Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall-runoff modelling under different input domains

HYDROLOGICAL PROCESSESHydrol. Process. 25, 175–193 (2011)Published online 25 August 2010 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/hyp.7831

Comparative study of neural network, fuzzy logic and lineartransfer function techniques in daily rainfall-runoff

modelling under different input domains

Anil Kumar Lohani,1* N. K. Goel2 and K. K. S. Bhatia3

1 Scientist E1, National Institute of Hydrology, Jal Vigyan Bhawan, Roorkee-247667, India2 Professor, Department of Hydrology, Indian Institute of Technology, Roorkee-247667, India

3 Director, Vira College of Engineering, Delhi Road, BIJNOR (UP), India

Abstract:

This paper compares artificial neural network (ANN), fuzzy logic (FL) and linear transfer function (LTF)-based approaches fordaily rainfall-runoff modelling. This study also investigates the potential of Takagi-Sugeno (TS) fuzzy model and the impact ofantecedent soil moisture conditions in the performance of the daily rainfall-runoff models. Eleven different input vectors underfour classes, i.e. (i) rainfall, (ii) rainfall and antecedent moisture content, (iii) rainfall and runoff and (iv) rainfall, runoff andantecedent moisture content are considered for examining the effects of input data vector on rainfall-runoff modelling. Usingthe rainfall-runoff data of the upper Narmada basin, Central India, a suitable modelling technique with appropriate model inputstructure is suggested on the basis of various model performance indices. The results show that the fuzzy modelling approachis uniformly outperforming the LTF and also always superior to the ANN-based models. Copyright 2010 John Wiley &Sons, Ltd.

KEY WORDS neural network; fuzzy logic; linear transfer function; antecedent moisture content; clustering; Gaussianmembership function

Received 22 March 2009; Accepted 21 June 2010

INTRODUCTION

The rainfall-runoff process is highly nonlinear, time-varying, spatially distributed and not easily describedby simple models. A number of investigators havetried to relate runoff with the different characteristicswhich affect it (Rodriguez-Iturbe and Valdes, 1979; Chowet al., 1988; Singh, 1989; Beven, 2000). Various attemptshave been made to address this modelling issue eitherusing knowledge-based models or data-driven models. Aknowledge-based model aims to reproduce the systemand its behaviour in a physically realistic manner andare generally called physically based model. The physi-cally based models generally use a mathematical frame-work based on mass, momentum and energy conserva-tion equations in a spatially distributed model domain,and parameter values that are directly related to catch-ment characteristics. For the purpose of rainfall-runoffprocess simulation, conceptual and physical-based mod-els are widely used. However, simulating the real-worldrelationships using these rainfall-runoff models is not asimple task because the various hydrological processesthat involve the transformation of rainfall into runoff arecomplex and variable. Many of the conceptual modelswidely used in rainfall-runoff modelling are lumped oneand the factors in generating runoff are not represented

* Correspondence to: Anil Kumar Lohani, National Institute of Hydrol-ogy, Jal Vigyan Bhawan, Roorkee-247667, India.E-mail: [email protected]; akl [email protected]

clearly by these models. The time required to constructthese models is enormous and thus an alternative mod-elling technique is sought when detailed modelling is notrequired in cases such as streamflow forecasting. The lin-ear regression or linear time series models such as ARMA(Auto Regressive Moving Average) have been developedto handle such situations because they are relatively easyto implement. However, such models do not attempt torepresent the nonlinear dynamics inherent in the hydro-logic processes, and may not always perform well. Inrecent years, data-driven technique, e.g. artificial neuralnetwork (ANN) has gained significant attention. Manyrainfall-runoff and flood forecasting models using ANNhave been reported in the literature (Dawson and Wilby,1998; Tokar and Johnson, 1999; Tokar and Markus, 2000;Agarwal and Singh, 2004). Recently, fuzzy rule-basedapproach (Zadeh, 1965) has been successfully appliedin hydrological modelling (Nayak et al., 2005; Tayfurand Singh, 2006; Lohani et al., 2005a,b, 2006, 2007a)and water resources management (Sahoo et al., 2006).Hundecha et al. (2001) demonstrated the applicabilityof a fuzzy logic (FL)-based approach to rainfall-runoffmodelling (Hundecha et al., 2001; Ozelkan and Duck-stein, 2001) and its comparison to ANN (Tayfur andSingh, 2003) and classical regression models (Sen andAltunkaynak, 2003) have been demonstrated by variousresearchers.

This study presents the development of intelligencemodels based on ANNs and Takagi-Sugeno (TS) fuzzy

Copyright 2010 John Wiley & Sons, Ltd.

176 A. K. LOHANI, N. K. GOEL AND K. K. S. BHATIA

inference system for prediction of runoff. The fuzzyrelations between the input and output variables wereinferred from the measured data and they are laid out inthe form of IF-THEN statements. Performance of runoffprediction models were also evaluated by introducing theantecedent moisture condition (AMC) in the input vector.The performance of the developed models is comparedwith linear transfer-based models.

