The flood can be defined from various points of view such as:
Geomorphology:
The flood is a flow that overtops the natural or artificial banks of a stream
Water management:
The flood is the interference of the flow excess with human activity. It is measured by actual or potential economic damage and danger to human life.
Hydrology:
The flood is a discharge wave moving downstream in a channel whereby the water level, water surface slope, velocity and discharge are changing with distance and time.
The magnitude of a flood can be related to several physical features of the flood wave such as

High flood level -the maximum elevation of water level attained during the flood at a fixed point or cross-section.

Peak discharge - the maximum rate of flow during the flood.

Flood volume - the volume of water in the flood wave above a given discharge.

Flood duration - the period during which the discharge does not drop below a certain limit.
Floods are basic elements of the water cycle. Natural sources are rainfall and water released by melting from snow and ice. Besides the runoff generation in the hillslopes areas, the runoff concentration in tributaries and, as a last element, the discharge wave transmission in the large streams, sometimes altered by dams and reservoir operation, are the basic elements of runoff proliferation for floods. Bank erosion, rising of river beds by deposition of silt and changing of the
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river course may result to flooding.
At special occasions, short-term rise in water level may result from ice, debris and boulders blocking the cross-section of a stream. Under normal circumstances, the floods in big river basins result from great amount of rainfall within several days, while for the small watersheds the floods may result from precipitation of local extent.
The flood wave in a stream is governed by temporal and spatial distribution of precipitation, and the storage and retention capacities of the land cover, soil, relief, river beds besides artificial reservoirs.
The coincidence of flood waves of main river reaches and tributaries also may augment flood water level in the main river. Man made influences on water courses may have possible influence of the flood's water level in a river as well.
Severe floods occur when natural interception and retention capacities are reached, with rapid response to precipitation. Man kind is responsible in changing the natural conditions of basin storage through the following interventions.

Sealing by urbanization

Changing landuse

Changing forest to arable land

Straightening and impoundment of water courses

Reduction of natural flooding areas.
The hydrological design is important for the safety, economy and proper functioning of the hydraulic structures. The purpose of the hydrologic design is to estimate the maximum, average or minimum flood, which the structure is expected to handle. the estimate has to be made rather accurately in order that the project functions properly. For example in the design of a spillway, one has to bear in mind that the flood water to be released through the spillway should not create flooding down stream. Thus a balance has to be worked out between the economy, efficiency in regard to flood moderation and safety.
Several methods are available by which the estimate of design flood can be made. Some of the methods are purely empirical and some are based on statistical analysis of the previous records. The commonly used methods can be categorized into two groups. These are

Frequency analysis which involves fitting observed flood data to statistical distributions such as the Gumbel distribution.

Rainfall-runoff modeling which involved estimation of runoff from rainfall. Simple models like the rational method and the unit hydrograph method are used to determine flood peak discharges from measurement of rainfall.
The frequency analysis approach only is discussed in the next section as a means to estimate the design flood for hydraulic structures.
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9.4.1 Introduction
The objective of flood frequency analysis is to estimate a flood magnitude corresponding to a required return period of occurrence. This information is required in the design of hydraulic structures such as dams, bridges, culverts and flood control structures, and further more for flood plain zoning.
Flood magnitude is generally designated by the peak discharge (Q). The return period of occurrence
(T), also known as recurrence interval, is an expression of the frequency of occurrence of a flood of a certain magnitude. Return period is defined as the average time interval between the occurrences of an event equal to or greater than.
The Q -T relationship, i.e., the relationship between the flood magnitude (Q) and its return period
(T) can be estimated from the analysis of observed flood peaks. For the analysis to be valid, the flood peaks, under consideration, must be of random magnitude and mutually independent. Given the usual shortness in observed flood records, extrapolation of frequency functions is quite uncertain so that the larger the flood and the greater the hazard it represents, the less reliable is the estimate of the frequency with which the flood is likely to occur.
9.4.2 Choice of flood model
The basic input in flood frequency analysis is a time series of flow data at a location or region of interest. Two types of flow data are mainly considered in the analysis of flood flows.

Annual Maximum (AM) series

Partial Duration (PD) series
Annual maximum series
The annual maximum (AM) series consists of the peak flow of each year, and is the most frequent used among the two common series. One of the aspects in favor of the AM series is the reasonable assumption that the data series is not serially correlated, i.e., successive values are independent provided that the ‘hydrological year’ or ‘water year’ is carefully chosen. This is an important prerequisite for the subsequent statistical treatment of data. A disadvantage of AM series is that the second or third, etc, highest events in a particular year may be higher than the maximum event in another year and yet they are totally disregarded.
Partial duration series
The disadvantage of disregarding some of the significant high events in any particular year in AM series is remedied in the partial-duration (PD) series sampling method in which all events above a certain base magnitude are included in the analysis. Further more, when only a limited period of record is available, the use of series over a threshold (POT series) has advantages of including a
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large number of flood peaks in the analysis. The base is usually selected low enough at least one event in each year is included. It is important that each event that is included in the partial-duration series must be separate and distinct.
9.4.3 Frequency estimation
There are two basic approaches for estimating frequency curves. These are graphical and analytical.
Graphical approach
Graphically, frequencies are evaluated simply by arranging observed values in the order of magnitude and considering that a smooth curve suggested by the array of values is representative of future probabilities.
The plotting position formulae are applied to compute the probability plotting positions for observed events. In other words, plotting position formulae express the relationship between order number of ordered statistics and corresponding average frequency value of that statistic over a large number of samples. Probability plotting of hydrological data requires that individual observations or data points be independent of each other and that the sample data be representative of the population.
Probability plotting positions are used to produce graphical display of annual maximum flood series and serve as estimates of the probability of exceedance of those values. Probability plots allow a visual examination of the adequacy of the fit provided by alternative parametric flood frequency models. They also provide a non-parametric means of forming an estimate of the data's probability distribution. An example of a probability plot is shown in Figure 9.1.
Figure 9.1 Probability plot of flood flows.
9-4

The common plotting position relationships used in flood frequency analysis are listed below.
California
Hazen
F(i) = i/N
F(i) = (i-0.5)/N
Weibull
Gringorten
Blom
Chegodayev
F(i) = i/(N+1)
F(i) = (I-0.44)/(N+0.12)
F(i) = (i-3/8)/(N+1/4)
F(i) = (i-0.44)/(N+0.40)
On the basis of comparative study of several plotting position relationships, Cunnane (1978) suggested the use of the unbiased plotting formulae as follows:
(a) for Normal probability paper
3
F

8
1
4
(9.1)
(b) for Gumbel and Exponential probability paper
F i
 i
N


0 .
44
0 .
12

 (9.2)
A Probability paper is a special designed paper which will plot a cumulative distribution function as a straight line. If a set of observed data plot as a straight line on probability paper, the data can be said to be distributed as the distribution corresponding to the probability paper used. The probability plot on Gumbel probability paper is shown on Figure 9.2.
Analytical Approach
In analytical approach, the concept of theoretical distributions is employed. A distribution is set of values that occur under fixed conditions in an infinite amount of times. If each element of a population has a given value then the distribution describes the constitution of the population as seen through the elements of the population. This is usually achieved through the description of the statistical properties of the population elements i.e. the mean, standard deviation and skewness.
A distribution can also give the relative frequency in the population in the same way that a histogram gives that information about a sample. These relative frequencies may also be considered as probabilities. The distribution therefore tells the probability, Pr(X<=x),that the X value on an element drawn randomly from the population would be less than a particular value x. Knowing
Pr(X<=x) for all x values , the laws of probability may then be used to deduce the probability of any proposition about the behaviour of a random sample of X values drawn from the population.
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Because of this probability interpretation, a relative frequency distribution is also called a probability distribution and the curve describing it is called a probability density function (p.d.f.), whose cumulative function is called the cumulative distribution function (c.d.f.).
The cumulative distribution function (c.d.f.) expresses probability of non-exceedance Pr(X
 x). Its largest value is unit. The complement of F(x) is called the exceedance probability of x, 1-F(x). The reciprocal of the exceedance probability is the return period T, i.e. 1/(1-F(x)) = T or F(X
T
) = 1-1/T, where X
T
has return period = T. Figure 9.3 shows a typical shape of (p.d.f.) and (c.d.f.).
Figure 9.2 Flood frequency curve plotted on Gumbel probability paper.
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Figure 9.3 A typical shape of (p.d.f.) and (c.d.f.)
Statistical distributions
Many statistical distributions have been proposed for flood frequency analysis todate. Various distributions have been proposed because of their ability to model different shapes of histogram of flood peaks. Typical shapes of histograms of flood peaks are shown in Figure 9.4.
9-7

