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Flow Routing in Open Channels: Some Recent Advances

Vijay P. Singh Department of Civil and Environmental Engineering

Louisiana State University Baton Rouge, LA 70803-6405

Abstract: Our understanding of the dynamics of flow in open channels has significantly increased in recent years. This can largely be attributed to the collection of both experimental and field data, new computational tools, and new analysis techniques developed in mathematics and statistics, and coming together of seemingly disparate water and environment-related areas. A snapshot of these ideas is presented here. 1. Introduction Flow routing in channels has been a subject of much discussion for over half a century and more especially since the advent of digital computers. Flow routing in open channels is a technique for determining the propagation of flow from one point in the channel to another. The term channel is used in a broad sense and includes rivers, streams, bayous, brooks, creeks, canals, sewers, partially flowing pipes and tunnels, gutters, borders, and furrows. Because of its ubiquitous application in hydraulic design and water resources management, flow routing is as old as hydraulics and has vast literature. The flow variables whose propagation characteristics are of interest are discharge, velocity, depth, cross-section, volume, and duration. The propagation characteristics of interest are peak, time to peak, duration of the hydrograph, and attenuation. The bulk of the literature on flow routing is related to the routing of discharge. Although routing can be from upstream to downstream or from downstream to upstream, the emphasis has, most of the time, been on routing from upstream to downstream.

Flow routing in open channels entails wave dispersion, wave attenuation or amplification, and wave retardation or acceleration. These wave characteristics constitute the hydraulics of flow routing or propagation and are greatly affected by the geometric characteristics of channels, the characteristics of sources and/or sinks, as well as initial and boundary conditions. In general, channels are heterogeneous and non-uniform with regard to geometric, morphologic and hydraulic characteristics. In other words, these characteristics vary longitudinally as well as transversely in space. Furthermore, they also vary in time, especially over macro scales. The objective of this note is to briefly reflect on the practice of flow routing in open channels and discuss some new and emerging concepts which are being brought to bear on the development of new flow routing technology. 2. A Short Review of Literature There is a multitude of flow routing models which can be broadly classified into hydrologic and hydraulic models. Hydrologic models are based on a spatially lumped form of the continuity equation, often called water budget or balance, and a flux relation expressing storage as a

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function of inflow and outflow (Singh, 1988). Since coupling of these two equations leads to a first order ordinary differential equation, only an initial condition is needed to solve this equation. This equation does not explicitly involve any spatial variability and expresses the flow routing variable as a function of only time. Hydraulic models are based on the St. Venant equations or simplifications thereof. A vast amount of literature dealing with applications of these equations or their simplifications to flow routing is available (Singh, 1996; ASCE, 1996). In a watershed there usually is a network of river channels and tributaries, that is, each river may have a number of tributaries. For purposes of applying these equations, a given river may be divided into a number of reaches. The hydraulic equations are applied to each reach and the system of equations corresponding to all the reaches are solved simultaneously. When the full St. Venant equations are applied, the computational demands may be formidable and the solution may be inefficient and may incur a large accumulated error. This may explain the reason for the increasing popularity of simplified hydraulic models. These simplified models include kinematic wave, diffusion wave, gravity wave, and quasi-steady state. Linearized forms of the St. Venant equations are also popular for flow routing (Dooge, 1980). Woolhiser and Liggett (1967) showed that kinematic wave models would be adequate if the kinematic wave number was greater than or equal to 20. This criterion is satisfied if the channel has a moderate slope and the flow is unsteady gradually varying and has little backwater effects (Ponce and Simons, 1977; Ponce et al., 1978; Hunt, 1984; Singh, 1996). The diffusion wave or non-inertia wave models are an improvement over kinematic wave models because they are capable of accommodating backwater effects (Akan and Yen, 1977; Hager and Hager, 1985; Dooge and Napiorkowski, 1987; Singh, 1996). The gravity wave models perform well when inertial effects dominate over slope terms (Singh, 1996). These simplified models are adequate in most cases of practical interest (Yen, 1979, 1982; Marsalek et al., 1996; Singh, 1996). For most flow routing problems of practical interest, analytical solutions for either full St. Venant equations or their simplified forms are not tractable and numerical solutions are therefore employed. Numerical methods to obtain solutions can be classified as: explicit finite difference, implicit finite difference, finite element, and boundary fitted coordinate (Singh, 1996). In each class there are many different types of methods. Different classes of methods are useful for different conditions. 3. Mathematical Formulation for Flow Routing A mathematical formulation of flow routing in open channels involves specification of geometry, governing equations, sources and sinks, and initial and boundary conditions. 3.1 Geometric Representation The geometry of an open channel entails cross-sectional shape, bed form, longitudinal form and branching. Depending on its characteristics, a channel can have a cross-section varying from a simple rectangular shape to a complicated form. The channel bed can vary from a smooth surface to a highly irregular one encompassing dunes, antidunes, riffles, and so on. Longitudinally, the

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channel can be straight, meandering, and braiding. The channel can be single, forked, or can have branches representing a network. These channel features are shown in Figure 1. Further complicating, these geometric characteristics is the spatial and temporal variability of these features. In mathematical representations, the channel is represented by a simple cross-sectional shape, such as rectangular, triangular or trapezoidal, and is assumed prismatic, at least over the channel reach under consideration. 3.2 Governing Equations Flow routing in open channels is, in general, unsteady, gradually varied turbulent flow but may also involve rapidly varied flow at certain locations. It is governed by the laws of conservation of mass, momentum and energy which are expressed as continuity, momentum and energy equations. The momentum and energy equations have the same form, except for correction factors and therefore only the momentum equation will be expressed here. Strictly speaking, flow routing is 3-dimensional in nature, primarily because of spatial heterogeneities and nonuniformities in the horizontal and vertical planes. Therefore, the governing equations are also 3-dimensional. However, because of the lack of data on the spatial variability of roughness, sources and sinks, and initial and boundary conditions and the difficulties encountered in solving them, a one-dimensional form is often employed. Assuming the channel to be prismatic and the flow to be one-dimensional, unsteady, and gradually varied, the continuity equation can be expressed as

r sA Q q qt x

∂ ∂+ = −

∂ ∂ (1)

where A is the flow cross-sectional area, Q is the flow discharge = uA, u is the cross-sectionally averaged velocity, qr is the lateral inflow (such as rainfall) per unit length of the channel, qs is the lateral outflow (such as infiltration or seepage) or per unit length of the channel, x is the longitudinal distance, and t is time. In general, the right side of equation (1) is assumed zero, meaning that the rainfall contribution is negligible and the loss of water due to seepage is small as compared with channel flow and can therefore be neglected. The momentum equation can be expressed as

