PRINT VERSION MODULE

Module Objectives

Introduction

Flood Forecasting Techniques

References
Contributors
Acknowledgement

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MODULE OBJECTIVES

 To get familiarized with a few Statistical parameters To grasp difference between empirical vs. theoretical frequency distribution To understand & perform various tests to ensure fitness of data for flood frequency analysis To learn how to plot confidence band and its significance To grasp the meaning and significance of confidence band; confidence limit; outliers; expected probability etc.

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INTRODUCTION

 The goal of flood forecast is to issue advance warning about water level or discharge large enough that threatens safety of structures and flood plain activities. As observed in previous module, an advance warning of this nature help authorities adopt a series of measures to contain adverse impacts of flood. Unlike several other disasters, approaching flood can be forecast ahead of its occurrence with advance collection of hydro-meteorological data, and its transformation into flood water level or flood hydrograph. Succeeding paragraphs of this module unfolds a range of commonly employed models in India.

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FLOOD FORECASTING TECHNIQUES

According to the various concepts used in developing models, the models can be classified into five categories.

 a) Based on correlation/coaxial diagrams between two variables or even more; b) Mathematical equations developed using regression/multiple linear regression techniques which combines independent variable with one or more than one variable; c) Hydrological models c.1 Rainfall run-off model i) Lumped ii) Quasi-distributed iii) Distributed c.2 Routing techniques i) Lumped, & Distributed d) Hydraulic models i) Dynamic Wave routing e) Data driven hydrological models i) Artificial Neural Networks ii) Fuzzy expert system design for FF iii) ANFIS (Adaptive Neuro-Fuzzy Inference System) models

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a. Correlation/Co-axial Diagrams

 Forecasters in India have developed a large set of correlation, and coaxial diagrams which display the pattern of correlation exhibited by two or more variables. Such charts are relatively less complex, and are quite popular among its users. Nevertheless, they need periodical updating to account for constant alteration in catchment characteristics and river regime. One out of several such diagrams used in India is shown here. When a number of tributaries affect the water level at the forecasting station, the variation in water level at base station (base station is a location upstream of forecast station) on the main river as well as base stations on the tributariesare considered to prepare co-axial diagrams. One such diagram developed for formulation of forecast at Patna (Gandhighat) on river Ganga is shown in Fig.1. In this diagram, water level fluctuation at Patna takes into account the variation in water level at Buxar on river Ganga; Darauli on river Ghaghra; Chopan on river Sone; and Rewaghat on river Gandak. This concept can also be extended to account for rainfall in upland area. Fig.2 is for formulating the forecasts at Khowang on river Brahmaputra considering rise and fall in water level at Naharkatia site. Additionally, rainfall observation at Naharkatia is also accounted for to incorporate its likely influence to the water level at Khowang. However, these charts carry limitations in that they provide only peak flow or water level information, and drop no hint about the shape of likely flood hydrograph at forecast site. This aside, there is absence of statistical test to measure the strength of correlation between dependent and independent variables. Nevertheless, such diagrams are proved quite useful in absence of fully developed network of hydro-meteorological stations; skilled personnel to operate sophisticated models; and seamless flow of data from remote locations to forecast centre.

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b. Mathematical Equations

 Unlike previous method, this method defines relationship mathematically among variables by 'regression/multiple regression techniques'. The strength of such pattern is easily determined by correlation coefficient, 'r', and thus subjective judgment of a person in drawing a best-fit line is eliminated. Mathematical equations offer much ease in calculation of dependent variable, and in turn speed up forecast process. Chart at Fig. 3 displays an equation that estimates water level at downstream location, Mahemdabad, Gujarat with change in water level at upstream site. A respectable degree of r2 as 0.9854 is achieved by introducing a time lag/shift of 4 hrs between two sets of data. The arrival of this time lag is based on output obtained through cross-correlogram technique. With no time lag, two sets of data are poorly correlated. Another approach is to develop a mathematical model relating forecast station water level with water level of a tributary joining in-between base and forecasting station, and of base station. This method is elaborated by an example comprising three stations. Location of stations may be visualized as shown in Fig.4. Table 1 lists water levels observed at these locations. A linear multiple regression equation with X1 as dependent variable and X2, X3 independent variables can be expressed as below. The coefficients a, b and c are estimated by the method of least square. Three equations noted down below help solve three unknowns. This can also be arranged in matrix form. Below is an arrangement of three matrices; where [X1], [R] and [A] are 3*1, 3*3, 3*1 matrix Substituting all summation terms, we get Matrix [A] containing all three unknown coefficients is solved by multiplying [R]-1 matrix with [X1] matrix. [A] = [R]-1 [X1] Solving the matrix in MS Excel results regression equation as below: X1 = - 223.017 + 2.71 X2-0.0003 X3 Computed/estimated X1 with the help of this equation appears in the last column of first table. Correlation coefficient, r for the defined equation is 0.99 suggesting higher degree of correlation among variables and can be adopted as forecast model. Standard Deviation (SD) and Sest are needed to estimate r2. To determine SD, reader may consult flood frequency module. Sestis determined by _/ S(Xi, obs- Xi,comp)2/(n-2). Finally, r2 is determined by following equation. Equation of the type X1 = a . X2 b1. X3 b2 .. can also be evaluated by converting them into a linear form by logarithmic transformation. Secondly, in the current example, two independent variables are water level. Reader can substitute it by other variables or add more variables to this equation. Solution of coefficients follows similar steps. Another relationship derived by multiple regression technique determines the change in water level at forecast site bases on the variations recorded at two upstream sites, commonly known as base stations. While preceding equation relates water levels of two sites, this equation correlates variation in water levels at different sites. A Mathematical Model using Muskingum Outflow Equation (after Hydrology by H M Raghunath) According to Muskingum method, outflow and inflow at two time steps, (t+1)& trelated to by equation (I). For a few initial time steps of observed inflow hydrograph, such as I1, I2 ---, I4, and O1, O2 ---, O4 of outflow hydrograph, a set of equations, with the help of eq. (I) can be written as below Now, assuming that discharge and water level curves at either location a straight line; and denoting water level at upstream and downstream sites as H & G respectively (Fig. 5), we can say that Replacing discharge component of equations (II) & (III) with this, we get Or, we can simply write these equations as Equation (VI) combines change in water level at downstream site with changes in water level at upstream site. At this stage, reader may please note thatsubscripts 4, 3, 2 in above equation denotes difference water level at time (t+1), t & (t-1) at respective stations. A set of equations,like this, may be obtained by suitably picking up data from observed hydrographs to estimate coefficients x1, x2, & x3by matrix method as elaborated earlier. While doing so, it is highly recommended to check'r2' value to ensure that model is worth for the purpose it is defined. This sort of equation can be developed for rising and falling stages separately. Further refinement is possible by dividing stages into two or three ranges with each range represented by unique equation (please see Fig.5). Additionally, equation (VI) considers only one station/site in the upstream. In case, water level at downstream site happens to be affected by more than one site, a modified Muskingum equation can be written as Where, H, H', H''represent water level at three upstream sites. Number of equations formed in this manner need to be solved for coefficients by matrix method. The steps involved in the process with same set of data used in previous example are illustrated below: Step 1 With two independent variables (water level at two upstream sites) and one dependent variable (water level at forecast station), equation (VI) takes following form. Step 2 With known water levels at respective sites (Fig. 4 & Table 1), a set of values each representing change in water level at various time interval is tabulated next in matrix form Values of coefficients are determined by solving the matrix by an equation given below. Here, equations are more than number of unknown. Therefore, H is a rectangular matrix.In MS excel, standard deviation, matrix multiplication, inverse and transpose commands are defined by STDEV, MMULT, MINVERSE, and TRANSPOSE respectively. Cell (s), where user intends to get result needs to be selected before executing the function. With coefficients indicted in above table, a mathematical equation takes following form for use in flood forecast. While seeking to define an equation by this approach, caution is needed toward the inherent assumption associated with the method, i.e. a plot between water level and discharge should closely follow a linear trend in that range for which user/forecaster intends to relate parameters. A comparison between observed vs. computed water level at X1 and relevant statistical parameters which measure the strength of model are presented in Table 2.

