrugt 2010; Pushpalatha *et al.* 2012). The characteristic results in the high sensitivity of *NSE*/SLS schemes to high flows. Meanwhile, as pointed out by Schaefli *et al.* (2005); Schaefli & Gupta (2007) and Muleta (2012), in the case of strong seasonal flows, the *NSE* overestimates the model performance, while in the case of weak seasonal flows, the *NSE* underestimates model performance. This result suggests that variance alone is not a perfect benchmark for calibrating hydrological models (Legates & McCabe 2013).

### Heteroscedastic maximum-likelihood estimator

*HMLE*, which is calculated as follows: where represents the weight depending on the flow magnitude at each time step and is computed as follows: where

*f*(

*t*) represents either

*q*

_{sim}(

*t*),

*q*

_{obs}(

*t*), or

*αq*

_{obs}(

*t*) +

*βq*

_{sim}(

*t*), with

*α*+

*β*= 1 (Sorooshian & Dracup 1980; Sorooshian & Gupta 1983; Sorooshian

*et al.*1993; Yapo

*et al.*1996). In this study, we used

*q*

_{obs}(

*t*), as recommended by Sorooshian

*et al.*(1993). As discussed previously, the unknown

*λ*value is estimated with an iteration procedure (see Sorooshian

*et al.*1993), which gives the

*HMLE*the ability to reduce heteroscedasticity.

### Heteroscedastic and symmetric efficiency

*HSE*) is proposed here: where

*q*

_{mean}represents the mean of the observed and predicted flows: The proof for the

*HSE*as a metric is provided in Appendix B (available with the online version of this paper). The

*HSE*is designed in a manner similar to the concept of the Canberra distance (Lance & Williams 1966), which is used to compute the similarity of two datasets. The

*HSE*is expected to reduce the heteroscedasticity of the SLS method but to a lesser degree compared to the

*HMLE*method, with an additional parameter (

*λ*). The 1/ means the weights of

*HSE*.

*HSE*is that

*q*

_{obs}(

*t*) ≥ 0 and

*q*

_{sim}(

*t*) ≥ 0, but at most only one of them could be zero at any time step

*t*, while the application condition of the

*HMLE*is

*q*

_{obs}(

*t*) > 0. Thus, the applicable scope of the

*HSE*is slightly broader than that of the

*HMLE*. The

*HSE*varies within the bounded range of −1 to 1 according to the following equation:

*HSE* = 1 indicates a perfect match between the simulated and observed flows, and *HSE* = −1 implies simulated flows with constant zero series. In other words, there is no effective information extracted from the model simulations if *HSE* = −1. The *HSE* calculation at a certain time step is independent from other calculations, which is referred to as temporal independence. As a result of the boundedness and temporal independence, the influence of a single time step on the *HSE* is limited within 2/*N* and, therefore, the *HSE* is a robust criterion suitable for cases with outliers.

### The White test for heteroscedasticity

*et al.*2006). Compared to the Breusch–Pagan test (Breusch & Pagan 1979), this test does not rely on the assumptions of normality or the linear form of heteroscedasticity. In this study, the White test for a single-independent variable was used to test the independence between residual variances and observed flows. The auxiliary regression of the White test is expressed as follows: where

*e*

^{2}(

*t*) represents either squared residuals in the

*NSE*method or squared weighted residuals in the

*HMLE*and

*HSE*methods. The greater the determination coefficient of the auxiliary equation, the more heteroscedastic

*e*

^{2}(

*t*) is against the observed flows. If the White statistic (

*NR*

^{2}) satisfies Equation (12), the null hypothesis of homoscedasticity is true; otherwise, it is rejected. where

*N*represents the sample size;

*R*

^{2}represents the coefficient of the determination of the auxiliary regression;

*α*represents the significance level (set to 0.01 in this study); and

*k*represents the degree of freedom of the auxiliary regression, with

*k*= 3 (including the constant term

*k*

_{0}, the linear coefficient

*k*

_{1}, and the quadratic coefficient

*k*

_{2}).

## MODEL AND DATASET

### Description of the abcd model

A commonly used monthly water balance model (i.e., the *abcd* model) (Thomas 1981) is selected in this study because it has been applied over a variety of hydroclimatic regions (e.g., Martinez & Gupta 2010). The model structure is illustrated in Figure 1, and the parameters with their descriptions are listed in Table 1. Detailed information on the *abcd* model can be found in Thomas (1981), and a brief description of this model is provided in Appendix C (available with the online version of this paper).

