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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 the Source Calibration Wizard are listed in the table below Table 1, including useful references for further information. Refer to the Bivariate Statistics User Guide entry for the objective function equations and further information . on their interpretation.
Table 1. List of Source calibration objective functions.Anchor Table 1 Table 1
| Objective Function Name | Description | Reference | |
|---|---|---|---|
| NSE Daily | Maximise the NSE of daily flows | Vaze et al. (2011), Section 6 | |
| NSE Monthly | Maximise the NSE of monthly flows | Vaze et al. (2011), Section 6 | |
| NSE Log Daily | Maximise the NSE of the logarithm of daily flows | ||
| Absolute Bias | Minimise the Absolute value of the relative bias | Vaze et al. (2011), Section 6 | |
| NSE Daily & Bias Penalty | Maximise the NSE of daily flows and bias penalty | Vaze et al. (2011), Section 6 | |
| NSE Log Daily & Bias Penalty | Maximise the NSE of the logarithm of daily flows and bias penalty | ||
| NSE Monthly & Bias Penalty | Maximise the NSE of monthly flows and bias penalty | Vaze et al. (2011), Section 6 | |
| NSE Daily & Flow Duration | Maximise the NSE of daily flows and the NSE of the flow duration | Vaze et al. (2011), Section 6 | |
| NSE Daily & Log Flow Duration | Maximise the NSE of daily flows and the NSE of the flow duration of log flows | Vaze et al. (2011), Section 6 | |
| Square-root Daily, Exceedance and Bias | Minimise a combination of the bias, daily Flows and daily exceedance (flow duration) curve | Lerat et al., 2013 |
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. The descriptions of the objective function equations assume that the observed In other words, the calibration objective function are calculated using observed and modelled data has been filtered to include only:
- data from within the calibration period, and
- data for time steps with complete data pairs.
Nash Sutcliffe Coefficient of Efficiency (NSE)
NSE Daily
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.
The NSE tends to produce solutions that match high and moderate flows very well but often will produce poor fits to low flows. It will also tend to favour solutions that provide a good match to the timing and shape of runoff events (Vaze et al., 2011).
The traditional formula for the NSE is:
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where:
Qobsi is the observed flow on day i,
Qsimi is the modelled flow on day i,
N is the number of days
An alternative, but equivalent, formulation of the NSE is:
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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.
NSE Log Daily
This objective function uses the same equation as for the NSE of daily flows (equation (1)), but applies it to log transformed data:
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where c is a small constant equal to the maximum of 1 ML and the 10th 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.
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:
- If the model is run on a daily time step, Qobsi becomes the sum of the observed flows for month i and Qsimi becomes the sum of the modelled flow for month i. The NSE calculation ignores observed and modelled data for all months where there are one or more days of missing data in the observed flow series.
- If the model is run on a monthly time step, then the monthly values are unchanged.
The NSE of monthly flows can be useful for initial calibration because it tends to find solutions that will match the overall movement of water through the conceptual stores in the rainfall-runoff model, without being influenced by the timing of individual runoff events (Vaze et al., 2011).
Flow Duration
Flow Duration
The flow duration objective function sorts the observed and modelled data values in increasing order and then calculates the NSE of the sorted data.
Log Flow Duration
This objective function 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:
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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.
Bias Penalty
The bias penalty objective function is described in Viney et al. (2009). The equation is given by:
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where B is the absolute value of the relative bias, as defined in equation (4).
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
NSE Daily and Bias Penalty
This objective function is a weighted combination of the daily NSE and the bias penalty developed by Viney et al. (2009), and the aim is to maximise its value.
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where
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).
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NSE Log Daily & Bias Penalty
This objective function is given by:
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NSE Log Daily & Bias Penalty = NSE Log Daily – Bias Penalty
where
NSE 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 NSE Daily 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 the bias penalty (Viney et al., 2009), and the aim is to find its maximum value:
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where
NSE Monthly is 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 given by:
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where:
a is user-defined weighting factor (0 ≤ a ≤ 1); and
NSE Daily is defined in equation (1)
Flow Duration is defined above
NSE Daily and Log Flow Duration
For this case the aim is to maximise the objective function, where:
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where:
a is user-defined weighting factor (0 ≤ a ≤ 1); and
NSE Daily is defined in equation (1)
Flow Duration is defined above
Combined Bias, Daily Flows and Daily Exceedance (Flow Duration) Curve (SDEB)
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- .
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where:
α is a weighting factor whose value can be set by the modeller (0 ≤ α ≤ 1).
RQobsk is the k’th ranked observed flow of a total of N ranked flows,
RQsimk is the k’th ranked modelled flow of a total of N ranked flows, and
Other terms are as defined previously.
As explained by Lerat et al. (2013), this function combines three terms: (i) the sum of squared errors on power transform of flow, (ii) the same sum on sorted flow values and (iii) the relative simulation bias.
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.
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.
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Black, D.C., Wallbrink, P.J., Jordan, P.W., Waters, D., Carroll, C., and Blackmore, J.M. (2011). Guidelines for water management modelling: towards best practice model application. eWater Cooperative Research Centre, Canberra, Australia. September. ISBN: 978-1-921543-46-3. Available via: www.ewater.com.au.Coron, L., Andrassian, V., Perrin, P., Lerat, J., Vaze, J., Bourqui, M., and Hendrickx, F. (2012) Crash testing hydrological models in contrasted climate conditions: an experiment on 216 Australian catchments. Water Resources Research, 48, W05552, doi:10.1029/ 2011WR011721.
Duan, Q., Sorooshian, S. and Gupta, V. (1992). Effective and Efficient global optimization for conceptual rainfall-runoff models. Water Resources Research, 28(4), 1015-1031.
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Vaze, J., Jordan, P., Beecham, R., Frost, A., Summerell, G. (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.com.au.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.