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c. Hydrological Models

c.1 Rainfall-Runoff Models

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i) Lumped Models

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ii) Quasi Distributed Model

 Following illustration demonstrates, with the help of Fig. 10, application of UH in conjunction with MUSKNIGUM routing method to estimate magnitude of flood and time of its occurrence. The set of data inputted here remains the same as for previous case. According to procedures illustrated in earlier example, UH considered for analysis represents an area of 8570 sqkm, and its convolution is based on average areal rainfall over the region. This concept of convolution of UH runs a risk of overestimating the flood because of departure from one of its fundamental assumption that rainfall is uniformly distributed over the region for a specified time. This may not be true for an area as large as 8570 sqkm. Additionally, this approach ignores likely impact of channel storage on flood attenuation. In order to adhere to this basic assumption, UH concept is usually applicable for an area less than or upto 5000 sqkm. Example quoted in the manual (p 244-253) overcomes this violation by sub-dividing the entire basin into three sub-basins A, B & C of area 2040 sqkm, 3470 sqkm and 3060 sqkm respectively, and assuming contribution of rain gauges a, b & c to respective sub-basins only. Accordingly, ordinates of UH has also been altered by a ratio between area of the respective part to the total area. Routing of flow along the reach is done by MUSKINGHUM method. Parameters K, X have been taken from the example and its stability is ensured by adhering to constraint, such as 2KX should be less than T. Additionally, equal contribution of 100 cumec as base flow from three parts is assumed. A basin delineated into three parts with two routing reaches is presented below. As discussed earlier, rainfall recorded at rain gauge 'c' contributes to sub-basin 'C', and therefore, its flood appears at basin outlet having propagated through reach -1 and reach -2. Similarly, sub-basin 'B' receives rainfall observed at rain gauge 'b', and resulting runoff travels through reach- 2 only. Sub-basin 'A' responds to rainfall at 'a', and its effect is visible at outlet (no routing is involved in this case). In agreement with discussion in preceding paragraph, UH for each basin& Muskingum parameters for two reaches are given in Tables 4 & 5. Rainfall excess at each station is at Table 6. A model (Fig. 11) duplicating three sub-basins and its reaches is created in HEC-HMS followed by data entry. HEC-HMS generated runoff at Outlet (Junction-2) appears at Fig. 12. Also shown there is change in water level according to fluctuation in discharge at this point of basin. Option is also available to mark warning level to distinguish critical period when there will be heightened risk because of swelling river. Table7 lists ordinates of flood hydrograph against time and compares with the result as presented in Manual. Options are also available in the HEC-HMS to observe resulting hydrographs for each and every element shown in model. Points to note Regardless of duration of rainfall and its distribution over time, UH of known duration, say t-hr is to be fed in the HEC-HMS keeping its ordinates spaced at t-hr apart. For example, if UH of 1mm rainfall derived is of 3hr duration, ordinates of UH must be entered at 3hr interval. Software automatically converts this UH to duration according to rainfall distribution over catchment. This process rid us of steps needed for conversion of a UH from one duration to another. A relationship between stage/water level and corresponding discharge at forecast station is best represented by a rating curve or a power equation of the type Q = c*(G-G0)^b. Caution is required here to feed latest rating curve of the site in the software; which is best estimate of the prevailing river regime. Secondly, fitting a rating curve does need some technical skill. HYMOS software performs this task with ease.

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iii) Distributed Model

 Forecast estimated by applying hydrological models such as one presented in preceding paragraphs tends to vary widely from real values, where assumptions in unit hydrograph or routing models are violated by prevailing hydro-meteorological conditions over catchment. For example, rainfall is non-uniform over the basin; it is not stationary and moving across the basin; rainfall is concentrated in one pocket and leaving holes elsewhere. Apart from this, soil type and land-use pattern also vary over the catchment/basin that govern the rising and falling limb of resulting hydrograph. In lumped model, these characteristics are represented by a single SCS CN (curve number) applicable for entire area under study. Scenario,where spatial and temporal variations are dominating factors, demands application of distributed model to accurately capture the basin response. Presented here is a distributed model developed and analyzed using Water Modeling System (WMS) and HEC-HMS. WMS software developers, on request, provide one time license key for 14 days for software evaluation purpose only. For its continued use, one needs to buy it. In this model, WMS software first delineates a watershed for an outlet point selected by the user, and thereafter creates grid (Fig. 13 & 14). For each grid, it determines CN values according to its soil type and land-use cover. Once land component is over, software prepares a gridded precipitation database based on rainfall input provided by the user. A set of these gridded information are subsequently exported to HEC-HMS for simulation run. HEC-HMS calculates the rainfall excess and route it to outlet by MODCLARK method (Fig. 15). A couple of screenshots display model set-up and results obtained at the end. WMS software can be downloaded by visiting site http://www.aquaveo.com/downloads.

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c.2 Routing Techniques

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i) Lumped, & Distributed Routing
 (a) Storage equation The Muskingum method of stream flow routing is most frequently used because of its simplicity, as it works with known inflow hydrograph and some fitted parameters without seeking additional information. However, in order to get high degree of accuracy, this method should be for gradually varied flow and not in cases where reach is often affected by backwater or unsteady flow condition. The two fundamental equations for stream flow routing by Muskingum method are:

S = Prism storage + Wedge storage

= K.Q0 + K.X.(Q1-Q0)

 Where, S = Total Storage K= A constant in time unit denotes the time of travel of flood wave through the reach. So, if flood wave velocity or celerity is C, C equals L/K, where L is reach length. X = A dimensionless factor which defines the relative weights given to inflow and outflow in determining storage. (Mostly varies between 0.1 & 0.3 and ranges between 0 & 0.5) Q1 or I = Inflow rate. Q0 orO = Outflow rate.

 (b) Continuity Equation The coefficients are connected by the relation Co + C1 + C2 = 1 Equation (I) with known coefficients, Co, C1& C2 computes outflow with inflow and outflow at time t & t+1. However, accurate estimation/selection of K, X, Dt and subdivision of river reach is central to successfulMuskingum routing. That is why these parameters are also termed as 'tuning knobs' of the model and merit due attention at the time of their estimation. Determination of K and X Even if the feasible range for the parameter X is (0, 0.5), there are other constraints apply to selection of X. With Muskingum routing, the distance step, Dx, is defined indirectly by the number of steps into which a reach is divided for routing. We will dig into example data set presented below (Table 8) to estimate these values. Later with HEC-HMS, optimized value for these parameters will be extracted. As with other models, Dx/Dt is selected in a manner to approximate c, where c = average wave speed (also celerity) over a distance increment Dx. If total reach length is L, and travel time is K, Wave speed, C is c = L/K = L/n.Dt If there are n sub-reaches, and each sub-reach requires Dt time for discharge to flow past, K = n.Dt So, the number of steps, n =K/Dt. For current example, distance between two stations is around 112 km;an estimated value of K is assessed about 34 hour with table 1(time interval between inflow peak and outflow peak- a rough estimation for K to begin with). Inflow flood hydrograph ordinates are at 2 hours interval, the routing reach should be divided in 17 steps, i.e. 34/2 to get the outflow hydrograph 112 km below. This leads to less attenuation as compared to routing carried out in a single step for the entire reach. Secondly, the parameters K, X and computational time step ?t must also be selected in a manner so as to ensure that the Muskingum model/its coefficients must be rational. This implies that the parenthetical terms of the coefficients C1, C2 and C3 must be non- negative. To maintain this, values of K and X must be so chosen so that the combination falls within the shaded region shown below. In other words, K, X &Dt must satisfy a condition given by 2KX

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d. Hydraulic Models

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e) Data driven hydrologic models

 Sometimes,it is argued that deterministic, reductionist models are inappropriate for real-time forecasting because of the inherent uncertainty that characterizes river catchment dynamics and the problems of model overparameterization. The advantages of alternative, efficiently parameterized data-based mechanistic models, identified and estimated using statistical methods, are discussed. Neuromorphic modelling techniques are now well established methods for describing physical processes occurring in the aquatic environment. The development in information technology over the last decade has presented opportunities of extended computational ability together with improved data manipulation, storage and retrieval. As a result, the Neuromorphic models are now being used more extensively in the management, design and operation of water based assets. The reason behind this is that in many areas of applications pertaining to the complex flow systems, the demands on computing time are of such a magnitude, which is far from acceptable. An elementary brief part of ANN has been added here in the distance learning course for easy insight, though there are advanced architectures like recurrent neural network (RNN), radial basis function (RBF),self-organizing map (SOM) and othersused in flood forecasting

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i) Artificial Neural Networks

 An artificial neural network is nothing but a collection of interconnected processing elements (PEs). The connection strengths, also called the network weights, can be adapted such that the network's output matches a desired response. Fig. 38depicts a typical multilayer perceptron, which has been used in this research, which resembles a black box model, where a set of a data like x1, x2, x3…xn are fed directly to the network through the input layer, and subsequently produces expected result y in the output layer. The output is determined by the architecture of the network. In multi-layered perceptron, hidden layer means a third layer of processing elements or units in between the input and output layers that increases computational power. In principle, the hidden layer can be more than one layer. In practice the number of neurons in this layer is evaluated by trial and error. Hornik et al. (1989) proved that a single hidden layer containing a sufficient number of neurons can be used to approximate any measurable functional relationship between the input data and the output variable to any desired accuracy. In addition, De Villars and Barnard (1993) showed that an ANN comprising of two hidden layers tends to be less accurate than its single hidden layer counterpart. In this, a single hidden layer ANN has been used. Each input xi (i =1,…,n) is attenuated by a factor wij, more commonly called a weight of the network, which is associated with the connection linking input xi to hidden neuron j (j = 1,….,k), where, k is the number of neurons in the single hidden layer. The weighted sum of the incoming signals entering a neuron is fed via an activation function, which is non-linear, producing a value that in turn, act as an input signal sent to the output layer. This is repeated for the output weights. The following expression gives the output value of the network. Where, the sigmoidal activation function Y is given by This function given at eq. 2 is a continuous function that varies gradually between asymptotic values 0 and 1 or -1 and +1. Where, a is the slope parameter, which adjusts the abruptness of the function as it changes between the two asymptotic values. Sigmoid functions are differentiable, which is an important feature of neural network theory. To obtain the best approximations, it is needed to determine the optimum set of weights wij and aj that will yield the least mean square value of the desired response. Thus the following performance criterion needs to be satisfied. The configuration chosen for the ANN models are shown in Fig. 39, where the bias inputs have the effect of lowering or increasing the net result of the activation function. The activation used is the sigmoidal function, which has the purpose of limiting the permissible amplitude range of the output values to some finite value. Normalization of the data It is mentioned that the sigmoidal function can take the values ranging in the (0, 1) domain, a normalisation of the values of the input variables are done. The standard procedure in neural network theory gives the normalisation equation used for this purpose. Where: X - actual value of a numeric column, Xmin - minimum actual value of the column, Xmax - maximum actual value of the column, SRmin - lower scaling range limit , SRmax - upper scaling range limit and SF - scaling factor, Xp-pre-processed value. Interpretation of results Numerous goodness of fit statistical criteria are proposed in the literature for evaluating hydrological modelling results. Here, only two of these are considered in this study, namely RMSE, and Nash - Sutcliffe coefficient (1970). RMSE can take any positive value but the closer it is to zero, the better the model performs. When the Nash value is between 0 and 1, the forecast model does better than simply forecasting using yo. The closer the Nash index is to one, the better. These performance criteria are used as basis of comparison to select the best model. Nash-Sutcliffe coefficient is defined by Where, Yo = Observed daily gauge of the catchment on day i; Yp = Predicted daily gauge of the catchment on day i; As could be seen from Fig 40, the modelproved their capability in predicting the data, especially the stage data, which shows a high correlation of the observed and predicted data.