Figure 9.4 Typical shapes of histogram of flood peaks
The statistical distributions provide the essential basic formulae to model a flood magnitude Q in terms of its exceedance probability or the inverse of the exceedance probability termed return period T.
Many statistical distributions have been proposed to date. In this section only three commonly used distributions are described. These are:
Extreme value type 1 (EV1)
Lognormal (LN)
(Gumbel,1941)
(Hazen,1914)
Log Pearson type 3 (LP3) (USWRC,1967)
Methods of parameter estimation
Fitting a distribution to data sets provides a compact and smoothed representation of the frequency distribution revealed by the available data, and leads to a systematic procedure for extrapolation to frequencies beyond the range of the data set. When flood data from a given site is assumed to follow a certain distribution, the next step is to estimate the parameters of that distribution so that
9-8

the required flood magnitudes can be calculated with the fitted model. Several general approaches are vailable for estimating the parameters of a distribution. The most commonly used estimation methods are:
(i) Method of moments (MOM)
The method of moments is one of the most commonly and simply used method for estimating the parameters of a statistical distribution. In most cases the first three central moments , i.e., the mean, variance and skewness are adequate to estimate the required parameters.
(ii) Maximum likelihood (ML)
Parameters estimated by maximum likelihood are determined by maximizing the sample log likelihood function. The unknown parameters may be obtained by setting each of the partial derivatives with respect to each parameter equal to zero and solving equations simultaneously.
These equations unfortunately do not often take a simple closed form and the numerical solutions have to be used. Sometimes there is a failure to obtain proper solutions to the equations due to complexity of log likelihood functions, particularly when sample size is small or when the distribution has more than two parameters.
(iii) Probability weighted moments (PWM)
This is a statistical estimation procedure based on the calculation of probability weighted moments
(PWM). This method is simple, unbiased and stable.
Application of statistical distributions
(i) Extreme Value type 1 (EV1)
The extreme type I (EV1) distribution is also known as Gumbel distribution. The c.d.f., F(x) of
Gumbel distribution is defined as
F
 e
 e
 x u

(9.3) where μ = Location parameter
α = Scale parameter
Estimation of parameters (MOM)
mean : μ = u + 0.57728α
St. deviation: s = 1.28α
skewness: g = 1.14
The parameters u and α of the distribution are determined from the sample data.
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The flood magnitude X
T of return period T is estimated from the expression
X
T
= u +
(9.4)
α y
T where y
T is a the EV1 reduced variate corresponding to a given return period T. The values of y
T
are provided in Tables 9.1 for selected values of return period.
(ii) Lognormal Distribution
The probability density function (p.d.f.), f(x) and the distribution function (d.f.), F(x) of the lognornal distribution are defined as
2 f
 e

1
2 log e
 x
  y y
(9.5)
 x
1
2

2
F

 x
1
2


0
X e

1
2 log e
 x
  y y dx (9.6) where y = log e x
 y
= Location parameter of y series
 y
= Scale parameter of y series
Estimation of parameters ( MOM)
Mean = E(y) =
 y
varience = E(y-E(y))
2
=
 y
2
Application of the Log-Normal distribution involves transforming annual floods to logarithmic values (y = log e x) and then computing the following statistics:
Mean =μ y
St. deviation =σ y
The flood magnitude X
T
of return period T is estimated from the expression y
T

 y
  y
.
Z
T
(9.7) where Z
T is a standardised reduced variate corresponding to a given return period T. The values of
Z
T
for some selected return periods are provided in Table 9.1. The values of Z
T
for any given nonexceedance probability can be determined from Table 9.2, which gives the area under the normal curve.
The untransformed value of the flood magnitude X
T
= 10 yT
.
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(iii) Log Pearson type III Distribution
The probability density function (p.d.f.), f(x) and the distribution function (d.f.), F(x) of the Log-
Pearson type III distribution are defined as f
   log e x
 x
0
 



1 e
 


 log e
 x
 x
0


(9.8)
F
  x x
0
 log e x
 x
0
 



1 e
 


 log e
 x
 x
0

 dx (9.9)
Where x
0
= Location parameter
β = Scale parameter
γ = shape parameter
Estimation of parameters ( MOM)
mean , μ = x
0
+ βγ
varience, σ
2
= β
2 γ
skewness, g = 2/√γ
Application of the Log-Pearson type 3 involves transforming annual floods to logarithmic values (z
= log
10
X) and then computing the following statistics:
Mean = μ z
St. deviation =σ z
Skewness = g
The flood magnitude z
T
of return period T is estimated from the expression z
T
  z

K
T
.
 z
(9.10) where K
T
= f(T,g), a function of both recurrence interval and skewness. These values are provided in Table 9.3.
The untransformed value of the flood magnitude, X
T

