1. Applied Hydrology Assessing Hydrological Model Performance Using Stochastic Simulation
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University

2. INTRODUCTION
Very often, in hydrology, the problems are not clearly understood for a meaningful analysis using physically-based methods.
Rainfall-runoff modeling
Empirical models – regression, ANN
Conceptual models – Nash LR
Physical models – kinematic wave
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

3. Regardless of which types of models are used, almost all models need to be calibrated using historical data.
Model calibration encounters a range of uncertainties which stem from different sources including
data uncertainty,
parameter uncertainty, and
model structure uncertainty.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

4. The uncertainties involved in model calibration inevitably propagate to the model outputs.
Performance of a hydrological model must be evaluated concerning the uncertainties in the model outputs.
Uncertainties in model performance evaluation.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

5. ASCE Task Committee, 1993
“Although there have been a multitude of watershed and hydrologic models developed in the past several decades, there do not appear to be commonly accepted standards for evaluating the reliability of these models. There is a great need to define the criteria for evaluation of watershed models clearly so that potential users have a basis with which they can select the model best suited to their needs”.
Unfortunately, almost two decades have passed and the above scientific quest remains valid.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

6. SOME NATURES OF FLOOD FLOW FORECASTING
Incomplete knowledge of the hydrological process under investigation.
Uncertainties in model parameters and model structure when historical data are used for model calibration.
It is often impossible to observe the process with adequate density and spatial resolution.
Due to our inability to observe and model the spatiotemporal variations of hydrological variables, stochastic models are sought after for flow forecasting.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

7. A unique and important feature of the flow at watershed outlet is its persistence, particularly for the cases of large watersheds.
Even though the model input (rainfall) may exhibit significant spatial and temporal variations, flow at the outlet is generally more persistent in time.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

8. Illustration of persistence in flood flow series
A measure of persistence is defined as the cumulative impulse response (CIR).
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

9. The flow series have significantly higher persistence than the rainfall series.
We have analyzed flow data at other locations including Hamburg, Iowa of the United States, and found similar high persistence in flow data series.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

10. CRITERIA FOR MODEL PERFORMANCE EVALUATION
Relative error (RE)
Mean absolute error (MAE)
Correlation coefficient (r)
Root-mean-squared error (RMSE)
Normalized Root-mean-squared error (NRMSE)
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

11. Coefficient of efficiency (CE) (Nash and Sutcliffe, 1970)
Coefficient of persistence (CP) (Kitanidis and Bras, 1980)
Error in peak flow (or stage) in percentages or absolute value (Ep)
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

13. Coefficient of Efficiency (CE)
The coefficient of efficiency evaluates the model performance with reference to the mean of the observed data.
Its value can vary from 1, when there is a perfect fit, to A negative CE value indicates that the model predictions are worse than predictions using a constant equal to the average of the observed data.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

14. Model performance rating using CE (Moriasi et al., 2007)
Moriasi et al. (2007) emphasized that the above performance rating are for a monthly time step. If the evaluation time step decreases (for example, daily or hourly time step), a less strict performance rating should be adopted.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

15. Coefficient of Persistency (CP)
It focuses on the relationship of the performance of the model under consideration and the performance of the naïve (or persistent) model which assumes a steady state over the forecast lead time.
A small positive value of CP may imply occurrence of lagged prediction, whereas a negative CP value indicates that performance of the considered model is inferior to the naïve model.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

16. Examples of flood forcating
Model forecasting
CE=
ANN model
observation
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

17. Model forecasting CE=0.68466 CP= -0.3314 Naive forecasting CE=0.76315
Which forecast do you prefer?
Model forecasting
CE=
CP=
Naive forecasting
CE=
ANN model
observation
Naïve model
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

18. Model forecasting CE=0.94863 12/2/2018
Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

19. Model forecasting CE=0.94863 CP= -0.218 Naive forecasting CE=0.95783
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

20. Model forecasting CE=0.90349 12/2/2018
Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

21. Model forecasting CE=0.90349 CP= -0.1875 Naive forecasting CE=0.91873
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

29. Why the naïve forecasting is often superior to ANNs, especially for floods with higher lag-1 autocorrelation coefficients.
The Markov property
Larger watersheds
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

30. Bench Coefficient
Seibert (2001) addressed the importance of choosing an appropriate benchmark series with which the predicted series of the considered model is compared.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

