The document describes research on using wavelet regression (WR) models and artificial neural network (ANN) models to forecast river stages of the Kosi River in India. WR models were developed using discrete wavelet transform to decompose historical river stage data into wavelet components, which were then used as inputs to autoregressive models. The WR models achieved higher coefficients of correlation and lower root mean square errors than ANN models when evaluating predictions against verification data, indicating WR models more accurately forecasted river stages of the Kosi River.

1. RIVER STAGE FORECASTING USING
WAVELET TECHNIQUE
By-
1. Rajeev Ranajan Sahay, Assistant Professor,
Department of Civil Engineering,
BIT, Mesra.
2. Vinit Sehgal, BE
Department of Civil Engineering,
BIT, Mesra.

2. FLOOD PRONE RIVER KOSI
 River Kosi,a tributary of Ganga River, is often
called the ‘Sorrow of Bihar ‘ because of the
widespread destruction caused by the river
due to frequent floods in it.
 About 90% of the catchment area of River
Kosi is flood prone.During floods,water stage
of Kosi increases to about 18 times its
average value.

3.  Floods generally cause great devastation due
to their poor and late estimation. A recent
flood occurred in River Kosi in Monsoon
season of 2008 that lead to heavy loss of
man and material.
 In event of occurring of a flood, authorities
and people of the area are not forewarned of
the incoming high flood, thereby, not allowing
them enough time for taking up appropriate
flood-fighting measures.

4. BASIN MAP OF KOSI RIVER (NORTH BIHAR)

5. RIVER STAGE FORECASTING
 To predict river stage accurately and timely,
several mathematical models are employed.
 These are based on several techniques. Some
of them include:
 Auto Regression(AR)
 Artificial Neural Networks(ANN)
 Genetic Algorithm(GA)
 Genetic Programming (GP)
 Discrete Wavelet Transform(DWT) etc.

6. DATA ANALYSIS
 For Flood forecasting, 231 monsoon stage
data of year 2005 and 2006 is used as
derivation set data.
 120 stage data of year 2007 is used as
verification set data for verifying the ANN and
DWT models.
 The developed models are applied to
forecast one-day ahead flood levels of Kosi
River.

7. PARAMETERS OF STAGE DATA
 The statistical parameters of stage data is
given in the given Table.
Parameter Verification data
(m)
Derivation data
(m)
Min. Stage 73.70 72.88
Max. stage 74.95 74.98
Mean stage 74.39 74.22
Std. Dev. 0.24 0.31
Range 1.25 2.10

8. WAVELET REGRESSION MODEL
 Wavelet Regression Model is a recent technique
to model complex and nonlinear hydrological
processes without underlying physics being
explicitly provided.
 Wavelet models have a strong generalization
ability, which means that once mother wavelet is
properly selected, they are able to provide
accurate results even for cases they have never
seen before.

9.  Wavelet analysis is multi resolution analysis in time and
frequency domain .
 Wavelets are simple oscillatory function of finite duration with a
mean value of zero. If φ(t) represents the mother wavelet,

 At each step of analysis, correlation of wavelet to the input signal
is measured. When the full series is covered, a set of wavelet
coefficients is generated having same consistency in time as that
of original signal.
 Wavelet analysis is the breaking up of a selected most suitable
mother signal into shifted and scaled versions of wavelets. This
convolution process is called continuous wavelet transform and is
given by:
 a denotes wavelet dilation and is called scaling or frequency
factor, b denotes time shift of wavelet and is hence called the time
factor, R denotes all real numbers and * Symbol denotes complex
conjugate

10. Figure showing scaled and translated wavelet on a given signal.
•

11.  Since most of the natural time series are
discreet in nature, Discrete Wavelet
Transform is to applied for decomposition
and reconstruction of time series.

12. DWT FILTERS
 DWT operates as set of two functions or
filters i.e. high pass filter and low pass filter.
 The original time series is decomposed
through a process consisting of a number of
successive filtering steps giving
 a) Approximation(Low frequency terms)
 b) Details(High frequency terms)

13. Flow Chart for Decomposition of Time Series Wave

14. MATLAB DECOMPOSITION OF TIME SERIES

15. DWT PROCESS
 The derivation set data was decomposed upto
three levels of decomposition to obtain
d1,d2,d3,a3 (i.e. the details and the
approximation).
 d1,d2,d3,a3 were arranged in various lags and
their correlation with the original derivation set
data was obtained.

16. TABULATION OF CORRELATION
coif5 Average value of R
d1
0.053
d2
0.144
d3
0.155
a3
0.79
 Average correlation for d1 is
lowest . This indicates that it
may induce irregularity in
the mathematical model.
Hence, it is eliminated from
further analysis.

17. FORMATION OF WR MODELS
 d1 component was
neglected and the d2, d3
and a3 so obtained,
were fed to three AR
models separately as
independent inputs. The
outputs of these AR
models were added to
obtain the predicted river
stage Sp. This provided
wavelet regression
model 1 (WR1 model).
 Another wavelet
regression model had
slightly different
methodology. A modified
series, obtained by
adding d2, d3 and a3 (MD)
was used as input to Auto
regression model. This
provided Wavelet
regression model 2 (WR2
model).