Parameters . | Units . | Description . | Ranges . | Initial value . |
---|---|---|---|---|

a | – | Propensity for flow to occur before the soil is completely saturated | (0,1) | 0.99 |

b | mm | Upper soil zone water storage capacity | (0,5000) | 400 |

c | – | Groundwater recharge coefficient | (0,1) | 0.2 |

d | – | Groundwater discharge constant | (0,1) | 0.8 |

Parameters . | Units . | Description . | Ranges . | Initial value . |
---|---|---|---|---|

a | – | Propensity for flow to occur before the soil is completely saturated | (0,1) | 0.99 |

b | mm | Upper soil zone water storage capacity | (0,5000) | 400 |

c | – | Groundwater recharge coefficient | (0,1) | 0.2 |

d | – | Groundwater discharge constant | (0,1) | 0.8 |

*Note*: The initial values were determined according to the median values of parameters in the 138 basins in a rough beforehand calibration.

### Study region and dataset

The Model Parameter Estimation Experiment (MOPEX) dataset (Duan *et al.* 2006) (available at ftp://hydrology.nws.noaa.gov/pub/gcip/mopex/US_Data/) provides the daily precipitation, potential evapotranspiration, streamflow, and maximum/minimum air temperatures of 438 basins located in the continental United States. In the US MOPEX dataset, only 138 basins (Figure 2) have complete time series of monthly flow data from 1948 to 2000, with *NSE* ≥ 0.5 (Moriasi *et al.* 2007), *HSE* ≥ 0.4, and |*λ*| ≤ 1, by using the *abcd* model (see the next section for details on the model calibrations). The analysis of this study focused on the ensemble results of the 138 basins. Furthermore, detailed model results of two illustrative examples are also presented to illustrate the similarity and differences among the *NSE*, *HMLE*, and *HSE*. The first catchment is the Quinebaug River Basin in Jewett City, CT (QRB; USGS: 01127000), and the other is the Current River Basin in Van Buren, MO (CRB; USGS: 07067000). Detailed calibration results of all 138 basins, including the simulated flows and optimal parameters with the three criteria, are provided in Supplementary Material (available online).

### Calibration and validation

Martinez & Gupta (2010) compared the ranges of the four parameters of the *abcd* model derived in their study with previously reported values (Alley 1984; Vandewiele *et al.* 1992). In their studies, the parameter *a* of *abcd* model can presumably range from 0.0 to 1.0, and the values of parameter *b* <1,900 (mm/month) are more common for no-snow basins; therefore, in this study, the parameters *a*, *c*, and *d* were set to vary from 0.0 to 1.0, and parameter *b* was set to vary from 0 to 5,000 (mm/month). The shuffled complex evolution method developed at the University of Arizona (denoted as SCE-UA) was employed as the calibration algorithm (Duan, 1991). The model warm-up period was 1948–1950, the calibration period was 1951–1980, and the validation period was 1981–2000. The model parameters were, respectively, calibrated based on the criteria of *NSE*, *HMLE*, and *HSE*.

## RESULTS AND DISCUSSION

### Effectiveness of variance stabilization

In this section, the White test was used to verify the variance-stabilizing abilities of the *NSE*, *HMLE*, and *HSE*. The results of the White test for the heteroscedasticity of the (weighted) residuals in all 138 basins and two illustrative basins are displayed in Figure 3. The *NSE* residuals in all 138 basins show significant heteroscedasticity based on the White test results (*NR*^{2} ranges from 37.6 to 332.8, with an average of 150.0). Meanwhile, the *HMLE-* and *HSE-*weighted residuals in 95.7% (132 out of 138) and 51.5% (71 out of 138), respectively, passed those of the White test at a significance level of *α* = 0.01 (*NR*^{2} > *x*_{0.01}(2) = 9.21). Compared to the White statistics of the *NSE* residuals, the *HMLE*- and *HSE-*weighted residuals reduced the heteroscedasticity, on average, by 96.9% and 87.7%, respectively. The scattergrams of the two illustrative basins also show that the *NSE* variances were positively correlated with the flow magnitude. In comparison, the *HMLE-* and *HSE-*weighted residuals were more homoscedastic, supporting their heteroscedastic assumptions on the relationship between the variances of raw residuals and the corresponding flows. In summary, both the *HMLE* and *HSE* methods were shown to be effective for stabilizing variances for different hydrological models, while the effectiveness of the *NSE* for stabilizing variances was rather weak.