DATA USED FOR THE STUDY

Validated and processed data of Narmada catchment up toManot gauging site covering an area of 4300 sq. km. havebeen selected for rainfall-runoff modelling. Validated andprocessed data of daily rainfall at Narayanganj, Bichhia,Baihar, Palhera, Manot, Gondia and Nimpur stationsand daily discharge at Manot gauging site have beenconsidered. The available data were divided into twosets, one for calibration and the other for validation. Thedaily rainfall and discharge data from June to September(monsoon period) of the years 1993 and 1996 wereused for calibration of the ANN model because these4 years of data represent the extreme values of rainfalland discharge. The data of year 1997 and 1998 were usedfor the validation of the model.

DEVELOPMENT OF RAINFALL-RUNOFF MODELS

Fuzzy model for rainfall-runoff dynamics

Selection of the input and output variables is thefirst step in development of a fuzzy rule-based rainfall-runoff model. Runoff at the outlet of a catchment isa function of previous rainfall and runoff values, aswell as of the meteorological, topological and soil andvegetative conditions of the catchment. Theoretically,a nonlinear and time-varying storage function may beuseful to express the rainfall-runoff process. There areinherent difficulties in defining such functions particularlywhen sufficient data are not available and estimation

of catchment response is only relying on availablerainfall data. Therefore, in the case of a rainfall-runoffmodel with minimum available data, the output variabledescribes the runoff that is to be predicted and possibleinput variables are measured rainfall and runoff data. Indaily rainfall-runoff modelling, AMC plays an importantrole. Therefore, in addition to daily rainfall values,the AMC is also introduced in the input vector of adaily rainfall-runoff model. The number of precedingday’s rainfall suitable for the computation of AMChas been decided by the cross-correlation analysis ofrunoff at Manot site and AMC(n) values computed forpreceding n days. The correlation matrix between runoffand AMC(n) is developed for n D 3 to 9 days. Thecorrelation analysis (Figure 1) between AMC and runoffsuggests that AMC values computed using 7 days rainfallshows maximum value of correlation, i.e. 0Ð835 and canbe suitably considered in the input vector of daily rainfall-runoff model of Manot catchment. The following 11combinations of input data vectors have been considered:

1. Only rainfall as input

M1 Qt D f�Pt, Pt�1, Pt�2, Pt�3�

2. Rainfall and AMC as input

M2 Qt D f�Pt, Pt�1, Pt�2, Pt�3, AMC�

3. Rainfall and runoff as input

M3 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1�

M4 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1, Qt�2�

M5 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1, Qt�2, Qt�3�

4. Rainfall, runoff and AMC as input

M6 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1, AMC�

M7 Qt D f�Pt, Qt�1, AMC�

M8 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1, Qt�2, AMC�

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

3

Value n considered in AMC(n) (Day)

Cor

rela

tion

coef

ficie

nt

4 5 6 7 8 9

Figure 1. Correlation coefficient between AMC(n) and runoff

Copyright 2010 John Wiley & Sons, Ltd. Hydrol. Process. 25, 175–193 (2011)

COMPARATIVE STUDY OF ANN, FL AND LTF IN RAINFALL-RUNOFF MODELLING 177

M9 Qt D f�Pt, Qt�1, Qt�2, AMC�

M10 Qt D f�Pt, Pt�1, Pt�2, Pt�3, Qt�1,

Qt�2, Qt�3, AMC�

M11 Qt D f�Pt, Qt�1, Qt�2, Qt�3, AMC�

where Qt and Pt are the runoff and precipitation at timet respectively.

The linguistic fuzzy model maps the characteristics ofinput data to input membership functions, input mem-bership function to rules, rules to a set of output char-acteristics, output characteristics to output membershipfunctions and the output membership function to a single-valued output or a decision associated with the output(Jang et al., 2002). Whereas, a fuzzy rule-based modelsuitable for the approximation of many systems and func-tions is the (TS fuzzy model in which the consequentsare expressed as (crisp) function of the input variables(Takagi and Sugeno, 1985). It is defined as:

Ri : IF x1 is Ai1 AND . . . . AND IF xn is Ain

THEN yi D ai1x1 C ai2x2 C . . . .ainxn C bi �1�

Where Ri�i D 1, r� indicates a set of r rules derivedby partitioning the input clustering approach proposedby Chiu (1994). x1, x2, . . . xn are the input variable in ndimensional input vector and yi 2 < is the consequentof the ith rule. In the consequent, ai is the parametervector and bi is the scalar offset. Ain is the (multivariate)antecedent fuzzy set of the ith rule.

