Uncertainty analysis of open channel ow: Stochastic ...scientiairanica.sharif.edu/article_4154_d919e51c5aa17be737cf3f8cf4946fcb.pdfStochastic solution. Abstract. In uid mechanics applications,

Scientia Iranica A (2017) 24(5), 2262{2270

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Uncertainty analysis of open channel ow: Stochasticapproach to advection-di�usion equation

H. Khorshidi�, N. Talebbeydokhti and Gh. Rakhshandehroo

Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran.

Received 9 October 2015; received in revised form 18 April 2016; accepted 18 June 2016

KEYWORDSAdvection-di�usionequation;Karhunen-Loeveexpansion;Monte Carlosimulation;Open channel ow;Stochastic solution.

Abstract. In uid mechanics applications, transport occurs through the combination ofadvection and di�usion. This paper presents a stochastic approach to describe uncertaintyand its propagation based on Advection-Di�usion Equation. To assess the uncertaintyin initial water depth, random initial condition is imposed on the framework of 1D opencannel ow. Karhunen-Loeve Expansion is adopted to decompose the uncertain parameterin terms of in�nite series containing a set of orthogonal Gaussian random variables.Eigenstructures of covariance function associated with the random parameter, which play akey role in computing coe�cients of the series, are extracted from Fredhulm's equation. The ow depth is also represented as an in�nite series of its moments, obtained via polynomialexpansion decomposition in terms of the products of random variables. Coe�cients ofthese series are obtained by a set of recursive equations derived from the ADE. Resultshighlight the e�ect of various statistical properties of initial water depth. The mean valueand variance for the ow depth are compared with Monte Carlo Simulation as a reliablestochastic approach. It is found that when higher-order approximations are used, KLEresults would be as accurate as the results of MCS, however, with much less computationaltime and e�ort.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Given the heterogeneous nature of many uid ows anddi�culties associated with understanding this hetero-geneity accurately, ow characteristics are often treatedas random functions, leading to governing equationsof stochastic types. Hence, it is no surprise thatstatistical estimation of such stochastic processes hasreceived considerable attention as an active �eld in real-world simulations. Although there has been continuouse�ort to develop stochastic models in various �elds of

*. Corresponding author. Fax: +98 713 647 3161Email addresses: [email protected] (H. Khorshidi);[email protected] (N. Talebbeydokhti);[email protected] (G. Rakhshandehroo)

doi: 10.24200/sci.2017.4154

uid ows, their application in open channel ows hasreceived less attention.

Advection-Di�usion Equation (ADE) arises usu-ally in transport modeling �elds of applications and inany disturbance analysis of surface water ows. In such ows, �eld parameters are often in uenced by uncer-tainty due to the lack of understanding of natural openchannel properties including roughness coe�cient, bedslope, and initial or boundary conditions.

A conventional method to solve Partial Di�er-ential Equations (PDEs) stochastically is PolynomialChaos Expansion (PCE). It was put forward byGhanem and Spanos [1], with application to trans-port in heterogeneous media [2,3] and di�usion prob-lems [4]. PCE is applied to model the uncertaintypropagation from the beginning of a waterhammerwith random system parameters and internal boundaryconditions [5]. This technique includes representing

H. Khorshidi et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2262{2270 2263

the random variables in terms of polynomial chaosbasis and deriving appropriate discretized equationsfor the expansion coe�cients via Galerkin technique.PCE allows for high-order approximation of randomvariables and possesses fast convergence under certainconditions. However, the deterministic coe�cients ofPCE are governed by a set of coupled equations, whichare di�cult to solve when the number of coe�cientsis large. PCE is based on the expansion of variablesby products of polynomial coe�cients and orthogonalchaos bases. It is needed to treat a system of equationsnumerically. PCE applications to stochastic shallowwater ows were reported by Ge et al. [6] and Liu [7].A comprehensive review of PCE approach is presentedby Debusschere et al. [8].

Karhunen-Loeve Expansion (KLE) is a exibleapproach to solve PDEs stochastically, leading tohigh-order moments with relatively small computa-tional e�orts. PCE [9,10], probabilistic collocationmethod [11,12], and KLE [9,13,14] have been utilizedto illustrate random processes in porous media. Thismethod is applied to decompose the solution to Boussi-nesq equations for the velocity, density, and pressure�elds [15].

