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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 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 - Insight - SRG for information).

Source implements five different basic types of objective function:

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Overview information on the four optimisation techniques in Source is available in Vaze et al. (2011).  Further information is in textbooks and papers, particularly for the genetic algorithm and uniform random sampling[DB1] .  Publications on the shuffled complex evolution method include papers by Duan et al. (1992) and Sorooshian et al. (1993).  Publications on the Rosenbrock method include the paper by Rosenbrock (1960).

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Availability

Provided with Source.

Some of these objective functions can be combined to create composite objective functions. For composite objective functions, the user is often able to enter a weight that determines the relative contribution of each objective function component to the 

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

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).

Structure & processes

Background

As the optimisation techniques and statistical measures of calibration performance used in Source are well established, they are not re-described here. However, as the objective functions used in the optimisation techniques have been customised for Source, further information on these follows and as many of them rely on the Nash Sutcliffe Coefficient of Efficiency (NSE), its formulation is restated below.

The choice of any particular objective function will depend on the intended application. Each of the pre-defined objective functions are formulated to put emphasis (reproduce as closely as possible) on different flow characteristics (Vaze et al, 2011).

The discussion below assumes that the objective functions are being applied to streamflow data but they can be applied to any time series data.

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Minimise a combination of the bias, daily Flows and daily exceedance (flow duration) curve

Implementation

Background

The optimisation techniques and statistical measures of calibration performance used in Source are well established and are not described in detail here. 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).

The choice of an appropriate objective function for calibration depends on the intended application of the model. Different objective functions are designed with the intention of emphasizing the fit of modelled flow to different aspects of the observed hydrograph (Vaze et al., 2011). The objective functions available in Source are listed in the table below, including useful references for further information. 

Since some of the objective functions have been customised for use in Source, the objective function equations are defined in the following sub-sections. The discussion assumes that the objectives are being applied to streamflow data, but the equations are applicable to any time series data.

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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Log Flow Duration

This objective function calculates the flow duration objective function using the uses log transformed flows, as described in Equation (3). The log transformed observed and modelled flows are sorted increasing order and the NSE is calculated on the sorted data.

Absolute Bias

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:

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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The bias penalty objective function is described in Viney et al. (2009). The equation is given by:

where B is the absolute value of the relative bias, as defined in equation (34).

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

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

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NSE Daily is defined in equation (1)

Bias Penalty is defined in equation (5)

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.

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This objective function is given by:

where 

NSE

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Log Daily is defined in equations (1) and (3)

Bias Penalty is defined in equation (5)

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 Daily 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 the weighted combination of the monthly NSE and a logarithmic function of the bias penalty (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.

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:

where 

NSE Monthly is as defined above

Bias Penalty is defined in equation (5)

NSE Daily and Flow Duration

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

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NSE Daily is defined in equation (1)

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:

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above

NSE Daily and Log Flow Duration

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

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NSE Daily is defined in equation (1)

Flow Duration is defined in equation ()above

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

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 1011	Image Modified

where:

α is a weighting factor whose value can be set by the modeller (0 ≤ α ≤ 1).

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Using values of power transform of less than 1 has the effect of reducing the weight of the errors in high flows, where the flow data are known to be less accurate.  Lerat et al. (2013) found that a power transform of ½ led to the best compromise between high and low flow performance in their project. This value has been adopted in Source.

Data

Input data

Details on data to be input by the modeller are provided in the Source User Guide.  Requirements for data series inputs to the various objective functions are included in the descriptions of each objective function, above.

Parameters or settings

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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adopted in Source.

Data

Input data

Details on data to be input by the modeller are provided in the Source User Guide. Requirements for data series inputs to the various objective functions are included in the descriptions of each objective function, above.

Parameters or settings

Modellers have the option of selecting one optimisation technique, two optimisation techniques (in series), or manual 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:

Output data

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

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Viney, N.R., Perraud, J-M., Vaze, J., Chiew F.H.S., Post, D.A. and Yang, A. (2009). The usefulness of bias constraints in model calibration for regionalisation to ungauged catchments.  In: 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation, July 2009, Cairns: Modelling and Simulation Society of Australian and New Zealand and International Association for Mathematics and Computers in Simulation: 3421-3427.