CHAPTER 3
Hydrological Forecasting
Introduction
in the previous chapters the various physical processes involved in the hydrologic
cycle have been enumerated and examined in detail. Methods of evaluating each
process have been suggested and often explained, and techniques discussed that can
be used to provide quantitative answers to many questions.
The remaining problem that must now be tackled is how to use this knowledge to
predict from existing data, however meagre it may be, what will happen in future.
This is a fundamental problem of all engineering design, since the engineer designs
and constructs work to provide for future needs, whether he be a structural engineer
designing an office block, an electrical engineer designing power systems to meet
future electrical demand, or a hydraulic engineer designing
reservoirs to meet future demand for water.
There is one major difference in these three cases. The structural designer is
working with homogeneous materials whose behaviour is known vithin nnnw limits.
His buildings will be used by people whose spacing, dimensions, weight and
behaviour, en masse, can be predicted quite accurately. He has to cope with natural
events only in the form of wind loads, which usually form a small proportion of the
total load, and earthquakes. For both these eventualities there are codes of practice
and recommendations available to him.
Flood formulae:
Catchment area formulae. The particular random variable of river flood discharge
has been of interest to engineers and hydrologists from the earliest days of hydrology
and many formulae have been proposed to define the ‘maximum flood’ that could
occur for a particular catchment. The formulae are empirical by nature, derived from
observed floods on particular catchments and usually of the form
Where:
Q = flood discharge in m3/s (or ft.3/s)
A = catchment area in km2 (or mile2)
1
n = an index usually befween 0.5 and 1.25
C = a coefficient depending on climate, catchment and units.
An early example of such a formula, due to Dickens, was developed in India :
with Q in ft.3/s and a in square miles; but since the formula takes no account of soil
mcirturc, rainfall, slope, altitude etc., it is clearly of vpt Jitt1 vq1U in general
application. This is true of all such formulae although they are frequently used to
obtain a quick first estimate of the order of ‘maximum flood’ that can be expected.
For such purposes Morgan [1] proposed the formula for a catastrophic flood in
Scotland and Wales of
where Q is in ft.3/s and M is catchment area in square miles, and a the sophistication
of a recurrence period T(in years) by quoting
design flood = catastrophic flood x (T/500)4
for cases where the adoption of the catastrophic flood was not justified by danger to
human life or the safety of a dam. A similar formula of the same type, due to Fuller,
has been widely used in the USA:
Where:
A is catchment area in square miles
C is a coefficient often taken as 75
Qav is average value of annual flood discharge in ft,/s.
The value of Qav is then substituted in the formula
Qm = Qav (1+ 0.8 log T)
where T is a return period in years and Qm is the ‘most probable’ annual
maximum flood .
Probability of the N-year event:
The term recurrence interval (also called the return period), denoted by T, is the
time that, on average, elapses between two events that equal or exceed a particular
level. Putting it another way, the N-year event, the event that is expected to be
equalled or exceeded, on average, every N years, has a recurrence interval, T of N
years.
As mentioned previously there is no implication that the N-year event occurs
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cyclicafly. It does, however, have a probability of occurrence in any particular period
under consideration.
Let the probability
represent the probability that x will not be equalled or
exceeded in a certain period of ti-me.
Then
will represent the probability that x will not be equalled or
exceeded in n such periods.
For an independent series and from the multiple probability rule:
Therefore
Now
Therefore
So, for example, the probability of
, where x is the value of a flood with a
return period of 20 years , occurring in a particular 3-year period is
Table 9:1 shows the probability of the N-year flood occurring in a particular period.
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For example, it can be seen from the table there isa 1 per cent chance of the 200year flood occurring in the next 2 years and an 8 per cent chance that it will not occur
for the next 500 years.
If the probability
is defined by a policy ruling, the value of n, the
design period, can be found from
TABLE 9.1 Percentage probability of the N-year flood occurring in a
particular period * :
Number
Of years
In period
1
2
3
5
10
20
30
60
100
200
500
1000
N = average return period Tr (years)
5
10
20
50
100
200
500
1000
20
36
49
67
89
99
99.9
-
10
19
27
41
65
88
96
99.8
-
5
10
14
23
40
64
78
95
99.4
-
2
4
6
10
18
33
45
70
87
98.2
-
1
2
3
5
10
18
33
45
70
87
99.3
-
.5
1
1.5
2.5
5
10
14
26
39
63
92
99.3
.2
.4
.6
1
2
4
6
11
18
33
63
86
.1
.2
.3
.5
1
2
3
6
10
18
39
63
* Where no figure is inserted the percentage probability > 99.9
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Therefor
Example 9.1
How long may a cofferdam remain in a river, with an even chance of
not being overtopped, if it is designed to be secure against a 10-year
flood ?
Here, the policy ruling is that there should be an even chance, so :
and
Then
:
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Determining the magnitude of the N-year event by plotting:
Having listed a series of events (for example, maximum floods) they are then each
accorded a ranking m, starting with m = 1 for the highest value, m = 2 for the next
highest and so on in descending order. The recurrence interval Tr can now be
computed from one of a number of formulae, which have been reviewed by Cunnane
[7].
That most frequently used in the past is the Weibull formula [8]
where :
m = event ranking and n = number of events, but there are objections to its use
because of the bias it introduces to the largest events in a short series.
Other formulae used are the Californian [9]
and Hazen’s [10]
about both of which there are reservations. One of the more satisfactory, due to
Cringorteh [11] is
For a single, simple compromise, Cunnane recommends
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The probabiLity P of an N-year event of return period Tr is
so that once the series has been ranked, its various events can be plotted on grphs
connecting the variable Q and either Tr or P .
