Improving reliability of Wavelet based hydrological ...

Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection Vinit Sehgal†Kinshuk Tripathi and Prof. Rajeev Ranjan Sahay Department of Civil Engineering, Birla Institute of Technology- Mesra, Ranchi, Jharkhand, India

SUMMARY:

Wavelet based hybrid models are very popular in hydrological modeling. However, the performance

of these models depends on the suitable selection of mother wavelet for carrying out discrete wavelet transform (DWT). This paper relates the vanishing moment of the mother wavelet to the performance of the wavelet based hydrological model. Study reveals that higher the vanishing moment of a wavelet, better is the performance of the wavelet based hybrid model. A case study is carried out at Kamla River of North Bihar, India; using wavelet based Auto Regressive models (WR) to forecast 1- day ahead water levels. varying duration. On comparison, data-based artificial

1. INTRODUCTION

intelligent models have recently gained popularity in

Recent floods all over the world have raised the issue of flood prediction and its control. Timely and accurate forecast of an impending flood is critical to water diversion and allocation, drought prevention, flood forecasting, dam and human safety, and ecosystem sustainability. Heavy downpours in upper catchments of North-Eastern Indian Rivers rush toward the main land with great velocity and

hydrological applications due to their rapid development, fewer

data

requirement

and

ease

of

real-time

implementation (Adamowski, 2008). Artificial neural networks (ANN) Auto Regression (AR) are examples of most widely used artificial intelligent models. However, these models are found inadequate when dealing with transitory or nonstationary data.

destroy lives, infrastructure, agriculture and industrial have

Wavelet transform, on the other hand are useful for

traditionally been used for flood forecasts, but their

analyzing variations, periodicities and trends in a time

dependency on too many exogenous inputs involving

series. A wavelet transformed time series captures useful

complicated differential equations make them unsuitable

information on various resolution levels which leads to

for modeling monsoon river levels that are characterized by

improvement in predictive ability of a model. Hence it has

irregularly spaced spiky large events and sustained flows of

been used in diverse hydrological problems like drought

production.

Conceptual

and

physical

models

“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1 forecasting (Kim and Valdes, 2003), streamflow analysis

3. Wavelet Analysis

(Admowski, 2008; Coulibaly and Burn, 2004; Kucuk and Wavelet analysis is a multi-resolution analysis in time and Agiralioglu, 2006; Smith et al 1998), precipitation analysis frequency domain. Scaled versions of a suitable wavelet, (Kim, 2004; Lu, 2002; Partal and Kisi, 2007; Xingang et also called the mother wavelet, are shifted forward in steps al., 2003), rainfall- runoff relationship (Labat et al., 2000), along the full length of a time series measuring at each step prediction of river discharge (Zhou et al., 2008); analysis of correlation between the wavelet and the time series (Figure suspended sediment load (Rajaee et al., 2010); estimation of unit hydrographs (Chou and Wang, 2002) and various other hydrological predictions (Wang and Ding, 2003).

1). When the full series is covered, a set of wavelet coefficients having same consistency in time as that of the original series is generated. The process is repeated with

2. OBJECTIVE

changed

scales.

This

convolution

process,

called

continuous wavelet transform (CWT), is defined for time Wavelet based models are known to perform better than the series, (t), with respect to a mother wavelet, ψ(t), as conventional models. However selection of mother wavelet plays crucial role in determining the predicting ability of a ,

wavelet based models. The performance of the wavelet

(1)

based models depends heavily on the mother wavelet used

Where W

for Discrete Wavelet Transform (DWT). Some wavelets are

shift b, and shows correlation between the wavelet and the

better suited for capturing the information in time series.

original signal.Symbolψ * (

a ,b

is the CWT coefficient for scale a and time-

𝑡−𝑏 𝑎

)denotes conjugate wavelet

Vanishing moment is a property of wavelet which indicates functions derived from a common mother wavelet function the maximum degree of polynomial a wavelet can

ψ(0,0) (t) by scaling (ordilating) it by a and translating it by

represent. This paper relates the performance of Wavelet b.Convoluting signals in continuous timeframe is enormous based auto regression models (WR) with the vanishing and time consuming process. Moreover, flood time series moment of the mother wavelets. Sixteen WR models were are discrete in nature and in analysing such series, discrete formulated using different wavelets for predicting 1 day wavelet transform is found more suitable. ahead water level of Kamla River of North Bihar, India and the relative performance of these models was compared with the vanishing moments of the mother wavelets chosen for DWT in each model. The study reveals that higher the

For a discrete time series xi, DWT is defined as N 1

W ' m , n  2  m / 2  xi  ( 2  m i  n ) i 0

vanishing moment of the mother wavelet, higher is the accuracy of the prediction models.

1 1

(2)

“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1 where scale a=2mand translation b =2m n, m and n being

DWT as envisaged in Mallat’s algorithm operates as set of

positive integers, N is the data length of the time series and

two functions or filters i.e. high pass filter and low pass

is aninteger power of 2, i.e., N=2M. This gives the ranges of

filter. The original time series is decomposed into

m and n as 0 < n < 2M-m -1 and 1 < m < M, respectively.

‘approximations’ and ‘details’ when passed through the

This power oftwo logarithmic scaling of a and b is known

low pass filter and the high pass filter, respectively. Hence,

as dyadic grid arrangement. In dyadic scheme, scale and

‘details’ follow the rapidly changing characteristics of the

step spacing betweenwavelets are increased by a factor of

signal where as ‘approximation’ follow the coarse and

two at each step, i.e., in the first step, we would use

slowly changing characteristics of the signal. The

wavelets of scale 1 and space them 1sample apart, and in

decomposition process can be iterated, with successive

the second step, they are scaled to two samples and spaced

decomposition of ‘approximations’ so as to break the signal

apart by two sample points.

into many lower-resolution components.