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1 2 3 4 0 51 1 ( / ) 1( ) [ ( ) ]f rx r sx s r s

Q Q A h Qa a a a S S a U q U q q qgA t gA x x gA A

∂ ∂ ∂+ + = − − − − −

∂ ∂ ∂ (2)

where g is the acceleration due to gravity, h is the flow depth, S0 is the channel bed slope, Sf is the friction slope, Urx is the x-component of lateral inflow joining the channel flow, Usx is the x-component of the lateral outflow leaving the channel flow, and ai, i =1, 2, 3, 4, and 5, are indices taking on values of either 0 or 1, introduced for defining different approximations of equation (2). The left side of equation (2) has three terms. The first term from the left denotes local acceleration, the second term denotes convective acceleration, and the third term denotes the pressure gradient. The right side comprises two parts. The first part within parenthesis is the

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difference between bed slope and friction slope. The second part denotes the contribution by lateral inflow and outflow to convective acceleration. 3.3 Sources and Sinks In channel flow routing source and sink terms are normally neglected. The source term is rainfall and the amount of rainfall in the channel is much small in comparison with channel flow and is often neglected. If the amount is not negligible, then rainfall is represented as a histogram and can be mathematically expressed using the Heaviside function as:

0 1 1 1 11

( , ) ( ) ( ) ( ( ) ( ))( ( ) ( ) [ ( 1) ]m

r j j mj

q x t q x u t q x q x u t jD q x u t m D−=

= + − − + − +∑ (3)

where u1(t) is the Heaviside function, D is the time interval, m is the number of pulses, qr(x), j=0, 1, 2, …., m, are intensities of rainfall qr(x), as shown in Figure 2.

Likewise, if the sink term is neglected then the assumption is that the channel bed is saturated and the loss of water due to seepage is negligible. However, if the channel is ephemeral and has an abstracting bed, this assumption will not be valid and the seepage term will have to be defined. One approximation for one-dimensional (longitudinal) seepage can be expressed as

1/ 2( , ) ( ) ( )s sq x t f x s x t−= + (4) where fs is the steady part of seepage varying along the channel, and s is a parameter analogous to sorptivity of soil as a function of x. 3.4 Initial and Boundary Conditions The initial condition for flow routing in open channels specifies flow depth, velocity or discharge at every point along the channel reach before the commencement of unsteady flow. In terms of discharge it is expressed as

0( ,0) ( )Q x Q x= (5) Frequently, Q0(x) is taken as constant. Depending on the nature of flow, the boundary conditions are expressed at the upstream end or the downstream end or both. The upstream boundary condition can be expressed in terms of discharge, flow depth, velocity, or the relation between flow depth and discharge.

(0, ) ( )uQ t Q t= (6) For a sloping channel, the velocity at the upstream boundary is frequently taken as 0: u(0, t) = 0 but the flow depth is non-vanishing: h(0, t) >0. Morris (1979) has used the zero depth gradient condition: ∂h/∂x = 0. This influences both subcritical and supercritical flows.

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In a similar vein, the downstream boundary condition can be expressed as

( , ) ( )dQ L t Q t= (7) where L is the channel reach length. The downstream condition has been expressed in three ways. First, the channel reach is continuing or is indefinite. Second, the flow is critical at x=L:

( , ) ( , )u L t gh L t= (8) where h is the flow depth, and u is the cross-sectionally averaged velocity. Third, the gradient of the flow depth at x=L is zero:

( , ) 0h L tx

∂=

∂ (9)

It is implied that the channel flow is subcritical.

For supercritical flow the conditions need to be satisfied only at the upstream boundary and at x=L the following condition is always satisfied at:

( , ) ( , )u L t gh L t> (10) 4. Simplifications of Governing Equations Equations (1) and (2) are difficult to solve even numerically. Various approximations of these equations are therefore employed in flow routing. In all of these approximations equation (1) remains in tact and only equation (2) is simplified. These approximations are:

(a) If a1 = a2 = a3 = a5 = 0 and a4 = 1 in equation (2), then the resulting approximation is the kinematic wave approximation.

(b) If a1 = a2 = a5 = 0 and a3 = a4 = 1, then the resulting approximation is the diffusion

approximation. It is also called as the non-inertia approximation. (c) If a1 = a2 = a3 = 1 and a4 = a5 = 0, then the resulting approximation is the gravity wave

approximation.

(d) a1 = 0 and a2 = a3 = a4 = a5 =1, then the resulting approximation is the quasi-dynamic wave approximation.

If a1 = a2 = a3 = a4 = a5 = 1, then the resulting representation is the full dynamic wave representation. Equations (1) and (2) are also simplified by linearization using the method of perturbation or freezing the coefficients associated with nonlinear derivative terms (Singh, 1996). Equation (1) is linear and equation (2) can be linearized by treating the unsteady open

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channel flow as perturbation from the steady condition and neglecting the second and higher order terms as.

