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Source includes a number of optimisation techniques and statistical measures for automated model calibration and to assist modellers with the evaluation of the quality of calibration. These are mainly intended for application when calibrating catchment rainfall-runoff models in Source, but are also applicable when calibrating river system models (e.g. see Lerat et al., 2013). Optimisation techniques available The available automatic optimisation algorithms are:
- Shuffled complex evolution
- Genetic algorithms
- Uniform random sampling
- Rosenbrock method
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Automated calibration requires the use of an objective function to direct the optimisation process. The Source calibration tool implements single objective function optimisation which reduces the comparison between the observed and modelled data during the calibration period to a single number to be optimised (Multiple objective optimisation is also available: see Multi-objective optimisation/trade-off analysis - Insight - SRG for information). The following nine forms of objective function are available in Source:
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Further information on the first seven of these objective functions is available in Vaze et al. (2011), Section 6. Guidance on model calibration is available in many publications, including various eWater Best Modelling Practice Guidelines (Black et al., 2011; Vaze et al., 2011; Black and Podger, 2012; and Lerat 2012).
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The traditional formula for NSE is:
| Equation 1 |
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where:
Qobsi is the observed flow on day i,
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N is the number of days
Alternatively,
| Equation 2 |
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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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This objective function will produce a match on the overall volume of flow generated but often will produce a poor fit to the timing of flows (Vaze et al, 2011). It has the following form:
| Equation 3 |
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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.
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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.
| Equation 4 |
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where:
B is the bias; and
| Equation 5 |
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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.
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For this case the aim is to maximise the objective function, where:
| Equation 6 | Objective function = A * NSE daily daily flows + (1 - A) * NSE daily FDC |
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where:
A is a weighting factor whose value can be set by the modeller (0 ≤ A ≤ 1); and
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For this case the aim is to maximise the objective function, where:
| Equation 7 | Objective function = A * NSE daily flows + (1 - A) * NSE log10(daily FDC) |
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where:
Ais a weighting factor whose value can be set by the modeller (0 ≤ A ≤ 1);
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This objective function is given by:
| Equation 8 | Objective function = NSE(logarithms of daily flows) – Bias Penalty |
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NSE(logarithms of daily flows) is calculated using value pairs of ln(Qobsi+c) and ln(Qsimi+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:
| Equation 9 |
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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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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:
| Equation 10 |
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where:
α is a weighting factor whose value can be set by the modeller (0 ≤ α ≤ 1).
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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:
| 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 |
Output data
Outputs include results of the evaluation of the selected objective function and other calibration performance statistics.
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