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ii) Fuzzy expert system design for flood forecasting

 Linguistic terms are chosen to describe the input variable stage and the results. Further refinement of the models could not be achieved by adding extra membership functions. Gaussian membership functions (the function is generally suited for Indian rivers) can be used. Applying a similar method of data classification, membership functions are determined for the output variable discharge. Rule definition Some years of average hourlystage data and expert knowledge are used to create a rule base for the fuzzy logic model. Rules are defined for both the high and low extreme conditions, with regard to actual occurrences, because of the physical nature of the relationships. Depending on number of membership functions for each input variable; the minimum rule base is created. For each data point, all rules are evaluated. Fuzzy model construction The platform selected for the fuzzy logic expert system is MATLAB and MATLAB'S Fuzzy Logic Toolbox. The variables are combined into rules using the concept of 'AND'. The fuzzy operator 'minimum' is applied as the 'AND' function to combine the variables. No weightings are applied, which means no rule is emphasized as more important in respect to estimating the discharge. Implication is performed with the minimum function, and aggregation is performed with the maximum function. The centre of gravity method is applied as a means of defuzzification of the output membership functions to determine a crisp set. Based on this structure a baseline model fuzzy logic expert system for stage-discharge relationship is constructed for the G&D stations. Alternate functions for the expert system are investigated through sensitivity analysis. Sensitivity analysis A sensitivity analysis is performed for the fuzzy logic operator AND, and for methods of implication, aggregation and defuzzification. The results of changing a single operator or method while the rest of the model is held constant are compared with the results from the baseline model. The results are evaluated on the basis of correct linguistic matches. Based on this sensitivity analysis, the AND operator 'minimum' and the implication method 'minimum' are found to perform better than the product method. The fuzzy logic and ANN models are evaluated based on their ability to predict the discharge.

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iii) ANFIS (Adaptive Neuro-Fuzzy Inference System) models

 The hybrid system of learning has been attempted at combining ANN and fuzzy logic for developing the stage-discharge relationship to achieve a faster rate of convergence by controlling the learning rate parameter with fuzzy rules. The objective is to get a minimizer, which has a low computing cost and a large convergence domain. This learning ability is achieved by presenting a training set of different examples to the network and using learning algorithm, which changes the weights in such a way that the network reproduces a correct output with the correct input values. The main dissimilarity between fuzzy logic system (FLS) and neural network is that FLS uses heuristic knowledge to form rules and tunes these rules using sample data, whereas NN forms "rules" based entirely on data. Sugeno type ANFIS can be used. Gaussian membership functions can be used with rule bases, because of their low computational requirements and as it has the important properties of smooth mapping, universal approximation, normal distributions can be approximated well by this type of functions.Learning rate control by fuzzy logic has been depicted at fig 41. ( FLC - fuzzy logic controller, MLP - multilayer perceptron Validation and comparison of results The ANN, fuzzy and neuro-fuzzy models thus developed is validated and compared with the observed data points and the statistical measures of goodness-of-fit of the neuromorphic models. Numerous goodness of fit statistical criteria are proposed in the literature for evaluating hydrological modelling results. Goodness of fit can be tested from standard statistics literature as has been shown in the aforesaid ANN paragraph.Fig 42 shows the validation and comparison of models with observed data. Conclusions As could be seen in preceding paragraphs, advance warning about the incoming flood peak and its probable time of occurrence can be achieved by several models. However, selection of a particular method or model, and its accuracy for a given site is largely governed by threefactors - data availability; forecaster's knowledge of, and his experience with the basin; and forecaster's familiarity with software to be used in the forecast process. The illustrated texts mentioned in this module are just the trail of a beginning and more of the subject and in-depth precision knowledge base, the readers are suggested to refer to advanced literature layouts.