(11)

where Q0 is the initial steady state uniform flow used as reference flow, Q* is the perturbation from the reference flow, A0 is cross-sectional area corresponding to the reference flow, and A* is the perturbation from the reference area. Neglecting sources and sinks and applying equation (11) to equations (1) and (2), the following is obtained:

* * 0A Qt x

∂ ∂+ =

∂ ∂ (12)

2* * *

0 0 0 0 * *2 (1 ) [ ]f fS SQ Q Au gh F gA Q At x x Q A

∂ ∂∂ ∂ ∂+ + − = − −

∂ ∂ ∂ ∂ ∂ (13)

where 2

0 0 00 0 3

0 0

( , ) ;( , )

A x t Q Th FT x t g A

= = ; the derivatives of friction slope with respect to discharge

and cross-sectional area are evaluated at the reference conditions. Different forms of second-order coupled equation can be derived by coupling of equations (1) and (2) and these forms have analytical solutions (Singh, 1996). 5. Applicability of Simplified Representations Different approximations lead to different types of flow routing models which are applicable under different conditions. Various criteria have been derived which shed light on the applicability of these models. The bases of the applicability criteria are different and they sometimes refer to different flow characteristics. In general, these criteria can be classified as point form, integrated form, temporally varying error equation, and spatially varying error equation. The point form criteria are derived by comparing flow or wave properties of different flow routing models at every point and time. Using linear perturbation theory, Ponce and Simons (1977) and Ponce et al. (1978) compared wave propagation characteristics and derived point form criteria. Mishra and Seth (1996) employed dimensionless hypothesis to define point form criteria. These criteria can be integrated into lumped form and are useful in choosing a flow routing model. The lumped (or integrated) form criteria are derived by comparing spatially and temporally flow quantities, such as a hydrograph at the outlet, obtained from different routing models at a specified point in space and time. Examples of these criteria are those of Woolhiser and Liggett (1967), Daluz (1983), Fread (1985), Price (1985), Ferrick (1985), Pearson (1989), Marsalek et al. (1996), and Moussa and Bocquillon (1996). These criteria give an idea as to how different models compare in an overall sense but do not provide any indication of the model

0 * 0 *( , ) ( , ) ( , ) ( , )Q x t Q Q x t A x t A A x t= + = +

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accuracy as a function of space and time. Thus it is not clear if a particular model is valid for the entire flow hydrograph or a portion thereof. Error equations have been derived for kinematic (KW) and diffusion wave (DW) approximations which specify errors as a function of time for space-independent flows and errors as a function of space for time-independent flows. Singh (1996) showed that the error equations for KW and DW approximations are Riccati equations. These equations contain a parameter which is a generalized form of the kinematic wave number derived by Woolhiser and Liggett (1967) and has since been used commonly. Under steady state conditions, Parlange et al. (1990) derived errors equations for KW and DW approximations. Similarly, Singh and Aravamuthan (1997) derived generalized error equations for KW and DW approximations for time-independent flows. These equations specify errors as a function of space and are generalized forms of the Riccati equation. For most cases of practical interest, the approximate models of flow routing are found to be adequate. Their adequacy depends on the peculiarities of a particular application. 6. Solution of St. Venant Equations Analytical solutions of the St. Venant equations are not yet tractable except under very simplified conditions. Therefore, numerical methods are the only resort for solving flow routing problems in most practical cases. These methods can be classified as: explicit, implicit, characteristic, finite element, and boundary-fitted coordinate. Examples of explicit numerical schemes are the MacCormack, lambda, Gabutti, diffusive, leapfrog, and Lax-Wendroff schemes. Popular implicit schemes include iterative, Preissmann, and mixed methods. Implicit methods contain difference equations which are solved using any of several methods, including Gaussian elimination, iterative, Newton-Raphson, secant, steepest-descent, and double sweep methods. The characteristic methods are classified as characteristic-grid, fixed-grid, explicit characteristic, implicit characteristic, first-order finite difference, and second-order difference methods. Finite element methods include different variations of the Galerkin scheme. The boundary fitted coordinate method is another numerical method which is especially suited for complex channel geometries. 7. Data Acquisition The data needed for flow routing are geometric, hydraulic, and flow roughness. The geometric data are channel reach length, channel width, bed forms, meandering dimensions, braided forms and dimensions, and channel cross-section shape. Flow hydraulics includes data on flow depth, velocity and discharge at the upstream and downstream boundaries and along the channel. Data on roughness at each point along as well as across the channel are needed. Clearly such data are rarely available and hence many simplifications are made with regard to channel characteristics. It is commonly assumed that the channel reach is uniform and homogeneous over the reach or at least over its sub-reaches. Major advances have been made in recent years in data acquisition technologies, such as remote sensing, space technology, digital elevation and terrain modeling. Using these technologies, it is now possible to get information beforehand on channel roughness, channel morphology and boundary conditions.

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8. Emerging Technologies in Flow Routing In recent years black-box methods developed in biological sciences as well as in information sciences and mathematics have received considerable attention in hydrology and hydraulics. These include artificial neural networks (ANN), fuzzy logic (FL), genetic algorithms (GA), and their combinations. These black-box methods provide a direct mapping between inputs and outputs and have also been employed for flood forecasting. It is interesting to note that hydrologic methods or systems approaches of flow routing are also black-box type and received considerable attention in the decades of 1950s to 1980s but lost favor in recent years. The argument for using the newly developed black-box methods is the same as for the systems based methods. 8.1 Artificial Neural Networks An artificial neural network (ANN) is a form of artificial intelligence mimicking the functioning of the human brain and nervous system. An ANN connects a system output (represented as a layer) to its input (represented as a layer) through a network of nodes (represented as a hidden layer). An ANN has the ability to learn from examples, recognize the pattern in the data, adapt solutions, and process information rapidly.