KLE approach has proved to be e�cient for un-certainty analysis in groundwater hydraulics [9,13,16].Contrary to PCE, the coe�cients associated withKLE appear in uncoupled equations, from which therequired statistical moments can be extracted. How-ever, this method has received little attention in openchannel applications.

A reliable tool usually used as a reference forsolving stochastic PDEs is Monte Carlo Simulation(MCS), which consists of three steps; 1) generatingseveral realizations of the uncertain parameter basedon its distribution function, 2) solving the governingequation by means of an appropriate deterministicscheme for the generated parameter, and 3) takingstatistical moments on the entire realizations obtainedin previous steps. MCS is simple to implement;however, it requires considerable computational e�ortdue to large number of realizations needed. Applicationof MCS in open channel ow has been reported byGates and Alzahrani [17,18] for Colombia River in US.The uncertainties in geometrical properties and bedslope are investigated via their distribution functionsand, consequently, statistical moments are evaluatedfor the ow �eld. A virtual sampling MCS is proposedto address uncertainty quanti�cation in ood modelingon a real test case for Tous dam break in Spain [19].Dutykh et al. [20] adopted MCS to quantify the e�ectof bottom roughness on maximum run-up height byresorting to nonlinear shallow water equations. Multi-level MCS is applied to uncertainty quanti�cation forporous media ow [21].

In the present work, KLE approach is applied to

1-D Advection-Di�usion equation considering uncer-tainty in the initial condition for a synthetic case inopen channel ow. Consequently, ow �eld parameterhas appeared as a random variable, too. Validity of theproposed model has been ensured through comparingit with MCS results for various spatial variability andcorrelation lengths.

2. ADE governing equation

On the Advection Di�usion equation method for wa-ter wave propagation, transport occurs through thecombination of advection and di�usion. ADE is de-rived mostly from the incorporation of advection intodi�usion equation, and could be utilized in di�erentgeometries and conditions. The derivation of the ADErelies on the principle of superposition: Advection anddi�usion can be added together if they are linearly in-dependent; the only way they can be dependent is whenone process feeds back into the other. One-dimensionalADE could be written for an incompressible uid as:

@H@t

+ U@H@x�D@2H

@x2 = 0; (1)

in which H(x; t) is ow depth, U is ow velocity(or advective velocity), and D is di�usive coe�cient(or di�usivity), subject to the initial and boundaryconditions given by:

H(x; 0) = h0 + asech2

"s3a4h3

0

�x� L

2

�#; x 2 D;

(2)

H(0; t) = H(L; t) = h0; t > 0; (3)

where L is channel length, H(x; 0) or h(x) is initialwater depth (treated hereafter as a random variable),h0 is undisturbed uniform water depth, a is initial waveheight, and D is spatial domain in x direction. Theinitial condition corresponds to a �rst-order solitarywave propagating in the positive x-direction [22]. Therandom nature of h(x) converts deterministic equations(Eqs. (1)-(3)) into stochastic ones, the solution towhich is sought in the form of statistical moments.The length of the channel is assumed su�cientlylarge compared to the characteristic length of solitarywave [23]. This justi�es the validity of Eq. (3), implyingthat the boundary values remain una�ected by theinitial wave form.

3. KLE stochastic representation of ADE

In KLE approach, initial water depth, h(x), is consid-ered a random variable due to many factors includingthe uncertainty inherent in measurements. It maybe decomposed to the mean term, < h >, and the

2264 H. Khorshidi et al./Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2262{2270

uctuation term, h0. KLE expresses h0(x) in terms ofeigenstructure for covariance function, Ch(x1; x2), ofthe random �eld as follows [1]:

h0(x; !) =1Xn=1

�n(!)p�nfn(x); (4)

where x and ! are indices of real and probabilityspaces, respectively; �n(!) is an orthogonal Gaussianrandom variable with zero mean; and �n and fn(x) areeigenvalue and eigenfunction associated with the givencovariance function, respectively. With a covariancefunction for the exponential distribution as:

Ch(x1; x2)�2hexp

��jx1 � x2j

�

�; (5)

the eigenstructures are obtained analytically fromFredholm's equation [13] as:

�n =2��2

h�2w2

n;

fn(x) =[�wn cos(wnx) + sin(wnx)]p

0:5L(�2w2n + 1) + �

; (6)

where �2h and � are variance and correlation length of

the random variable h(x; !), respectively. It is worthmentioning that a similar problem has been treated byZhang and Lu [13] when modeling groundwater ow ina random porous medium. In the above expression, wnrefers to positive roots of the characteristic equation:

(�2w2 � 1) sin(wL) = 2�w cos(wL):

For notational convenience, the functionp�nfn(x) is

replaced with f�n(x) hereafter.

