PROBABILITY OF HYDROLOGIC EVENTS

14.2 PROBABILITY OF HYDROLOGIC EVENTS

Since most hydrologic events are represented by continuous random variables, their density functions denote the probability distribution of the magnitudes. Some of the frequently used density functions used in hydrologic analysis are given below.

(a) Normal distribution. The density function of normal probability distribution is given by

f (x) =

exp −

HG 2 σ 2 p KJ

,–∞<x<∞

where σ p and µ are the two parameters, which affect the distribution, Fig. 14.2. In this distribution, the mean, mode and median are the same and the area under the

curve is unity. (b) Gamma distribution. The density function of this distribution is given by

xe a − xb /

f (x) =

for 0 < x < ∞

ab !

elsewhere

STATISTICAL AND PROBABILITY ANALYSIS OF HYDROLOGICAL DATA

where a and b are the two parameters, which affect the distribution (Fig. 14.3). A change of the parameter b merely changes the scale of the two axes.

Variate x

Fig. 14.3 Gamma distribution Here µ = b (a + 1) and σ 2 p 2 =b (a + 1)

(c) Poisson distribution. If n is very large and y is very small, such that y . n = m is a positive number, then the probability density function which is in the Poisson distribution is given by

m − x − e m y = f (x) =

Under abnormal skewness, the Poisson distribution is useful. The statistical param-

eters are σ p = µ and skewness =

(d) Lognormal distribution. A random variable x (variate) is said to be in log-normal distribution if the logarithmic values of x is distributed normally. The density function in this distribution is given by

f (x) =

exp −

HG 2 σ y KJ

where y = ln x, x = variate, µ y = mean of y, σ y = standard deviation of y. This is a skew distribu- tion of unlimited range in both directions. Chow has derived the statistical parameters for x as

2 µ = exp (µ 2

y + σ y /2), σ = µ exp σ y − 1 ...(14.21)

v = exp σ y − 1 ,

C s = 3C v + C v 3

(e) Extremal distribution This is the distribution of the n extreme values (largest or the smallest), each value being selected out of p values contained in each of n samples, which approaches an asymptotic limit as p is increased indefinitely. Depending on the initial distri- bution of the n.p values, three asymptotic (types) extremal distributions can be derived.

HYDROLOGY

(i) Type-I distribution In this distribution, the density function is given by

I , –∞ < x < ∞

c NM HG c KJ HG c KJ QP

where x is the variate and a, c are the parameters. By the method of moments the parameters have been evaluated as

a = γc – µ, c =

where γ = 0.57721 ... Euler’s constant. The distribution has a constant C s = 1.139. The Gumbel distribution used in flood frequency analysis is an example of this type.

(ii) Type-II distribution In this type, the cumulative probability is given

...(14.24) where θ and k are the parameters. (iii) Type-III distribution. In this type, the cumulative probability is given by

F(x) = exp [– (θ/x) k ], –∞ < x < ∞

x −∈ I O

F (x) = exp M −

P , –∞ < x ≤ ε

N M HG θ −∈ KJ Q P

where θ and k are the parameters. Weibull distribution used in draught-frequency analy- sis is an example of this type.

(f ) Pearson’s Type-III distribution This is a skew distribution with limited range in the left direction, usually bell shaped. The Pearson curve (Fig. 14.4), is truncated on one side of the axis of the variate, i.e., below a certain value of the variate the probability is zero, but it is

infinite converging asymptotically to the axis of the variate. This means even values infinitely large (or small) have a certain probability of occurrence.

(f) ,

F requency

Variate (x)

median

Fig. 14.4 Pearson’s Type-III distribution

STATISTICAL AND PROBABILITY ANALYSIS OF HYDROLOGICAL DATA

(g) Logarithmic Pearson Type-III distribution. This distribution has the advantage of providing a skew adjustment. If the skew is zero, the Log-Pearson distribution is identical to the log-normal distribution.

The probability density function for type III (with origin at the mode) is

f (x) = f F x 0 I 1− exp (– cx/2)

HG ...(14.26)

a KJ

µ 2 = the variance,

µ = third moment about the mean = σ 6 3 g

e = the base of the napierian logarithms Γ = the gamma function n = the number of years of record

g = the skew coefficient σ = the standard deviation

The US Water Resources Council (1967) adopted the Log-Pearson Type-III distribution (to achieve standardisation of procedures) for use by federal agencies. The procedure is to convert the data series to logarithms and compute.

n Σ (log x − log ) x 2

Standard deviation:

σ log x =

...(14.27 a)

n Σ (log x − log ) x 3

Skew coefficient:

The values of x for various recurrence intervals are computed from

...(14.28) and the frequency factor K is obtained from Table 14.2 for the computed value of ‘g’ and the

log x = log x +Kσ log x

desired recurrence interval (see Example 15.1).

HYDROLOGY

Table 14.2. K-Values for the Log-Pearson Type-III Distribution

Skew

Recurrence interval (T-yr)

coefficient 2 5 10 25 50 100 200 (g)

Per cent chance (annual probability of occurrence P(%))