Hsu et al. (1995) and Minns and Hall (1996) applied ANNs to rainfall-runoff modeling, and Raman and Sunilkumar (1995) used them for synthetic inflow generation. Dawson and Wilby (1998) used an ANN for river flow forecasting and Jain et al. (1999) used it for reservoir inflow prediction. Liong et al. (2002) applied an ANN for river stage forecasting in Bangladesh. Chau and Cheng (2002) employed an ANN with improved back propagation algorithms for real-time prediction of water stage. 8.2 Fuzzy Logic Fuzzy logic (FL) is another method which has an ability to describe the knowledge in a descriptive human-like manner in the form of simple rules using linguistic variables. FG has been employed in a range of applications. Russell and Campbell (1996) used fuzzy programming to develop reservoir operating rules. Fortane et al. (1997) used a fuzzy set theory for planning reservoir operations. Cheng and Chau (2001) employed a fuzzy operation method for reservoir flood control operations. Dubrovin et al. (2002) used fuzzy similarity for real-time reservoir operations. Tilmant et al. (2002) compared reservoir operating policies due to fuzzy and non-fuzzy stochastic dynamic programming. Poonambalam et al. (2002) used a fuzzy system to minimize the variance of reservoir operation benefits.

Ozelkan and Duckstein (2000) used an FL-based method to deal with parameter uncertainties related to data and/or model structure. Yu et al. (2000) combined gray and fuzzy methods for rainfall forecasting. Xiong et al. (2001) used fuzzy logic in flood forecasting, and recommended it as an efficient system for flood forecasting. Yu and Yang (2000) pointed out that the fuzzy multi-objective function can lead to improved simulation of a wide range of flow stages.

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8.3 Genetic Algorithms Similar to the biological processes of natural selection, inheritance, mutation, and evolution, genetic algorithm (GA) is an evolutionary technique. Holland (1975) and Goldberg (1989) provided a comprehensive review of genetic algorithms. Wang (1991) and Savic et al. (1999) applied it to rainfall-runoff modeling. Olivera and Loucks (1997) applied a GA for operation of multi-reservoirs systems. In a similar vein, Wardlaw and Sharif (1999) employed a GA for optimal reservoir operation. Cheng et al. (2002) used a GA for calibrating conceptual rainfall-runoff models, whereas Chau (2002) for calibrating flow and water quality models. 8.4 Combination Methods It is also possible to combine ANNs and GA as well as FL and develop a more versatile method for flood forecasting and routing. Thus, a GA-based ANN can be employed for flow routing. This combined method can also be integrated with fuzzy logic and hence an adaptive fuzzy inference system can be developed. 9. Uncertainty Analysis The factors and processes affecting flow routing in open channels are subject to stochastic variability. This is reflected through data as well as model structure and parameters. For example, there is uncertainty in the description of channel geometry, channel roughness, and specification of sources and sinks, and initial and boundary conditions. The stochastic variability is caused by heterogeneities and nonuniformities in channel morphology, and random irregularities in sources, sinks and initial and boundary conditions, as well as errors in model hypotheses. The uncertainty in flow routing can be described in several ways. The uncertainty of a flow variable is described by its probability density function (PDF). The uncertainty of a physical process can be described statistically in terms of the uncertainty of each contributing factor. Some of the popular statistical methods to that end are Monte Carlo Simulation (MCS), Latin Hypercube Simulation (LHS), Direct Integration Method (DIM), First Order Second Moment (FPSM) Method, and Advanced First Order Second Moment (AFOSM) Method. If there is only one variable of interest then its PDF can be derived either phenomenologically or empirically. If the number of variables of interest is more than one, say flow peak and flow volume or flow peak and duration or flow velocity or stage, then joint the PDF of the variables is needed. Traditional techniques for derivation of joint PDFs have limited potential but newly developed techniques are efficient and extremely useful. 9.1 Univariate Distributions from Impulse Response Functions

It is hypothesized that if a flow variable is described by a linear or linearized governing equation, then the solution of this equation for the unit impulse function can be interpreted as a probability density function for describing the probabilistic properties of the variable. This hypothesis is

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tantamount to mapping from the unit impulse response function plane to the probability density function plane. For example, the impulse response of a linearized diffusion model of channel flow can be interpreted as a probability distribution for frequency analysis of floods. The impulse response of a diffusion equation for pollutant transport provides a probability distribution for pollutant concentration in storm water or river flow. In this vein, the impulse responses of physically-based equations applicable to flow routing are proposed as probability distributions which can be tested using field or laboratory data.

Thus, the methodology entails the following elements: (1) derivation of the governing equation, (2) linearization of the governing equation if it is nonlinear, (3) derivation of the solution of the linearized equation for the unit impulse function, (4) mapping the instantaneous unit response function (IURF) onto the probability plane, (5) testing the validity of the UIRF using field or experimental data, and (6) interpreting the meaning of parameters and their determination using physically measurable characteristics.

Coupling equations (12) and (13) leads to a linearized form St Venant equations:

tQe

xQd

tQc

txQb

xQa

∂∂

+∂∂

=∂∂

+∂∂

∂+

∂∂

2

22

2

2 (14)

where a, b, c, d and e are parameters as functions of channel and flow characteristics at the reference steady state condition. If all three of the second-order terms on the left-hand side of equation (1) are neglected, the linear kinematic wave model is obtained. If the second and the third second-order terms are expressed in terms of the first on the basis of the linear kinematic wave approximation, the result is the kinematic diffusion (KD) model (Singh, 1996; Strupczewski and Napiorkowski, 1990). The KD model is parabolic form, and satisfactorily fits the solution of the complete linearized St. Venant equation only for small values of the Froude number and slow rising waves. If the diffusion term is expressed in terms of two other terms using the kinematic wave solution, one gets the rapid flow (RF) model which is also of parabolic form (Strupczewski and Napiorkowski, 1990) and provides an exact solution for the Froude number equal to one and can consequently be used for large values of the Froude number. Although RF and KD models correspond to quite different flow conditions, the structure of their UIRFs, h(x, t), is similar (Strupczewski and Napiorkowski, 1990). For small Froude numbers, h(x, t) can be expressed as:

2

3

( )( , ) exp[ ]44

x x uth x tDtDtπ

−= − (15)

where x is the reach length, D is the hydraulic diffusivity, u is the convective velocity. Denoting / 4x D α= and /(4 )xu D β= , and renaming t as y one gets a 2-parameter PDF:

2

3

( )( ; , ) exp[ ]

yf y

yy

βαα αα βπ

−= − (16)

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Equation (16) is PDF and is an inverse Gaussian PDF. It can be extended to a three parameter distribution.

