FLOOD FREQUENCY STUDIES
8.4 FLOOD FREQUENCY STUDIES
When stream flow peaks are arranged in the descending order of magnitude they constitute a statistical array whose distribution can be expressed in terms of frequency of occurrence. There are two methods of compiling flood peak data—the annual floods and the partial duration series. In the annual floods, only the highest flood in each year is used thus ignoring the next highest in any year, which sometimes may exceed many of the annual maximum. In the par- tial duration series, all floods above a selected minimum are taken for analysis, regardless of the time-interval, so that in some years there may be a number of floods above the basic stage, while in some other years there may not any such flood at all. The disadvantage of the partial duration series is that the data do not furnish a proper frequency (true distribution) series and so a reasonable statistical analysis cannot be made. But all the larger floods are used in this analysis, which is an advantage while in annual flood series some big floods are omitted be- cause they were not the highest floods in any year considered. Usually the basic stage is as- sumed sufficiently low so that as many peaks (4 or 5) as possible each year are above this stage. The two series give very nearly the same recurrence interval for the larger floods, but the partial series indicates higher floods for shorter recurrence intervals. For information about floods of fairly frequent occurrence, as is required during the construction period of a large dam (say, 4-5 years), the partial series are the best, while for the spillway design flood the
HYDROLOGY
annual series are preferable, since the flood should not be exceeded in the dam’s life time, say 100 years.
Annual Flood Series The return period or recurrence interval (T) is the average number of years during which a flood of given magnitude will be equalled or exceeded once and is computed by one of the following methods.
California method (1923):
Allen Hazen method (1930):
...(8.11 a)
Weilbul method (1939):
n +1
...(8.11 b)
where n = number of events, i.e., years of record m = order or rank of the event (flood item) when the flood magnitudes (items) are ar-
ranged in the descending order (m = 1 for the highest flood, m = n for the lowest flood)
T = recurrence interval (T = n-yr for the highest flood, T = 1 yr for the lowest flood, by California method)
The probability of occurrence of a flood (having a recurrence interval T-yr) in any year, i.e., the probability of exceedance, is
or the percent chance of its occurrence in any one year, i.e., frequency (F) is
...(8.12 a)
and the probability that it will not occur in a given year, i.e., the probability of non-exceedance (P′), is
...(8.12 b) One interesting example of the application of statistics to a hydrologic problem (i.e.,
P′ =1–P
stochastic hydrology), is Gumbel’s theory of extreme values. The probability of an event of magnitude x not being equalled or exceeded (the probability of non-occurrence, P′), based on the argument that the distribution of floods is unlimited (i.e., for large values of n, say n > 50),
...(8.13) and the probability of the event x being equalled or exceeded (i.e., probability of occurrence, P) is
e − e − P′ y =
...(8.13 a) where e = base of natural logarithms
− P − = 1 – P′ = 1 – e e y
y = a reduced variate given by
(x – x + 0.45 σ),
for n > 50
x = flood magnitude with the probability of occurrence, P x = arithmetic mean of all the floods in the series
FLOODS-ESTIMATION AND CONTROL
σ = standard deviation of the flood series
n − 1 n = number of items in the series, i.e., the number of years of record. and the recurrence interval of the event of magnitude x (See Chapter-15)
If the event x, of recurrence interval T-yr, is x T , then from Eq. (8.14)
for n > 50 ...(8.17) or in terms of flood discharge items. Q T = Q + σ (0.78 log e T– 0.45), for n > 50
T = x x + σ (0.78 y – 0.45),
...(8.17 a) Q =Q T , when 0.78y = 0.45, or y = 0.577 which corresponds to T = 2.33 yr. The final plot of ‘flood
items (x) versus recurrence interval (T)’ can be made on probability or semi-logarithmic paper. In the Gumbel-Powell probability paper, the plotting paper is constructed by laying out on a linear scale of y the corresponding values of T given by Eq. (8.16) (after Powell, R.W., 1943) and is given in Table 8.3. It is sufficient to calculate the recurrence interval of two flood flows, say mean flood x (or Q with T = 2.33 yr) and x 150 or Q 150 obtained from Eq. (8.17 a) putting T = 150, and to draw a straight line through these points. A third point serves as a check, Fig. 8.2. This straight line can then be extrapolated to read the flood magnitude against any desired return period (T). The Gumbel distribution does not provide a satisfactory fit for par-
tial duration floods or rainfall data. Actual observations of flood data reveal that there are a greater number of floods below the mean than those above it and variations above the mean are greater than those below the mean. Therefore, a curve which fits the maximum 24-hour annual flood data on a log-log paper will not be a symmetrical curve, but a ‘skew curve’ which is unsymmetrical, i.e., the points do not lie on a straight line but the line bends off. The general slope of this curve is given by the coefficient of variation C v , and the departure from the straight line is given by the coefficient of skew C s , While plotting the skew probability curves, three parameters have to be calculated from observed flood data as
(i) Coefficient of variation, C v = ...(8.18)
(ii) Coefficient of skew, C s =
(iii) Coefficient of flood, C f = 0.8
A /2.14
where A = area of the catchment in km 2
HYDROLOGY
Table 8.3 Gumbel’s probability data
Reduced variate
Probability of exceedance (y)
Recurrence interval
% Chance or probability of flood > Q, in any one year Frequency F = 1 ´ 100 %
50 6.25 Q 0.67 0.5 150
B B Q Q 150 150 Observed Observed 40
93-yr flood 93-yr flood
Q Q 25 25 Q Q 25 25
C (check) C (check)
30 line line
cumec Q,
frequency frequency 20 Flood Flood Gumbel Gumbel
flood peak
Q (or x) – –
A A Q mean Q mean (T = 2.33) (T = 2.33)
ual 10 Ann
0 3.902 y 3.902 y – 1 .53 – 1 .53 –1 – .475 – .475
1.17 1.17 Reduced Variate y (to linear scale) Reduced Variate y (to linear scale)
(Recurrence interval T-yr:) T =
1 1–e –e–y
Fig. 8.2 Gumbel-Powell probability paper
FLOODS-ESTIMATION AND CONTROL
When the flood items are tabulated in terms of mean flood, C v = σ. If the annual flood is x and the mean flood is x , then the annual flood in terms of mean flood is x/ x . The coefficient of variation for annual precipitation data is equal to the standard deviation of the indices of wetness.
While a small value of C v indicates that all the floods are nearly of the same magnitude,
a large C v indicates a range in the magnitude of floods. In other word C v represents the slope of the probability curve, and the curve is horizontal if C v = 0. The actual length of records available has a very little effect on the value of C v , i.e., C v for a 20-yr record varies very little from that for a 100-yr record.
The coefficient of skew, C s is seriously affected by the length of record and will be too small for a short period; C s is then modified to allow for the period of record (n) by multiplying by a factor (1 + k/n) where the constant K = 6 to 8.5. If even this adjusted C s does not give a curve to fit actual observed data, an arbitrary value of C s will have to be assumed to fit the curve for the given annual flood data. From this theoretical curve can then be read off the probability or the percentage of time, of a flood of any given magnitude occurring, usually,
...(8.21) The coefficient of flood indicates the general magnitude of the floods in the particular
C s ≈2C v
stream; hence, it fixes the height of the curve above the base. Using C f , the mean flood of a stream, for which no flood data are available, can be got, as
A 0.8
Mean flood = C f ×
The exponent 0.8 is the slope of the line obtained by plotting the mean annual flood against water-shed area for a number of streams. Almost all observed data till to-date confirm this value originally obtained by Fuller.