Journal Hydrology

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A non-linear rainfallrunoff model using an articial neuralnetwork

N. Sajikumar a,*, B.S. Thandaveswara b

aGovernment Engineering College, Thrissur 680 009, Kerala, Indiab Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Received 17 April 1998; accepted 19 October 1998

Abstract

A rainfallrunoff model that can be successfully calibrated (i.e., yielding sufciently accurate results) using relatively shortlengths of data, is desirable for any basin in general, and the basins of developing countries like India, in particular, for whichscarcity of data is a major problem. An articial neural network paradigm, known as the temporal back propagation neuralnetwork (TBP-NN), is successfully demonstrated as a monthly rainfallrunoff model. The performance of this model in ascarce data scenario (i.e., the effects of using reduced calibration periods on the performance) is compared with Volterra-type Functional Series Models (FSM), utilising the data of the River Lee (in the UK) and the Thuthapuzha River (in Kerala,India). The results conrm the TBP-NN model as being the most efcient of the black-box models tested for calibration periodsas short as six years. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Non-linear rainfallrunoff modelling; Scarce data situation; Temporal back propagation neural network; Functional seriesmodels

1. Introduction

A model of the rainfallrunoff relationship is anessential component in the process of evaluation of water resource projects for which, most of the time, asufcient length of record ofow may not be availablewhereas the rainfall data may be available. Thisindeed is a major problem for developing countrieslike India where the use of a well-equipped measuringsystem is of recent origin. Even when such concurrentdata are available, the length of record is generallyrather limited. Hence a rainfallrunoff model thatcan yield sufciently accurate results with such shortlengths of data is desirable and useful.

Different types of rainfallrunoff transformation

models were proposed and in vogue, ranging frompurely empirical simple models, such as the rationalmethod, to highly sophisticated distributed physicalprocess models dened by partial differential equa-tions, such as SHE (Syste m Hydrologique Europee`n)model (Abbott et al., 1986). Based on the degree of representation of the involved physical processes, themodels are classied, with the increasing degree of representation, as black-box models, conceptualmodels, and physically-based distributed models.Black-box models are often used because they avoidhaving to address the problems of the spatial andtemporal variability of the inputs and parameters,and the complexities of the involved physicalprocesses. The unit hydrograph is one such linearblack-box model and has been widely accepted as apractical tool. However, the limitation of the unit

Journal of Hydrology 216 (1999) 3255

HYDROL 3690

0022-1694/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S0022-1694( 98)00273-X

* Corresponding author.


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hydrograph in representing the rainfallrunoff rela-tionship is not only because of the restrictions of line-arity and time invariance but is also because of theuncertainties in the determination of the effectiverainfall and the separation of baseow (Brathand Rosso, 1993). Hence, non-linearity was intro-duced as Volterra integral series (VIS). Amorochoand Orlob (1961) were probably the rst to haveexploited the VIS for the analysis of hydrologicsystems (Singh, 1988) introducing the functionalseries models (FSM). A detailed discussion on themethodology of functional series, as applied to hydro-logic system, is available in Amorocho (1973). Eventhough the application of functional series to rainfallrunoff modelling is widely reported in the literature

(Muftuoglu, 1984; Xia, 1991), its application in theprediction of monthly ows are limited. In fact,monthly ows are sufcient and indeed preferablefor economic reasons when the principle objective isto determine the feasibility and storage requirementsfor a proposed water resources project (Muftuoglu,1991).

The functional series is a general mathematicalmodel for a non-linear black-box system,whichproduces a single output from a serial input (Muftuo-glu, 1991). The systems approach, in general, and the

non-linear functional analysis in particular, arecurrently undergoing a kind of renaissance, throughthe development of new tools such as the articialneural network (ANN) and the genetic algorithm(Minns and Hall, 1996). Among the plethora of applications of ANN, the systems approach ingeneral may be categorised into the pattern mappingproblem, i.e., inputoutput mapping as in the caseof black-box modelling; for which one of the bestsuitable architectures is a feed-forward network. (Inneural network terminology, a pattern is an inputoutput pair.) Hornik et al. (1989) established that afeed-forward network could be considered as ageneral non-linear approximator . This shared prop-erty of generality prompted the investigators tocompare the ANN model with the functional seriesmodel.

The major advantage of an ANN is its ability torepresent the non-linearity by means of smallernumber of parameters and to learn from examples(i.e., from its environment). ANN learns about itsenvironment through an iterative procedure known as

training which in turn adjusts the parameters(weights) of the network. In contrast, the applicationof functional series requires the kernel to be evaluatedby means of methods such as the BrandstetterAmorcho methods (Amorocho and Brandstetter,1971), and the least-squares method. However, thefunctional series can be calibrated adaptively,although the extra effort in doing so may not beworth it in terms of results. Moreover, the applicationof an ANN in rainfallrunoff modelling does notrequire any a priori assumption regarding theprocesses involved. The advantages of an ANN, asagainst the conventional models, are discussed indetail by French et al. (1992).

Even though the development of the feed-forward

ANN in its present form was foreshadowed by thework of Werbos (1974) and reinvented separatelybyRumelhart et al., 1986, its application in rainfallrunoff modelling is of recent origin. Karunanidhi et al.(1994) used a cascade correlation algorithm forpredicting the ow at a location in a river by usingthe ows at different locations along the river andalong its tributaries, as input. A ve day window of each input is used to account for the time dependenceof the phenomenon. In a discussion of this work,Heggen (1995) stated that the ANN model should

have been compared with a more sophisticatedmodel rather than a primitive power model. Zhu andFujitha (1994) compared the performance of fuzzyreasoning in rainfall runoff modelling and a feed-forward ANN model in predicting a 3-hour lead-runoff. Here again, the time dependence of thephenomenon was taken into account by using awindow of rainfall inputs.

Smith and Eli (1995) used a feed-forward network to predict the runoff peak value and the time to peak for spatially distributed rainfall. However, the training(calibration) was attempted using simulated data.Even though this exercise demonstrates ANNscapability in learning the rainfallrunoff relationshipthrough the training procedure, it is of limited practi-cal use as the information stored in the weights (i.e.the parameters) of the network is that of the metho-dology used in simulating the outow, namely, acascade of non-linear reservoirs. Hsu et al. (1995)demonstrated the use of an Ann as a rainfallrunoff model. They compared the performance of the modelwith that of an Auto-Regressive Moving Average

N. Sajikumar, B.S. Thandaveswara / Journal of Hydrology 216 (1999) 3255 33


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eXogenous input (ARMAX) time series model (Boxand Jenkins, 1976) and the Sacramento soil moistureaccounting model of U.S. National Weather Service(Burnash et al., 1983). In addition, they proposed anew strategy known as the linear least squares simplexfor training (i.e. calibrating) the network. This strat-egy is proved to be of better performance by them,though the strategy requires the manipulation of theentire weight matrix which effectively eliminates thelocal and recursive nature of the weight adaptation. Thetime dependency of the rainfallrunoff relationship is

accounted by using a window of rainfall and runoff asinput.

Minns and Hall (1996) used a feed-forward ANN topredict runoff from rainfall data. However, thepurpose of that paper was to demonstrate the learningability of the network, namely, learning fromexample, rather than applying it to a practical situa-tion, where the learning was achieved using simulateddata which were free from data errors. In this context,a practical application of an ANN in predicating themonthly runoff, in a scarce data scenario, whereby

N. Sajikumar, B.S. Thandaveswara / Journal of Hydrology 216 (1999) 325534

Fig. 1. (a) A feed forward neural network; (b) A recurrent neural network.


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the effect on model performance of different selec-tions of calibration record lengths, is investigated inthe present study. The input to the ANN consists of the time series of resealed effective rainfall, thenetwork output being rescaled values of the corre-sponding storm runoff. The performance is of thismodel is compared with those of second orderVolterra-type functional series models (FSM), whichinclude the linear model (LM) as a special case(Muftuoglu, 1984, Muftuoglu, 1991). Primary empha-sis is given here to evaluating the ability of thesemodels in prediction, with shorter record lengths of calibration data.