Identification of rules by manual inspection has its lim-itations. The data-driven approach based on subtractiveclustering has shown promising results in various hydro-logical modelling applications (Lohani 2005a,b). The pur-pose of subtractive clustering is to identify natural group-ing of the data from a large data set and finally to producea concise representation of a systems behaviour (Lohani2006, 2007a,b). The subtractive clustering approach isused in the present study to determine the number ofrules and antecedent membership functions by consid-ering each cluster centre as a fuzzy rule. The antecedentfuzzy set is defined by the Gaussian membership function(mi�x�):

mi�xi� D exp

(�

∥∥xi � xŁi

∥∥2

�ra/2�2

)�2�

Where, mi�x� : <n ! [0, 1], ra is a positive constantcalled cluster radius, xŁ

1 is the ith cluster centre.For the input x the output y of the TS model is defined

by the fuzzy mean formula:

y Dr∑

iD1

[a1x1 C a2x2 C . . . ..anxn C an C 1].

n∏jD1

mij�xj�

r∏iD1

mi�x�

�3�The evaluation of a set of fuzzy rules (or rule base)

in a fuzzy rule-based model for the determination of

the runoff value is an important task. The basis of FLis to consider hydrologic variables in a linguisticallyuncertain manner, in the form of subgroups, each ofwhich is labeled with successive fuzzy word attachmentssuch as ‘low’, ‘medium’, ‘high’ etc. In this way, thevariable is considered not as a global and numericalquantity but in partial groups which provided better roomfor the justification of sub-relationship between two ormore variables on the basis of fuzzy words (Sen andAltunkayank, 2003). As rainfall-runoff relationship, ingeneral, has a direct proportionality feature, it is possibleto write the following rule base for the description of TSfuzzy rainfall-runoff model.


M1 Rule Ri : IF�Pt, Pt�1, Pt�2, Pt�3� is Ci THEN

Qt D a1iPt C a2iPt�1 C a3iPt�2 C a4iPt�3 C ci �4�

2. Rainfall and AMC as input

M2 Rule Ri : IF�Pt, Pt�1, Pt�2, Pt�3,

AMC� is Ci THEN

Qt D a1iPt C a2iPt�1 C a3iPt�2

C a4iPt�3 C a5i AMC C ci �5�



Qt�1� is Ci THEN

Qt D a1iPt C a2iPt�1 C a3iPt�2 C a4iPt�3

C a5iQt�1 C ci �6�


Qt�1, Qt�2� is Ci THEN


C a5iQt�1 C a6iQt�2 C ci �7�


Qt�1, Qt�2, Qt�3� is Ci THEN


C a5iQt�1 C a6iQt�2 C a7iQt�3 C ci �8�



Qt�1, AMC� is Ci THEN


C a5iQt�1 C a6i AMC C ci �9�

M7 Rule Ri : IF�Pt, Qt�1, AMC�

is Ci THEN

Qt D a1iPt C a2iQt�1 C a3i AMC C ci �10�




Qt�1, Qt�2, AMC� is Ci THEN


C a5iQt�1 C a6iQt�2 C a7i AMC C ci �11�

M9 Rule Ri : IF�Pt, Qt�1, Qt�2, AMC�

is Ci THEN

Qt D a1iPt C a2iQt�1 C a3iQt�2

C a4iAMC C ci �12�

M10 Rule Ri : IF�Pt, Pt�1, Pt�2, Pt�3, Qt�1,

Qt�2, Qt�3, AMC� is Ci THEN


C a5iQt�1 C a6iQt�2 C a7iQt�3


M11 Rule Ri : IF�Pt, Qt�1, Qt�2, Qt�3,

AMC� is Ci sTHEN

Qt D a1iPt C a2iQt�1 C a3iQt�2 C a4iQt�3


where aji and ci are the parameters of the consequentpart of rule Ri.

Using the linear consequent part of the fuzzy rainfall-runoff model, subtractive clustering-based identificationmethod has been applied. The model performance isexamined by means of Nash Sutcliffe (NS) efficiency(Nash and Sutcliffe, 1970) and root mean square error(RMSE) criteria. In order to find the optimal model, theparameters of the subtractive clustering algorithm werefinalized after a number of trial runs. In the trials, theparameters of subtractive clustering were varied from0Ð5 to 2 for quash factor and 0Ð1 to 1 for the clusterradius (ra), accept ratio and reject ratio with steps of0Ð01. The cluster centres and thus the Gaussian mem-bership function identified for each case were used tocompute consequent parameters through a linear leastsquare method and finally a TS fuzzy model was devel-oped. The developed model gives crisp output value for agiven input data. Fuzzy model developed form the actualdata sets have different rules ranging from 4 to 7. Perfor-mance indices such as RMSE between the computed andobserved runoff, correlation coefficient and NS efficiencywere used to finalize the optimal parameter combina-tion of the model. The effect of error in peak and lowobservations are taken care by the criteria viz. correla-tion coefficient and NS efficiency. The error in time topeak is another criterion which is not considered hereas it is normally considered in storm studies. In rainfall-runoff modelling, accurate estimation of total volume isan important aspect. Therefore, another criterion knownas volumetric error (Kachroo and Natale, 1992) has beenconsidered in this study to hydrologically evaluate the

performance of the models under consideration. The vol-umetric error (Ver) is expressed as:

Ver D

n∑iD1

�Qci � Qoi�

n∑iD1

Qoi

ð 100 �15�

where Qci, Qoi and n are computed runoff, observedrunoff and number of data sets.