10
Z
T .
Table 9.1 Frequency factors for use in the Log-Normal and EV1 distributions.
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Return Period, T
2
5
10
25
50
100
1000
Reduced variate, z
T
(for log-normal distribution)
0.00
0.84
1.28
1.75
2.05
2.33
3.09
Reduced variate, y
T
(for EV1 distribution)
0.37
1.50
2.25
3.20
3.90
4.60
6.90
The complete elimination of floods is an impossible task, but damages can be reduced by taking preventive measures. Whatever measures which have to be considered to control floods, they have to be guided by the principle to minimise human interferences on the river regime. Further more, the different measures must be integrated into all interdisciplinary water resources management plans of the river basin. The final decision in adopting a flood damage reduction programme is a political issue which will take into consideration economic, social and cultural values and environmental effects. Possible measures which can be considered to reduce flood damage are presented below.
Maintain water retention areas
Restoration of inundation plains and natural river system, maintaining and promoting small-scale structures for water retention in the land scape.
Maintain the natural water ways
To secure discharge capacity of the water course, flooded areas and flow paths must be maintained and for ecological reasons be enlarged.
Flood protection
Construction of dikes, walls, retention basins and reservoirs may provide protection against the dangers of floods up to a certain design flood level.
Flood zoning
Technical flood protection does not guarantee absolute safety. A detailed flood risk map must be prepared. In order to reduce the flood damage potential, the flood zoning map should indicate activities allowed to be undertaken on the map. For example no construction permits should be issued in flood-prone areas.
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Catchment management
Conservation of natural forests and afforestation has the effect of increasing infiltration because vegetation cover retards surface flow, giving the water additional time to enter the soil. This results in reducing the discharge peaks.
Increase the awareness of flood risk
The society has to be informed of the threat of floods as part of the awareness of the natural conditions on water courses and where necessary be advised to construct houses which are flood proof.
Flood warning system
The flood warning system is a tool which can be used to give a warning of an extreme flood event.
The forecast is usually made on the basis of rainfall-runoff transformation model using raingauge measurements or else by means of models simulating flood wave propagation using the measurement of water levels in some significant hydrometric sections. Forecasts of short lead time like one day forecast is more accurate but it is less useful because it doesn’t give much time for preparation to evacuate people from a floodplain for example. The development of the flood forecasting system therefore should aim to try to achieve greater levels of accuracy in forecasts of higher lead time so as to appropriate measures to be taken to reduce flood damage in an event of a big flood.
Cunnane, C., 1978: Unbiased plotting positions – a review. J. Hydrol. 379/4), 205-222.
Gumbel, E.J., 1941: The return periodof flood flows, Annals of Mathematical Statistics, 12(2),
163-190.
Hazen, A., 1932: Flood flows. John Wiley, New York.
USWRC, 1967: A uniform technique for determining flood flow frequencies, Bulletin 15,
Hydrology Committee, Water resources Council, Washington.
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Question 1
The annual maximum discharge values from river Mbarali at Igawa, Tanzania, for the years
1965 - 1986 are as follows, in m
3
/sec,
826.84, 402.57, 630.67, 575.73, 467.79, 263.01, 121.83, 263.01, 317.89, 197.06, 324.34,
151.22, 244.15, 278.67, 286.69, 132.11, 276.69, 98.13, 100.37, 156.94, 69.65, 62.50
(a) Rank the data, compute the probability plotting positions for the observed data and then prepare a probability plot on the Gumbel probability paper.
(b) Calculate the mean, Q and St. deviation,

Q
of the observed AM series and then estimate the parameters u and

assuming the data are EV1 distributed.
(c) Show the line Q = u +
 y on the probability plot.
(d) Read from the fitted line, the flood magnitude, Q, which has non-exceedance probability of
0.9.
(e) Calculate the flood magnitude, Q, which has non-exceedance probability of 0.9 from the expression determined in (c) and compare the result with the one determined in (d).
Using data for Chimala at Chitekelo in Tanzania, for the period 1976 – 1992, calculate estimates of flood magnitudes, Q
T
for T = 2,5,10,20 and 50 years, under the assumptions of
(a) 2 parameter log-normal
(b) Log-Pearson type 3 distribution
Peak discharge data (m
3
/sec)
36.78, 67,62, 81.07, 58.58, 70.21, 78.28, 56.23, 23.55, 91.77, 115,31, 83.92, 64.25, 157.37,
19.84, 48.78, 176.73, 26.57, 87.80
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Question 3
Given a series of peak floods at a gauging site are assumed to be EV1 distributed with mean =
200 m
3
/sec and standard deviation = 60 m
3
/sec. Calculate the peak flow which has
(a) Non-exceedance probability of 0.8
(b) Non-exceedance probability of 0.15
(c) Exceedance probability of 0.10
(d) What value of exceedance probability has Q = 275 m
3
/sec
(e) What value of non-exceedance probability has Q = 150 m
3
/sec
9-15