Introduction
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- Nash-Sutcliffe Efficiency (NSE)
- NSE of Log Data (NSE Log)
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- Pearson's Correlation Coefficient
- NSE of Flow Duration (Flow Duration)
- NSE of Flow Duration of Log Data (Log Flow Duration)
- Sum of Daily Flows, Daily Exceedance (Flow Duration) Curve and Bias (SDEB)
- Kling-Gupta Efficiency (KGE)
Source also offers a range of composite statistics that combine the NSE with other metrics. These are discussed in the section on Composite Bivariate Statistics Involving the NSE:
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There are three variants of the composite statistic that combine the KGE with Bias Penalty and are discussed in the section on Composite Bivariate Statistics Involving the KGE:
- Trotter
- Split Trotter
- Split Trotter Weighted
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The NSE is a normalised metric that measures the relative magnitude of the model error variance compared to the measured data variance (Nash and Sutcliffe, 1970). It is defined as:
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where:
Qobs,i is the observed flow for time step i
Qmod,i is the modelled flow for time step i
N is the number of time steps
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An alternative, but equivalent, formulation of the NSE is:
Equation 2 |
This formulation obviates the necessity to calculate the average of the observed flows before evaluating the denominator in the traditional version.
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NSE Log is the standard NSE metric (Equation 1) applied to the logarithm of flow data, based on a form proposed by Croke et al (2005) and Croke et al (2006):
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where:
c is a positive constant equal to the maximum of 1 ML/d (megalitre/day) (after Lerat et al, 2013) and the 10th percentile (90% flow exceedance) of the observed non-zero flows (after Croke et al, 2006). Other terms are as defined in Equation 1.
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The relative bias measures the magnitude of the model errors relative to the magnitude of the observations. Use of this statistic is described in Croke et al (2005) and Moriasi (2015). It has the form:
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where:
Qobs,i is the observed flow for time step i
Qmod,i is the modelled flow for time step i
N is the number of time steps
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- as a percent - called Volume Bias % in Source. Volume Bias % can be found in the Statistics tab in Results Manager. The bias in the modelled values expressed as a percent of the observed flow volume is defined as:
Equation 5 |
- an absolute value - called Minimise Absolute Bias in Source. Minimise Absolute Bias is available as an objective function for calibration and as one of the statistics in the Statistics tab in Results Manager. The absolute value of the relative bias is defined as:
Equation 6 |
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where B is the relative bias, as defined in Equation 4.
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Pearson's correlation coefficient measures the linear correlation between two variables and is available in Source in the Results Manager statistics tab for Bivariate Statistics. The Pearson's correlation coefficient is given by:
Equation 8 |
where:
xi is the value of time series x at time step i
yi is the value of time series y at time step i
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where:
RQobs,k is the k'th ranked observed flow of a total of N ranked flows RQsim,k is the k'th ranked modelled flow of a total of N ranked flows
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The NSE of the Flow Duration of log data is based on the same equation as the NSE of log data (Equation 3) but is calculated applying the Flow Duration to the log transformed data
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where c is calculated as in the NSE of Log Data
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- the sum of errors on power transformed flow,
- the same sum on sorted flow values and
- the relative simulation bias.
The SDEB equation is:
Equation 11 |
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where:
α is a weighting factor set to 0.1
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The Kling-Gupta Efficiency (KGE, after Gupta et al., 2009) is increasingly being used for model calibration and evaluation . The KGE is given by:
Equation 12 |
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where 𝑟 is the linear correlation between observations and simulations (Pearson’s Correlation Coefficient), 𝛼 is the measure of the flow variability error and 𝛽 represents the bias.
𝛼 is given by:
Equation 13 |
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where 
is the standard deviation and Qsim and Qobs are the simulated flows and observed flows respectively.