2. Articial neural network

2.1. General

The speed and efciency with which the humanbrain processes information in certain tasks has beenfascinating scientists for quite a long time. The questto understand these processes and to solve the asso-ciated types of problems have led to the developmentof articial intelligence (AI). Among the many eldsof AI, the ANN is gaining a status prime importance

even in other elds of engineering, owing to itsinteresting properties such as learning fromexamples, the ability to represent non-linearity bymeans of a smaller number of parameters and theleast requirement of information regarding the processto be modelled (Yegnanarayana, 1994; Pal andSrimani, 1996). Haykin (1994) dened the ANN asfollows:

An articial neural network is a massively paralleldistributed processor that has a natural propensity for storing the experiential knowledge and making it available for use. It resembles the brain in twoaspects:

1. Knowledge is acquired by the network through alearning process.

2. Inter-neuron connection strengths known as synap-tic weights are used to store the knowledge.

The common tasks that can be accomplished bothby the brain and the ANN may be categorised intodifferent groups such as pattern association, mapping

and clustering. The pattern association is the processof linking an input or an output, depending on whetherit is an auto-associative or a hetero-associative neuralnetwork, to the corresponding input, by means of theinter-neuron connection strengths. (In this context theword networkrepresents neural network). Incontrast, in pattern mapping, the under-lying inputoutput relationship is captured by means of a suitablelearning strategy. The pattern clustering deals withgrouping of patterns based on the proximity of eachpattern with others (Thandaveswara and Sajikumar,1998).

The black-box type modelling of the rainfallrunoff relation can be classied under the categoryof pattern mapping. Two types of networks, namely,a

feed-forward multi-layer perceptron (MLP) network (Rumelhart et al., 1986) and Counter propagationnetwork (CPN) (Hecht-Nielsen, 1987) are usuallyused for pattern mapping problems. Of the twotypes of networks (MLP and CPN), the rst one isselected for application in this study as the parameter(weights) estimation of the network has close similar-ity with that of functional series.

An MLP network (Fig. 1(a)) consists of a set of sensory units that constitute the input layer, one ormore hidden layers of computational nodes (neurons)

and an output layer of computational nodes. In a feed-forward network, information passes only in onedirection, i.e., from the neurons of a layer to theneurons of succeeding layer. Thus, all input to aneuron in a particular layer is from the precedinglayer and these unidirectional connection strengthsare known as weights. A gradient descent procedureknown as generalised error-back propagation isusually employed for training (i.e., calibrating) theMLP network. Hence the MLP network is alsoknown as a back-propagation network (BPN). Theerror-back-propagation consists of two passes: aforward pass and a backward pass. In the forwardpass, an activity pattern is applied to the input layerand its effect is propagated through network, layer bylayer. The activity (i.e. the net effect) at a neuron iscomputed as the weighted sum of the outputs of theneurons of the previous layer. The output from theneuron (node) is a non-linear function of the thus-computed activity. The sum of squared deviation of the output from the target value at the nodes of theoutput layer denes the error signal that is to be



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propagated back to the previous layers such that theparameters are adjusted to minimise the error infurther computation. This backward pass is a part of the learning strategy rather than part of the network and hence does not contradict the initial part of thisparagraph.

The major shortcoming in applying the standardBPN to rainfall runoff modelling is that the network does not incorporate storage elements, so does not takeaccount of temporal dependence. In a BPN, the outputat a particular instant totally depends on the input atthat instant, which is not true in the case of the rainrunoff modelling. To circumvent this problem, one of the following three methods could be used;

1. The method of using a window of input series: i.e.,P (t), P (t 1),, P (t m), as input at an instant. Inwhich P (t) represents one of the inputs at time t andm the memory length (Karunanidhi et al., 1994;Smith and Eli, 1995; Hsu et al., 1995; Minns andHall, 1996). Hence, a single real input requires minput units in the BPN network and the total

number of input neurons increases with the lengthof memory used.2. The recurrent networks method: The recurrent

network (Fig.1(b)) has feed-back connections inaddition to the feed-forward connections, i.e., theneurons of a layer can receive input from neuronsof preceding, succeeding and the layer itself. Thefeed-back connections make the network dynamicin nature, in the sense that output of a neuron is animplied function of previous and present inputs.This type of network requires a very small learningrate (a factor, which controls the speed of learn-ing process; the higher the learning rate thehigher the speed of learning) in order to maintainthe stability of the learning process (Werbos,1990). Moreover, the weight updating (parameterestimation during the process of learning) is notlocal, unlike that of the standard BPN, in thesense that updating of a single weight requiresthe manipulation of the entire weight matrixwhich in turn increases the computational effortand time.


Fig. 2. (a) Dynamic neuron and its structure; (b) Signal ow diagram in a single connection (Haykin, 1994).


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3. The temporal back-propagation neural network (TBP-NN) method: The standard BPN uses asingle value to represent the connection strength(weight) between two neurons of consecutivelayers. In contrast, the TBP-NN uses a niteimpulse response (FIR) lter to represent theweight, i.e., the connection strength is an arrayrather than a single value. Elements of the arrayare the weights for present and previous inputs tothe neuron. Thus, the activity from a connection iscomputed as the sum of the product of elements of the weight with the corresponding elements of theinput series. The activity (i.e., the net effect) at aneuron is computed as the sum of thus-computedactivity for all the connections to the neuron from

the preceding layer (Fig. 2(a)). Hence, the arrays of weights accomplish the time dependence effects of storage by means of internal delays at everyneuron. In the case when the input array to thenetwork is the most recent rainfall pattern, overthe memory length of the catchment, the storageeffects are implicitly accounted for by using inter-nal delays. The use of internal delays to representthe time dependence, rather than using a window of inputs as mentioned earlier, is established ashaving better performance by Waibel et al.

(1989) in the case of the phoneme recognitiontask. The TBP-NN method is used in this studynot only because of the advantages referred tobut also because of the similarity between anarray of weights and its hydrologic counter-part,namely, the response functions (i.e. in the linearform, the unit hydrograph). Hence, the incorpora-tion of physical information into the network at alater stage (not attempted in this paper), if required,would be easier (Sajikumar, 1998).

2.2. Temporal back propagation neural network (TBP-NN)

To make the MLP network dynamic in nature, Wan(1990) proposed the use of the nite impulse response(FIR) linear lter to represent the synaptic weights.The network resembles the standard back-propagationnetwork (Fig. 1(a)) except for having a linear FIRlter for each connection. This difference is high-lighted in Fig. 2(a) by focusing on the details of asingle neuron. Fig. 2(a) shows the dynamic neuron

and its structure of the weights. Fig.2(b) representsthe signal ow diagram through a connection, whichin turn, indicates that the inputs at a neuron are movedthrough a system having a memory length. (in which z 1, analogues to the backward shift operator, repre-sents the time delay.) The elements of the weightarray are multiplied by the corresponding previousand present inputs to compute y1tj. The input arraymoves on through time and thus making the presentinput as the previous inputs in the subsequent steps.The aforementioned procedure is different from themethod (i) presented in the previous section as theprocess of moving the input array through a timewindow is represented at the neuron level ratherthan at the input layer alone.

The coefcients for a synaptic lter can be repre-sented by a weight vector and similarly the output of any neuron. The following expressions indicate theprocesses in a neuron:

W lij wlij 0 ; w

lij 1 ; w

lij M l

T 1

X l j k xli k ; x

li k 1 ; x

li k M l

T 2

sl 1ij k Wlij

T X li k 3

vl 1 j k N l

i 1sl 1ij

N l

i 1W lij T X li k 4

xl 1i k f vl 1 j k 5

in which W lij is the array of weights for the connectionbetween neuron i in layer l to neuron j in the nextlayer; M l is the delay in the layer l; X

l j(k ) is the

input vector at neuron j in layer 1; Sij1 1 (k ) is theweighted sum for a connection; v1 j (k ) is the weightedsum (activity) at the jth neuron of the ( l 1) th layer; N l is the number of neurons in the lth layer; xil 1(k ) isthe output at the present time step from neuron j inlayer l for the k th pattern, which is an input to the ( l1)th layer and f () denotes the neuron transfer function.(It may be noted that the superscript l for X and W and(l 1) for s, v and y are not shown in Figs. 2(a) and2(b) for the sake of clarity in representation). Theletter l represents the layer number of the neuron,i.e. that to which the neuron belongs.

The neuron transfer function f () is usually a non-linear output function. There are different forms for



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the output functions such as linear, sigmoid (logistic)and hyperbolic tangent (Haykin, 1994). Of thesethree, the sigmoid function is the most widely usedand the function value is bounded in the range [0,1].Hence, the network output is bounded in this rangeand this would necessitate the scaling down of theactual observed output so as to be within this range.This is important not only for the purpose of calibrat-ing the network, but also for comparing the outputwith the observed outputs. In contrast, the inputsmust also scaled down to the same range [0,1], inorder to avoid the saturation of the output function.The saturation of the output function means that theoutput value will always be very near to unity. If thescaling down is not carried out, owing to the large

value of the weighted sum computed by Eq. (4) andasymptotic nature of the output function near theboundary values, the output from the neuron willalways be near to unity. Saturation of the output func-tion has a detrimental effect on the training of thenetwork because the training uses a gradient descentprocedure to approach the minimum error. The train-ing procedure is given in Appendix.

2.3. Issues related to scaling down

Ooyen and Nichhuis (1992) reported that theconvergence is slow and the learning process is inef-cient if the output data items contain values near tozero or unity. The reason behind this behaviour couldbe identied from the expansion of the derivative of the output function which is utilised in a gradientdescent method (see Appendix). When the sigmoidfunction is used as the output function, the derivativeterm can be reduced as

f H xl j xl j 1 x

l j 6

in which x1 j is the output at the jth neuron. A closerlook at the derivative suggests that the derivativeapproaches zero very fast when the target value is inthe neighbourhood of 0 or 1. This results in inefcientlearning. Ooyen and Nichhuis (1992) suggested amethod to overcome this difculty by using a differentform of error function. However, the usual error func-tion (expressed as the sum of squared deviation of theoutput from the target value) is of a very generalnature and is extensively used in least-squares analy-sis and in automatic calibration procedures of

rainfallrunoff processes. Hence changing the errorfunction from the usual one is not consideredadvisable. Moreover, it is the experience of thepresent authors that the pattern with target values inthe neighbourhood of zero will not be learnt properly.The scaling down procedure employed by differentinvestigators (Smith and Eli, 1995; Minns and Hall,1996) is the division of input or output values bythe corresponding maximum value present in thepattern. The knowledge about the process underconsideration can some times dene the lowerand upper limit of the variables. In the absenceof knowledge about the process, the best estimateof the process behaviour may be the observedmagnitudes. This restricts the maximum value

which can be predicted to be the maximumvalue present in the output of the training set,i.e., extrapolation is not permitted. Extrapolationin the neighbourhood of limits is not a formidabletask but is an essential one. To alleviate suchdifculties, a scaling down procedure is adoptedwhich not only avoids the values in the neighbour-hood of zero and one, but also allows for limitedextrapolation.

The input and target values are scaled down into arange FMIN to FMAX, rather than zero and one,

(FMIN 0 and FMAX 1), such that X n FMIN

X u fact minfact max fact min

FMAX FMIN7

X u and X n represent the variable to be scaled down andits scaled down values respectively. fact max and fact min are the maximum and minimum valuepresent in the X vector. One should be careful inselecting the values of FMIN and FMAX as, on theone hand, reduction of the range to a very small valuewill have a negative inuence on the training while, incontrast, the amount of allowed extrapolation shouldnot exceed a certain limit.

3. Functional series models (FSM)

For the sake of completeness, the functional seriesmodels are discussed briey. The functional series,truncated to second order, can be represented



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(Muftuoglu, 1984;Muftuoglu, 1991) as:

Qmm

i 1

P ihim

i 1

m

j i

P iP jhij 8

in which Q is the runoff, P is the effective rain-fall, h i and h ij are the rst and second orderresponse function ordinates respectively and m isthe memory length. It is to be noted that the orderof the response function ordinates are depicted inopposite direction, for the sake of easy representa-tion. When the second order term is neglected, itreduces to the linear model. Muftuoglu (1984)modied the second order functional series modelby incorporating the behaviour of physicalprocesses, namely, that the non-linear effect existsin the immediate output and the linear effect persistsin the delayed output. The modied form can bewritten as:

Qml

i 1P ihi

n

i 1

n

j i

P i lP j lh y 9

in which l and n represent the linear and non-linearmemory lengths, respectively. It may be noted that

the linear part starts after the non-linear part endsand the total memory length of the catchment isequal to the sum of the non-linear and linearmemory lengths (i.e. m l n). To obtain theresponse functions, the normal least-squares equa-tions are derived by minimising the sum of thesquared deviations between the predicted andobserved runoff and these linear normal equationsare solved using the lowerupper triangle (LU)decomposition method (Press et al., 1993). Muftuoglu(1991) assumed that all the response function ordi-nates are positive. In the case of occurrence of nega-tive ordinates, normal equations are created with anextra number of response functions (in addition to thenegative ordinates) and these extra response functionsare minimised using the linear programming (LP)technique (Muftuoglu, 1991). However, the secondorder response function ordinates can take negativevalues (Singh, 1988). Hence, only the linear responseordinates are constrained to be positive in the presentstudy. The Muftuoglu modication of the FSM isreferred to in this study as the MFSM.

4. Data used

4.1. Data from the river Lee (in the UK)

The data from the river Lee in the UK (Muftuoglu,1991) and the Thuthapuzha river, Kerala, India havebeen utilised for the present study. The River Lee hasa catchment area of 1419 km 2, on which snowfalls arerare and negligible. Another distinct feature of thisbasin is that the river is continuously fed by thegroundwater aquifers. The effective rainfall andrunoff data are originally available in the form of graphs in Muftuoglus paper (Muftuoglu, 1991).These graphs have been blown up to a larger scaleand they have been digitised by the present authors,

using a digitiser. In order to check the validity of thedigitised data, the results of the Muftuoglus paper(Muftuoglu, 1991) are reproduced. The resultsmatched well (with a maximum discrepancy of the order of 1 mm depth of runoff). This resultproves the accuracy of digitisation of the data. Theperennial nature of the river and the availability of long data length accounted for to its selection inthe present study as it gives the opportunity tostudy the effect of scarcity of the data by usingdifferent independent smaller periods of the data for

calibration.

4.2. Data from the Thuthapuzha river

The Thuthapuzha river, which is a tributary of the Bharathapuzha river basin in Kerala, has acatchment area of 1030 km 2 and is situated in atropical region. The main criteria used in theselection of this basin are the data availability andthe uncontrolled nature of ow. Moreover, theThuthapuzha river, unlike the River Lee, is an ephem-eral river and hence the testing of the models in ahydrologically different situation, which is desirable,becomes possible. Thus, it ideally suits the purposesof the present study. For the Thuthapuzha river, theow data are available for one gauging station and therainfall data for three stations. Of the three raingauging stations, one lies just outside the boundaryof the basin. Owing to the lack of rainfall data in thelower reach of the river, data from this external stationare also utilised.



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4.2.1. Determination of effective rainfall and memorylength

Nine years of rainfall and runoff data (1984 to1992) are available for this basin. The rainfall datawere missing for two of the stations for one yeareach. The missing data are estimated using a normalratio method (Tung, 1983). The areal averaged preci-pitation is computed using the Theisson polygonmethod (Singh and Chowdhury, 1986). Climatologi-cal data required for the computation of potentialevapotranspiration (PET) are not available at a stationwithin the basin, but such data are available for astation located very close to the boundary, fromwhich station the rainfall data is also utilised. Theavailable PET data also spans over the same period

of 9 years (1984 to 1992). The PET is computed usingthe Thornthwaite method (Jain and Sinai, 1985;Singh, 1989). This method is suited to the basinbecause the basin is located in a highly humid region.Thus, it is assumed that the moisture is available tosatisfy the actual evapotranspiration at the PET rate.The effective rainfall is set to zero if the rainfall is lessthan PET. It is also assumed that the replenishment of the moisture takes place in the months of June andJuly at a rate proportional to that of the rainfall, as themajor portion of rainfall occurs during this period and

is preceded by a dry season. Hence, the differencebetween the dry season evapotranspiration demandand the actual rainfall is accumulated and is reducedfrom the rst two months of the rainy season (Juneand July) in proportion to the rainfall of these months,representing the replenishment process of moisture.

The memory length is derived from the cross corre-logram of rainfall and runoff. It is assumed that thenon-linear memory does not exceed two months. Byconducting different trial, it is found that the memorylengths of 3 and 2 months are suitable for the linearand non-linear parts respectively of the MFSM.Hence, a total memory length of 5 months is used.

5. Performance criteria

The performance of a model can be evaluated interms of several characteristics. The three importantcharacteristics of a good model are accuracy, consis-tency and versatility (Kachroo, 1992). The term(model) accuracy refers to the ability of the

model to reduce the calibration (i.e., the model output)error, to the observed ows of the calibration period.In contrast, (model) consistency is used for repre-senting the characteristics of the model whereby thelevel of accuracy and the estimate of the parametervalues persist through different samples of the data. Aversatile model is dened as the model which is accu-rate and consistent when used for the diverse applica-tions involving model evaluation criteria not directlybased on the objective function used during the cali-bration of the model (Kachroo, 1992). In the presentinvestigation, of the three characteristics, only the rsttwo characteristics are considered.

5.1. Model evaluation criteria

A variety of verication criteria which could beused for the evaluation and inter-comparison of differ-ent models were proposed by World MeteorologicalOrganisation (WMO) and other investigators (WMO,1975; Nash and Sutcliffe, 1970; Aitken, 1973;Kachroo, 1992). They are grouped as graphical andthe numerical performance indicators. WMO haslisted four indicators as graphical, of which, thoseindicators suited to the present objective are as listedas follows:

1. A linear scale plot of the simulated and observedhydrograph for both calibration and the vericationperiods.

2. Double mass plots of the simulated and observedows for the verication period.

3. A scatter plot of the simulated versus observedows for the verication period.

Of the several numerical indicators (WMO, 1975),suitable ones for the present study are chosen. Theyare the root mean square error (RMSE) and the R2

efciency (Nash and Sutcliffe, 1970). These aregiven by Eqs. 1013.

RMSE

K k 1

Qk ^

Qk 2

K

vuuuut 10

R2F o F

F o11



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T a b l e 1

P e r f o r m a n c e o f t h e T B P - N N m o d e l a n d f u n c t i o n a l s e r i e s m o d e l s i n t e r m s o f R o o t m e a n s q u a r e e r r o r ( R M S E ) i n m m ( R i v e r L e e )

C a s e s

M o d e l 3

P e r i o d 5

L i n e a r m o d e l ( L M l i n e a r

m e m o r y l e n g t h i s 7 )

F S M ( l i n e a r a n d n o n - l i n e a r

m e m o r y l e n g t h s a r e 5 )

M F S M ( l i n e a r a n d n o n - l i n e a r

m e m o r y l e n g t h s a r e 2 a n d 7

r e s p e c t i v e l y )

T B P - N N m o d e l

( m e m o r y l e n g t h s

i n l a y e r s 1 a n d 2

a r e 7 a n d 2


C a l i b r a t i o n p e r i o d

V e r i c a t i o n P e r i o d

( a ) a

( b ) b

( a )

( b )

( a )

( b )

( a )

( b )

i

0 1 7 4

1 7 5 3 4 8

4 0 . 0

3 1 . 7

2 1 . 8

3 0 . 9

2 1 . 5

2 7 . 7

2 1 . 4

2 3 . 5

i i

1 7 5 3 4 8

0 1 7 4

3 0 . 0

4 1 . 3

2 4 . 1

2 9 . 6

2 3 . 2

2 7 . 7

2 4 . 7

2 1 . 4

i i i

0 3 6

1 7 5 3 4 8

3 6 . 3

3 5 . 5

1 9 . 3

5 4 . 5

1 2 . 2

2 7 . 9

1 6 . 3

2 2 . 6

i v

3 7 8

4

1 7 5 3 4 8

2 4 . 4

3 1 . 1

1 4 . 9

3 0 . 8

1 9 . 5

2 6 . 4

1 4 . 9

2 4 . 7

v

8 5 1

3 2

1 7 5 3 4 8

4 2 . 9

3 7 . 7

1 3 . 7

4 4 . 3

1 9 . 6

3 1 . 2

1 8 . 3

2 6 . 2

v i

1 3 3 1 7 4

1 7 5 3 4 8

4 0 . 6

4 3 . 4

1 2 . 2

4 5 . 2

1 8 . 9

3 4 . 9

2 3 . 7

2 6 . 6

v i i

1 7 5 2 2 8

0 1 7 4

2 0 . 9

4 2 . 9

1 8 . 9

3 8 . 7

2 1 . 9

3 3 . 6

2 0 . 6

2 9 . 2

v i i i

2 2 9 2 8 8

0 1 7 4

2 2 . 0

4 1 . 4

1 6 . 9

4 1 . 2

2 0 . 5

3 0 . 8

2 0 . 8

2 8 . 7

i x

2 8 9 3 4 8

0 1 7 4

3 0 . 4

4 2 . 5

2 2 . 9

3 7 . 0

2 0 . 9

2 5 . 7

2 2 . 6

2 3 . 8

a

C a l i b r a t i o n R M S E

.

b

V e r i c a t i o n R M S E

.

T a b l e 2

P e r f o r m a n c e o f t h e T B P - N N m o d e l a n d f u n c t i o n a l s e r i e s m o d e l s i n t e r m s o f t h e R

2

m o d e l e f c i e n c y i n d e x ( R i v e r L e e )

C a s e s M o d e l 3

P e r i o d 5

R u n o f f s t a t i s t i c s

o f c a l i b r a t i o n

d a t a ( m m )

L i n e a r m

o d e l

( l i n e a r

m e m o r y

l e n g t h i s 7 )

F S M ( l i n e a r & n o n - l i n e a r

m e m o r y l e n g t h s a r e 5 )

M F S M ( m e m o r y

l e n g t h s a r e 2 a n d 7


T B P - N N m o d e l ( m e m o r y l e n g t h s

a r e 7 a n d 2 r e s p e c t i v e l y )

C a l i b r a t i o n p e r i o d

V e r i c a t i o n p e r i o d M e a n

S D c

( a ) a

( b ) b

( a )

( b )

( a )

( b )

( a )

( b )

i

0 1 7 4

1 7 5 3 4 8

1 0 4 . 3 2

9 5 . 2

4 0 . 8 3 2 0 . 8 4 6 0 . 9 5 3

0 . 8 5 0

0 . 9 5 1

0 . 8 8 2

0 . 9 5 3

0 . 9 0 6

i i

1 7 5 3 4 8

0 1 7 4

1 1 2 . 8 5

7 9 . 9

2 0 . 8 6 0 0 . 8 2 2 0 . 9 0 8

0 . 9 1 5

0 . 9 1 6

0 . 9 2 0

0 . 9 0 5

0 . 9 5 2

i i i

0 3 6

1 7 5 3 4 8

1 1 4 . 7 7

9 8 . 8

9 0 . 8 7 2 0 . 8 0 9 0 . 9 7 3

0 . 5 7 8

0 . 9 8 5

0 . 8 7 8

0 . 9 7 4

0 . 9 2 0

i v

3 7 8

4

1 7 5 3 4 8

1 0 9 . 2 9

8 0 . 7

8 0 . 9 1 0 0 . 8 5 0 0 . 9 7 1

0 . 8 5 1

0 . 9 4 3

0 . 8 9 0

0 . 9 6 7

0 . 9 0 5

v

8 5 1

3 2

1 7 5 3 4 8

8 9 . 9

5

1 0 2 . 1 3 0 . 8 1 9 0 . 8 1 0 0 . 9 8 0

0 . 7 2 0

0 . 9 6 4

0 . 8 5 9

0 . 9 6 7

0 . 9 0 2

v i

1 3 3 1 7 4

1 7 5 3 4 8

9 3 . 7

0

9 3 . 3

3 0 . 8 4 6 0 . 7 2 6 0 . 9 8 5

0 . 7 0 2

0 . 9 6 8

0 . 8 1 9

0 . 9 3 2

0 . 8 9 7

v i i

1 7 5 2 2 8

0 1 7 4

1 1 7 . 2 3

6 7 . 6

6 0 . 9 0 5 0 . 8 0 8 0 . 9 2 2

0 . 8 5 5

0 . 8 9 6

0 . 8 8 1

0 . 8 9 5

0 . 8 6 7

v i i i

2 2 9 2 8 8

0 1 7 4

1 1 1 . 1 9

6 9 . 1

1 0 . 8 9 9 0 . 8 2 0 0 . 9 3 8

0 . 8 3 5

0 . 9 1 4

0 . 9 0 0

0 . 9 1 0

0 . 9 1 4

i x

2 8 9 3 4 8

0 1 7 4

8 7 . 0

3

6 3 . 3

7 0 . 7 7 5 0 . 8 1 3 0 . 8 9 5

0 . 8 6 9

0 . 8 9 3

0 . 9 3 2

0 . 8 7 4

0 . 9 4 2

a

C a l i b r a t i o n R

2

e f c i e n c y .