31. The bench coefficient provides a general form for measures of goodness-of-fit based on benchmark comparisons.
CE and CP are bench coefficients with respect to benchmark series of the constant mean series and the naïve-forecast series, respectively.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

32. The bottom line, however, is what should the appropriate benchmark series be for the kind of application (flood forecasting) under consideration.
We propose to use the AR(1) or AR(2) model as the benchmark for flood forecasting model performance evaluation.
A CE-CP coupled MPE criterion.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

33. ASYMPTOTIC RELATIONSHIP BETWEEN CE AND CP
Given a sample series { }, CE and CP respectively represent measures of model performance by choosing the constant mean series and the naïve forecast series as benchmark series.
The sample series is associated with a lag-1 autocorrelation coefficient
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

34. [A]
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

35. Given a data series with a specific lag-1 autocorrelation coefficient, we can choose various models for one-step lead time forecasting of the given data series.
Equation [A] indicates that, although the forecasting performance of these models may differ significantly, their corresponding (CE, CP) pairs will all fall on a specific line determined by .
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

36. Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

37. The asymptotic CE-CP relationship can be used to determine whether a specific CE value, for example CE=0.55, can be considered as having acceptable accuracy.
The CE-based model performance rating recommended by Moriasi et al. (2007) does not take into account the autocorrelation structure of the data series under investigation, and thus may result in misleading recommendations.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

38. Consider a data series with significant persistence or high lag-1 autocorrelation coefficient, say 0.8. Suppose that a forecasting model yields a CE value of 0.55 (see point C). With this CE value, performance of the model is considered satisfactory according to the performance rating recommended by Moriasi et al. (2007).
However, it corresponds to a negative value of CP (-0.125), indicating that the model performs even poorer than the naïve forecasting, and thus should not be recommended.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

39. Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

40. 1= 0.843 CE=0.686 at CP=0 1= 0.822 CE=0.644 at CP=0 1= 0.908
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

41. For these three events, the very simple naïve forecasting yields CE values of 0.686, 0.644, and respectively, which are nearly in the range of good to vary good according to the rating of Moriasi et al. (2007).
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

42. In the literature we have found that many flow forecasting applications resulted in CE values varying between 0.65 and With presence of high persistence in flow data series, it is likely that not all these models performed better than naïve forecasting.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

43. A nearly perfect forecasting model
Individual events
CE=
CE=
CE=
CE=
CE=
CE=
CE=
CE=
Artifactual series
CE=
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

44. Individual events CP=0.193442 CP= -0.0089 CP= -0.17602 CP=0.137917
Artifactual series
CP=
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

45. A CE-CP COUPLED MPE CRITERION
Are we satisfied with using the constant mean series or naïve forecasting as the benchmark?
Considering the high persistence nature in flow data series, we argue that performance of the autoregressive model AR(p) should be considered as a benchmark comparison for performance of other flow forecasting models.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

46. From our previous experience in flood flow analysis and forecasting, we propose to use AR(1) or AR(2) model for benchmark comparison.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

47. The asymptotic relationship between CE and CP indicates that when different forecasting models are applied to a given data series (with a specific value of 1, say *), the resultant (CE, CP) pairs will all fall on a line determined by Eq. [A] with 1= * .
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

48. In other words, points on the asymptotic line determined by 1= 
In other words, points on the asymptotic line determined by 1= * represent forecasting performance of different models which are applied to the given data series.
Using the AR(1) or AR(2) model as the benchmark, we need to know which point on the asymptotic line corresponds to the AR(1) or AR(2) model.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

49. CE-CP relationships for AR(1) model
[B]
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

50. CE-CP relationships for AR(1) and AR(2) models
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

51. Example of event-1 1=0.843 AR(2) model Data AR(2) modeling
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

52. Assessing uncertainties in (CE, CP) using modeled-based bootstrap resampling
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

53. Assessing uncertainties in MPE by bootstrap resampling (Event-1)
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

54. Assessing uncertainties in MPE by bootstrap resampling (Event-1)
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

55. Conclusions
Performance of a flow forecasting model needs to be evaluated by taking into account the uncertainties in model performance.
AR(2) model should be considered as the benchmark.
Bootstrap resampling can be helpful in evaluating the uncertainties in model performance.
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

56. Seibert (2001)
“Obviously there is the risk of discouraging results when a model does not outperform some simpler way to obtain a runoff series. But if we truly wish to assess the worth of models, we must take such risks. Ignorance is no defense.”
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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU