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Missing Data

It is common for observed time series of hydrological processes to contain missing values. Also, the observed and modelled time series may have different start and end dates. The Source calibration tool calculates the objective function values using only data from those time steps for which both observed and modelled data is available.

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where B is the absolute value of the relative bias, as defined in equation (3).

In Source, the Bias Penalty is always used in combination with other objective functions and is not available on its own.

Combinations of the NSE, Flow Duration and Bias Penalty Objective Functions

The following nine forms of objective function are available in Source:

1. Minimise Absolute Bias between Observed and Modelled Flows (calculated using daily flows)
2. Match to NSE of Daily Flows but Penalise Biased Solutions
3. Match to NSE of Monthly Flows
4. Match to NSE of Monthly Flows but Penalise Biased Solutions
5. Combined Match to NSE and Match to Flow Duration Curve (Daily)
6. Combined Match to NSE and Match to Logarithm of Flow Duration Curve (Daily)
7. Combined Match to NSE of Logarithms of Daily Flows with Bias Penalty
8. Combined Bias, Daily Flows and Daily Exceedance (Flow Duration) Curve (SDEB)

3. Match to NSE of Daily Flows but Penalise Biased Solutions

This objective function is a weighted combination of the daily NSE and a logarithmic function of bias based on Viney et al (2009), and the aim is to find its maximum value. 

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where:

B is the bias; and

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NSE Daily and Bias Penalty

This objective function is a weighted combination of the daily NSE and a logarithmic function of bias based on Viney et al (2009), and the aim is to find its maximum value. 

where 

NSE Daily is defined in equation ()

Bias Penalty is defined in equation ()

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.

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).

4. Match to NSE of Monthly Flows

5. Match to NSE of Monthly Flows but Penalise Biased Solutions

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.

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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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.

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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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.

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

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This objective function is based on the function introduced by Coron et al. (2012) and has been successfully applied in a number of projects (e.g. Lerat et al., 2013).  It has the following equation:

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The coefficient α and the power transform are used to balance the three terms within the objective function.  The weighting factor α is used to reduce the impact of the timing errors on the objective function. This type of error can have a significant effect on the first term in the equation, where a slight misalignment of observed and simulated peak flow timing can result in large amplitude errors. Conversely, the second term is based on sorted flow values, which remain unaffected by timing errors. By way of example, Lerat et al. (2013) in their study of the Flinders and Gilbert Rivers in Northern Australia used values of α of 0.1 for the Flinders calibration and 1.0 for the Gilbert calibration.

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Modellers have the option of selecting one optimisation technique, multiple optimisation techniques (in parallel), or combinations two optimisation techniques (in series), or not using optimisation.  Modellers can also select which objective function they wish to use.  The other parameters the modeller can input are described in the following table:

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Weighting factor for the objective function in case 9

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Parameter	Description	Units	Default	Range
A	Weighting factor for the objective function in cases 6 and 7	Dimensionless	0.5	0 ≤ A ≤ 1
α	Weighting factor for the objective function in case 9	Dimensionless	0.5	0 ≤ α ≤ 1

Objective Function	Paramater	Parameter Description	Units	Default	Range
NSE Daily
NSE Monthly
NSE Log Daily
Absolute Bias
NSE Daily & Bias Penalty
NSE Log Daily & Bias Penalty
NSE Monthly & Bias Penalty
NSE Daily & Flow Duration
NSE Daily & Log Flow Duration
Square-root Daily, Exceedance and Bias

Output data

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

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