Write a brief note on frequency factor and its estimation in

Write a brief note on frequency factor and its estimation in Gumbel\'s method.

Solution

As a simple method, frequencies (or probabilities), P(X x), of the observed flood peaks could be calculated. The curve of probabilities versus flood peaks (f Vs. x) is then plotted on log-probability paper and a smooth curve is fitted covering all points. By extrapolation of the curve extreme values could be obtained.

Since the observed data is usually short it may not represent the population and hence we can not entirely keep reliance on the curve obtained from observed data.

Now considering that recorded data constitutes a random sample of their parent population a theoretical frequency distribution suitable for the data could be fitted.

Once the distribution is fitted properly to the observed data extrapolation to calculate required probabilities can be easily done.

The Gumbel method of frequency analysis is based on extreme value distribution and uses frequency factors developed for theoretical distribution. The method utilises general equation given for hydrologic frequency analysis which is stated as below.

x = x + x …(0)

Where x is magnitude of flood of some given probability (P) or return period (7)

x is mean of floods on record

x is departure of variate from the mean.

x depends on dispersion characteristics, recurrence interval (T) and other statistical parameters. It can be expressed as

x = S K

where S is standard deviation of the sample and K is frequency factor Thus, equation (i) above can be expressed as

x = x + KS

Table 5.6 gives theoretically derived values of the frequency factor if for various sample sizes and return periods.

Steps Involved in Frequency Analysis:

Various steps involved in frequency analysis by the Gumbel method are as follows:

(i) List and arrange annual floods (x) in descending order of magnitude.

(ii) Assign rank ‘m’, m = 1 for highest value and so on.

(iii) Calculate return period (T) and/or probability of exceedence (P) by equations n + 1/m and m/n +1 respectively. These values together with respective flood magnitude give plotting positions.

(iv) Using tabular form calculate x2 and x and Ex2.

(v) Now calculate mean x; squared mean x2; mean of squares x2 and standard deviation S.

(vi) From the Table 5.6 of frequency factors for Gumbel method read if values for desired return periods (7) and the available sample size.

(vii) Using relation x = x + KS calculate flood values for various return periods.

(viii) Using the extreme value probability paper plot the x values against respective return periods or P values and join the points to obtain the required frequency curve.