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Y.K. Tung
Symposium Proceedings WWRC-89-30 1989
In
Channel Flow and Catchment Runoff
Proceedings of the International Conference
for Centennial of Manning's Formula and
Kuichling's Rational Formula
Yeou-Koung Tung, Assoc. Professor
Wyoming Water Research Center
and Statistics Department
University of Wyoming
Laramie, WY 82071
ABSTRACT
Channel flow routing provides information on the movement of flood wave as it travels downstream. From the flood forecast viewpoint, it is desirable to have some knowledge about the expected arrival time of peak flow. However, many parameters involved in the flow routing model, in reality, can not be quantified with certainty. In this paper, a probabilistic analysis using the Mellin transform is performed to assess the uncertainty in computing the travel time of flood wave based on the kinematic wave model. Information about the uncertainty of the arrival time of flood peak may have a potential impact on decision making relating to various contingency plans in attempting to alleviate the potential flood damage.
INTRODUCTION
Channel flood routing is an important hydraulic engineering practice because it deals with the modeling of movement of flow along the channel over time. Results of channel flow routing provide information regarding to the temporal and spatial distributions of flood wave which is essential for flood warning and protection.
Mathematical models used in channel flood routing range from very sophisticated 3D distributed- parameter models such as the St. Venant full dynamic wave equation to the simple lump-parameter Muskingum model (Chow et al., 1988). Intuitively, a sophisticated model not only is expansive and difficult to use, but also it requires a large amount of data with reasonable accuracy in order to produce reliable results. The selection of an appropriate model to use would generally be dictated by a number of factors such as objectives of the study, accuracy requirements, data availability, budget and time constraints, and others. The solution to a channel flood routing model is a function of hydrologic characteristics of the channel such as roughness, slope, channel geometry and boundary conditions.
Deterministic analysis of channel flood routing would assume that all model parameters, boundary conditions and initial conditions are known with certainty. Such idealizations, in general, are not true because the hydraulic characteristics of channels in the model constantly vary both in time and in space. Therefore, the uncertainty in model parameters would be carried through the computation and appear in the solution.
Uncertainty in hydraulic computation for water flowing through or passing a hydraulic structure can broadly be classified into three types (Tung and Mays, 1980): model uncertainty, parameter uncertainty, and uncertainty in operational conditions of hydraulic structures. Model uncertainty results from the use of a simplified hydraulic equation to describe a complex flow phenomenon. Parameter uncertainty arises from the inaccuracy associated with the quantification of model parameters. Uncertainty in operational conditions of hydraulic structures relates to the progressive change over time of hydraulic characteristics of the structures due to sedimentation and clogging that might change the effective opening size of hydraulic structure. For flood routing in a natural channel, the uncertainty in operational conditions could be lumped into the category of parameter uncertainty.
Uncertainty analyses of hydraulic computations in channel flood routing are mainly concerned with the assessment of the uncertainty feature of the computation results. In channel flood routing, the results of primary interest are the travel time of flood peak, the magnitude of peak and corresponding water surface profile along with the area of inundation. This paper examines the uncertainty of the travel time derived from using the kinematic wave routing procedure. The method of uncertainty analysis employed is called the Mellin transform.
KINEMATIC WAVE MODEL
The kinematic wave model describes flood wave motion without considering the influence of mass and hydrodynamic force. It is a simpler version than the full dynamic wave model but still is regarded as a distributed-parameter model. The 1D kinematic wave model is defined by the following coupled equations:
Continuity: dQ/dx + dA/dt = q (1a)Momentum: So = Sf (1b)
where Q and A are discharge and flow cross-sectional area, respectively, which are function of both spatial location x and time t, q is the lateral inflow, and Soand Sf are channel bottom slope and frictional slope, respectively. Sometimes, the momentum equation can also be expressed in the form
A = a Qb (2)
Many of the well-known hydraulic equations such as Manning's formula can be expressed as Eq. 2. Utilizing Eq.2, the continuity equation, although originally involving two dependent variables, A and Q, can be reduced to
dQ/dx + (abQb-1) dQ/dt = q (3)
In theory, the kinematic wave should advance downstream with its rising limb getting steeper. However, the size of the wave does not become longer, or attenuated. Therefore, the peak of flood stays the same (Chow et al., 1988).