Network architecture for ANN s model

Three-layered feed forward neural network was con-sidered for the design of the ANN model for the rainfall-runoff process in this study. The network structure isformulated by considering single output neuron in outputlayer corresponding to the predicted runoff at time t. Asdescribed through Equations 4–14, computed areal rain-fall, preceding runoff at Manot gauging site and AMCconstitutes the input neurons in the input layer. Thedata are normalized between 0 and 1 before the start ofthe training of the ANN model. The learning algorithmadopted here was error back propagation algorithm basedon the generalized delta rule. After the normalization ofdata the next step in the development of ANN model wasthe determination of the optimum number of neurons inthe hidden layer. The optimum number of neurons in thehidden layer was identified using a trial and error pro-cedure by varying the number of neurons in the hiddenlayer from 2 to 10. A number of trial runs were madebefore the finalization of the number of neurons in thehidden layer of the network. The number of neurons inthe hidden layer of the networks of models M1 to M11was finalized and they vary from 4 to 6. In the processof model development the initial weights were randomlyassigned and the activation functions such as sigmoidand linear functions were used for the hidden and out-put nodes, respectively. The mean square error (averagesquared error between the network outputs and the tar-get outputs) was used to measure the performance of atraining process. During training the weights and biasesof the network were adjusted using gradient descent withmomentum weight and bias learning function. As dis-cussed in the previous articles, a number of trials havebeen made until consistent results are obtained. Further-more, the developed model was simultaneously checkedfor its improvement on testing data on each iteration toavoid over training. Therefore, an ANN with n input neu-rons, k hidden neurons and 1 output neurons (n-k-1) wasadopted as the best structure combination to capture therainfall-runoff relationship inherent in the data sets underconsideration. Optimized values of hidden neurons (k)are presented in Table I.

Linear transfer function model

Box and Jenkins (1976) described a linkage betweentwo time-dependent variables of a discrete linear system



Table I. Optimum number of neurons in hidden layer

ANNModel

Number ofinputs

Number ofneurons in

hidden layer

Number ofoutputs

M1 3 4 1M2 4 4 1M3 4 4 1M4 5 4 1M5 6 4 1M6 5 4 1M7 3 4 1M8 6 5 1M9 4 4 1M10 7 6 1M11 5 4 1

by the following mathematical expression:

yasZt D easXt �16�

where Xt, Zt are input and output variables at time t.as is back shift operator. yas and eas are polynomialsof order (1 � a1as1 � a2as2 . . . .apasp) and (b0 C b1as1 Cb2as2 . . . .bqasq).

Equation (16) can be rewritten as,

�1 � a1as1 � a2as2 . . . .apasp� Ð Zt D �b0

C b1as1 C b2as2 . . . .bqbsq� Ð Xt �17�

or

�1 � a1Zt�1 � a2Zt�2 . . . .apZt�p� D �b0Xt C b1Xt�1

C b2Xt�2 . . . .bqXt�q� �18�

or

Zt Dp∑

kD1

ak Ð Zt�k Cq∑

kD0

bk Ð Xt�k �19�

Shifting the base time for variable X from 1 to q,Equation (19) can be expressed as:

Zt Dp∑

kD1

ak Ð Zt�k Cq∑

kD1

bk Ð Xt�kC1 �20�

where p and q are time memory or response for input(Xt) and output (Zt) variables. a and b are parameters orconstants.

Equation (20) produces a set of ‘t’ linear equations tobe solved for (p C q) number of constants. The linearequations can be represented in matrix form as:

Z D Y Ð M C �< �21�

where Z D output vector D [Z1, Z2, . . . .Zt]T �22�

Y D input matrix

0 0 0 . . 0 X1 0Z1 0 0 . . 0 X2 X1

Z2 Z1 0 . . 0 X3 X2

Z3 Z2 Z1 . . 0 X4 X3

Z4 Z3 Z2 . . Z5�p X5 X4

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .Zt�1 Zt�2 Zt�3 . . Zt�p Xt Xt�1

0 . . 00 . . 0

X1 . . 0X2 . . X4�qC1

X3 . . X5�qC1

. . .

. . .

. . .

. . .Xt�2 . . Xt�qC1

�23�

M D time memory vector

D [a1, a2, . . . .ap, b1, b2, . . . . . . bq]T �24�

�< D error vector

D [e1, e2, . . . .et]T �25�

The least square method which minimizes the sumof square of difference between observed and estimatedvalues is used to solve the set of above defined ‘t’ linearequations.