Following the above line of logic, a linear reservoir gives rise to an exponential probability density function widely used in modelling the inter-arrival times of floods, and the rainfall depth, intensity, and duration. The Muskingum model yields a two-parameter probability distribution function as a weighted sum of two functions: a delta function and an exponential function, found useful for modeling floods from ephemeral streams. The cascade of n-equal linear reservoirs yields the gamma probability density function.

9.2 Entropy-Based Univariate Probability Distributions In search of an appropriate probability distribution for a given random flow variable, there is some information available on the random variable which the distribution must satisfy. The entropy theory is ideally suited to that end (Singh, 1998). The chosen probability distribution should then be consistent with the given information. There can be more than one distribution consistent with the given information. From all such distributions, we should choose the distribution that has the highest entropy. To that end, Jaynes (1957) formulated the principle of maximum entropy (POME) according to which the minimally prejudiced assignment of probabilities is that which maximizes entropy subject to the given information, i.e., POME takes into account all of the given information and at the same time avoids consideration of any information that is not given.

Let p(x) be the probability distribution of X that is to be determined. The information on X is available in terms of constraints as:

1,01

=≥ ∑=

n

iii pp (17)

and

mrapxg r

n

iiir .....,..........,2,1)(

1

==∑=

(18)

where m is the number of constraints, m + 1 ≤ n, gr is the r-th constraint.

According to POME, there will be only one distribution which will correspond to the maximum value of entropy and this distribution can be determined using the method of Lagrange multipliers which will have the following form:

nixgxgxgp immiii ..,..........,2,1)],(......................)()(exp[ 22110 =−−−−= λλλλ (19)

where λi, i = 0, 1, 2, ………………, m, are Lagrange multipliers which are determined using the information specified by equations (17) and (18).

Substitution of equation (19) in equation (17) yields

])([exp)(exp1 1

0 ∑ ∑= =

−==n

i

m

jijj xgZ λλ (20)

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where Z is called the partition function, and λ0 is the zeroth Lagrange multiplier. The Lagrange parameters are obtained by differentiating equation (20) with respect to Lagrange multipliers:

mjgEa jjj

.........,..........,3,2,1],[0 ==−=∂∂λλ

][20

2

jj

gVar=∂∂λλ

(21)

],[02

kjkj

ggCov=∂∂

∂λλλ

][330

3

jj

gµλλ

−=∂∂

where E[.] is the expectation, Var [.] is the variance, Cov [.] is the covariance, and is the third moment about the centroid, all for gj.

When there are no constraints, then POME yields a uniform distribution. As more constraints are introduced, the distribution becomes more peaked and possibly skewed. In this way, the entropy reduces from a maximum for the uniform distribution to zero when the system is fully deterministic. 9.3 Joint Probability Distributions Joint probability distribution can be obtained by either conventional multivariate statistical analysis techniques or the copula method. Comparing to the conventional approach, Copula, which avoids the limitations of the multivariate distribution approximation by the conventional statistical approach, is efficient and useful for representing multivariate distributions.

Copulas are functions that couple multivariate probability distributions to their one-dimensional marginal probability distributions. The probability distribution for observations (x11, x21,…, xN1),…, (x1n, x2n,…, xNn) from a multivariate population of X1, X2,…, Xn, with non-normal multivariate distribution H, may be obtained by expressing H in terms of its marginals and its associated dependence function, C, defined through the identity C(F1(x1), F2(x2),…, FN(xN) ) = H(x1, x2,…, xn). Then C is a mapping which is uniquely determined on the unit square whenever Fi(xi) are continuous, and captures the essential features of the dependence between the random variables. Thus, a logical approach to analyze multivariate data consists of estimating: (1) the marginal distributions (or marginals) separately and (2) the dependence function. This two-step approach allows the study of the multivariate probability distributions with different marginal distributions and permits the investigation of the dependence structure regardless of the marginal distributions.

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Multivariate distributions can be derived using the copula method developed by Genest and Rivest (1993). The dependence structure (multivariate distribution) of most multivariate data can be represented by one parameter Archimedean copulas, θC , which can be expressed as:

{ } 1,0 ,)()(),( 1 <<+= − vuvuvuC φφφθ (22)

where )(•φ is the copula generator which is a convex decreasing function satisfying (1) 0φ = ; subscript θ of copula C is a parameter hidden in the generating function φ ; u = F(x) and v = F(y) are uniformly distributed random variables; and )(1 •−φ is equal to 0 when )0(φ≥v . The Archimedean copula representation permits reducing a multivariate formulation in terms of a single univariate function. As an example, the following one parameter Archimedean copula has been found to be useful to represent the positively dependent bivariate random variables:

( ) ( )( )( ) [ )∞∈−+−−=== −− ,1,lnlnexp),())(),((),(/111 θθθθ vuyxHyFxFCvuC (23)

where θ is a parameter of the generating function ( ) ( ln )t t θφ = − , with t = u or v as a uniformly distributed random variable varying from 0 to 1, 11τ θ −= − which is Kendall’s coefficient of correlation between random variables X and Y. Note that ( ln ) ( )u uθ φ− = and ( ln ) ( )v vθ φ− = in the above equation. 9.4 Point Estimation Methods Many times the PDF of a flow variable is not available. Therefore the uncertainty of the variable can be expressed in terms of its statistical moments. To that end, point estimation methods are frequently employed. These methods are computationally straightforward and can be employed for determining statistical moments of any order of function involving several variables correlated or uncorrelated. A short discussion of these methods is given here. 9.4.1 Rosenblueth’s Method Consider a variable y as a function of variables xi, i= 1, 2, 3, …, n; y = f(x1, x2, x3, ….., xn). The Rosenblueth method bases the probability distribution of y on the first three moments of independent variables xi, i= 1, 2, 3, …, n. The probability distribution of each independent variable is approximated by concentrating the entire probability mass at two points xj- and xj+, each having a specific weight p- and p+ on the distribution. The moment of m-th order of the distribution of y can be expressed as:

