Application of Grey System Theory in the Development of a Runoff Prediction Model

ARTICLE IN PRESS Biosystems Engineering (2005) 92 (4), 521–526 doi:10.1016/j.biosystemseng.2005.09.005 SW—Soil and Water

Application of Grey System Theory in the Development of a Runoff Prediction Model H.V. Trivedi; J.K. Singh Department of Soil and Water Conservation Engineering, G.B.P.U.A. & T., Pantnagar 263 145, India; e-mail of corresponding author: [email protected] (Received 10 January 2005; accepted in revised form 13 September 2005; published online 28 October 2005)

The mathematical models used for modelling hydrological processes generally require relatively long-time data series. For developing countries like India, one of the major problems is unavailability of reliable hydrological data for longer durations. An attempt has been made in present study to model the rainfall–runoff process by using grey system theory, a relatively new approach in the field of hydrology. Grey system theory advances an approach to investigate the relationship of an input–output process with unclear inner relationships, uncertain mechanisms and insufficient information. Sixteen storm events for Kothuwatari watershed of Tilaiya dam catchment, Jharkhand, India, collected for the years 1993–1997 have been used for development and prediction of differential hydrological grey models. A least-squares method has been employed for the estimation of model parameters. Performance of the developed model has been evaluated through the qualitative and quantitative statistical indices. Lower values of various error indices and higher values of correlation indices confirm the ability of the model to predict storm runoff with reasonable accuracy for the study area. r 2005 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

1. Introduction The extent to which a rainfall–runoff model represents the response to a given rainfall has a major influence on the reliability of hydrological forecasts. A mathematical model provides a simplified representation of a complex process/system in which the behaviour of the system is represented by a set of equations, perhaps together with logical statements, expressing relations between variables and parameters. Mathematical modelling in hydrology has progressed to such a degree that today a hydrologic engineer can access several algorithms and choose to simulate the future behaviour of a river in order to plan the optimal use of water based on appropriate research or developmental priorities. In general, the mathematical models used for modelling hydrological processes require relatively long-time series of historical data. These models are effectively being adopted in developed countries where hydrological data banks are relatively massive; however, their acceptance in developing countries like India has been 1537-5110/$30.00

hindered by the non-availability of adequate hydrological data. Therefore, the development of a rainfall–runoff model which could be used for prediction of flow rates by utilising a very few hydrological data sets will be of much practical utility. The grey system theory provides an approach to investigate the relationships of input–output process with unclear inner relationships, uncertain mechanisms and insufficient information. Without restoring to forming a knowledge base, the grey modelling scheme constructs a differential equation to characterise the controlled system; therefore, the next output from the model can be obtained by simply solving the differential equation. In view of the grey characteristics widely existing in hydrosystem, grey system theory can be applied by regarding the random process in a hydrosystem as a grey process. This data-driven technique can minimise the noise in the model because it does not consider the intermediate hydrological processes while predicting the lumped model output. Consequently, an attempt has been made in the present study to develop 521

r 2005 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

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Notation h i k N Q Q0 Q1 q0 q1 R0 R1 Rs r0

order of grey model summation variable time index time index runoff matrix raw runoff data series first-order accumulated runoff data series raw data of runoff first-order accumulated data of runoff raw rainfall data series first-order accumulated rainfall data series special correlation coefficient raw data of rainfall

runoff prediction model on a storm basis using the grey system theory. A grey system is a partially known and partially unknown system. The objective of grey system theory and its application is to bridge the gap existing between the social and natural sciences. In situations where a large sample set is not easily available, a system may be considered in the status of poor, uncertain and incomplete information and is known as a grey system, such as the human body, agriculture, hydrology, economy, etc. (Deng, 1989). The raw data time series from an unknown system may be random; however, its degree of randomness may be reduced after subjecting it to accumulated generating operation (AGO) (Deng, 1989). A once or twice accumulation of raw time series is normally enough to support the differential equation (Yu et al., 2000). The analysis of system characteristics is normally based on statistical models, which find the statistical properties between data in large sample sets. However, the aims of grey system theory are to provide theory, techniques, notions and ideas for analysing latent and intricate systems to establish a non-functional model instead of a regressive model, to find real-time techniques instead of statistical model, to obtain an approach to modelling with few data (as few as four) instead of searching for data in quantities (Deng, 1989).