Now, instead of finding the exact solution of Equation(21), a search is to be made for M D OM which minimizesthe sum of squared error defined by:

E�M� Dt∑

iD1

�Zi � mTi M�2 D �Z � M Ð Y�T�Z � M Ð Y�

�26�where �< D Z � Y Ð M is the error vector produced bya specific choice of M. E�s� has a quadratic form andhas a unique minimum at M D OM . The squared errorin Equation (26) is minimized when M D OM , called theleast square estimator which satisfies the normal equation:

YT Ð Y Ð OM D YT Ð Z �27�

orOM D [YT Ð Y]�1 Ð YT Ð Z �28�

RESULTS AND DISCUSSION

Runoff prediction models for the rainfall-runoff hydro-logical process have been developed using linear transferfunction (LTF), ANNs and fuzzy rule-based modellingsystem for Manot catchment using different combinationsof input data vectors. The model performance evaluationhas been carried out through statistical and hydrological



performance evaluation criteria, viz. RMSE, correlationcoefficient, NS efficiency and volumetric error (Ver).

Linear transfer function runoff prediction model

The LTF models have been developed for the predic-tion of runoff. The LTF models are trained using the sameinput data set as used for the ANN and fuzzy rule-basedmodels, to enable a direct comparison. Considering var-ious combinations of input vectors (Equations 4–14) 11separate models have been developed and mathematicallyexpressed as follows:


Qt D 10Ð42Pt C 5Ð89Pt�1 C 3Ð35Pt�2 C 2Ð5Pt�3

�29�2. Rainfall and AMC as input

Qt D 8Ð95Pt C 3Ð04Pt�1 C 0Ð68Pt�2

C 0Ð07Pt�3 C 1Ð75AMC �30�



C 0Ð52Pt�3 � 0Ð61Qt�1 �31�


C 0Ð03Pt�3 C 0Ð547Qt�1 C 0Ð08Qt�2 �32�


C 0Ð001Pt�3 C 0Ð528Qt�1 C 0Ð09Qt�2

C 0Ð07Qt�3 �33�



C 0Ð007Pt�3 C 0Ð522Qt�1 C 0Ð67AMC �34�

Qt D 7Ð76Pt C 0Ð526Qt�1 C 0Ð68AMC �35�

Qt D 4Ð67Pt C 0Ð28Pt�1 C 0Ð15Pt�2 C 0Ð02Pt�3

C 0Ð52Qt�1 C 0Ð04Qt�2 C 0Ð67AMC �36�

Qt D 7Ð34Pt C 0Ð511Qt�1 C 0Ð29Qt�2

C 0Ð63AMC �37�


C 0Ð09Pt�3 C 0Ð514Qt�1 C 0Ð10Qt�2

C 0Ð02Qt�3 C 0Ð34AMC �38�

Qt D 0Ð75Pt C 0Ð532Qt�1 C 0Ð11Qt�2

C 0Ð22Qt�3 C 0Ð31AMC �39�

Performance indices of the LTF models developed inthe form of mathematical Equations 29–39 are presentedin Table II. A comparison of the developed model hasbeen carried out on the basis of model performanceindices viz. RMSE, coefficient of correlation and NSefficiency. The LTF runoff prediction models for Manotbasin show the values of coefficient of correlation and NSefficiency in the range of 0Ð54 to 0Ð766 and 0Ð400 to 0Ð658respectively for the developed models during calibration.Values of coefficient of correlation and NS efficiency varyin the range of 0Ð539 to 0Ð758 and 0Ð347 to 0Ð584 duringthe model validation. RMSE varies from 163Ð0 to 284Ð8during calibration and 128Ð6 to 299Ð5 during validation.An improvement in the model performance is observedby including AMC in the input vector of the LTF models.In the models where only rainfall is the input, ModelM2 is the best model with the values of correlationcoefficient, NS efficiency and RMSE as 0Ð563, 0Ð473 and266Ð8 respectively during calibration and 0Ð565, 0Ð402and 281Ð0 respectively during validation. Models M3 to

Table II. Statistical performances indices of linear transfer function models

Model Calibration Validation

Coefficientof

correlation

NS RMSE Volumetricerror (Ev)

Coefficientof

correlation

NS RMSE Ev

Only rainfall as input1 0Ð540 0Ð400 284Ð8 35Ð5 0Ð539 0Ð347 299Ð5 37Ð7Rainfall and AMC as input2 0Ð563 0Ð473 266Ð8 33Ð8 0Ð565 0Ð402 281Ð0 35Ð8Rainfall and runoff as input3 0Ð698 0Ð575 193Ð7 27Ð2 0Ð627 0Ð451 156Ð7 –29.04 0Ð702 0Ð592 189Ð4 �26Ð9 0Ð631 0Ð522 154Ð4 �28Ð75 0Ð700 0Ð589 190Ð3 �27Ð0 0Ð629 0Ð508 155Ð3 �28Ð9Rainfall, runoff and AMC as input6 0Ð699 0Ð588 192Ð4 �27Ð1 0Ð628 0Ð501 156Ð0 �29Ð07 0Ð693 0Ð564 193Ð4 �27Ð7 0Ð615 0Ð500 157Ð0 �29Ð58 0Ð766 0Ð658 163Ð0 �21Ð1 0Ð758 0Ð584 128Ð6 �22Ð79 0Ð764 0Ð656 163Ð6 �21Ð3 0Ð755 0Ð582 134Ð5 �22Ð910 0Ð761 0Ð656 163Ð6 �21Ð5 0Ð754 0Ð580 131Ð4 �23Ð211 0Ð761 0Ð655 164Ð0 �21Ð5 0Ð753 0Ð581 132Ð7 �23Ð2

Ev, error variation.