........ ....... ......... ....... .... ........[ ] ( ) ( ) ( )m m m mp p p p p pE y p y p y p y+++ +++ −++ −++ −−− −−−= + + (24)

where .... 1 1 1 2 2 2( , ,........, )J J J

p p p py f x x x x x xσ σ σ+++ + + += + + + ,

.... 1 1 1 2 2 2( , ,......, )J J Jp p p py f x x x x x xσ σ σ−++ − + += − + + + , and

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....... 1 1 1 2 2 2( , ,.........., )J J Jp p p py f x x x x x xσ σ σ−−− − − −= − − + − , and so on. The values of J

ix + and Jix − are

calculated as:

211 ( )2 2

Jix

γγ+ = + + and 1

J Ji ix x γ− += − , 1γ = skewness coefficient of the random variable xi,

1

1 21 1 11

( , ,....., ) ( )i

n n n

n ib i j iji ji

p b b b p b b a−

= = −=

= +∑ ∑∏ (25)

where bi indicates the +/- signs and aij is given by

21

1

12

(1 )4

ijij n n

i

aρ

γ

=

=

+∏ (26)

in which ρij is the correlation coefficient for random variables xi and xj. 9.4.2 Harr’s Method This method assumes that the entire probability mass distribution of an independent variable xi is distributed between two points, xi- and xi+. The m-th moment of the probability distribution of y is calculated as

1

1

[ ]

nm

i im i

n

ii

yE y

λ

λ

=

=

=∑

∑ (27)

where yi is the mean of yi+ and yi-, yi =(yi+ + yi-)/2= [f(xi+) +f(xi-)]/2, and λi are the eigenvalues obtained as follows. The correlation matrix ρ of variables decomposed using the orthogonal transformation method into an eigenvector matrix (w1, w2, w3 ,……, wn), W, its transpose WT and a diagonal matrix ∆ containing the eigenvalues 1 2, ,....., nλ λ λ :

TW Wρ λ= (28) where superscript T denotes the transpose of the matrix. The uncorrelated standardized coordinates of the vectors of the n random variables x+ and x- are generated as:

i ix n D wµ− = − and i ix n D wµ+ = + (29) where µ is the vector of the expected values of the random variables 1 2, ,......., ,nx x x D is the diagonal matrix of the variance of the random variables, and wi is the eigenvector associated with the eigenvalue λi.

15

Chang et al. (1995) modified the Harr method by evaluating y as:

1[ ]

nmi

m iy

E yn

==∑

(29)

in which yi is calculated as before. The weighting factor for each independent variable xi is considered for the modified uncorrelated standardized coordinates in the eigenspace as

( )i ix DW neµ− = − ∆ and ( )i ix DW neµ+ = + ∆ (30) W is the eigenvector matrix, ∆ is the diagonal matrix of the eigenvalues and ie is a unit vector with i-th element equal to 1 and 0 everywhere else. The Harr method is computationally more efficient than the Rosenblueth method because it reduces the computational runs from 2n to 2n and uses only the first and second moments of each stochastic variable. 9.4.3 Li’s Method This method assumes that the entire probability mass of a random variable is concentrated at three points xi- , x+ and µ, having, respectively, the probability values of p-, p+ and p0. The probability distribution of y is obtained from the first four moments of independent variables. The m-th moment of the probability distribution of y is calculated as

01 1

3[ ] (1 ) ( 1) )2 2

n n n nm m m m

i i i i i i ij iji i j i j

nE y p p y p y yη η η+ + − −= = <

= − + + + − + + +∑ ∑ ∑∑ (31)

where η is the sum of all the ηi, ηi is the sum of all ηij with respect to i and ' '/( )ij ij i jx xη ρ + += . Note that 1iiη = . The points xi-, xi+ , and µ are computed as

2 21' ' ' ' '

1 0

4 3 4 3, , , ,

2 2i i i i i

i i i i i i i i i

k kx x x x x x x

γ γ γ γµ σ µ σ µ+ + − − + −

+ − − −= + = + = = =

where k is the coefficient of kurtosis. The weight of each point is given as

0' ' ' ' ' '

1 1, , 1( ) ( )i i i i

i i i i i i

p p p p px x x x x x+ − + −+ + − − − +

= = = − −− −

(32)

This method is efficient and accurate.

16

9.4.4 Modified Rosenblueth’s Method Tsai and Franceschini (2003) modified the Rosenblueth method for cases involving more than three stochastic variables. The m-th moment of the probability distribution of y is calculated as

1

1 1 1( ) [(1 ) ] [( 1) ]

2

N N Nm m m m

i i i ij iji i j i

NE y n y p y yη η η−

+ += = = +

−= − + + − + +∑ ∑ ∑ (33)

This modification preserves the capabilities of the original Rosenblueth method and is an improvement at the same time.