4. Moment equations in KLE

As mentioned, h(x) is considered as a random variableand other terms as deterministic ones. KLE, as a per-turbative expansion technique, expands the dependentvariable H(x; t) by the following series:

H(x; t) = H(0) +H(1) +H(2) + ::: (7)

Substituting the above expansion and h(x) =< h >+h0 in Eqs. (1)-(3) and considering only the zero-orderterms, the governing equation and related conditionswill take the form:

@H(0)

@t+ U

@H(0)

@x�D@2H(0)

@x2 = 0; (8)

H(0)(x; 0)=h0+�asech2

"s3a4h3

0

�x�L

2

�#; x 2 D;

(9)

H(0)(0; t) = H(0)(L; t) = h0; t > 0; (10)

in which �a is mean value of a. The above equationmay be solved for H(0) in a deterministic manner(Appendix A). Similarly, one may obtain the followingexpression for any higher order term (m) [13]:

U@@x

(H(m))�D @2

@x2 (H(m))

+mXk=0

(�1)k

k![h0(x)]k

@H(m�k)

@t= 0; (11)

H(m)(x; 0) = 0; x 2 D; (12)

H(m)(0; t) = H(m)(L; t) = 0: (13)

Various components of Eq. (7) can now be expandedby suitable polynomial expansions in terms of theorthogonal Gaussian random variable � as illustratedin Table 1. where H(1)

i , H(2)ij , and H(3)

ijk (for i; j; k =1; 2; :::) are deterministic coe�cients obtained fromthe associated governing equation, numerically. Notethat the above governing equations are derived viasubstituting the expansions of h0(x) and H(m)(x; t)with m = 1; 2; 3 in Eqs. (11)-(13) and simplifying theresulting expressions in view of orthogonality of therandom variable, �. The equations listed in Table 1 aretreated recursively in a numerical manner (Appendix).

Table 1. Expansions for H(1), H(2), and H(3) in terms of the orthogonal Gaussian random variable �, and the governingequation for any term.

Term Governing equation

First term H(1)(x; t) =P1i=1 �iH

(1)i (x; t) @H(1)

i@t � f�n(x) @H

(0)

@t + U @@x (H(1)

i )�D @2

@x2 (H(1)i ) = 0

Second term H(2)(x; t) =P1i;j=1 �i�jH

(2)ij (x; t)

@H(2)ij@t � 1

2f�i (x)

@H(1)j@t � 1

2f�j (x) @H

(1)i@t + 1

2f�i (x)f�j (x) @H

(0)

@t

+U@H(2)

ij@x �D @2H(2)

ij@x2 = 0

Third term H(3)(x; t) =P1n=1 �nH

(3)n (x; t)

+P1i;j;k=1 �i�j�kH

(3)ijk(x; t)

@H(3)ijk@t � 1

3

PPijk

f�i (x)@H(2)

jk@t + 1

6

PPijk

f�i (x)f�j (x) @H(1)k@t

� 16

PPijk

f�i (x)f�j (x)f�k (x) @H(0)

@t + U@H(3)

ijk@x �D @2H(3)

ijk@x2 = 0


Table 2. Statistical moments of the ow depth.

Flow depth H(x; t) �P3i=0 H

(i)(x; t)

Mean value hH(x; t)i �P3i=0hH(i)(x; t)i = H(0)(x; t) +

P1i=1H

(2)ii (x; t)

Perturbation term H 0(x; t) = H(x; t)� hH(x; t)i �P3i=1H

(i)(x; t)�P1i=1H(2)ii (x; t)

Cross-covariance between initialwater depth and ow depth

ChH(x; y; �) =p�nfn(x)H 0(x; !) = f�n(x)H 0(x; !) =

P1n=1 f

�n(x)H(1)

n (y; �)+3P1i;j=1 f

�i (x)H(3)

ijj (y; �)

Flow depth covarianceCH(x; t; y; �) =

P1i=1H

(1)i (x; t)H(1)

i (y; �) + 2P1i;j=1H

(2)ij (x; t)H(2)

ij (y; �)+3P1i;j=1 H

(1)i (x; t)H(3)

ijj (y; �) + 3P1i;j=1H

(1)i (y; �)H(3)

ijj (x; t)