2. Study area The present study was carried out in Kothuwatari watershed of Tilaiya dam catchment in upper Damodar Valley, Jharkhand, India. The Tilaiya dam catchment is

r1 T t U x a1 h
a1 ;
a2 ;
a3   
an ;
b0 ;
b1 ;
b2   
bn ^ a1 ;
^ b0 ;
^ b1
Superscripts m,n 4

first-order accumulated data of rainfall transpose operator time rainfall–runoff matrix variable first-order inverse accumulated generating operator parameter vector grey parameters estimated grey parameter

orders of differential equation estimation

one of the five sub-catchments of Damodar–Barakar catchment in upper Damoder Valley. The Kothuwatari river is an important north flowing river, which originates from south-eastern part of the watershed, runs to a distance of about 10 km and then joins the Mahyaghat river near the village Karso, situated outside the watershed. A number of tributaries join the Kothuwatari river on both its sides. The watershed is situated within the geographical co-ordinates of 241120 2700 to 241160 5400 northern latitudes and 851240 1800 to 851280 1000 eastern longitudes. The Kothuwatari watershed was selected for Watershed Management Programme of Indo–German Bilateral Project (IGBP) in the year 1991 for assessing the effect of various soil conservation measures. Recorded rainfall and runoff data at Karso gauging station under this project by Damodar Valley Corporation, Hazaribagh, India, have been used for this study.

3. Model development Watershed processes being dynamic in nature may be represented more correctly by a differential hydrological grey model (DHGM). As per the grey system theory, the raw data series can systematically be transformed to AGO series of different orders as shown in Table 1. It can be seen from the table that the data of a particular order AGO series may conveniently be expressed in terms of previous order AGO series data. According to Xia (1989), the nth-order causal relationship between rainfall and runoff over a catchment area, governed by the grey differential equation,

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APPLICATION OF GREY SYSTEM THEORY

Table 1 Transformation of raw data series to accumulated generating operation (AGO) series of different orders Raw data series, r0 r01 r02 r03 | | | | r0N

First-order AGO (1AGO) series, r1

Second-order AGO (2AGO) series, r2

---

pth-order AGO (p-AGO) series, rp

r11 ¼ r01 ¼ r01 þ r02 r13 ¼ r01 þ r02 þ r03 | | | | r1N ¼ r01 þ r02 þ    þ r0N

r21 ¼ r11 ¼ r11 þ r12 r23 ¼ r11 þ r12 þ r13 | | | | r2N ¼ r11 þ r12 þ    þ r1N

-------

rp1 ¼ r1p1 ¼ r1p1 þ rp1 2 rp3 ¼ rp1 þ r2p1 þ r3p1 1 | | | | p1 rpN ¼ r1p1 þ r2p1 þ    þ rN

r12

r22

may be expressed as dn p dn1 Q ðtÞ þ
a1 n1 Qp ðtÞ þ    þ
an Qp ðtÞ n dt dt dm p dm1 ¼
b0 m R ðtÞ þ
b1 m1 dt dt  Rp ðtÞ þ    þ
bm Rp ðtÞ p

---

rp2

depict the rainfall–runoff relationship for a watershed. They also suggested that the first-order differential equation should be preferred to avoid mathematical complications to solve it. Equation (1) as governed by a first-order differential grey model may be expressed as ð1Þ

p

where: Q (t) and R (t) are the pth-order time series of runoff and rainfall, respectively, such that

d 1 d Q ðtÞ þ
a1 Q1 ðtÞ ¼
b0 R1 ðtÞ þ
b1 R1 ðtÞ dt dt

(6)

Qp ðtÞ ¼ ½qp ð1Þ; qp ð2Þ; qp ð3Þ    qp ðkÞ    qp ðtÞ

(2)

Applying Laplace transformation and convolution integral to Eqn (6) yields

Rp ðtÞ ¼ ½rp ð1Þ; rp ð2Þ; rp ð3Þ    rp ðkÞ    rp ðtÞ

(3)

Q1 ðtÞ ¼ ½Q1 ð0Þ 
b0 R1 ð0Þe
a1 t þ
b0 R1 ðtÞ Z t e
a1 t R1 ðt  tÞ dt ð7Þ þ ½
b1 
b0
a1 

and where qp and rp are the pth-order AGO (pth-AGO) series of runoff and rainfall and are defined, respectively, as qp ðtÞ ¼

t X

qp1 ðiÞ

(4)