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REFERENCES

 Flood forecasting Manual, (1989), Central Water Commission, New Delhi Raghunath, H M., (2006), Book on Hydrology principles analysis and design, New Age International Technical references Manual, (2004), HEC-HMS, USACE Technical references Manual, (2004), HEC-RAS, USACE Wheater, H. S., Jakeman, A. J. &Beven. K. J. (1993) Progress and directions in rainfall-run-of modelling. In Modelling change in environmental systems (ed. A. J. Jakeman, M. B. Beck &M. J. McAleer), pp. 101-132, Wiley.

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MORE LITERATURES OM NEURAL NETWORK AND FUZZY LOGIC CAN BE FOUND AT THE FOLLOWOING REFERENCES

1. Dastorani, M.T. and Wright., N.G., 2002, Artificial neural network based real-time river flow prediction, Hydroinformatics, Proc. of 5th Int. Conf., Cardiff, UK.
2. Haykin, Simon, (1999) Neural Networks: A Comprehensive Foundation, Prentice Hall.
3. Karunanithi, N., Genny, W.J., and Whitley, D 1994. Neural networks for river flow prediction., journal of Computational Civil eng, 8(2),201-220.
4. McCulloch, W.S., Pitts, W. 1943. A logical calculus of the ideas imminent in nervous activity.Bull.Math.Biosphys.5, 115- 133.
5. Nash, J.E., and Sutcliffe, J.V., 1970. River flow forecasting through conceptual models, part-1- A discussion of principles. Journal of Hydrology, 10(3), 282-290.
6. Rajsekharan, S. and G. A Vijayalakshmi Pai, 1996. Neural networks, fuzzy logic and genetic algorithm, Prentice Hall of India, pp-34-86.
7. Rumelhart, D.E, Hinton, G.E, and Williams, R.J, 1986. "Learning internal representation by back propagation errors.", Nature 323, 533-536.
8. Sankhua, R. N, 2006, Application of artificial neural network for daily river stage forecast in the Brahmaputra River, Water and Energy International Journal, CBIP, New Delhi, Vol. 63, No 3, pp.55-62, Jul.-Sep.
9. Sankhua, R. N, 2006, Monitoring of Morphological Changes of Indian Rivers through ANN Based Spatio-Temporal Model- An approach, National Seminar on silting of rivers in India, February, 12-13, New Delhi.
10. Sankhua, R. N, 2006, Spatio-Temporal Modeling of Hydrological Variability for the river Brahmaputra using Artificial Neural Network, proc. International Symposium on Role of Water Sciences in Transboundary River Basin Management, Ubon Ratchathani, Thailand, March 10-12,pp-25-31
11. Zhu, M.L, and Fujita, M (1994), "Comparisons between fuzzy reasoning and neural networks methods to forecast runoff discharge." J. of Hydroscience, Hydraulic eng., 12(2), 131-141.

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CONTRIBUTORS

• Dr. R N Sankhua, Director, National Water Academy, CWC, Pune
• A K Srivastava, Director, National Water Academy, CWC, Pune

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ACKNOWLEDGEMENT
 Contributors of this module hereby acknowledge the invaluable support received from Shri D S Chaskar, Director, National Water Academy, CWC, Pune in presentation of this module in current shape.