Flow depth variance �2H(x; t) =

P1i=1

hH(1)i (x; t)

i2+ 2

P1i;j=1

hH(2)ij (x; t)

i2+ 6

P1i;j=1H

(1)i (x; t)H(3)

ijj (x; t)

IndexPPijk(:) is found by a substitution manner, i.e.:X

Pijk

rf�i :rH(2)jk =rf�i :rH(2)

jk +rf�j :rH(2)ik

+rf�k :rH(2)ij :

For the trivial solutions to exist, H(3)(x; t) should beexpanded in terms of �n and �i�j�k simultaneously [13].Manipulating the third-order approximation of H(x; t)(Eq. (7)) mathematically, one may compute highermoments of the ow depth as shown in Table 2. Itis important to note that the same approach is chosento solve the governing equation of H(0) to H(3)

ijk becauseof the diversity in homogeneity property. Despite theexistence of analytical solution to Eqs. (8)-(10) (i.e.,convolution integral), QUICKEST approach (as a �nitedi�erence method expressed in Appendix) is utilizedto treat all the governing equations to have the samesolution process.

5. Hypothetical test problem

KLE approach is applied to a hypothetical channel(Figure 1) to compute higher-order ow depth mo-ments, and veri�ed by comparing its results with thoseof MCS. A hypothetical channel of length L = 100 mis considered with a �rst-order solitary wave withmaximum height of a within normal distribution andmean value of �a = 0:05 m centered at x = L=2.Moreover, the water depth is kept constant at h0 = 1 mover the channel ends, advective velocity U = 2:5 m/s,and di�usion coe�cient D = 1 m2/s. Schematicof the initial condition over the channel and wavepropagation sketch at di�erent times are shown inFigure 1. E�ects of di�erent correlation lengths, �h,and various degrees of spatial variability, �2

h, on owdepth variance, �2

H , have been investigated. MCS isexamined for about 1000 realizations and the moments

Figure 1. (a) Schematic of the initial condition. (b)Wave propagation sketch at di�erent times.

of ow depth are computed for di�erent correlationlengths and variances of the input random variable.

6. Results and discussion

A su�cient number of terms to be incorporated inH(x; t) and, subsequently, in expansions of H(m) areinvestigated. Furthermore, e�ects of correlation lengthand spatial variability of random variable h(x) on vari-ance of the random function �2

H have been discussed.


6.1. Number of terms to be incorporated inH(x:t)

KLE approach is applied to compute mean ow depthpro�le at di�erent times, and compared with MCSones as shown in Figure 2. Incorporation of the �rsttwo terms in Eq. (7) leads the results to a goodagreement with the results of MCS (Figure 2(a)).Incorporating two more terms, i.e., H(2) and H(3),only slightly improves H(x; t) (Figure 2(b) and (c)).Indeed, numerical values of subsequent terms decreaseby one order of magnitude. For example, H(1)(x; t)(with a certain number of terms considered in itsexpansion as illustrated in the next section) takesthe value of 0.01 m; however, the value of 0.001 mis obtained in estimation of H(2)(x; t) with su�cient

Figure 2. Comparison of mean ow depth pro�lescomputed by KLE method (incorporating 2, 3, and 4terms) and those derived by MCS (for 1000 realizations)at (a) t = 5 s, (b) t = 10 s, and (c) t = 15 s.

number of terms considered in its expansion. Moreover,H(3)(x; t) is obtained at the order of 0.0001 m, i.e.two orders of magnitude smaller than H(1)(x; t). Onemay conclude that the more the number of termsincorporated in H(x; t), the more accurate the resultswould be; however, incorporation of four terms (H(0) toH(3)) is deemed su�cient to expand H(x; t), becauseof the size of the disturbance, 0.05 m, caused by thesolitary wave.

6.2. Number of terms in expansion of H(m)

Number of terms considered in expansion of H(m)

has a direct e�ect on the accuracy of results in KLEapproach. Figure 3 depicts magnitudes of H(m)

i1;i2;:::;imfor m = 1; 2; 3 on the center of domain at t = 5 s.The advantage of KLE largely depends on the numberof terms required to approximate the m-order termH(m). On the other hand, the magnitude of H(m)

Figure 3. Values of H(m)i1;i2;:::;im on the center of domain

at t = 5 s for (a) H(1)i , (b) H(2)

i , and (c) H(3)ijk.


