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The traditional formula for NSE is:

Equation 1	Image Added

where:

Qobs_i    is the observed flow on day i,

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N           is the number of days

Alternatively,

Equation 2	Image Added

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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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:

Equation 3	Image Added

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.

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	Image Added

where:

B is the bias; and

Equation 5	Image Added

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.

The NSE of monthly flows and NSE-Bias of monthly flows (below) can be useful for initial calibration because they tend 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).

5. Match to NSE of Monthly Flows but Penalise Biased Solutions

This objective function is the weighted combination of the monthly NSE and a logarithmic function of bias (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.

6. Combined Match to NSE and Match to Flow Duration Curve (Daily)

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

where:

A is a weighting factor whose value can be set by the modeller (0 ≤ A ≤ 1); and

NSE daily FDC is calculated using ranked value pairs of Qobs_i and Qsim_i.

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).

7. Combined Match to NSE and Match to Logarithm of Flow Duration Curve (Daily)

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

where:

Ais a weighting factor whose value can be set by the modeller (0 ≤ A ≤ 1);

NSE log₁₀(daily FDC) is calculated using ranked value pairs of log₁₀(Qobs_i+c) and log₁₀(Qsim_i+c).

c is the maximum of 1 ML and the 10^th 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.

8.  NSE Log Daily & Bias Penalty Objective Function

This objective function is given by:

NSE(logarithms of daily flows) is calculated using value pairs of  ln(Qobs_i+c) and ln(Qsim_i+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	Image Added

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.

9. 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 10	Image Added

where:

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

RQobs_k is the k’th ranked observed flow of a total of N ranked flows,

RQsim_k 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.

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:

Output data

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

Reference list

Aitken, A.P. (1973). Assessing systematic errors in rainfall-runoff models. J. Hydrol, 20, 121–136.

Black, D.C. and Podger, G.M. (2012). Guidelines for modelling water sharing rules in eWater Source: towards best practice model application. eWater Cooperative Research Centre, Canberra, Australia. July. ISBN: 978-1-921543-74-6.  Available via: www.ewater.com.au.

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.

Lerat, J. (2012). Towards the adoption of uncertainty assessment in water resources models: the eWater Source uncertainty guideline.  Proceedings of the 34^th Hydrology and Water Resources Symposium, 19-22 November 2012, Sydney, NSW. 

Lerat, J., Egan, C. A., Kim, S., Gooda, M., Loy, A., Shao, Q., and Petheram, C. (2013). Calibration of river models for the Flinders and Gilbert catchments. A technical report to the Australian Government from the CSIRO Flinders and Gilbert Agricultural Resource Assessment, part of the North Queensland Irrigated Agriculture Strategy. CSIRO Water for a Healthy Country and Sustainable Agriculture flagships, Australia.

Nash, J.E. and Sutcliffe, J.V. (1970). River flow forecasting through conceptual models, I, A discussion of principles. J. Hydrol, 10, 282–290.

Rosenbrock, H.H. (1960). An automated method of finding the greatest of least value of a function.  The Computer Journal, 3, 303-307.

Sorooshian, S., Duan, Q. and Gupta, V. (1993). Calibration of rainfall-runoff models: application of global optimization to the Sacramento Soil Moisture Accounting Model.  Water Resources Research, 29(4), 1185-1194.

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.