18. -0.3
-0.15
0
0.15
0.3
0 100 200
-0.3
-0.15
0
0.15
0.3
0 100 200
-0.3
-0.15
0
0.15
0.3
0 100 200
73
74
75
0 50 100 150 200
73
74
75
1 101 201
73
74
75
0 50 100 150 200
(f) ORIGINAL RIVER STAGE DATA(e) MD (a3+ d3+ d2)
(d) a3(c) d3
(b)d2(a) d1
Decomposed wavelet sub time series components

19. STRUCTURE OF WR1 AND WR2 MODEL

20. RESULTS FOR WR MODELS
DERIVATION PERIOD VARIFICATION PERIOD
WR1 CC RMSE (m) CC RMSE (m)
MD t-1 0.925 0.129 0.912 0.104
MD t-1, MD t-2 0.959 0.096 0.945 0.082
MD t-1, MD t-2, MD t-3 0.959 0.096 0.947 0.0817
MD t-1, MD t-2, MD t-3, MD t-4 0.963 0.091 0.951 0.079
MD t-1, MD t-2, MD t-3, MD t-4, MDt-5 0.963 0.091 0.951 0.079
WR2 CC RMSE (m) CC RMSE (m)
MD t-1 0.893 0.140 0.901 0.110
MD t-1, MD t-2 0.917 0.124 0.914 0.104
MD t-1, MD t-2, MD t-3 0.939 0.107 0.932 0.092
MD t-1, MD t-2, MD t-3, MD t-4 0.943 0.104 0.938 0.089
MD t-1, MD t-2, MD t-3, MD t-4, MDt-5 0.945 0.103 0.943 0.085

21.  The model for river stage for Kosi River using the
best performing WR1model is given as:
St = 0.707+(1.319 MDd2(t-1)-1.99 MDd2(t-2) +1.30 MDd2(t-3)
-0.95 MDd2(t-4) +0.193 MDd2(t-5) )+(2.83 MDd3(t-1)-3.47 MDd3(t-2)
+1.93 MDd3(t-3)-0.27 MDd3(t-4)-0.164 MDd3(t-5) )+(3.87 MDa3(t-1)
-5.95 MDa3(t-2)+4.437 MDa3(t-3)-1.54 MDa3(t-4)+0.179 MDa3(t-5)
 Similarly for WR2 model, the equation
describing river stage with best results is given
by:
St = 2.36+ 2.26 MDt-1 - 2.67 MDt-2 + 2.15 MDt-3 -1.12 MDt-4 + 0.34 MDt-5

22. EFFECT OF INCREASING THE NUMBER OF LAGS
 In the models for river stage of Kosi River, the river stage
values only up to last five days are involved. The models
involving higher number of previous stage data were also
tested but it was observed that there was no significant
increase in the result of the models in terms of coefficient of
correlation.
0.87
0.88
0.89
0.9
0.91
0.92
0.93
0.94
0.95
0.96
1 2 3 4 5 6 7 8 9 10
Rvalue
No. of previous day's data involved

23. EFFECT OF USING DIFFERENT WAVELETS
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
haar db3 db5 db10 sym9 coif5 bior6.8 rbio6.8 dmey
Rvalue
Wavelet used in model formation
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
haar db3 db5 db10 sym9 coif5 bior6.8 rbio6.8 dmey
RMSE
Wavelet used in model formation

24. ANN PROCEDURE
 As in the case of the WR models, five input
combinations based on previous Daily River stages
were used as inputs to the ANN models to estimate
the current stage value.
 The output node consisted of stage S t for the current
day.
 The three input nodes in this network represent the
stages at days t - 1, t - 2, and t – 3 (S t-1, S t-2, and S
t-3), and the unique output represents the stage (St) to
be predicted.

25. RMSE & R STATISTICS OF ANN
MODEL
Model Inputs
ANN
STRUCTURE
DERIVATION
PERIOD
VARIFICATION
PERIOD
CC RMSE (m) CC RMSE (m)
S t-1 (1,4,1) 0.815
0.179
0.787
0.152
S t-1, S t-2 (2,6,1) 0.819
0.297
0.821
0.147
S t-1, S t-2, S t-3 (3,8,1) 0.819
0.180
0.827
0.146
S t-1, S t-2, S t-3, S t-4 (4,8,1) 0.822
0.179
0.829
0.146
S t-1, S t-2, S t-3, S t-4, St-5 (5,10,1)
0.823 0.179 0.831 0.146

26. COMPARISON OF PERFORMANCE
 Performance indices considered for comparing results from
different models are the root mean square error (RMSE),
coefficient of correlation (CC) and the discrepancy
ratio.(DR).RMSE, CC, DR are defined as:

27. ACCURACY OF MODEL
 If the accuracy of the model can be defined as
percentage of DR values falling between -0.001 to
0.001i.e. Predicted values of river stage lying
between 99.8% and 100.2% of the measured
values.

28. COMPARISON OF PERFORMANCE
Name of the
model
RMSE (m) CC DR Range Accuracy in %
WR 1 0.079 0.95 -0.0016 to 0.0012 97.41
WR 2 0.085 0.94 -0.0014 to 0.0015 95.65
ANN 0.153 0.83 -0.002 to 0.0027 81.74

29. 0
5
10
15
20
25
30
35
40
45
ANN WR2
WR1
COMPARISON OF DISCREPANCY RATIO FOR VARIOUS MODELS.

30. COMPARISON OF RESULT OF DIFFERENT MODELS
73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
No. of days in verification period
Original River stage Stage predicted using ANN model

31. 73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
No. of days in Verification period
Original River stage Stage predicted using WR1 model

32. 73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
No. of daya in verification period
Original River stage Stage predicted using WR2 model

33. 73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
Original River stage Stage predicted using ANN model
73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
Original River stage Stage predicted using WR1 model
73.70
73.90
74.10
74.30
74.50
74.70
74.90
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
RiverStage(m)
Original River stage Stage predicted using WR2 model

34. CONCLUSION
 The WR model has been found to perform
better than ANN models.
 WR model has high value of R, low RMSE
and greater accuracy as compared to ANN.
 It was also observed that more number of
previous days’ stage data did not give much
increment to the performance of any model.
Hence models involving only up to five
previous days’ stage data were studied.

35. THANK YOU..