It is often assumed that such series are normally distributed, in which case the
plotted points on normal probability paper would lie on a straight line. This seldom
happens for flood series and shallow curves more often result, making extrapolations
more difficult. To overcome this difficulty the variate Q is sometimes plotted
]ogarith-rn.icafly, which requires logarithmic-normal probability, or log-normal
paper.
Table 9.2 is a listing of the annual maximum mean daily flows of the River Thames at
Ted dinglon Weir for the years 1883—1988. This is a true annual series with return
periods and probabilities .
Table 9.2 data can be plotted in a variety of ways and figures 9.2 to
9.5 illustrate the most common:
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8
9
(a) Q against T using linear co-ordinates (figure 9.2). Extrapolation of the curve
to high values of Tr depends critically on the few highest points. Circled
points are corresponding positions of equation 9.1 for two highest values.
Figure 9.2 Annual maximum mean daily flow of the River Thames at
Tiddington , 1883-1988
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(b) Q (linear) against Tr (logarithmic) (figure 9.3). This yields a straight line fitted to
all but the lowest -values. Although extrapolation is simpler, unless T follows a
logarithmic law, eitrapolation is not necessarily more accurate than for figure 9.2.
Figure 9.3 Annual maximum mean daily flow of the River Thames
Tiddington , 1883-1988(semi-log)
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(c) Q. (linear) against probability (per cent) (figure 9.4). As often flood series points
lie on a shallow curve on probability paper (where a normal distribution of
probability is assumed).
Figure 9.4 Annual maximum mean daily flow of the River Thames at
Teddington1 2883—1988 (normal probability)
(d) Q (logarithmic) against probability (per cent) (figure 9.5). The curve of figure 9.4
is now transformed to a straight line,
A variation of the approach in figure 9.4 is to assume that the logarithm of the variate
Q is normally distributed, leading to the use of iogarit.LLLinormal distribution (or
log-normal paper)—first used by Whipple [11].
(e) Other investigators have proposed methods assuming other theoretical frequency
distributions. Gumbel [13, 14, 15] used extreme-value theory (EV1) to show that in a
series of extreme values X1, X2 . . .Xn where the samples are of equal size and X is
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an exponentially distributed variable (for example the maximum discharge observed
in a year’s gauge readings), then the cumulative probability P` thaCany of the ii
values will be less than; a particulaf’aIue X (of return period T) approaches the value
Where
e is the natural logarithm base
and
Figure 9.5 Annual maximum mean daily flow of the River Thames at
Teddington, 1883-1988 (log-normal) .
That is, P` is the probability of non-occurrence of an event X in Tyears, or
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(Note that this argument refers to Gumbel’s method. The reader should not confuse
this with. the normal usage of T = 1/P where P = probability of occurrence.)
The event X, of return period Tyears, is now defined as QT and
Where
Qav = average of all.values of ‘annual flood’ Qznax
a = standard deviation of the series.
Thus
Where
n = number of years of record = number of Qmax valuses
= sum of the squares of n values of Qmax
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Table 9.3 gives values of y as a function of T.
Powell [16] suggested that if plotting paper is prepared in which:the horiiontal lines
are spaced linearly and the vertical lines’ spacing is made proportional toy, QT and T
will plot as straight lines. This is the basis of Gumbel—Powell probability paper, used
to plot the River Thames data in figure 9.6. The return period T has been computed,
as before, from Tr .
The straight line on this figure has been drawn between the two points Qav and Q200.
Qav from occurs when 0.78y =0.45 or = 0.577, which corresponds
to T =2.33 years. QT holds for large values of n, say
n> 50, when Qav at 2.33 years is included on the line through the points. The other
point Q200, represents the ‘200-year flood’ and is found by inserting the appropriate
vaiues in QT
Q200 = Qav + 133.8 (0.78 x 4.6 - 0.45) = 329.7 + 492.9 = 823 m3/s and similarly
Q100 = Qav + 133.8(0.78 x 5.30 —0.45) = 329.7 + 419.9 = 750 m3/s
The correspondence between the plotted data and Gumbel’s theoretical line is
demonstrated. Gumbel paper should not be used for partial series, which usually plot
better on semi-log paper as used in figure 9.3.
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From the plots presented in figures 9.2 to 9.6, it may seem there is little to choose
between the particular plotting papers available. This is very often the case and
investigators should use whichever distribution makes their particular job of fitting
and extrapolation simplest and the line apparently of best fit.
The foregoing is a necessarily brief résumé of the methods of plotting flood events, in
current use. For the underlying theories the reader should refer to original papers and
more comprehensive treatments available [5, 7, 17, 18].
Figure 9.6 annual maximum mean daily flow of river thames at
Teddington , 1883 – 1988 ( gumbel distribution )
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Determining the magnitude of the N-year event by calculation :
Although the use of a normal probability distribution has been used above to plot
events, and hence to extrapolate for rarp values that may be used in design, values of
particular probabilities can be calculated since a normal distribution curve is defined
by only two parameters, the mean and standard deviation.
Accordingly, to determine the specific discharge associated with a particular
probability of occurrence r in an annual series that is normally distributed, it is
necessary to compute only .
where
= standard deviation and K is listed in table 9.4.
Example 9.1
Determine by calculation the mean daily discharge of the River Thames at
Teddinaton with a 100-year.retum period, assuming the annual eiiès is noThiafly
distriSuted ?
From table 9.2
Qav = 329.7 rn3/s and
a = 133.8 m3/s. For Tr = 100 years,
P is 1.0 per cent; and from table 9.4, K = 2.33. Therefore
Q100 = 329.7 + (2.33 x 133.8)
= 641 m3 / s
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