b

V e r i c a t i o n R

2

e f c i e n c y .

c

S t a n d a r d d e v i a t i o n


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in which

F oK

k 1

Qk Q 2 12

F K

k 1Qk

^

Qk h i2 13

k is the dummy time variable for runoff; K is the umberof data elements in the period for which the computa-tions are to be made, Qk and

^

Qk are the observed andthe computed runoffs at the k th time interval respec-tively, and Q is the mean value of the runoff for thecalibration period (Kachroo, 1992).

Though the RMSE can indicate the relative perfor-mance of different models for the same lengths of calibration periods and verication periods, it cannot really indicate the performance of the modelsfor different calibration record lengths. The R2

model efciency criterion is a better choice in sucha situation. However, the RMSE can give a quantita-tive indication of the model error in terms of a dimen-sioned quantity. Hence, the RMSE is also presentedalong with the R2 efciency. The ideal value of theRMSE is zero, in which case the value the R2 modelefciency index is unity.

6. Results and discussion

In the present study, the emphasis has been directedtowards (i) comparing the temporal back-propagationneural network (TBP-NN) model with similar black-box models and (ii) evaluating the performance of these models when the calibration data length issmall. As there is no analytical procedure available,other than the standard model performance indicatorssuch as RMSE and R2, for evaluating the suitability of a model for operational purposes, the only way is tocompare the values of the performance indices for themodel with those of established alternative models,having similar data and operational requirements.Hence, three black-box models, namely, the linearmodel (LM), the second order functional seriesmodel (FSM) and the Muftuoglu form of functionalseries model (MFSM; Muftuoglu, 1991) are chosenfor comparison with the TBP-NN model.

6.1. Application of the TBP-NN and the three selected models to the river Lee

The data from the reference (Muftuoglu, 1991)comprise 29 years of monthly effective rainfall andrunoff. In this study, nine cases are analysed, as shownin Tables 1 and 2. The calibration and vericationperiods are indicated in columns 2 and 3 respectivelyof these tables. The months are listed serially startingfrom October 1936 (as the rst month). The perfor-mance of all the four models, in terms of RMSEvalues and R2 efciency, are presented in Tables 1and 2 respectively. The mean and standard deviationof the calibration data are also given in columns 4 and5 of Table 2. The mean and standard deviation for

each verication period can also be obtained bychoosing the appropriate case from Table 2. Therst two cases are considered in order to comparethe results of the proposed TBP-NN model withthose obtained by Muftuoglu (1991). In case (i), therst half of the 29 years of the data is used for cali-brating the models and the second half is used forverifying and vice versa in case (ii).

In order to study the effect of different independentsets of calibration data of nearly the same but shorterlengths (i.e. varying between 3 and 5 years) on the

performance of the models, seven cases, namely,cases (iii)(ix) inclusive, are investigated. In cases(iii) to (vi), the calibration periods are sub-sets of the initial half of the data period whereas the verica-tion data period is the entire latter half of the data, i.e.,the same verication period as in case (i). In contrast,in cases (vii) to (ix), the calibration periods are sub-sets of second half of the data and the verication dataperiod is the entire rst half, i.e., the same vericationperiod as used in case (ii). Thus, the verication datalength of cases (iii) to (ix) is maintained at 14.5 yearsin order to facilitate the comparison with the results of cases (i) and (ii). The calibration period lengths forcases (iii) to (ix) are 3, 4, 4, 4.5, 4.5, 5, and 5 yearsrespectively. It is also noted that, in Table 2, the veri-cation values for cases (i), (iii), (iv), (v) and (vi) arethe calibration values for case (ii) and that the veri-cation values for cases (ii), (viii) and (ix) are the cali-bration values of case (i).

Thus, it may be seen from Tables 1 and 2 that therst half to the data is split into four calibration setsand the second half into three. Attempts were made to



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split the data in the second half into four calibrationsets (as had been done in the rst half of the data set),but it resulted in high verication errors for all themodels. As all the four models (LM, FSM, MFSM,TBP-NN model) behaved similarly in that scenario, itindicates that the information available in a periodless than 4.5 years of the second half of the entiredata set may not be sufcient to calibrate the modelsproperly. This behaviour emphasises the signicanceof data length on model calibration.

However, it cannot be concluded that a longer cali-bration period necessarily results in better modelperformance. This may be evident from the compar-ison of the verication error for case (iv) for all themodels with those of case (i). The verication RMSE

of all the models for this set of calibration data arecomparable with those of cases (i), though the calibra-tion period for case (iv) is only 4 years as against 14.5years for case (i). The R2 efciency values (Table 2)also conrm that verication errors are almost equalor slightly less than that of case (i). This may beattributed to two reasons, namely, (i) the data usedfor calibration may contain sufcient informationcovering the entire data space or/and (ii) this part of the data may be subjected to smaller amount of dataerrors compared to the remaining part of the data.

While computing the RMSE and the R2

efciencyof the calibration and verication periods, the errorscorresponding to the warm-up period of themodels (equal to the total memory length) for bothcalibration and verication have been excluded forall the cases, so that the effect of error in the assumedinitial conditions dies out. In fact, the initial condi-tions for the verication periods of cases (iii) to (vi)are known as these periods are later than the calibra-tion periods, but this has not been considered in orderto have consistency for comparison purposes and tomaintain the exact chosen calibration lengths. Such anexclusion for all the cases has been adopted to suit theeld situation where the available calibration data andthe data for verication may not be a part of a contin-uous record, that is, chronologically the vericationdata may not lie immediately after the calibrationperiod or vice versa. The performances of the chosenmodels are now discussed in the following sections.While comparing the corresponding values fromTables 1 and 2, it may be noted that the averagevalue used in the computation of the R2 model

efciency index is the average value of the calibrationdata.

6.1.1. Linear model (LM)The results reported by Muftuoglu (1991) indicate

that the optimum performance of the LM wasobtained with a memory length of 7 among theemployed memory lengths (7, 8, and 9). However,the possibility of using smaller lengths of memorywas explored by the authors, but it was found thatthe memory length of seven months is suitable forthe Lee basin. Therefore, a total memory length of 7months was used for the LM in this study to obtain theoptimum performance of the model. The model wasrun for all the nine cases indicated earlier and the

response functions were obtained. Some of the ordi-nates of the response functions corresponding to cases(v) and (vi) were found to have negative values andhence the linear programming technique, as detailedearlier, is used to obtain physically realisableordinates.

It may be seen that the verication errors of the LMare, in general, increased as the calibration data lengthis reduced, but the increase is only marginal. Incontrast, in cases (iv), a lesser verication error thanthat of cases (i) for the LM has resulted, for which the

verication data are the same. This could also be seenfrom Table 2 that case (iv) gives the highest R2 ef-ciency (even when compared to case (i)). This beha-viour emphasises that a mere increase in thecalibration data length will not necessarily increasethe accuracy and consistency of the model. Incontrast, the R2 efciency of calibration of case (ix)is less than that of all other cases, but the correspond-ing RMSE value is not the highest. This behaviour isattributed to the lowest mean value of runoff obtainedfor that calibration period (Table 2). Hence, the modelerror (mist of the computed runoff with observedrunoff) is more, though the RMSE value is not thehighest. Thus, R2 efciency is a better indicator thanRMSE to represent the model error, if the calibrationor verication periods are different. However, if thecalibration and verication periods are same fordifferent models, (as in the case of a row in Table1), then it can indicate the relative model perfor-mance. The case (vi) gives the most inconsistent resultfor the LM, in the sense that it has the highest veri-cation error, both in terms of RMSE and R2 efciency.