The discrepancy between observed and computed y can be expressed as

1

1 1 1

3( ) [( 1) ]2

N N Nm m m m m m

i i i i i ij iji i j i

NE y y y p y p y yη η η−

+ + − −= = = +

−− = + − + + +∑ ∑ ∑ (34)

9.4.5 Characteristics of Point Estimation Methods The various point estimation methods can be compared based on the moments to be used, intensity of computation and the capability to deal with variables. These are summarized in the table below. Characteristics Rosenblueth’s

method Harr’s method

Modified Harr’s method

Li’s method Modified Rosenblueth’s method

Moments needed Intensity of computation Capability to consider correlated variables Capability to consider asymmetric variables

3 2N

yes yes

2 2N yes yes

2 2N yes yes

4 (N2 +3N+2)/2 yes yes

3 (N2 +3N+2)/2 yes yes

10. Conclusions New flow routing techniques currently being developed are either based on ANN, FL, or GA, or are part deterministic and part stochastic. River flow is inherently spatial and complex, and our understanding of the systems governing it is less than complete. Many of the systems are either

17

fully stochastic or part-stochastic and part-deterministic. Their stochastic nature can be attributed to randomness in one or more of the following components that constitute them: (1) system structure (geometry), (2) system dynamics, (3) forcing functions (sources and sinks), and (4) initial and boundary conditions. As a result, a stochastic description of these systems is needed, and the statistical techniques are available which enable development of such a description. Stochastic techniques are based on either point estimation methods or probability distribution functions. Copulas have tremendous potential in describing dependence between flow routing variables. References Akan, A.O. and Yen, B.C., 1977. A nonlinear diffusion wave model for unsteady open channel flow. Proceedings of the 17th IAHR Congress, August, Baden-Baden, Germany. ASCE, 1996. Handbook of Hydrology. ASCE Manuals and reports on Engineering Practice No. 28, American Society of Civil Engineers, New York. Chang, C., Tung, Y. and Yang, J., 1995. Evaluation of probability point estimate methods. Applied Mathematical Modeling, Vol. 19, No. 2, pp. 95-105. Chau, K.W., 2002. Calibration of flow and water quality modeling using genetic algorithms. Lecture Notes in Artificial Intelligence, Vol. 2557, 720-720. Chau, K.W. and Cheng, C.T., 2002. Real-time prediction of water stage with artificial neural network approach. Lecture Notes in Artificial Intelligence, Vol. 2557, 715-715. Cheng, C.T. and Chau, K.W., 2001. Fuzzy iteration methodology for reservoir flood control operation. Journal of American Water Resources Association, Vol. 37, No. 5, pp. 1381-1388. Cheng, C.T., Ou, C.P. and Chau, K.W., 2002. Combining a fuzzy optimal model with a genetic algorithms to solve multi-objective rainfall-runoff model calibration. Journal of Hydrology, Vol. 268, No. 3, pp. 72-86. Daluz, V.J.H., 1983. Conditions governing the use of approximations for the Saint-Venant equations for shallow surface water flow. Journal of Hydrology, Vol. 60, pp. 43-58. Dawson, C.W. and Wilby, R., 1998. An artificial neural network approach to rainfall-runoff modeling. Hydrological Sciences Journal, Vol. 43, No. 1, pp. 47-66. Dooge, J.C.I., 1980. Flood routing in channels. Unpublished notes, Department of Civil Engineering, University College, Dublin, Ireland. Dooge, J.C.I. and Napiorkowski, J.J., 1987. Applicability of diffusion analogy in flood routing. Acta Geophysica Polonica, Vol. 35, No. 1, pp. 66-75.

18

Dubrovin, T., Jolma, A. and Turunen, E., 2002. Fuzzy model for real-time reservoir operation. Journal Water Resources Planning and Management, ASCE, Vol. 123, No. 3, pp. 154-162. Ferrick, M.G., 1985. Analysis of river wave types. Water Resources Research, Vol. 21, No. 2, pp. 209-220. Fortane, D.G., Gates, T.K. and Moncada, M., 1997. Planning reservoir operations with imprecise objectives. Journal of Water Resources Planning and Management, ASCE, Vol. 123, No.3, pp. 154-162. Fread, D.L., 1985. Applicability criteria for kinematic and diffusion routing models. Laboratory of Hydrology, National Weather Service, NOAA, U.S. Department of Commerce, Silver Spring, Maryland. Genest, C. and Rivest, L.-P., 1993. Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, Vol. 88, pp. 1034-1043. Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimisation and Machine Learning. Addison Wesley, Reading, Massachusetts. Hager, W.H. and Hager, K., 1985. Application limits of the kinematic wave approximation. Nordic Hydrology, Vol. 16, pp. 203-212. Holland, J.H., 1975. Adaptation in Neural and Artificial Systems. An Arbor Science Press, Ann Arbor, Michigan. Hsu, K.-L., Gupta, H.V. and Sorooshian, S., 1995. Artificial neural network modeling of the rainfall-runoff process. Water Resources Research, Vol. 31, No. 10, pp. 2517-2530. Hunt, B., 1984. Asymptotic solution for dam break on sloping channel. Journal of Hydraulic Engineering, ASCE, Vol. 109, No. 12, pp. 1689-1706. Jaynes, E.T., 1957. Information theory and statistical mechanics, I. Physical Review, Vol. 106, pp. 620-630. Li, K.S., 1992. Point estimate method for calculating statistical moments. Journal of Engineering Mechanics, ASCE, Vol. 118, No. 7, pp. 1506-1511. Liong, Y.S., Lim, W.H. and Paudyal, G.N., 2000. River stage forecasting in Bangladesh: neural networks approach. Journal of Computing in Civil Engineering, ASCE, Vol. 14, No. 1, pp. 1-8. Marsalek, J., Maksimovic, C., Zeman, E. and Price, R., editors, 1996. Hydroinformatics Tools for Planning, Design, Operation, and Rehabilitation of Sewer Systems. NATO-ASI Series, Kluwer Academic Publishers, Dordrecht, The Netherlands.