M5 with only rainfall and runoff as inputs show that themodel M4 performs better than the other two models.The correlation coefficient, NS efficiency and RMSEof Model M4 are 0Ð702, 0Ð592 and 189Ð4 respectivelyduring calibration and 0Ð631, 0Ð522 and 154Ð4 duringvalidation. It is observed that the inclusion of 3-dayprevious runoff in the model M5 reduces the modelperformance. Models M6 to M11 consider rainfall, runoffand AMC in the model input vector. In this category ofmodels, M8 is the best model with values of coefficient ofcorrelation, NS efficiency and RMSE as 0Ð766, 0Ð658 and

163Ð0 respectively during calibration and 0Ð758, 0Ð584and 128Ð6 respectively during validation. In general, theperformance indices of model calibration and validationindicate that inclusion of Qt�3 in the model input vectorreduces the model performance. Figures 2–12 illustratethe time series of observed runoff and model predictedrunoff for the 11 LTF models for the validation period1997 and 1998. It is observed that the model M1and M2 estimates zero runoff values when there is norainfall during the periods selected in the input modelstructure. Inclusion of previous day discharge values in

0

500

1000

1500

2000

2500

1 51 101 151 201

Time (Day)

Run

off (

m3 /

s)

Observed Discharge

Linear Transfer Function (M1)

Figure 2. Time series of observed runoff and model predicted runoff-Linear Transfer Function Model (M1)

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m3 /

s)

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m3 /

s)

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Linear Transfer Function(M3)




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off (

m3 /

s)

Observed DischargeLinear Transfer Function (M6)


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the input model structure has direct impact on the modelperformance.

ANN runoff prediction model

Table III presents the performance indices of all the11 models (M1–M11) developed using ANNs. Modelperformance evaluation criteria RMSE, coefficient ofcorrelation and NS efficiency were used to evaluate theperformance of the developed model. It is observed thatthe values of coefficient of correlation and NS efficiencyvary in the range of 0Ð661 to 0Ð884 and 0Ð569 to 0Ð770respectively for the developed models during calibration.During the model validation, values of coefficient ofcorrelation and NS efficiency vary in the range of 0Ð671to 0Ð868 and 0Ð563 to 0Ð745. RMSE varies from 133Ð2to 241Ð2 during calibration and 94Ð2 to 217Ð9 duringvalidation. ANN models developed for three differentcases, i.e. (i) only rainfall in the input vector, (ii) rainfalland AMC in the input vector and (iii) rainfall, runoff andAMC in the input vector. Models developed under thesethree classes show distinct performance. The model resultindicates that the antecedent moisture content valuesconsidered in the input vector show an improvement inthe daily rainfall-runoff model performance. Model M1with only rainfall in the input vector show the values ofcoefficient of correlation, NS efficiency and RMSE valuesas 0Ð661, 0Ð569 and 241Ð2 during calibration and 0Ð671,0Ð563 and 217Ð9 during validation respectively. ModelM2 is defined by including antecedent moisture contentin the input vector of model M1. Model M2 show thevalues of coefficient of correlation, NS efficiency andRMSE as 0Ð672, 0Ð572 and 218Ð6 during calibration and0Ð678, 0Ð574 and 216Ð2 during validation, respectively.This indicates that the inclusion of antecedent moisturecontent in the input vector has a direct relation withthe performance of the model. Now, improvement ofmodel M1 by inclusion of preceding day runoff is veryobvious and this is confirmed by the Models M3, M4

and M5. Inclusion of preceding 3 days of runoff in theinput vector, trim down the model performance. Valuesof coefficient of correlation, NS efficiency and RMSE ofmodel M4 (i.e. 0Ð869, 0Ð635 and 173Ð6) are better than themodel M5. This indicates that only previous two runoffvalues are required to predict runoff. Model M6 to M8are same as Model M3 to M5 except the term antecedentmoisture content in the input vector. Model M6 toM8 show an improvement in the model performanceby the inclusion of antecedent moisture content. It isdepicted from the performances indices of Models M3to M5 that the values of coefficient of correlation, NSefficiency and RMSE varies in the range of 0Ð801 to0Ð838; 0Ð649 to 0Ð662 and 113Ð3 to 114Ð7 respectivelyduring validation. Slightly better performance is obtainedfrom the models M6, M8 and M10. For these threemodels (M6, M8 and M10) values of coefficient ofcorrelation, NS efficiency and RMSE are found in therange of 0Ð822 to 0Ð868; 0Ð654 to 0Ð745, and 94Ð2 to115Ð1 respectively, during validation. In case of modelM6, M8 and M10 the performance of the model M10decreases when compared with model M8. Further, it isfound from model M6 and M9 that instead of previousday rainfall values previous day runoff values have moreeffect on model performance. This may be due to the factthat the preceding day runoff and antecedent moisturecontents provide sufficient memory to the daily rainfall-runoff model which is otherwise associated with rainfallvalues. Another important criterion, i.e. volumetric error(Ver) also indicates a pattern in tune with the correlationcoefficient. Figures 13–23 illustrates the time series ofobserved runoff and model predicted runoff for the 11ANN models for the validation period 1997 and 1998.

