Page Comparison

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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 

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

1. Match to 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)

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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 traditional formula for NSE is:

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where:

Qobs_i    is the observed flow on day i,

Qsim_i    is the modelled flow on day i,

N           is the number of days

Alternatively,

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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.

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).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.

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)

The traditional formula for the NSE is:

where:

Qobs_i    is the observed flow on day i,

Qsim_i    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.

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.

Absolute Bias

Combined 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)

1. Match to Nash Sutcliffe Coefficient of Efficiency (NSE) of Daily Flows

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