19

Minns, A.W. and Hall, M.J., 1996. Artificial neural networks as rainfall-runoff models. Hydrological Sciences Journal, Vol. 41, No. 3, pp. 399-418. Mishra, S.K. and Seth, S.M., 1996. Use of hysteresis for defining the nature of flood wave propagation in natural channels. Hydrological Sciences Journal, Vol. 42, No. 2, pp. 153-170. Morris, E.M., 1979. The effect of the small slope approximation and lower boundary conditions on solution of Saint Venant equations. Journal of Hydrology, Vol. 40, pp. 31-47. Moussa, R. and Bocquillon, C., 1996. Criteria for the choice of flood routing methods in natural channels. Journal of Hydrology, Vol. 186, pp. 1-30. Olivera, R. and Loucks, D.P., 1997. Operating rules for multireservoir systems. Water Resources Research, Vol. 33, No. 4, pp. 839-852. Ozelkan, E.C. and Duckstein, L., 2001. Fuzzy conceptual rainfall-runoff models. Journal of Hydrology, Vol. 253, No. 1-4, pp. 41-68. Parlange, J.Y., Hogarth, W., Sander, G., Rose, C., Haverkamp, R., Surin, A. And Brutsaert, W., 1990. Asymptotic expansion for steady state overland flow. Water Resources Research, Vol. 26, No. 4, pp. 579-583. Pearson, C.P., 1989. One dimensional flow over a plane: criteria for kinematic wave modeling. Journal of Hydrology, Vol. 111, pp. 39-48. Ponce, V.P. and Simons, D.B., 1977. Shallow wave propagation in open channel flow. Journal of Hydraulics Division, ASCE, Vol. 103, No. HY12, pp. 1461-1475. Ponce, V.M., Li, R.M. and Simons, D.B., 1978. Applicability of kinematic and diffusion models. Journal of Hydraulics Division, ASCE, Vol. 104, No. HY3, pp. 363-360. Price, R.K., 1985. Flood routing. in Developments in Hydraulic Engineering, Series 3, edited by P. Novak, Elsevier Applied Science Publishers, London, pp. 129-173. Raman, H. and Sunilkumar, N., 1995. Multivariate modeling of water resources time series using artificial neural networks. Hydrological Sciences Journal, Vol. 40, No. 2, pp. 145-163. Rosenblueth, E., 1981. Point estimates for probability. Applied Mathematical Modeling, Vol. 5, pp. 329-335. Russell, S.O. and Campbell, P.F., 1996. Reservoir operating rules with fuzzy programming. Journal of Water Resources Planning and Management, ASCE, Vol. 122, No. 3, pp. 165-170. Savic, D.A., Walters, G.A. and Davidson, J.W., 1999. A genetic programming approach to rainfall-runoff modeling. Water Resources Management, Vol. 13, pp. 219-231.

20

Singh, V.P., 1988. Hydrologic Systems: Vol. 1. Rainfall-Runoff Modeling. Prentice Hall, Englewood Cliffs, New Jersey. Singh, V.P., 1996. Kinematic Wave Modeling in Water Resources-Surface Water Hydrology. John Wiley & Sons, New York. Singh, V.P., 1998. The use of entropy in hydrology and water resources. Hydrological Processes, Vol. 11, pp. 587-626. Singh, V.P. and Aravamuthan, V., 1997. Accuracy of kinematic-wave and diffusion-wave approximations for time-independent flow with momentum exchange included. Hydrological Processes, Vol. 11, pp. 511-532. Strupczweski, W. and Napioprkwoski, J.J., 1990. Linear flood routing model for rapid flow. Hydrological Sciences Journal, Vol. 35, No. ½, pp. 149-164. Tilmant, A., Vanclosster, M., Duckstein, L. and Persoons, E., 2002. Comparison of fuzzy and nonfuzzy optimal reservoir operating policies. Journal of Water Resources Planning and Managemnt, ASCE, Vol. 128, No.6, pp. 390-398. Tsai, C. and Franceschini, S., 2003. An improved point estimate method for probabilistic risk assessment. Proceedings of World Water and Environmental Resources Congress, ASCE, Phildelphia. Wang, Q.J., 1991. The genetic algorithm and its application to calibrating conceptual rainfall-runoff models. Water Resources Research, Vol. 27, No. 9, pp. 2467-2471. Wardlaw, R. and Sharif, M., 1999. Evaluation of genetic algorithms for optimal reservoir system operation. Journal of Water Resources Planning and Management, ASCE, Vol. 125, No. 1, pp. 25-33. Woolhiser, D.A. and Liggett, J. A., 1967. Unsteady one-dimensional flow over a plane: the rising hydrograph. Water Resources Research, Vol. 3, No. 3, pp. 753-771. Xiong, L., Asaad, Y., Shamseldin, Y. and O’Connor, K.M., 2001. A non-linear combination of the forecast of rainfall-runoff models by first order Takagi-Sugeno fuzzy system. Journal of Hydrology, Vol. 254, No. 1-4, pp. 196-217. Yen, B, C., 1979. Unsteady flow mathematical modeling techniques. in Modeling of Rivers, edited by H.W. Shen, Wiley Interscience, New York, New York, pp. 13.1 to 13.33. Yen, B.C., 1982. Some measures for evaluation and comparison of simulation models. in Urban Stormwater Hydraulics and Hydrology, edited by B.C. Yen, Water Resources Publications, Littleton, Colorado.

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Yu, P.S., Chen, C.J. and Chen, S.J., 2000. Application of gray and fuzzy methods for rainfall forecasting. Journal of Hydrologic Engineering, ASCE, Vol. 5, No. 4, pp.339-345. Yu, P.S. and Yang, T.C., 2000. Fuzzy multi-objective function for rainfall-runoff model calibration. Journal of Hydrology, Vol. 238, No. 1-2, pp. 1-14. Yu, P.S., Yang, T.C. and Chen, S.J., 2001. Comparison of uncertainty analysis methods for a distributed rainfall-runoff model. Journal of Hydrology, Vol. 244, pp. 43-59. Yue, 2001) is missing

22

• River Network

• River Meandering

• River Braiding

• Bed forms

Figure 1. River Geometry.

θ

23

Figure 2. Lateral inflow composed of m pulses, each of duration D time units.

qr(t)

0 1 2 3 4 m-1 mt

q0 q1

q2

q3

qm-2

qm-1

qm