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Application of this objective function involves maximising the NSE (i.e. getting it as close to 1.0 as possible).   The calculation of the NSE is in accordance with Nash and Sutcliffe (1970) and uses observed and modelled daily flow data for all days within the calibration period for which observed daily flow data, including zero flow values (i.e. cease to flow), is available.

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N           is the number of days

Alternatively, An alternative, but equivalent, formulation of the NSE may be written asis:

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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where c is a small constant equal to the maximum of 1 ML and the 10^th percentile of the observed flowflows. The use of this constant is intended to de-emphasise very small flows, which tend to be unreliable, and overcome the problem of trying to take logarithms of zero flows.

NSE Monthly

This objective function uses the same equation as for the NSE of daily flows (equation (1)), but applies it to monthly rather than daily data:

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This formulation makes sure that the models are calibrated predominantly to optimise NSE while ensuring a low bias in the total streamflow. It avoids solutions that produce biased estimates of overall runoff, which can produce marginal improvements in low flow performance over the NSE objective function. However, NSE-Bias will still be strongly influenced by moderate and high flows and by the timing of runoff events, which can still often result in poor fits to low flows (Vaze et al., 2011).The evaluation of this objective function uses observed and modelled daily flow data for all days within the calibration period for which observed daily flow data, including zero flow values, is available.

NSE

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Log Daily & Bias Penalty

This objective function is

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6. Combined Match to NSE and Match to Flow Duration Curve (Daily)

For this case the aim is to maximise the objective function, where:

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where:

A is a weighting factor whose value can be set by the modeller (0 ≤ A ≤ 1); and

NSE daily FDC is calculated using ranked value pairs of Qobs_i and Qsim_i.

This objective function and the following objective function are hybrids that compromise between the fit to the timing of high and moderate flows from the NSE component and the fit to the shape of the whole flow duration curve (FDC).  The NSE-logFDC (below) will produce the closer fit to low flows (Vaze et al, 2011).

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given by:

NSE(logarithms of daily flows) is calculated using value pairs of  ln(Qobs_i+c) and ln(Qsim_i+c), where B and v are defined in the same way as above.  

This objective function captures the model’s ability to fit the shape of the observed daily flow hydrograph, with an emphasis on mid-range to low flows (in contrast to the arithmetic form of the NSE which tends to put an emphasis on medium to high flows), while ensuring a low bias in the total streamflow.

NSE Monthly and Bias Penalty

This objective function is the weighted combination of the monthly NSE and a logarithmic function of bias (Viney et al, 2009), and the aim is to find its maximum value.  The equation used is the same as for the case “Match to NSE of Daily Flows but Penalise Biased Solutions” above.  The NSE and Bias calculations ignore observed and modelled data for all months where there are one or more days of missing data in the observed flow series.

where 

NSE Monthly is as defined above

Bias Penalty is defined in equation ()

NSE Daily and Flow Duration

For this case the aim is to maximise the objective function, where:

where:

Aa is a user-defined weighting factor whose value can be set by the modeller (0 ≤ A  a ≤ 1); and

NSE log₁₀(daily FDC) is calculated using ranked value pairs of log₁₀(Qobs_i+c) and log₁₀(Qsim_i+c).

c is the maximum of 1 ML and the 10^th percentile of the observed flows.  The use of this constant is intended to de-emphasise very small flows, which tend to be unreliable, and overcome the problem of trying to take logarithms of zero flows.

8.  NSE Log Daily & Bias Penalty Objective Function

This objective function is given by:

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NSE(logarithms of daily flows) is calculated using value pairs of  ln(Qobs_i+c) and ln(Qsim_i+c), where B and v are defined in the same way as above.  The Bias Penalty is based on Viney et al (2009) and is: 

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This objective function captures the model’s ability to fit the shape of the observed daily flow hydrograph, with an emphasis on mid-range to low flows (in contrast to the arithmetic form of the NSE which tends to put an emphasis on medium to high flows), while ensuring a low bias in the total streamflow.

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Daily is defined in equation ()

Flow Duration is defined in equation ()

This objective function and the following objective function are hybrids that compromise between the fit to the timing of high and moderate flows from the NSE component and the fit to the shape of the whole flow duration curve (FDC).  The NSE-logFDC (below) will produce the closer fit to low flows (Vaze et al, 2011).

NSE Daily and Log Flow Duration

For this case the aim is to maximise the objective function, where:

where:

a is user-defined weighting factor (0 ≤ a ≤ 1); and

NSE Daily is defined in equation ()

Flow Duration is defined in equation ()

Combined Bias, Daily Flows and Daily Exceedance (Flow Duration) Curve (SDEB)

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Output data

Outputs include results of the evaluation of the selected objective function and other calibration performance statistics. 

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