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The reason for such a behaviour may be clear, oncethe analysis of all the models are complete. In general,the LM provided consistent results, though its accu-racy is less compared to other models.

6.1.2. Second order functional series (FSM)In the case of the functional series model, the mini-

mum error reported by Muftuoglu (1991) corre-sponded to a total memory length of 7 months. The

possibility of lesser error with a smaller number of ordinates had not been explored by him. Such anumerical experiment, with memory length varyingfrom 3 to 7 months, was conducted by the presentauthors and the results are presented in Table 3.Table 3 indicates the performance of the FSM withrespect to different memory lengths in terms of theRMSE. The memory lengths are indicated at the toprow. As the aim is to examine the effect of varying thememory length, the RMSE is sufcient to indicate theperformance because the performance is compared fordifferent memory lengths for each case and therebyfor the same calibration and verication periods. Itmay be observed from Table 3 that verication errorin terms of RMSE decreases for each case with lesscalibration data (cases (iii) to (ix)) as the memorylength increases up to ve and then it increases.Another interesting feature experienced is that nega-tive ordinates appear in the linear part of the responsefunction when the memory length is greater than vemonths. Negative ordinates in the linear part of themodel are not physically realisable but, in contrast,

the non-linear (second order) response functions of Eq. (8) can give rise to negative ordinates (Singh,1988). In addition, the model is relatively stable (i.e.consistent) with the memory length ve, whencompared to higher memory lengths. It is experiencedthat the use of even higher memory lengths, i.e.greater than seven months, result in a reduction of the calibration error to a very small value and asharp increase in the verication errors (not shown

in Table 3). This trend is also seen for cases (iii),(vi) and (vii) and even at the memory length of seven months (Table 3). This is attributed to theover-adaptability of the functional series model, i.e.,the model memorises the calibration data rather thancapturing the general relationship.

The best outcome of the functional series model,using a total memory length of ve months, is repro-duced in Table 1 for the purpose of comparison. Thereis a small difference in the RMSE obtained for cases(i) and (ii) between this study and that of Muftuoglu(1991) and this is as a result of different memorylengths used (5 instead of 7). As can be seen fromTable 1, the calibration error is less than the verica-tion error, indicating a lack of consistency in predic-tion by the FSM for the case of smaller calibrationlengths. This is clear from the R2 efciency valuesgiven in Table 2, which shows a larger differencebetween the calibration and verication R2 efcien-cies. As mentioned earlier, this lack of consistency islargely owing to the over-adaptability (i.e., over-para-meterisation) of the model. The verication error has


Table 3RMSE of the computed runoff (in mm) of the functional series model (FSM) with different memory lengths (River Lee)

Memory length in months used for linear and non-linear part3 4 5 6 7

Cases (a) a (b) b (a) (b) (a) (b) (a) (b) (a) (b)I 38.1 43.1 29.3 36.2 21.8 30.9 19.3 29.5 18.0 28.8ii 38.1 42.6 30.1 35.1 24.1 29.6 20.6 29.8 19.0 29.9iii 42.4 43.4 28.0 61.5 19.3 54.5 10.7 18.3 0.1 527.0iv 37.8 41.0 25.7 34.0 14.9 30.8 12.6 27.6 7.2 37.6v 29.6 47.1 18.0 51.5 13.7 44.3 06.8 62.2 1.5 73.5vi 22.6 51.5 14.8 45.2 12.2 45.2 8.20 68.5 4.2 398.0vii 29.9 59.0 22.5 48.4 18.9 38.7 14.0 48.6 9.5 67.6viii 34.0 48.4 24.4 41.2 16.9 41.2 10.6 45.0 8.6 46.9ix 42.0 43.2 29.4 48.3 22.9 37.0 18.0 43.7 13.9 47.4

a Calibration RMSE.b Verication RMSE.


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increased considerably for cases (iii) to (ix). A smaller

calibration length of data resulted in smaller calibra-tion error and corresponding higher verication errorfor this model. This indicates the lack of consistencyof the model for smaller calibration data lengths. i.e.,the over-adaptability is exposed more in the case of such smaller data lengths.

6.1.3. Modied functional series model (MFSM)This model gave consistency in prediction even for

the smaller calibration data lengths. Here the linear

and non-linear memory lengths are maintained as 7and 2 respectively, as advocated by Muftuoglu (1991),for the best performance of the model in case of thisdata set. It may be seen from the Tables 1 and 2 thatcomparable order of verication errors to those corre-sponding to cases (i) and (ii) are obtained for somecases (cases (iii) and (iv)) with smaller calibrationlengths. This behaviour is seen clearly on comparingthe performance of the MFSM in the different caseshaving a common verication period (i.e. its perfor-mance in cases (iii) to (vi) is compared with that incase (i) and that in cases (vii) to (ix) is compared withthat of case (ii). In case (iv), the error differencebetween calibration and verication is comparativelyhigher than that of the other cases. The vericationresults of all the other models for this particular caseare consistent with those obtained in calibrating theMFSM. It may be concluded that the lower data errorin the corresponding calibration data is the cause forthe above. It is also noted from Table 2 that the MFSMis accurate and consistent compared to the models sofar discussed.

6.1.4. Temporal back-propagation neural network

(TBP-NN) modelThe computer program for the temporal back-

propagation neural network, developed for the presentstudy, is capable of handling any realistic number of hidden layers, neurons and delays for each layer. Themodel was tested with different numbers of hiddenlayers for case (i), but the error reduction was notappreciable even when the number of hidden layerswas increased (up to 3). Normally, one or two hiddenlayers are adopted. In contrast, the training (calibra-tion) process slows down drastically by increasing the

depth (i.e., the number of hidden layers) of thenetwork. As it takes more time in learning for a neuralnetwork with more layers, without an appreciableincrease in the performance, a network architecturewith one hidden layer was chosen for the presentstudy. The optimum number of neurons is found tobe in the range of 5 to 7. Hence, with a view to parsi-mony, the number of neurons in the hidden layer ischosen as ve. The performance of this TBP-NNmodel is also given in Tables 1 and 2.

The learning rate h used for getting the best perfor-mance for the nine cases (Table 1) considered areshown in Table 4. Owing to the reason detailed inSection 2.3, the input and output series are scaleddown to a range of FMIN and FMAX. The values of FMIN and FMAX and used for the nine cases are alsogiven in Table 4. The optimum values of FMIN andFMAX are 0.05 and 0.95 respectively. However, thereare some exceptions, as in cases (v) and (vi). It isinteresting to see that the larger value of FMIN usedfor cases (v) and (vi) correspond to the dry spell of theclimate (see Table 2 for the mean values) and hence a


Table 4Learning rate h , FMIN and FMAX for the best performance of the TBP-NN model for the data from the River Lee

Cases Learning rate(n) FMIN (minimum factor) FMAX (maximum factor) Remark

i 0.9 0.05 0.95ii 0.9 0.05 0.95iii 0.9 0.10 0.90iv 0.7 0.05 0.95v 0.9 0.30 0.90 Dry spellvi 0.9 0.30 0.90 Dry spellvii 0.5 0.05 0.95viii 0.5 0.05 0.95ix 0.5 0.05 0.95 Dry spell but less SD a

a SD Standard deviation.


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larger number of data will be nearer to being zeroows and zero rainfall. This is consistent with thetheory presented in Section 2.3 regarding the slowingdown of the learning process in the case of valuesbeing in the neighbourhood of zero and one. However,case (ix) also falls in the dry spell of the climateexcept for the rst year, but 0.05 and 0.95 are foundto be the optimum values for this case. This may becaused by the smaller standard deviation for the calibra-tion data (Table 2) and hence a lesser spread of data.