𝛽 is given by:
Equation 14 |
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where 
represents the mean
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Definition
Equation 15 | NSE Daily & Bias Penalty = NSE Daily – Bias Penalty |
where:
NSE Daily is the NSE of daily flows as defined in Equation 1
Bias Penalty is defined in Equation 7
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Definition
Equation 16 | NSE Log Daily & Bias Penalty = NSE Log Daily – Bias Penalty |
where:
NSE Log Daily is the NSE of the logarithm of daily flows, as defined in Equation 3
Bias Penalty is defined in Equation 7
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Definition
Equation 17 | NSE Monthly & Bias Penalty = NSE Monthly – Bias Penalty |
where:
NSE Monthly is the NSE of monthly flows, as defined in Equation 1
Bias Penalty is defined in Equation 7
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Definition
Equation 18 | NSE Daily & Flow Duration = a * NSE Daily + (1 - a) * Flow Duration |
where:
a is a user-defined weighting factor (0 ≤ a ≤ 1)
NSE Daily is the NSE of daily flows as defined in Equation 1
Flow Duration is defined in Equation 9
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Definition
Equation 19 | NSE Daily & Log Flow Duration = a * NSE Daily + (1 - a) * Log Flow Duration |
where:
a is a user-defined weighting factor (0 ≤ a ≤ 1)
NSE Daily is the NSE of daily flows as defined in Equation 1
Log Flow Duration is defined Equation 9 and Equation 10
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The objective function Trotter (after Trotter et al., 2023) aims at solving the tendency of the NSE and KGE to preference match high flows by the inclusion of a low flow component. Therefore, Trotter is used to ensure capturing of both high-flow and low-flow aspects of hydrograph in a model as well as obtain minimal volumetric bias (Trotter et al., 2023). The objective function T is given by:
Equation 20 |
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Interpretation
The first part of the model efficiency E as in the above equation is a combination of the mean of KGE of direct flows and KGE of fifth root of KGEflows. The use of the fifth root of flows is intended to provide greater emphasis to small flows and is a better alternative than the more common inverse or log transformations for zero-flow conditions. The fifth root of KGE is calculated by calculating the fifth root of observed and simulated data. The second part of the equation consists of bias penalisation (after Viney et al., 2009) for reducing the efficiency value when the when the volumetric bias (B) between the observed and simulated flows deviates from zero. The objective function combines advantages of both KGE and Bias Penalty.
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The objective function Split Trotter (after Fowler et al., 2018) is a time-based meta-objective function (an objective function that considers different aspects of a flow regime and combines them into a single-objective function) that explicitly considers different subperiods of the calibration period. This function aims to rectify the tendency of objective functions with least squares to ignore dry years in the calibration process. This is achieved by calculating the objective function (Trotter) for each subperiod and averaging them over all subperiods. The Split Trotter function ST is defined as:
Equation 21 |
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where, Ti is the Trotter value for the ith subperiod, tsi is the number of time steps in the ith subperiod.
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The objective function Split Trotter Weighted aims to place additional emphasis on model fitting to drier years by assigning a weight to every year. The objective function STW is defined as:
Equation 22 |
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where, is the weight for the ith subperiod, calculated as:
Equation 23 |
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where, Qobsi is the aggregate flows or the subperiod i for the observed time series and the other variables are as explained above.
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B. F. W. Croke, F. Andrews, A. J. Jakeman, S. Cuddy and A. Luddy (2006) Redesign of the IHACRES rainfall-runoff model. Engineers Australia 29th Hydrology and Water Resources Symposium, 21–23 February 2005, CanberraLerat
Fowler, JK., CG. ACoxon, J. EganFreer, SM. KimPeel, MT. GoodaWagener, A. LoyWestern, QR. Shao Woods and CL. Petheram Zhang (2013) Calibration of river models for the Flinders and Gilbert catchments. A technical report to the Australian Government from the CSIRO Flinders and Gilbert Agricultural Resource Assessment, part 2018) Simulating Runoff Under Changing Climatic Conditions: A Framework for Model Improvement. Water Resources Research, 54, 9812–9832. https://doi.org/10.1029/2018WR023989
Gupta, H.V., H. Kling, K.K. Yilmaz and G.F. Martinez (2009) Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of Hydrology, 377, 80-91, doi: http://dx.doi.org/10.1016/j.jhydrol.2009.08.003
Knoben, W.J.M., J.E. Freer and R.A. Woods (2019) Technical note: Inherent benchmark or not? Comparing Nash-Sutcliffe and Kling-Gupta efficiency scores. Hydrology and Earth System Sciences, 23(10), 4323-4331, doi: https://doi.org/10.5194/hess-23-4323-2019
Lerat, J., C.A. Egan, S. Kim, M. Gooda, A. Loy, Q. Shao and C. Petheram (2013) Calibration of river models for the Flinders and Gilbert catchments. A technical report to the Australian Government from the CSIRO Flinders and Gilbert Agricultural Resource Assessment, part of the North Queensland Irrigated Agriculture Strategy. CSIRO Water for a Healthy Country and Sustainable Agriculture flagships, Australia.
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Nash, J.E. and J.V. Sutcliffe (1970) River flow forecasting through conceptual models part I — A discussion of principles. Journal of Hydrology, 10 (3), 282–290.
Trotter, L., M. Saft, M.C. Peel and K.J.A Fowler (2023) Symptoms of Performance Degradation During Multi-Annual Drought: A Large-Sample, Multi-Model Study. Water Resources Research, 59, e2021WR031845. https://doi.org/10.1029/2021WR031845
Vaze, J., P. Jordan, R. Beecham, A. Frost, G. Summerell (2011) Guidelines for rainfall-runoff modelling: Towards best practice model application. eWater Cooperative Research Centre, Canberra, ACT. ISBN 978-1-921543-51-7. Available via www.ewater.org.au.
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