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‫ﻛﻠﯿﺔ اﻟﮫﻨﺪﺳﺔ‬
‫ﻗﺴﻢ اﻟﮫﻨﺪﺳﺔ اﻟﻤﺪﻧﯿﺔ‬

‫‪Probability in Hydrology‬‬

‫اﻻﺳﻢ ‪ :‬ﻳﺎﺳﺮ ﻋﺎﻣﺮ ادرﻳﺲ ﻛﺎظﻢ‬

‫اﻟﻤﺮﺣﻠﺔ اﻟﺮاﺑﻌﺔ‬
‫‪C‬‬
Introduction
• Rainfall depths for specific durations and streamflow peaks occurring during
long periods of time are stochastic variables and can be analyzed as such.
Statistical and probabilistic analyses allow the development of probability
statements or estimates related to the magnitude of certain events. Such
estimates can be used for design purposes.Random or stochastic variables
may be discrete or continuous. Rainfall accumulation and streamflow are
generally considered continuous, since they may take any value on the real
axis (or at least at any positive value). To quantify or parameterize the
probability of occurrence of a continuous random variable, one can use a
standard probability density function (pdf) f(X). The probability of a random
variable, X, taking a value in the infinitesimal range [X, X + dX], is f(X) dX.
Given the pdf f(X), then the probability that the random variable X assumes
a value between x1 and x2 is
 Procedures for Flood-Frequency
Analysis
• Two broad classes of flood-frequency analyses can be identified, depending upon
whether or not stream-gaging records exist at or near the location of interest. Procedures
available for use at ungaged sites, however, either use directly, or depend on, results of
procedures designed for use at gaged sites; they offer no useful insights that are not
given more clearly by discussion of procedures designed for use at gaged sites. For this
reason, only those procedures designed for use with gage data are discussed here.In
order to promote correct and consistent application of statistical flood- frequency
techniques by the many private, local, state, and federal agencies having responsibilities
for water resource management, the U.S. Water Resources Council formulated a set of
guidelines for flood-frequency analysis known as Bulletin 17, or the WRC procedure
(Interagency Advisory Committee on Water Data, 1982). These guidelines, originally
issued in 1976, reflect an evaluation, selection, and synthesis of widely known and
generally recognized methods for flood-frequency analysis. The guidelines prescribe a
particular procedure but do permit properly documented and supported departures from
the procedure in cases where other approaches may be more appropriate. Federal
agencies are expected to follow these guidelines, and nonfederal organizations are
encouraged to do so.
• frequency analysis overall has six major steps:

1. systematic record analysis;

2. outlier detection and adjustment, including historical
adjustment andconditional probability adjustment;
3. generalized skew adjustmen
4. frequency curve ordinate computation;
5. probability plotting position computation; and
6. . confidence-limit and expected-probability computation.
Flood Recurrence Intervals and Flood
Risk
A widely used means of expressing the magnitude of an annual flood relative to
other values is the return period or the probability of exceedance. The 100-year or 1
percent chance flood is defined as that discharge having a 1 percent average
probability of being exceeded in any one year; this discharge can be estimated from
the probability distribution of annual floods. (Note that the 100-year flood is not a
random event like the flood that occurs in a particular year; rather it is a quantile of
the flood-frequency distribution.) If the occurrence of an annual flood exceeding the
100-year flood is called an exceedance or a success, and if annual floods are
independent of each other, then the probability of an exceedance on the next trial is
1 percent, regardless of whether the present trial resulted in success or failure. The
probability that the next trial will fail but the second one succeed is 0.99 × 0.01. The
probability that the next success will be on the third trial is 0.99 × 0.99 × 0.01, on
the fourth trial (0.99)3 × 0.01, and so on. Thus, the most likely time for the next
success or exceedance of the 100-year flood is on the next trial. The average time to
the next exceedance, however, works out to be 100 trials. This paradox is resolved by
noting that the probability distribution of in table D-1
Estimating the Return Period of the
PMF
• In this section we consider if it is possible to credibly estimate the probability that
a probable maximum flood (PMF) estimate will be exceeded, even to within an
order of magnitude, through examination of the rainfall-runoff relationship and
the probabilities of various events. The following section considers whether
statistical approaches are able to provide reliable and credible estimates of flood-
frequency curves out into the 10,000- to 1,000,000-year event range.It would be
useful if a reliable and credible estimate of the return period or, equivalently, the
exceedance probability of a PMF could be obtained by analysis of rainfall-runoff
processes. Thus, one would start with an analysis of the frequency of extreme
precipitation, as has been done in some studies Then one would need to consider
how such precipitation totals would be distributed in time and would interact with
winds and antecedent conditions within the basin (both moisture levels in the soil
and snowpack in some places). All possible combinations of these factors and the
probabilities they would occur jointly must then be determined to arrive at the
frequency distribution of extreme floods, as indexed by the surcharge over the dam
or the flood volume, maximum discharge rate, or jointly by both.
Frequency Analyses for Rare Floods
• provides uniform procedures for estimation of floods with
modest return periods, generally 100 years or less.
Extrapolation much beyond the 100-year flood using flood-
frequency relationships based on available 30- to 80-year
systematic records is often unwise. First, the sampling error in
the 2 or 3 estimating parameters is magnified at these higher
return intervals (Kite, 1977). Moreover, the available record
provides little indication of the shape of the flood-frequency
curve or confirmation of a postulated shape at the extreme
return periods of interest here.
Estimating the Probability Density
Function
• If F(q) is the probability of a flood less than q, then the probability
density function f(q) is the first derivative of F(q) with respect to
q. The best way to calculate f(q) is to develop an analytical
description of the flood-frequency curve yielding an analytical
expression for f(q). For example, if F(q) is obtained by a linear
extension on lognormal paper of the flood-frequency curve
through the PMF, then over that range
Glossary
• ONE-HUNDRED- YEAR (100- YEAR) EX- CEEDANCE INTERVAL
FLOOD ::
• The flood magnitude expected to be equaled or exceeded on the average of
once in 100 years. It may also be expressed as an exceedance frequency
with a 1 percent chance of being exceeded in any given year.

• Probability:: The likelihood of an event's occurring.

• Probability maximum flood ( PMF ) :: The flood that may be expected from
the most severe combination of critical meteorologic and hydrologic
conditions that are reasonably possible in the region. This term as official
documents of the Corps of Engineers identifies estimates of hypo
REFERENCES AND
BIBLIOGRAPHY
• Ahlberg, J. H., E. N. Nilson, and J. L. Walsh (1967). The Theory of Splines and
Their Application . New York: Academic Press
• Algermissen, S. T. (1969). Seismic risk studies in the United States. Paper
presented at 4th World Conference on Earthquake Engineering, Santiago, Chile.
• American Society of Civil Engineers (1973). Re-evaluating Spillway Adequacy of
Existing Dams. New York: Task Committee of the Committee on Hydrometeorology,
Hydraulics Division.
• Benjamin, J. R. and C. A. Cornell (1970). Probability, Statistics and Decisions for
Civil Engineers.New York: McGraw-Hill.
• Cohn, T. and J. R. Stedinger (1983). The use of historical flood records in flood
frequency analysis.Paper presented at American Geophysical Union Meeting, San
Francisco, California,December 1983.