The input information required for the calibrationand operation of the neural network consists of thedata pertaining to the structure of the network (i.e.the number of layers, the number of neurons in eachlayer, the delay to be used in each layer, etc.), the

parameter(s) which control(s) the learning process(learning rate), the parameters (FMIN and FMAX)for scaling down, details regarding the calibrationand verication data (length of calibration and veri-cation data) and the input and output series of calibra-tion and verication data. The output of theverication period is only used for computing theperformance indicators. In the present study, there isonly one input and one output and, hence, the inputlayer and the output layer contain only one neuroneach. It should be noted that the input to the network

is the rescaled effective rainfall and the output is therescaled runoff. As indicated before, one hidden layerwith ve neurons is used for this study. The sum of delays used will be equal to the total memory lengthof the catchment (e.g., the delays used for the Thutha-puzha basin between the rst and second layers andthe second and third layers are 3 and 2 respectively).

It may be observed from Table 1 that there is gener-ally an increase in the verication error compared tothe calibration error for most of the cases considered.However, the R2 efciency indicates that vericationerror is less, compared to the calibration error, forcases (ii) and (ix). The calibration RMSE for case(ix) is obtained as smaller than that of verication(though the actual t is not good), as the mean forthe calibration data is smaller than that of verication(87.03 mm as against 104.32 mm). As mentionedearlier, the R2 efciency is a better indicator thanthe RMSE in the case of different calibration datalengths. On comparing the verication results of cases (iii) to (vi) with case (i), all having the secondhalf of the entire data set as the verication period, it

is notable that the R2 value for case (iii) is marginallybetter than that of the case (i), while those of cases (iv)to (vi) are not signicantly different from that of case(i). Similarly, on comparing the results of cases (vii)to (ix) with case (ii), all of which have the rst half of the total set as the verication period, it is seen that theverication performance of case (ix) is marginallybetter than that of case (ii). Such improved verica-tion results for cases (iii) and (ix) are not unusual insuch modelling exercises.

In general, these results indicate that the TBP-NNmodel can provide the same level of accuracy withshorter lengths of calibration data. The calibrationdata lengths used in this study vary from 3 to 5years. Therefore, using the TBP-NN model, a calibra-

tion data length of 6 years (to be on conservative side),is considered sufcient for the monthly ow predic-tion for the River Lee. Hence, it may be concludedthat TBP-NN model gives accurate and reasonablyconsistent results for the River Lee data.

6.1.5. Comparison of the performance of the selected models on the River Lee data

For comparing the models, the qualitative term performance is evaluated based on two criteria,namely, accuracy and consistency. While comparing

the results, it is to be kept in mind that data lengths forcalibration and verication are not the same except forcases (i) and (ii). However, the verication data lengthis maintained the same for the sake of comparison. Infact, case (i) and cases (iii) to (vi) have same verica-tion data and similarly, case (ii) and cases (vii) to (ix)have the same verication data. It may be seen fromTable 1 that the optimum performance for the Leecatchment is obtained for TBP-NN model and theworst for the FSM. The models can be arranged, inascending order of performance in a data scarce situa-tion, as the FSM, the LM, the MFSM and the TBP-NNmodel. The performance of the LM is close to that of the FSM and likewise the performance the MFSM isclose to those of the TBP-NN model.

In a scarce data situation, the LM and the FSMare not suitable models as the LM is incapable of capturing the non-linear information while the FSMis over-adaptable. On comparing the performancesbetween the MFSM and the TBP-NN model, the veri-cation error of the TBP-NN model is consistentlyless than that of the MFSM proposed by Muftuoglu



16/24

(1984), Muftuoglu (1991)). In terms of accuracy andconsistency, the overall performance of the TBP-NNmodel is better than the other models. This conclusionis also obvious from the R2 efciency values (Table

2). However, there is a slightly larger differencebetween the verication errors and calibration errorsfor certain cases, but these verication errors are stillless than those of the other models.


Fig. 3. Computed runoff for the verication data using MFSM and TBP-NN and the observed runoff of case (i) for River Lee.


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In order to substantiate the discussion before, plotsof observed and computed runoff hydrographs of theMFSM and the TBP-NN model for a typical case (e.g.case (i)) are shown in Fig.3. The computed hydro-graphs using the LM and the FSM are not shown astheir performances are far worse. The scatter plot of computed versus observed runoff are also shown inFig. 4(a) for the MFSM. Similarly the scatter plot forthe TBP-NN model is given in Fig. 4(b). The scatterplot is well spread over the ideal line in the case of theTBP-NN model, whereas it is shifted to one side in thecase of the MFSM. This shift from the ideal line indi-cates the possibility of systematic errors (Aitken,

1973). The corresponding mass curve plots of theobserved and the computed runoffs are shown inFig. 5. The mass curve of the computed runoff usingthe TBP-NN model is nearer to the observed runoff than of that the MFSM. The plots show the verica-tion results of case (i) for the river Lee. These guresconrm that the TBP-NN model performs better thanthe MFSM on this catchment.

6.2. Application of the TBP-NN model and the other three selected models to the Thuthapuzha basin

As stated earlier, the data from the Thuthapuzhariver comprise of nine years data, of which the initialsix years are used for calibration and the rest for veri-

cation. The areal averaged precipitation is computedas described earlier. The plot of the areal averagedrainfall and the observed runoff, as shown in Fig. 6,reveals the non-linear nature of the rainfallrunoff relationship of this basin. Prominent non-linearity isevident in the fth year, which is preceded by a rela-tively dry year and also in the sixth year. The perfor-mance of the selected models also indicated thisprominent non-linear nature of the ow generationprocess by the resulting high verication error if thesixth year (year 1989) of data is not included in the

calibration of the model. In contrast, inclusion of thisyear of the data for calibration has resulted in theincrease of the calibration error in general and in thecalibration of the simple LM in particular. Anotherreason which may explain at least some of this devia-tion is the missing data in the record of the previousyear at one of the stations, even though they are esti-mated using the normal ratio method for use in thecomputation of the areal averaged rainfall.

The performances of the models, in terms of theRMSE and the R2 efciency values for the Thutha-puzha, are given in Tables 5 and 6 respectively.Columns 2 and 3 in Tables 5 and 6 gives the RMSEand R2 efciency values respectively for the calibra-tion and verication periods when PET information isused. In order to check the validity of the assumptionregarding the distribution of the backlog of the accu-mulated decit in rainfall (i.e., accumulated differenceof rainfall and PET in the dry season), a numericalexperiment is carried out with one, two and threemonths distributions. It is found that the backlogdistribution with two months gives better results


Fig. 4. (a) Scatter plot of computed runoff using MFSM againstobserved runofffor verication data of case (i) for the River Lee. (b)Scatter plot of computed runoff using TBP-NN against observedrunoff for the verication data of case (i) for the River Lee.


18/24


Fig. 5. Mass curves of the computed runoffs using MFSM and TBP-NN with observed runoff.

Fig. 6. Monthly areal averaged rainfall and runoff of the Thuthapuzha river basin.


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compared to the one and three months distributions.This is because of the fact that, for most of the years of

record, about 50% of the rainfall is received in thesetwo months.

On consideration of the performance of the LM, thecalibration error is higher than the verication error of the LM, indicating consistency in the result. (The termerror, rather than RMSE or R2 efciency, is usedhere as the RMSE and the R2 efciency values givethe same trend of result. The following discussionpertains to both Tables 5 and 6). However, the accu-racy of the LM is less compared to the other modelsexcept for the case of the MFSM. Surprisingly, for the

Thuthapuza basin, the MFSM gives the lowest perfor-mance in terms of accuracy and consistency. TheMFSM model structure, whereby the linear termsstart after the non-linear terms, led to the least perfor-mance in this case. As the total memory length used isthe same, the MFSM has therefore resulted in a smal-ler number of the linear component ordinates than thatof the LM (see Eqs. (8) and (9)). The effect of thisconstraint of adopting the same total memory length

for the MFSM as used for the LM may not be sostrong in the case of a perennial river, such as the

River Lee, where the total memory length of thecatchment is relatively larger. A total memory lengthgreater than ve months (total memory length of vemonths is presently used), in order to abandon theaforementioned constraint, is not used because theow in the river reaches a value in the neighbourhoodof zero, once the rainfalls of the months continuouslybecome zero, within a maximum of ve months. Thisbehaviour of the river indicates that the total memorylength is less than or equal to ve months and henceusing a larger memory length, for the sake of suiting

the model structure, is not appropriate as it is notphysically realisable.The FSM model yields better accuracy compared to

the other models, though the consistency of the modelperformance is less than that of the LM and of theTBP-NN model. The verication errors for all themodels except for the FSM are reduced, owing tothe reason mentioned at the beginning of the sectionin discussing the prominent non-linearity of the ow


Table 5Comparison of performances of different models based on the RMSE in mm, for the data from the Thuthapuzha river

RMSE error in (mm)

When PET information (with backlog distributedto June and July) is used

When No PET information is used

Model Calibration RMSE Verication RMSE Calibration RMSE Verication RMSE

LM 63.4 52.6 51.8 52.5FSM 46.0 53.6 44.2 62.2MFSM 78.0 59.1 55.3 43.4TBP-NN model 54.6 45.9 42.6 42.8

Table 6Comparison of performances of different models in terms of the R2model efciency index, for the data from the Thuthapuzha river

The R2model efciency index

When PET information (with backlogdistributed to June and July) is used

When No PET information is used

Model Calibration R2 Verication R2 Calibration R2 Verication R2

LM 0.606 0.697 0.737 0.698FSM 0.792 0.685 0.809 0.576MSFSM 0.404 0.617 0.700 0.794TBP-NN model 0.707 0.769 0.822 0.799


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season is not sufcient to satisfy the PET requirement.Hence, the ow in the river is almost entirelysustained by the base ow during the summer season,though there are months with the no ow condi-tion. Moreover, the monthly PET values computed forthe months of the rainy seasons are generally verysmall values while those of the dry season are veryhigh. Hence, it is perhaps reasonable to assume that

the monthly total rainfall is adequate for predictingthe ow from this river. In all of these trials, wherePET data are not used, the TBP-NN model gives thebest performance in terms of both accuracy andconsistency. Unlike the previous case (when PETinformation is used), the MFSM gives the secondbest performance. The bad performance of theMFSM in the previous case where PET informationis used may also be because of the larger number of zero rainfall values appearing in the effective rainfallseries, owing to higher PET than total rainfall, whichin turn has resulted in the apparent suppression of thelinear part of the mode.

Graphical indicators, for the case where no PETinformation is used, are provided in Figs. 79. In thesegures, the performance of the TBP-NN model iscomparedwith thatof the MFSMasthe MFSMprovidesthe second best performance when no PET informa-tion is used. Fig. 7 depicts the plots of the observedrunoff and the computed runoffs, using the TBP-NNmodel and the MFSM, as functions of the time, inmonths, marked serially from 1 to 108 with the rst

month as January 1984. The calibration and vericationperiods are marked on the gure itself. The computedrunoff using TBP-NN model tracks the observed runoff better than that of the MFSM. Fig. 8 shows the scatterplot of the computed runoff using the TBP-NN modeland the MFSM against the observed ows for theverication period. The scatter plot indicates that thespread in the values computed using the TBP-NN

model is less than that using the MFSM. Fig. 9 indi-cates the mass curves of the computed runoffs usingthe TBP-NN model and the MFSM along with that of the observed runoff for the verication period. It isseen that the computed mass curve of the TBP-NNmodel closely tracks that of the observed data.

The evaluation of the performance of the differentmodels indicates that the TBP-NN model could effec-tively be used as a non-linear rainfallrunoff black-box model for the Thuthapuzha catchment also. Asthe behaviour of the TBP-NN model is the bestamong the compared models for the two hydrologi-cally different (perennial and ephemeral) test basinslocated in different parts of the world, it might reason-ably be assumed that the TBP-NN model will behavein a similar manner for other basins.

7. Conclusions

ATBP-NN model, as a non-linear black-boxrainfallrunoff model, which can take care of the


Fig. 9. Computed and observed mass curves of the verication data for the Thuthapuzha river.


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variation of the runoff generation process with respectto time internally, is used in this study. The perfor-mance of the model is evaluated in a scarce datasituation by comparing its performance with that of the linear model and of the second order and theMuftuoglu modied form of second order functionalseries model. The following conclusions are drawnfrom this study:

1. On considering the different calibration data lengthsvarying from 3 to 5 years, it is established that theTBP-NN model can be calibrated using 5 or moreyears of the data with comparable accuracy andconsistency with that obtained using longer datalengths of about 15 years for the basin considered.

2. The MFSM also gave comparable accuracy andconsistency in the case of the River Lee, but failedto give this result in the case of the Thuthapuzhabasin when PET information is used. This may bebecause of the model structure of MSFM, wherebythe linear component starts only after the non-linear component ends, resulting in the linear partof the MFSM having a smaller number of ordinatesthan the LM. This model structure may be valid inthe case of a perennial river, but may not be so inan ephemeral river for which the memory length is

comparatively smaller. This is also caused by thelarge number of zero effective rainfall valuesappearing in the input series, owing to the PETbeing larger than the rainfall, which in turn hasresulted in the apparent absence or suppression of the linear part of the model. This line of thought isalso consistent with the superior performance of the MFSM when no PET information is usedwhere the number of zero rainfall values iscomparatively smaller.

3. The performance of the linear model in a scarcedata situation indicated that the model is consis-tent, but not as accurate as the other models (theFSM, the MFSM, and the TBP-NN model). Hence,it cannot be recommended for use in a basin withprominent non-linearity.

4. The performance of the FSM indicates that themodel is accurate in calibration, but is not consis-tent in scarce data situation. This is because of the over-parameterisation in the FSM. Hence, theFSM is not appropriate for use in a scarce datasituation.

5. Evapotranspiration has negligible effect in themonthly runoff prediction for the Thuthapuzhariver (located in a tropical area) using non-linearmodels such as the TBP-NN model and functionalseries models.

Perhaps a drawback in splitting the original calibra-tion period of cases (i) and (ii) for the River Lee datainto four and three respectively is that no wide rangeof calibration periods was tested, the proportion beingeither approximately a quarter or one third of theoriginal half of the total monthly data considered forcalibration purpose. However, this approach has atleast the advantage of independent calibrationsamples, though it was still restricted in the range of lengths tested.

In general, it is found that the TBP-NN modelperforms better than other three models in terms of accuracy and consistency for the case of two testbasins. Two test basins are used in the present studyas one can rarely draw rm conclusions from theresults of one catchment alone. However, the beha-viour of the test basins are hydrologically different,one river being perennial and the other being ephem-eral. Moreover, these basins are from the different partof the world. While these facts increase the efcacy of the results and conclusions of the study, it is of coursedesirable that further such studies be carried out for awider range of test catchments.

Acknowledgements

The authors wish to express their deep felt gratitudetowards the reviewers for their keen interest and criti-cal reviews, without which the paper would not havetaken its present form.

Appendix A. Training procedure for the TBP-NN Wan (1990) has derived two formulations for

weight adaptation for the FIR multilayer perceptronnetwork. The rst one uses a virtual neuron for eachdelay and can be thought as equivalent to the commontechnique of viewing the structure unfolded in time.But this structure loses the symmetry between forwardand backward propagation and the desired nature of distributed gradient computation. The loss of symme-try results in the lack of a general recursive equation



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for weight adaptation for arbitrary layers. In contrast,the second formulation alleviates all these drawbacks.Using this formulation, the weight adaptation recur-sive equation can be written as

W lij n 1 W lij n hd

l 1 j k X

li k A : 1

d l j k

e j k f H y L j k l L

f H yl j k N l 1

m 1Dl 1m k W

l 1 jm 1 l L 1

PTTTR

A : 2

in which e j(k ) is the error at the j th output neuron inthe k th sample, f Hthe derivative of output function, his the learning rate and

Dl 1m k d m k ; d m k 1 ; d m k 2 ; d m k M l A : 3

in which M l is the delay of the present layer. A carefulexamination of the equation reveals that the change inthe total error caused by a change in an internal state isa function of future values within the network. Thiscould be easily remedied by adding a nite number of simple delay operators into the network (Wan, 1990).Hence, this method is used in the present study. Eqs.(A.1) and (A.3) may also be modied to account forthe momentum factor a (as in the case of standardback-propagation) enabling faster convergence.

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