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The discussion below assumes that the objective functions are being applied to streamflow data but they can be applied 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 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 of Daily Flows
The traditional formula for the NSE is:
...
where:
Qobsi is the observed flow on day i,
Qsimi is the modelled flow on day i,
N is the number of days
Alternatively, the NSE may be written as:
...
This formulation obviates the necessity to calculate the average of the observed flows before evaluating the denominator in the traditional version.
NSE of Log Transformed Flows
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 flow.
NSE of Monthly Flows
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.
Flow Duration
Flow Duration of Daily Flows
The flow duration objective function sorts the observed and modelled data values in increasing order and then calculates the NSE of the sorted data.
Flow Duration of Log Transformed Flows
This objective function calculates the flow duration objective function using the log transformed flows in Equation (3).
Absolute Bias
The equation for the absolute value of the relative bias is:
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Bias Penalty
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 (3).
Combinations of the NSE, Flow Duration and Bias Penalty Objective Functions
The following nine forms of objective function are available in Source:
- Minimise Absolute Bias between Observed and Modelled Flows (calculated using daily flows)
- Match to NSE of Daily Flows but Penalise Biased Solutions
- Match to NSE of Monthly Flows
- Match to NSE of Monthly Flows but Penalise Biased Solutions
- Combined Match to NSE and Match to Flow Duration Curve (Daily)
- Combined Match to NSE and Match to Logarithm of Flow Duration Curve (Daily)
- Combined Match to NSE of Logarithms of Daily Flows with Bias Penalty
- Combined Bias, Daily Flows and Daily Exceedance (Flow Duration) Curve (SDEB)
- Match to Nash Sutcliffe Coefficient of Efficiency (NSE) of Daily Flows
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| Objective Function Name | Description |
|---|---|
| NSE Daily | Maximise the NSE of daily flows |
| NSE Monthly | Maximise the NSE of monthly flows |
| NSE Log Daily | Maximise the NSE of the logarithm of daily flows |
| Absolute Bias | Minimise the Absolute value of the relative bias |
| NSE Daily & Bias Penalty | Maximise the NSE of daily flows and bias penalty |
| 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 |
| NSE Daily & Flow Duration | Maximise the NSE of daily flows and the NSE of the flow duration |
| NSE Daily & Log Flow Duration | Maximise the NSE of daily flows and the NSE of the flow duration of log flows |
| Square-root Daily, Exceedance and Bias |
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 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:
| Equation 1 |
|---|
where:
Qobsi is the observed flow on day i,
Qsimi is the modelled flow on day i,
N is the number of days
Alternatively, the NSE may be written as:
| Equation 2 |
|---|
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:
| Equation 3 |
|---|
where c is a small constant equal to the maximum of 1 ML and the 10th percentile of the observed flow.
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 calculates the flow duration objective function using the log transformed flows in Equation (3).
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:
| Equation 3 |
|---|
The evaluation of this objective function uses observed and modelled daily flow data for all days within the calibration period for which observed daily
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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).
2. Minimise Absolute Bias between Observed and Modelled Flows
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.
...
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:
| Equation 4 |
|---|
where B is the absolute value of the relative bias, as defined in equation (3).
Combinations of the NSE, Flow Duration and Bias Penalty Objective Functions
The following nine forms of objective function are available in Source:
- Minimise Absolute Bias between Observed and Modelled Flows (calculated using daily flows)
- 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.
...
where:
B is the bias; and
...
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
This objective function works in the same way as for the case “Match to NSE of Daily Flows” except that monthly flows are used to evaluate the NSE instead of daily flows. 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.
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- Match to NSE of Monthly Flows
- Match to NSE of Monthly Flows but Penalise Biased Solutions
- Combined Match to NSE and Match to Flow Duration Curve (Daily)
- Combined Match to NSE and Match to Logarithm of Flow Duration Curve (Daily)
- Combined Match to NSE of Logarithms of Daily Flows with Bias Penalty
- 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.
| Equation 4 |
|---|
where:
B is the bias; and
| Equation 5 |
|---|
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
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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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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:
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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.
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