The kinematic wave model can be solved by using the finite difference scheme or the method of characteristics. For a problem with simpler boundary and initial conditions, kinematic wave routing can be performed analytically. For example, in case that q = 0 and a, b values are known, the outflow hydrograph at the downstream end from routing an inflow hydrograph can be easily obtained by shifting the inflow discharge by a time interval of L/ck, where ck, is the wave celerity, dx/dt. In other words, a discharge Q entering a channel of length L at time t0 will appear at the outlet at time
t = t0 + L/ck (5)
It can be easily proved that, using Manning's formula, the travel time T of a kinematic wave in a wide rectangular channel carrying a flow of Q is (Chow et al., 1988)
T = 3/5 [nB2/3/1.49 S1/2]3/5 Q- 2/5 L (6)
where B is the channel width, n is the Manning's roughness, and L is the length of channel reach.
UNCERTAINTY ANALYSIS OF TRAVEL TIME
In facing major flood events, the local authority in charge of natural disaster emergency management might consider issuing an evacuation warning. The timings of announcing the warning and of implementing the evacuation would indeed cause public inconvenience even if the threat is imminent. The prediction of the arrival time of a major flood can be made by solving a flood routing model. However, due to the existence of various hydraulic uncertainties as described previously, the arrival time of a flood is generally not certain. The purpose of uncertainty analysis of the travel time is to assess its random nature.
Uncertainty characteristics of the travel time can generally be defined by its statistical moments such as the mean, variance, and other higher moments if necessary. The complete derivation of the probability density function (pdf) as the function of pdf's of model parameters with uncertainty is generally very difficult, if not impossible. Using Eq. 6 as an example, the uncertainty of T is a function of uncertainties in n, B, s0, Q, and L which are related in the multiplicative fashion. One commonly used method for assessing the statistical moments in the uncertainty analysis is the first-order second-moment (FOSM) (Benjamin and Cornell, 1970; Ang and Tang, 1975; Yen et al., 1986). However, the FOSM analysis only gives approximations of the mean and variance instead of their exact values. When higher order moments are needed, the FOSM method would become computationally cumbersome as the order gets larger. There are evidences showing that when the nonlinearity of the functional form gets higher the accuracy of approximation by the FOSM method would get worse, especially for high order moments (Tung and Hathhorn, 1988).
Consider a general functional relationship between a dependent random variable Y and a number of independent random variables X = (X1, X2,...., Xk) as
Y = u (X) = u(X1, X2,...., Xk) (7)
If the functional relation Eq.7 satisfies two conditions, the exact moments of any order can be derived by the Mellin transform as the function of moments of X's without extensive simulation. The two conditions are: (1) function u(X) has a multiplicative form as Eq.6 and (2) random variables X's are independent, nonnegative. In general, the non-negativity condition of X's is not strictly required in the Mellin transform; but it would require some mathematical manipulations to find the Mellin transform of a function with its variables that can take negative values. The Mellin transform is particular attractive in uncertainty analysis of hydrologic and hydraulic problems because many equations and the variables involved satisfy the above two conditions.
MOMENTS AND MELLIN TRANSFORM
Statistical Moments:
The statistical moment of order r of a random variable X about any reference point X = x0 is defined as
mr0 = E[(x-x0)r] =
-¥ ¥ (x-x0)r f(x) dx (8)
where E [ ] is an expectation operator and f (x) is the pdf of random variable X. In general, moments in uncertainty analysis that are commonly used are central moments with reference point x0 = m and the moments about the origin with x0 = 0. In other words, the central moments and moments about origin can be defined, respectively, as
m r = E[(X-m )r] (9)m 'r = E[Xr] (10)
where m = E[X] = expectation of random variable X. It can be easily shown, through the binomial expansion, that the central moments are related to the moments about the origin as
mr = åri=0 rCi (-1)i mi m !r-i (11)
where r Ci = r!/ [ (r-i) ! i!]. More specifically, the second, third, and fourth central moments can be expressed as
s 2 = E(X2) - m2 (12a)in which s 2 is the variance. From Eqs. 12b and 12c, one can calculate the skew coefficient (g) and kurtosis (k) asm 3 = E(X3) -3mE(X2) + 2m3 (12b)
m 4 = E(X4) -4mE(X3) + 6m2E(x2) -3m4 (12c)
g = m3 / s 3 (13)k = m4 / s 4 (14)
The Mellin Transform
The Mellin transform of a function f (x), Mx(s), where x is positive, is defined as (Giffin, 1975; Springer, 1978)
Mx(s) = M[f(x)] =
Like Fourier and Laplace transforms, there exists a one-to-one correspondence between Mx (s) and f(x) . When f(x) is a pdf, one can easily identify the relationship between the Mellin transform of a pdf and the moments about the origin as
m's-1 = E (xs-1) = Mx(s) (16)
for s = 1,2,.. As can be seen, Mellin transform provides an alternative way to find the moments of any order of a random variable.
Operational Properties of Mellin Transform
Consider the product of two independent non-negative random variables, Z = XY. The pdf of Z, f(z), can be obtained as
f(z) =
where g( ) and h( )are the pdf's of X and Y, respectively. In fact, Eq.17 is exactly the definition of Mellin convolution. Therefore, similar to the convolutional property of Laplace and Fourier transforms, the Mellin transform of f(z) can be obtained as
Mz(s) = M[f(z)] = M[g(x) * h(y)] = Mx(s) * My(s) (18)
in which `*' is the convolution operator. From Eq. 18, the Mellin transform of the convolution of the pdf's of two random variables in the product form is simply the product of the Mellin transform of two individual pdf's. From this convolutional property of the Mellin transform and its relationship between statistical moments, one can immediately see the advantage of the Mellin transform as the tool in obtaining the moments of a random variable which is related to other random variables in a multiplicative fashion. In addition to the convolutional property, which is of primary importance, the Mellin transform also has several useful operational properties which are summarized in Table 1. These properties of Mellin transform can be easily derived from the basic definition given in Eq.15.
Applying the definition of the Mellin transform and its basic operational properties in conjunction with the convolutional properties, the Mellin transform of algebra of random variables in the form of products and quotients can be derived. Some useful results are summarized in Table 2.
Mellin Transform of Some Probability Density Functions
In uncertainty analysis, parameters with uncertainty are generally treated as random variables with the associated pdf. Given the functional relationship as Eq.7 in which X's are related in multiplicative fashion, the moments of Y can be obtained by Mellin transform of the pdf of X's. From the previous studies (Epstin, 1948; Park, 1988), the Mellin transform of some commonly used pdf's are tabulated in Table 3. Using the results in Tables 2 and 3, one can easily derive the exact moments of the dependent random variables.
APPLICATION
In this section, the Mellin transform is applied to Eq.6 to obtain the statistical moments of the travel time derived from a kinematic wave model. Since T is related to n, B, S0, Q, L in a multiplicative manner and all the parameters with uncertainty are non-negative, the Mellin transform is especially applicable. Based on Eq.6, the Mellin transform of the pdf of the travel time can be expressed as
E(Ts-1) = MT(s)
= cTs-1 Mn(0.6s+0.4) * MB(0.4s+0.6) * Ms0(-0.3s+1.3)
MQ(-0.4s+1.4) * ML(s) (19)
where cT = 0.6(1.49)-0.6. More specifically, the first four moments of travel time about the origin are
