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

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 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Uncertainties in model performance evaluation. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Illustration of persistence in flood flow series 1/31/ A measure of persistence is defined as the cumulative impulse response (CIR). Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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) 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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) 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

An example of river stage forcating 1/31/ Model forecasting CE= ANN model observation Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

1/31/ Model forecasting CE= CP= Naive forecasting CE= ANN model observation Naïve model Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

1/31/2014 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU 18 Model forecasting CE=

1/31/2014 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU 19 Model forecasting CE= CP= Naive forecasting CE=

1/31/2014 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU 20 Model forecasting CE=

1/31/2014 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU 21 Model forecasting CE= CP= Naive forecasting CE=

Bench Coefficient Seibert (2001) addressed the importance of choosing an appropriate benchmark series with which the predicted series of the considered model is compared. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ A CE-CP coupled MPE criterion. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

1/31/ [A] Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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 ( ), indicating that the model performs even poorer than the naïve forecasting, and thus should not be recommended. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

1/31/ = CE=0.686 at CP=0 1 = CE=0.644 at CP=0 1 = CE=0.816 at CP=0 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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). 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

A nearly perfect forecasting model 1/31/ CE= CE= CE= CE= CE= CE= CE= CE= CE= Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

A CE-CP COUPLED MPE CRITERION Are we satisfied with using the constant mean series or naïve forecasting as 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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

From our previous experience in flood flow analysis and forecasting, we propose to use AR(1) or AR(2) model for benchmark comparison. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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 = *. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

CE-CP relationships for AR(1) model AR(1) 1/31/ [B] Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

CE-CP relationships for AR(1) and AR(2) models AR(2) 1/31/ [C] Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Example of event-1 1/31/ AR(1) model AR(2) model Data AR(2) modeling Data AR(1) modeling 1 =0.843 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Assessing uncertainties in (CE, CP) using modeled-based bootstrap resampling 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Assessing uncertainties in MPE by bootstrap resampling (Event-1) 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

Assessing uncertainties in MPE by bootstrap resampling (Event-1) 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

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. 1/31/ Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU