SPRING 2000

CEE 360/548: Risk Assessment and Management

Prof. Erik VanMarcke

ASSIGNMENT #4

The problems below concern probabilistic modeling of extreme flood data, the basis for the hydrologic design of structures like dams. Table 1 provides a set of data of annual peak flood flows (in cubic feet per second -- cfs) for "Cypress Creek" for the period from 1945 through 1984. Both the flows and the logarithms of the flows are listed, along with a set of sample statistics such as means and variances.

PROBLEM # 1 -- ANALYSIS BASED ON THE NORMAL DISTRIBUTION

(a) Construct a histogram and cumulative frequency distribution for the flow data, using a class interval of 2000 cfs.

(b) Assuming the annual peak flow data are normally distributed, with mean and variance respectively equal to the sample mean and standard deviation of the annual peak flows, find:

(i) the peak flow of the "25-year flood"

(ii) the peak flow of the "50-year flood"

(iii) the peak flow of the "1000-year flood"

(Use the Standard Normal Distribution Table, as on page 20 in the first set of handouts). The "T-year flood" has, by definition, probability (1/T) of occurring in any one year, that is, P[X > x] = 1/T(x); T(x) is called the "(mean) return period" of flood level x.

PROBLEM # 2 -- ANALYSIS BASED ON THE LOGNORMAL DISTRIBUTION

Using the Cypress Creek data expressed as logarithms (base 10), proceed as in Problem #1, namely:

(a) Construct a relative frequency histogram and a cumulative frequency distribution for the "log-flow" data, using a class interval of 0.5 (log-base-10 cfs).

(b) Assuming the annual peak flow data are lognormally distributed (implying that the "log-flow" data are normally distributed) find :

(i) the peak flow of the "25-year flood"

(ii) the peak flow of the "50-year flood"

(iii) the peak flow of the "1000-year flood"

PROBLEM # 3 -- ANALYSIS BASED ON THE TYPE-I EXTREME VALUE DISTRIBUTION

Again applying the "method of moments" (i.e., equating the sample mean and variance to the true mean and variance) to a list of the annual flood flows, estimate the "25-yr, 50-yr and 1000-yr flood flows" based on theType-I Extreme Value distribution.

NOTE: The Type-I Extreme Value distribution (or the "Gumbel" distribution) is defined by F(x) = P[X < x] = exp{- exp{-a (x - u)]}, for any x-value. The mean of X equals (u + 0.577/a) and its variance is ¹²/(6a²).

PROBLEM # 4 -- Use the information from Problems #1, 2, and 3

(a) A classical flood-design standard was to build a structure to withstand twice the magnitude of the largest flood that occurred historically, which equals 15,600 cfs in the case of Cypress Creek based on the record covering the period 1945-1984. Estimate the "(mean) return period" (the reciprocal of the annual design-flood exceedance probability) associated with this design.

(b) To appreciate the pitfalls in the philosophy underlying the design standard in Part (a) above, consider the hypothetical case in which flood-data recording for Cypress Creek had started in 1950 instead of in 1945. To simplify the analysis, don't necessarily change the computed mean and variance or the assumed probability distribution, but evaluate the "(mean) return period" of a design based on twice the magnitude of the largest flood, assuming a record covering the period 1950-1984. Compare the implied annual probabilities of exceeding the "design level" for cases (a) and (b).

PROBLEM # 5 -- (Quick/Easy and Not Based on Answers to Preceding Problems)

A temporary dam, called a cofferdam, is being designed to protect a 5-year construction project. The cofferdam's design is based on "the 25-year flood". What is the chance that the cofferdam will be overtopped by a flood

(a) at no time during the project ?

(b) at least once during the project ?

(c) during the first year of construction ?