Fuzzy logic runoff prediction model

Fuzzy rule-based models developed using model struc-tures presented through Equations 4–14 were comparedusing various statistical model performance indices, e.g.

Table III. Statistical performances indices—ANN models


Coefficientof

correlation

NS RMSE Ev Coefficientof

Correlation

NS RMSE Ev

Only rainfall as input1 0Ð661 0Ð569 241Ð2 25Ð7 0Ð671 0Ð563 217Ð9 23Ð1Rainfall and AMC as input2 0Ð672 0Ð572 218Ð6 19Ð8 0Ð678 0Ð574 216Ð2 21Ð2Rainfall and runoff as input3 0Ð854 0Ð612 174Ð3 14Ð1 0Ð801 0Ð649 114Ð7 10Ð94 0Ð869 0Ð635 173Ð6 �12Ð8 0Ð838 0Ð662 113Ð3 �9Ð75 0Ð862 0Ð631 174Ð1 �13Ð4 0Ð834 0Ð656 114Ð5 �10Ð3Rainfall, runoff and AMC as input6 0Ð858 0Ð629 176Ð3 �13Ð8 0Ð822 0Ð654 115Ð1 �10Ð67 0Ð854 0Ð620 179Ð2 �14Ð1 0Ð819 0Ð649 118Ð4 �10Ð98 0Ð884 0Ð770 133Ð2 �11Ð4 0Ð868 0Ð745 94Ð2 �8Ð69 0Ð878 0Ð767 136Ð0 �11Ð7 0Ð860 0Ð742 98Ð6 �8Ð810 0Ð883 0Ð769 137Ð5 �11Ð6 0Ð860 0Ð743 99Ð4 �8Ð711 0Ð885 0Ð767 137Ð6 –s11.5 0Ð858 0Ð742 100Ð1 �8Ð6



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Figure 13. Time series of observed runoff and model predicted runoff-ANN Model (M1)

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RMSE, coefficient of correlation, NS efficiency and volu-metric error. Table IV presents these performance indicesof all the 11 model structure defined as M1 to M11. Themodels classified in three different groups were evalu-ated within the same group and than compared with themodels of other group. Inclusion of AMC in the inputvector Model M1 results a new model M2. Model results(Table IV) show that the model M2 performs better thanM1 model both during calibration (M1: 0Ð672, 0Ð584,237Ð2; M2: 0Ð696, 0Ð583, 206Ð4) and validation (M1:0Ð683, 0Ð577, 214Ð1; M2: 0Ð693, 0Ð585, 208Ð5). ModelM3 to M5 were developed for different combinations

of precipitation and runoff and they show coefficientof correlation in the range of 0Ð823 to 0Ð842, NS effi-ciency in the range of 0Ð663 to 0Ð675, RMSE in therange of 108Ð7 to 109Ð4 and volumetric error in therange of �9Ð66 to 10Ð3 during validation. A compari-son of fuzzy rule-based runoff prediction models M3 toM5 indicates that the model M4 is the best model inthis group with coefficient of correlation 0Ð842, NS effi-ciency 0Ð675, RMSE 108Ð7 and volumetric error �9Ð66.Models M6 to M11 falling in the fourth group also haveAMC term in the input vector. Statistical performanceindices of these models, i.e. coefficient of correlation,



Table IV. Statistical performances indices—fuzzy models


Coefficientof

correlation

NS RMSE Ev Coefficientof

correlation

NS RMSE Ev

Only rainfall as input1 0Ð672 0Ð584 237Ð2 23Ð6 0Ð683 0Ð577 214Ð1 21Ð2Rainfall and AMC as input2 0Ð696 0Ð583 206Ð4 17Ð6 0Ð693 0Ð585 208Ð5 19Ð0Rainfall and runoff as input3 0Ð870 0Ð634 170Ð1 11Ð7 0Ð823 0Ð663 109Ð2 10Ð34 0Ð878 0Ð647 168Ð7 �11Ð0 0Ð842 0Ð675 108Ð7 �9Ð665 0Ð871 0Ð642 169Ð3 �11Ð6 0Ð838 0Ð672 109Ð4 �10Ð2Rainfall, runoff and AMC as input6 0Ð868 0Ð640 172Ð1 �11Ð9 0Ð838 0Ð671 110Ð2 �10Ð57 0Ð863 0Ð631 172Ð8 �12Ð4 0Ð827 0Ð662 111Ð6 �10Ð98 0Ð906 0Ð776 130Ð5 �8Ð7 0Ð876 0Ð755 86Ð9 �7Ð59 0Ð896 0Ð770 133Ð9 �8Ð6 0Ð871 0Ð752 94Ð2 �7Ð4210 0Ð897 0Ð772 134Ð2 �8Ð9 0Ð870 0Ð753 94Ð2 �7Ð7511 0Ð901 0Ð771 134Ð5 �8Ð7 0Ð869 0Ð752 95Ð7 �7Ð58

