applied hydrology assessing hydrological model performance using stochastic simulation professor...

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

04/10/23 2Lab 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.


• 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.

04/10/23 4

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.


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.


• 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.


Illustration of persistence in flood flow series

04/10/23 8

A measure of persistence is defined as the cumulative impulse response (CIR).

1

1CIR

p

ii

1

t

p

iitit xx

10


• 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.


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)

04/10/23 10

obs

RMSENRMSE


• 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)

04/10/23 11

n

tt

n

ttt

m QQ

QQ

SST

SSECE

1

2

1

2

)(

)ˆ(11

n

tktt

n

ttt

N QQ

QQ

SSE

SSECP

1

2

1

2

)(

)ˆ(11


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.

04/10/23 13

n

tt

n

ttt

m QQ

QQ

SST

SSECE

1

2

1

2

)(

)ˆ(11


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.


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.

04/10/23 15

1 CP

n

tktt

n

ttt

N QQ

QQ

SSE

SSECP

1

2

1

2

)(

)ˆ(11


An example of river stage forcating

04/10/23 16

Model forecasting

CE=0.68466

ANN model

observation


04/10/23 17

Model forecasting

CE=0.68466

CP= -0.3314

Naive forecasting

CE=0.76315ANN model

observation

Naïve model


04/10/23 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Eng, NTU

18

Model forecasting

CE=0.94863


19

Model forecasting

CE=0.94863

CP= -0.218

Naive forecasting

CE=0.95783


20

Model forecasting

CE=0.90349


21

Model forecasting

CE=0.90349

CP= -0.1875

Naive forecasting

CE=0.91873

Bench Coefficient• Seibert (2001) addressed the importance of

choosing an appropriate benchmark series with which the predicted series of the considered model is compared.

04/10/23 22

n

ttbt

n

ttt

bench

QQ

QQG

1

2,

1

2

)(

)ˆ(1

n

tktt

n

ttt

QQ

QQCP

1

2

1

2

)(

)ˆ(1

n

tt

n

ttt

QQ

QQCE

1

2

1

2

)(

)ˆ(1


• 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.


• 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.

04/10/23 24

A CE-CP coupled MPE criterion.


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 .

04/10/23 25

ntxt ,,2,1,

1


04/10/23 26

[A]


• 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 .

04/10/23 27

1


Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.

04/10/23 28

6.01


• 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.


• 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.


Asymptotic relationship between CE and CP for data series of various lag-1 autocorrelation coefficients.


04/10/23 32

1= 0.843

CE=0.686 at CP=0

1= 0.822

CE=0.644 at CP=0

1= 0.908

CE=0.816 at CP=0


• For these three events, the very simple naïve forecasting yields CE values of 0.686, 0.644, and 0.816 respectively, which are nearly in the range of good to vary good according to the rating of Moriasi et al. (2007).


• In the literature we have found that many flow forecasting applications resulted in CE values varying between 0.65 and 0.85. With presence of high persistence in flow data series, it is likely that not all these models performed better than naïve forecasting.


A nearly perfect forecasting model

04/10/23 35

0

200

400

600

800

1000

1200

1400

1600

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323 337 351 365 379 393 407 421

CE=0.85599

CE=0.79021

CE=0.66646

CE=0.79109

CE=0.80027

CE=0.62629

CE=0.77926

CE=0.76404

CE=0.84652


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.


• From our previous experience in flood flow analysis and forecasting, we propose to use AR(1) or AR(2) model for benchmark comparison.


• 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= * .


• 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.


CE-CP relationships for AR(1) model

• AR(1)

04/10/23 40

144 2 CPCPCE [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)

04/10/23 41

31

4

1

84

1

422

22

222

CPCPCE [C]

tttt XXX 2211


Example of event-1

04/10/23 42

AR(1) model

AR(2) model

Data AR(2) modeling

Data AR(1) modeling

1=0.843


Assessing uncertainties in (CE, CP) using modeled-based bootstrap resampling


Assessing uncertainties in MPE by bootstrap resampling (Event-1)


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.


• 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.”


applied hydrology assessing hydrological model performance using stochastic simulation professor...

Documents

model calibration

ntu slide

model outputs

remote sensing hydrology

model parameters

hydrological model performance

model structure uncertainty

model performance evaluation