Further, Eqn (7) may be expressed in its discrete form as (Xia, 1989)

(5)

q1 ðt þ 1Þ ¼ ½q1 ð0Þ 
b0 r1 ð0Þe
a1 t þ
b0 r1 ðtÞ t X þ ½
b1 
b0
a1  e
a1 i r1 ðt  i þ 1Þ ð8Þ

i¼1

p

r ðtÞ ¼

t X

r

p1

ðiÞ

0

i¼1

The superscripts n and mðnXmÞ are the orders of the differential equations, and
a1 ;
a2 ;
a3   
an and
b0 ;
b1 ;
b2   
bm are grey parameters. The first-order differential equation has been adopted in the present study in view of the following two reasons. (1) Wen (2004) showed that the grey model, GM (h, N) for order hX2 does not exist, because according to grey system theory the ingredient of ðd 2 =dt2 Þx1 ðk þ 1Þ is equal to ½x1 ðk þ 1Þ; but the value of ½x1 ðk þ 1Þ cannot be found regardless of how the values of grey parameters (
a1 and
a2 ) are changed or not changed. This term, ½x1 ðk þ 1Þ; represents the (–1)th order accumulated data series of x up to (k+1)th data. (2) Yu et al. (2001) suggested that normally watersheds have rapid response to rainfall output; therefore, the first-order differential equation may be enough to

i¼1

Since in present study and in grey system theory, the data set starts from time t equal to 1 instead of 0. Therefore, the Eqn (8) may be represented as q1 ðt þ 1Þ ¼ ½q1 ð1Þ 
b0 r1 ð1Þe
a1 t þ
b0 r1 ðtÞ t X þ ½
b1 
b0
a1  e
a1 i r1 ðt  i þ 1Þ ð9Þ i¼1

The first-order differential grey model as shown in Eqn (9) contains three parameters, i.e.
a1 ;
b0 and
b1 : The least-squares method has been used for estimation of model parameters. According to grey system theory for discrete data, the whitening of grey derivatives of Eqn (6) can be

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H.V. TRIVEDI; J.K. SINGH

expressed as
d 1

Q ðtÞ
¼ a1 ½q1 ðkÞ ¼ q1 ðkÞ  q1 ðk  1Þ ¼ q0 ðkÞ dt t¼k

(10) 1

where: a is the first-order inverse AGO (1-IAGO) and k represents the time index. Similarly
d 1

R ðtÞ
¼ r0 ðkÞ (11) dt t¼k

estimated grey parameter vector, and T is the transpose operator. Thus for each storm event there will be a set of ^ b0 and
^ b1 : On substituting the ^ a1 ;
parameters, viz,
average values of parameters, considering all the storm events, in Eqn (9), the prediction equation for AGO runoff data in discrete from may be expressed as (Xia, 1989) ^

^ b0 r1 ð1Þe
a1 :k þ
^ b0 r1 ðkÞ q^ 1 ðk þ 1Þ ¼ ½q1 ð1Þ 
^ b1 
^ a1  ^ b0
þ ½


(12)

R1 ðtÞjt¼k ¼ r1 ðkÞ

(13)

Substitution of Eqns (10)–(13) in Eqn (6) results in the following equation
a1 1 q ðkÞ þ ½q ðkÞ þ q1 ðk  1Þ 2 ¼
b0 r0 ðkÞ þ
b1 r1 ðkÞ 0

^

e
a1 :i r1

i¼1

For discrete data, Q1 ðtÞ and R1(t) may be defined as (Lee & Wang, 1998) 1 Q1 ðtÞjt¼k ¼ ½q1 ðkÞ þ q1 ðk  1Þ 2

k X

 ðk  i þ 1Þ

ð18Þ

Thus the accumulated values of runoff q^ 1 ðtÞ; t ¼ 1; 2; 3; . . . ; k þ 1 can be estimated from Eqn (18). Having known the predicted values of accumulated runoff for different periods k, the predicted values of storm runoff were estimated by the following equation: ^ þ 1Þ ¼ q^ 1 ðk þ 1Þ2q^ 1 ðkÞ qðk

(19)

or ð14Þ

^ þ 1Þ qðk

^ b0 r0 ð1Þe
^ a1 k ½1  e
^ a1  ¼ ½q0 ð1Þ 
Since the data set starts from time t ¼ 1; Eqn (14) may ^ a1  ^ b1 
^ b0
^ b0 ½r1 ðkÞ  r1 ðk  1Þ þ ½
be expressed in a matrix form for k ¼ 2; 3; 4; . . . ; N as þ
" (Trivedi, 2004) k X ^ 3 3 2 2 0 e
a1 :i r1 ðk  i þ 1Þ  1 1 0 1 1=2½q ð1Þ þ q ð2Þ r ð2Þ r ð2Þ q ð2Þ i¼1 7 7 6 6 0 # 6 q ð3Þ 7 6 1=2½q1 ð2Þ þ q1 ð3Þ r0 ð3Þ r1 ð3Þ 7 k1 X 7 6 7 6 6 ^ 
:i 1 7 7 6 0  e a1 r ðk  iÞ ð20Þ 6 q ð4Þ 7 6 1=2½q1 ð3Þ þ q1 ð4Þ r0 ð4Þ r1 ð4Þ 7 7 i¼1 7¼6 6 7 7 6 6 .. 7 .. .. 6 .. 7 6 6 6 . 7 6 . 7 . . 7 5 4 4 5  1 1 0 1 0 1 2½q ðN  1Þ þ q ðNÞ r ðNÞ r ðNÞ q ðNÞ 2 3 4. Results and discussion
a1 6 7 7 6 ð15Þ The differential hydrological grey model (DHGM) as 4
b0 5 shown in Eqn (20) was developed and calibrated for
b1 study watershed by using grey system theory to predict The above matrix may be represented in simplified runoff on storm basis. Ten storm events (i.e. those on: form as 25 March 1993; 25 May 1993; 14 September 1993; 2 July Q ¼ Uh (16) 1994; 15 February 1995; 21 June 1996; 9 August 1996; 28 June 1997; 6 July 1997 and 6 September 1997) were Since U is a non-square matrix, the value of grey included for model development and calibration, parameter vector h can be identified by the least-squares whereas six storm events (i.e. those on: 5 September method as 1993; 28 June 1994; 7 March 1995; 10 February 1996; 8 2 3 ^ a1
July 1997 and 5 August 1997) were considered for model 6 ^ 7 T 1 T verification and validation purposes. The least-squares h^ ¼ 4
¼ ½U U U Q (17) 5 b0 method was applied for the estimation of the model ^ b1
parameters and the stormwise values so obtained are where: Q is the runoff matrix, U is the rainfall–runoff given in Table 2. The average values of the model ^ a1 ;
^ b0 and
^ b1 are estimators or identified parameters, i.e.
a1 ;
b0 and
b1 for the study area were matrix,
values of
a1 ;
b0 and
b1 ; respectively, h^ is the found to be 0299868, 0536347, and 2061873,

ARTICLE IN PRESS APPLICATION OF GREY SYSTEM THEORY

Table 2 Stormwise computed values of model parameters Date of storm

Grey parameters
a1


a0


a0

25 March 1993 25 May 1993 14 September 1993 2 July 1994 15 February 1995 21 June 1996 9 August 1996 28 June 1997 6 July 1997 6 September 1997

0373184 0070989 0259528 0264177 0600255 0313967 0320693 0272887 0245736 0291562

0279423 0268388 0011059 0321563 1398889 0951988 0037376 0301765 0386406 0388028

1641437 0572867 0815796 1199553 3596121 1565439 1571239 1265047 156488 1820996

Average value

0299868

0536347

2061873

525

respectively. On substituting the average value of the model parameters in Eqn (20), the developed model for study area for predicting the storm runoff was obtained and further verified for its performance. From the model response, it is concluded that all the three parameters vary with factors such as rainfall, runoff and watershed characteristics for each storm event. The parameter
a1 signifies the storage capacity and the physiographical factors of the grey process. Its positive value explains the grey response function in decay of the above watershed characteristics. Further,
b0 ; which is the most sensitive parameter, depends on the temporal and spatial variations of the rainfall. The negative value of the parameter
b0 signifies the convex nature of the rainfall hyetograph. Similarly,
b1 ; which is more stable parameter for flood analysis, is pertinent to the overall

Fig. 1. Observed (—) and predicted (- - - - -) runoff hydrographs for different storm events: (a) 5 September 1993; (b) 28 June 1994; (c) 7 March 1995; (d) 10 February 1996; (e) 8 July 1997; (f) 5 August 1997