Figure1: Correlating a time series with the shifting and translating wavelet.

4. Focus River

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1 Kamla

is

an

international

river

originating

from

the remaining 120 km is in India. During the monsoon the

Mahabharat Range of Nepal at an elevation of 1,200 m. Out

river carries huge flood causing heavy bank erosion,

of its total catchment area of 7232 km2 , 4488 km2 lies in

affecting a population of over 3.9 million. In the lower

Indian Territory (Bihar State) and the rest in Nepal.

reaches it follows the course of the Balan River and is

Flowing with steep slopes, it enters Indian Territory in

therefore also known as Kamala-Balan. Figure 2shows the

Madhubani district of Bihar State of India. The total length

basin map of Kamla River.

of the Kamala is 328 km of which 208 km is in Nepal and

Figure2: Basin map of Kamla River

for

the year

2007. Table1

summarizes statistical

information on the observed datasets.

5. Implementation of Models Formulated models were derived using 354 monsoon (June to Oct) water level data of the Kamla River for the years 2004-6. After the models were successfully derived, their performance was evaluated using the verification dataset which consisted of another 123 monsoon water level data

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1 A correlation study was carried out to determine the

current day water level.

correlation of the current water level time series with Inclusion of more antecedent time series would complicate antecedent day’s time series. The study revealed that time the models with no significant improvement in its series upto t-4 days possessed significant information of the predicting ability. Parameter

Training data(m)

Checking data(m)

Max. Level

53

52.36

Min. Level

47.45

47.81

Mean Level

48.85

49.07

Std. Dev.

0.813

0.832

Range

5.55

4.55

Table 1: Statistical parameters for the water level data of Kamla River To derive the models, first, the observed water level time

decomposed level series at three resolution levels using

series was decomposed into sub-time series at three

coif5 wavelet. D1, D2 and D3 indicate sub-time series

resolution levels by the dyadic DWT. These wavelet sub

corresponding to scales or periodicities of 2-days, 4-days

time series upto lag4, were then used as inputs to auto

and 8-days, respectively, and A3 is the approximation

regression models. In this study 16 WR models were

component corresponding to 8-days scale or periodicity.

formed for 1 day ahead water level forecasting each formed using different mother wavelets. Figure 3 shows

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1

Figure3: Multi-resolution analysis of the original time series

𝑅= All the developed models were successfully derived and

𝑆𝑜 − 𝑆𝑜 𝑆𝑝 − 𝑆𝑝 𝑆𝑜 − 𝑆𝑜

2

(𝑆𝑝 − 𝑆𝑝)

2

tested with satisfactory performance indices for 1 day , where, Sp and So are predicted and observed water levels, advance prediction.The performance of the models was respectively and n is the number of data points. measured by three statistical indices, e.g., the Nash Sutcliffe coefficient (E), the root mean square error (RMSE), the coefficient of correlation (R). They are defined as:

𝐸 =1−

𝑅𝑀𝑆𝐸 =

1 1

𝑛 2 𝑖=1 (𝑆𝑜−𝑆𝑝 ) 𝑛 (𝑆𝑜−𝑆𝑜 )2 , 𝑖=1

𝑆𝑝 − 𝑆𝑜 𝑁

2

“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1 Among all the models, WR models using db45 as mother

0.057 m and E of 99.53%. Table 2 shows the performance

wavelet performed best with R value of 0.998, RMSE of

of WR models with different mother wavelets.

Mother wavelet

Vanishing Moment

R

RMSE (m)

E (%)

bior 1.1

1

0.949

0.263

89.94

haar

1

0.949

0.263

89.94

coif1

2

0.956

0.244

91.36

db2

2

0.973

0.196

94.41

bior 3.3

3

0.982

0.159

96.34

db 5

5

0.985

0.142

97.07

bior 6.8 db 10

6 10

0.992 0.994

0.106 0.093

98.36 98.73

coif5 db 15

10 15

0.995 0.995

0.086 0.084

98.93 98.97

db 20

20

0.996

0.072

99.24

db 25

25

0.996

0.078

99.11

db 30

30

0.997

0.066

99.37

db 35

35

0.997

0.067

99.35

db 40 db 45

40 45

0.997 0.998

0.067 0.057

99.30 99.53

Table 2: Performance of WR models using various mother wavelets.

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1

Figure 4 plots performance of the WR models wrt the

the vanishing moment, better is the performance of WR

vanishing moment of the wavelets. It is evident that more

models. Figure 5 shows the scatter plots of observed v/s predicted levels for WR models using different wavelets.

Figure4: Performance of WR models v/s vanishing moment of the mother wavelets.

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1

Figure5: Scatter plots of observed v/s predicted water levels for WR models using different wavelets.

5. Conclusion

in increasing the reliability and performance of a hydrological model. WR models employing db45 with

In this study, sixteen WR models were formed to predict 1 vanishing moment 45 predicted 1 day ahead water level with day ahead water level of Kamla River. The models so formed an accuracy of 99.53%, coefficient of correlation of 0.998 used different mother wavelets having a specific vanishing and RMSE of 0.057 m where as the haar and bior1.1 moment. The study revealed that, in general, higher the wavelets with vanishing moment of 1 predicted water level vanishing moment of the mother wavelet, better is the with an accuracy of 89.94%, correlation coefficient of 0.949 performance of the wavelet based models. Vanishing moment is the indicative of the ability of a wavelet to capture the information of a time series. Hence, it can be concluded that wavelets with high vanishing moment would be helpful

1 1

and RMSE of 0.263 m.

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“Improving reliability of Wavelet based hydrological models by suitable mother wavelet selection” - V. Sehgal, K. Tripathi and Prof R. R. Sahay 1

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