should statistically decrease for increasing values of theindices i1; i2; :::; im. This condition is satis�ed for H(1),although it seems that convergence is not achieved(cf. Figure 3(a) regarding the plot of H(1) for �xedt = 5 s) when H(2) and H(3) show a non-decreasingbehavior (cf. Figure 3(b) and (c) regarding the sametime instant). In this computation, 100, 20, and 10terms are found su�cient when evaluating H(1)

i , H2ij ,

and H3ijk, respectively (Figure 3(a) to (c)), meaning

that index i in H(1)i operates up to 100, each index in

H(2)ij operates up to 20, and so on. Number of times,

Sm, necessary to solve governing equations for H(1) toH(3) is obtained from Sm = n(n+1):::(n+m�1)

m! , wheren is the number of deterministic coe�cients in anyexpansion procedure. When N = 15, as an example,it is required to solve governing equations 120 and 680times to compute H(2) and H(3), respectively; muchless than the few thousand times usually needed forMCS.

6.3. Dependency of ow depth variance (�2H)

on input correlation length (�h)Possible e�ects of input correlation length � on owdepth variance, as one of the most important propertiesof the random function H(x; t), are investigated. Forthis purpose, KLE approach is utilized to calculate thevariance for di�erent correlation lengths at t = 5 s,which is later compared to MCS results (Figure 4(a)to (c)). As shown in �gures, 1st and 2nd-order KLEyields almost identical results both revealing a fairagreement with MCS and overestimating the peakpoint by 12%. Similar �ndings achieved for di�erentcorrelation lengths suggest that ow depth variance isindependent from input correlation length. This maybe attributed to the fact that the roots of characteristicequation appear to be sensitive to correlation length.

6.4. E�ects of input variance (�2h) on ow

depth variance (�2H)

Figure 5 compares ow depth variances from 1st and2nd-order KLE approaches with those from MCS fordi�erent input water depth variances of 0.0025, 0.0064,and 0.0121 m2 and correlation length of 4 at t =5 s. As shown, for �2

h = 0:0025 m2, ow depthvariances computed by 1st and 2nd-order KLE areclose to the results of MCS with the 1st order mainlyoverestimating MCS with a maximum error of 9%(Figure 5(a)). As the variance increases, 1st-order KLEresults overestimate MCS again with a maximum errorof 9%; however, 2nd-order KLE underestimates MCSwith a maximum error of 23% (Figure 5(b)). Finally,for �2

h = 0:0121 m2, ow depth variances calculatedby the 1st-order KLE remain unchanged (with errorssimilar to those in the previous cases), but high errorsof up to 52% are observed for the underestimating 2nd-

Figure 4. Comparison of ow depth variances computedby (�rst and second-order) KLE and MCS for: (a)�h = 1 m, (b) �h = 4 m, and (c) �h = 10 m (�2

h = 0:01 m2

for all the cases).

order KLE (Figure 5(c)). It may be concluded that asthe input variance, �2

h, increases, 1st-order KLE resultsremain overestimated and unchanged; however, 2nd-order KLE results increasingly underestimate MCSresults. It is concluded that for higher input variances,unlike our expectations, ow depth variance will notimprove considerably by incorporation of higher order.

7. Conclusions

In this study, KLE approach was applied to solvestochastic ADE solution in a hypothetical channelconsidering uncertainties in the initial condition. Ran-dom variable, h(x), was decomposed to in�nite seriesbased on eigenstructures of its covariance function.The latter was obtained applying Fredholm's solutionto 1D exponential covariance function analytically.Then, ow depth H(x; t) was expanded to series ofH(m), each of which was further expanded to in�nite


Figure 5. Comparison of ow depth variances computedby 1st and 2nd-order KLE with MCS, when (a)�2h = 0:0025 m2, (b) �2

h = 0:0064 m2, and (c)�2h = 0:0121 m2 (�h = 4 m for all the cases).

series based on random variable �. Coe�cients of thelast expansions were deterministically obtained fromnumerical solutions to the governing equation. Arecursive method was utilized by KLE approach which,unlike PCE approach, did not require any couplingprocedure to derive ow depth moments.

KLE approach was adopted to calculate mean ow depth pro�le at di�erent times and comparedwith MCS results. It was concluded that the morethe number of terms incorporated in H(x; t), the moreaccurate the results would be; however, incorporationof four terms (H(0) to H(3)) was deemed su�cientfor the size of disturbance (0.05 m) considered inthis study. It was also found that 100, 20, and 10

terms were su�cient for computing H(1)i , H(2)

ij , andH(3)ijk, respectively, resulting in much smaller number

of times required to solve the governing equationsthan the few thousand times usually needed for MCS.Flow depth variance was found to be independent frominput correlation length. Flow depth variance did notsigni�cantly improve when higher input variances wereconsidered.