Note that the parameter *λ*, which represents the degree of heteroscedasticity in the *HMLE* procedure, was generally close to zero, with a mean value of −0.04 and |*λ*| ≤ 0.2 for 75% of all basins (Figure 4). Meanwhile, the weights of the *HSE* were significantly correlated with those of the *HMLE* in a positive manner (*R*^{2} = 0.92 ± 0.046), which indicates that the *HSE* method has a strong relationship with the *HMLE* method, even if the *HSE* does not use any additional parameters.

### Further analysis of residuals

Ideally, the model residuals should be independent of the flow magnitudes, which usually derivates from real situations. Therefore, we regressed the model residuals against the observed flows in all 138 basins (e.g., Figure 5), and the corresponding statistical parameters (slope, intercept, and *R*^{2}) of the linear regression are shown in Figure 6. The findings are listed below:

The slope was mostly negative (Figure 6(b)), and the intercept was mostly positive (Figure 6(c)), which indicated that the flows were generally overestimated in low flow regimes, and vice versa.

The

*HMLE*and*HSE*intercepts were considerably smaller than the*NSE*intercept, indicating much better performances at low flows using the*HMLE*and*HSE*than the*NSE*(Figure 7). Meanwhile, the performances of the*HSE*simulations and most of the*HMLE*simulations, where severe negative biases did not occur, were only slightly worse than the*NSE*simulations at high flows.When

*λ*≥ 0, the*NSE*,*HMLE*, and*HSE*residuals had similar*R*^{2}values against the observed flows. However, when*λ*< 0, the*HMLE*residuals depended more on the observed flows than the*HSE*and*NSE*residuals, and*R*^{2}values for the*HMLE*even reached up to 0.6–0.9, which was not expected in the model evaluation. Moreover, when*λ*< 0, the*HMLE*slope was more negative than the*NSE*and*HSE*slopes, which led to severe negative bias at middle and high flow levels (e.g., Figures 6(b) and 8(b)).

### Performance consistency between the calibration and validation periods

We compared the values of the *NSE*, *HMLE*, and *HSE* during calibration and validation. The *R*^{2} value for the *NSE* during the calibration and validation periods was 0.61, and the *R*^{2} values for the *HMLE* and *HSE* were 0.80 and 0.85, respectively (Figure 9). The criterion values for the *HMLE* and *HSE* (especially the *HSE*) showed great agreement between the calibration and validation periods, which is a good phenomenon in model validations. In addition, data quality, especially the rainfall input data, would influence the parameter estimation in model calibration and validation to some extent (Beven & Smith 2015). High-resolution datasets should be used to gain more precise performance.

## CONCLUSIONS

Due to different weighting schemes, the *NSE* puts more emphasis on high flows, while the *HMLE* and *HSE* do not emphasize any flow levels. If modelers would like to care more about the flood peaks, the *NSE* is suggested to be used; if modelers would like to obtain more uniform performances along with flow magnitude, either the *HMLE* or *HSE* is suggested.

For the two heteroscedastic criteria, the *HMLE* produced slightly better homoscedastic variances than the *HSE*, with the aid of an additional parameter (*λ*). However, this result is not accomplished without a cost to achieve variance stabilization. When *λ* < 0, the *HMLE* tends to allot excessive weights for low flows and insufficient weights for middle and high flows, which might cause severe bias at middle and high flow levels. Compared to the *HMLE-*weighting approach, the *HSE-*weighting approach is slightly less effective for variance stabilization but produces more robust simulations due to the use of mean flows (*q*_{mean}) instead of observed flows and setting *λ* as zero. The robustness of the *HSE* is also embodied in two other aspects. The first aspect is that the influence of a single time step on the *HSE* is limited within 2/*N*, and the second aspect is that the *HSE* during calibration and validation shows more agreement than the *NSE* and *HMLE*. Therefore, if modelers would like to stabilize the residual heteroscedasticity as much as possible, the *HMLE* is recommended for use; if the modelers would like to obtain a more robust criterion that could reduce the vast majority of residual heteroscedasticity, the *HSE* is recommended. It is suggested that future research into the proposed *HSE* and *HMLE* would be employed in different hydrological models.

## ACKNOWLEDGEMENT

This study was supported by the National Key R&D Program of China (2017YFA0603702) and National Natural Science Foundation of China (no. 41571019 and no. 51779009). We are very grateful to the editors and reviewers for their valuable comments and constructive suggestions that helped us to greatly improve the manuscript.

## REFERENCES

*A Global Optimization Strategy for Efficient and Effective Calibration of Hydrologic Models*

*PhD Thesis*