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Figure 24. Time series of observed runoff and model predicted runoff-Fuzzy Model (M1)

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NS efficiency, RMSE and volumetric error varies in therange of 0Ð827 to 0Ð876; 0Ð662 to 0Ð755; 86Ð9 to 111Ð6and �7Ð42 to �10Ð9 respectively. These values of per-formance indices are always higher than the performanceindices of group 3 models, i.e. models M3 to M5. Thisconfirms that the model performance improves when theAMC is also included in the input vector of fuzzy rule-based runoff prediction models. Further, a comparison

of all the 11 models show that the model M8 is thebest fuzzy rule-based runoff prediction model for thecatchment of Narmada up to Manot with coefficient ofcorrelation 0Ð876, NS efficiency 0Ð755, RMSE 86Ð9 andvolumetric error �7Ð5 during validation. Figures 24–34illustrates the time series of observed runoff and modelpredicted runoff for the 11 ANN models for the validationperiod 1997 and 1998.



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Figure 35. Comparison of correlation coefficients of ANN, Fuzzy and Linear Transfer Function Models-calibration data

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Figure 36. Comparison of correlation coefficients of ANN, Fuzzy and Linear Transfer Function Models-validation data

Comparison of different methods

In order to assess the ability of fuzzy rule-based runoffprediction models relative to that of neural networkand LTF models, ANN and LTF models were alsodeveloped using the same input vectors to that of fuzzymodels. The performance of ANN, LTF and fuzzy modelsare compared in terms of the performance indices ofthe developed models. Plots of correlation coefficient(Figures 35 and 36), NS efficiencies (Figures 37 and 38)and RMSE (Figures 39 and 40) of the developed ANN,FL (Model M1–M11) and LTF models are compared.It is observed from these figures that the inclusion ofAMC in the input vector has direct impact on model

performance. From the performance indices it is depictedthat the LTF model-M8, ANN model-M8 and the fuzzymodel-M8 are the best models in their respective groups.Further, a close comparison of the ANN models andthe fuzzy models indicates that the fuzzy M8 modelis the best rainfall-runoff model for the catchment ofNarmada upto Manot gauging site. Figures 35–40 andTables II–IV suggests that though the performance ofboth the ANN and the fuzzy models are similar duringtraining and validation, the fuzzy models shows a slightimprovement over the ANN. It is also evident from thesefigures that the fuzzy and the ANN models outperformthe LTF models. A significant improvement is observed



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Figure 37. Comparison of NS Efficiency of ANN, Fuzzy and Linear Transfer Function Models-calibration data

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Figure 38. Comparison of NS Efficiency of ANN, Fuzzy and Linear Transfer Function Models-validation data

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Figure 39. Comparison of RMSE of ANN, Fuzzy and Linear Transfer Function Models-calibration data

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Figure 40. Comparison of RMSE of ANN, Fuzzy and Linear Transfer Function Models-validation data



for the fuzzy model in the runoff volume computationcompared to ANN.

SUMMARY AND CONCLUSIONS

In this study, LTF, ANN and fuzzy rule-based tech-niques have been used to develop models for the predic-tion of runoff using rainfall-runoff models for Narmadacatchment upto Manot gauging site. Potential of fuzzyrule-based technique for modelling of rainfall-runoff pro-cess is investigated by comparing results of FL-basedrainfall-runoff models with the three-layered feed for-ward neural network and LTF-based models developedusing the same input vectors. The daily rainfall and runoffdata of the monsoon season (mid-June–September) from1993 to 1998 were considered for the development (cal-ibration and validation) of models. The rainfall andrunoff data required for the study were processed usingHYMOS, 2001 software. The concept of antecedentmoisture content in the fuzzy rule-based daily rainfall-runoff modelling has been introduced. Through cor-relation analysis between antecedent moisture contentof different periods and runoff, a suitable structure ofantecedent moisture content is decided. Rainfall-runoffmodels have been developed for four major inputs. Fur-ther, a detailed analysis and comparison of the mod-els of the same group, models of different groups andthe models developed using different modelling tech-niques have been carried out. The analysis of resultsindicates that the introduction of antecedent moisture con-tent is very useful for daily rainfall-runoff modelling asit improves the performance of the models. The studysuggests a suitable daily rainfall-runoff model structurefor the study area and concludes that the fuzzy rule-based approach outperforms both the ANN and LTFapproaches.

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Documents

Comparative study of neural network, fuzzy logic and linear transfer function techniques in daily rainfall-runoff modelling under different input domains