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24. Leonard, B.P. \A stable and accurate convective mod-elling procedure based on quadratic upstream interpo-lation", Comp. Meth. in Appl. Mech. and Engage., 19,pp. 59-98 (1979).

25. Abbott, M.B. and Basco, D.R. \Computational uiddynamics", Harlow, Essex: Longman Scienti�c andTechnical, New York, USA, pp. 146-151 (1989).

26. Szymkiewicz, R. \Numerical modeling in open channelhydraulics", 83, pp. 263-300, Springer Science andBusiness Media (2010).

Appendix

Numerical approachi) Homogenous form. Homogeneous form of the

governing equation (Table 1) was solved to obtainH(0)(x; t). Accordingly, QUICKEST (QuadraticUpstream Interpolation for Convective Kinematicswith Estimated Streaming Terms) scheme, as themost frequently applied �nite di�erence approachto solve ADE, was chosen [24-26]. QUICKESTutilizes up to the 3rd order of approximation andimplements explicit centered di�erences as:

Hj+1i =Hj

i +�Cd(1� Ca)� Ca

6

�C2a � 3Ca

+ 2��Hji+1 �

�Cd(2� 3Ca)

� Ca2

(C2a � 2Ca � 1)

�+�Cd(1� 3Ca)

� Ca2

(C2a � Ca � 2)

�Hji�1 +

�Cd:Ca

+Ca6

(C2a � 1)

�Hji�2; (A.1)

in which Hji is ow depth at the ith �t (spatial

step) and jth �t (time step), and Ca and Cd areadvective and di�usive Courant numbers, respec-tively:

Ca =U:�t�x

; Cd =D:�t�x2 : (A.2)

Regions of stability for QUICKEST scheme may bewritten as [24]:8>><>>:

Cd � (3�2Ca)(1�C2a)

6(1�2Ca) if Ca < 12

and Cd � 0Cd � (3�2Ca)(C2

a�1)6(2Ca�1) if Ca > 1

2

(A.3)

ii) Non-homogeneous form. The simplest form ofADE with a source term f may be written as [26]:

@H@t

+ U@H@x�D@2H

@x2 � f = 0: (A.4)


A splitting technique is applied to decomposeEq. (A.4) in the form:

@H@t

+ F (1) + F (2) = 0; (A.5)

where:

F (1) = U@H@x�D@2H

@x2 and F (2) = �f: (A.6)

At any time step, �t, Eq. (A.5) is solved in two stages.In the �rst stage:

@H(1)

@t+ U

@H(1)

@x�D@2H(1)

@x2 = 0;

H(1)(x; t) = H(x; t): (A.7)

The QUICKEST scheme solves the homogeneous formof ADE as outlined earlier. The source term is takeninto account via the second stage:

@H(2)

@t= f; H(2)(x; t) = H(1)(x; t+ �t); (A.8)

subject to the results already obtained in the �rst stageas the initial condition. Finally, results of the secondstage are considered as the values for new time stepsH(x; t+ �t) � H(2)(x; t+ �t).

Biographies

Hossein Khorshidi received a BS degree in Civil En-gineering from Yasouj Branch, Islamic Azad University,Yasouj, Iran, in 2004, and an MS degree in HydraulicStructures, in 2008, from Shiraz University, Iran, wherehe is currently a PhD-degree student in the Departmentof Civil and Environmental Engineering. His researchinterests include stochastic modeling in open channel ows and free surface hydraulics.

Nasser Talebbeydokhti is a Professor of Hydraulicand Environmental Engineering in the Department ofCivil and Environmental Engineering at Shiraz Uni-versity, Shiraz, Iran, where he is currently Head of theEnvironmental Research and Sustainable DevelopmentCenter. He has published more than 80 journal papersand 150 conference papers. His main areas of interestinclude hydraulic engineering, sediment transport, en-vironmental engineering, and hydraulic structures.

Gholamreza Rakhshandehroo is a Professor ofCivil Engineering in the Department of Civil and En-vironmental Engineering at Shiraz University, Shiraz,Iran, where he has published more than 40 journalpapers and 100 conference papers. His main areasof interest include hydraulic engineering, groundwatermathematical modeling, environmental engineering,and climate change.

Documents

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