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Table 3 Quantitative performance indices of developed model for Kothuwatari watershed Date of storm 5 September 1993 28 June 1994 7 March 1995 10 February 1996 8 July 1997 5 August 1997

MAD

RMSE

RTIC

ISE, %

056 093 040 053 100 068

067 105 046 059 139 082

038 038 023 033 063 045

544 353 288 417 628 555

CC

CV

Rs

0976 0982 0986 0987 0971 0967

022 014 012 002 028 025

0969 0989 0990 0996 0947 0964

APE, %

CE, %

1808 1251 1002 1658 2019 2051

8843 9642 9872 9186 8410 9114

MAD, mean absolute deviation; RMSE, root mean square error; RTIC, revised Theil inequality coefficient; ISE, integral square error; CC, correlation coefficient; CV, coefficient of variation; Rs, special correlation coefficient; APE, absolute prediction error; CE, coefficient of efficiency.

meteorological condition of the study area during the flood event. The goodness of fit of a predicted runoff hydrograph to an observed one was tentatively compared by qualitative evaluation. This procedure is based on visual comparison of observed and predicted storm runoff hydrographs. Observed and predicted storm runoff hydrographs for the storm events used for verification have been shown in Fig. 1. It is evident from the figures that the peak flow rates of predicted and observed runoff hydrographs are in fairly good agreement. A close agreement between rising and recession segments of the observed and predicted hydrographs can also be seen from figures. The quantitative performance is ascertained by a number of statistical indices such as absolute relative error (ARE), mean absolute deviation (MAD), root mean square error (RMSE), revised Theil inequality coefficient (RTIC), integral square error (ISE), special correlation coefficient Rs, absolute prediction error (APE), coefficient of variation (CV), correlation coefficient (CC) (Sarma et al., 1973; Yue et al., 2002). The quantitative performance indices for developed model are presented in Table 3. It can be observed from the table that the developed model yielded very low values of all the error indices considered in the study for all the storm events. The correlation values more than 096 and values of coefficient of efficiency higher than 84% speak of the suitability of the grey system theory-based runoff prediction model for the study area.

5. Conclusions The qualitative evaluation of the model performance on the basis of visual observations revealed a close agreement between observed and predicted runoff hydrographs with regard to various segments of flood hydrographs, peak runoff rates, time to peak and time to base. Under quantitative evaluation, the lower values of various error indices and higher values of correlation

indices also confirm the model’s ability to predict storm runoff with reasonable accuracy in the study area. Some deviations between observed and predicted ordinates of storm runoff, as evident from the values of different performance indices, may be due to errors in the collection of hydrological data and/or assumptions involved in formulation and development of the model. The developed model may also be applied for predicting the storm runoff with a good degree of accuracy for other watersheds, having similar hydro-meteorological conditions. Thus, the grey system theory may be recommended as a valuable tool in the field of hydrology especially for those watersheds, which possess scanty hydrological data. References Deng J (1989). Introduction to grey system theory. Journal of Grey System, 1(1), 1–24 Lee R H; Wang R Y (1998). Parameter estimation with colored noise effect for differential hydrological grey model. Journal of Hydrology, 208, 1–15 Sarma P B S; Delleur J W; Rao A R (1973). Comparison of rainfall–runoff models for urban areas. Journal of Hydrology, 18, 329–347 Trivedi H V (2004). Grey system theory based mathematical model for runoff prediction—a case study. M. Tech. Thesis, Department of Soil and Water Conservation Engineering, G.B. Pant University of Agriculture and Technology, Pantnagar, India Wen K L (2004). The grey system analysis and its application in gas breakdown and VAR compensator finding. Journal of Computational Cognition, 2(1), 21–44 Xia J (1989). Research and application of grey system theory to hydrology. Journal of Grey System, 1(1), 43–52 Yu P S; Chen C J; Chen S J (2000). Application of grey and fuzzy methods for rainfall forecasting. Journal of Hydrologic Engineering, 5(4), 339–345 Yu P S; Chen C J; Chen S J; Lin S C (2001). Application of grey model toward runoff forecasting. Journal of the American Water Resource Association, 37(1), 151–166 Yue S; Ouarda T B M J; Bobee B; Legendre P; Bruneau P (2002). Approach for describing statistical properties of food hydrograph. Journal of Hydrologic Engineering, 7(2), 147–153