Keywords

Gumbel Extreme Value Type-I , return period , Flood Frequency , Kaveri River

1 . INTRODUCTION

2 . STUDY AREA

Figure 1. Physiography of the Kaveri Basin and locations of discharge gauging site

3 . DATA AND METHODOLOGY

The AMS data from 1971 to 2019 recorded at gauge and discharge (G/D) sites have been acquired from the water year book of 2018-19 published by Central Water Commission (2020) and were used to accomplish Flood Frequency Analysis of the Kaveri River and its tributaries. The record lengths of the data are from 17 to 47 years. The data were available for the Kudige, Kollegal, Biligundulu, Urachikottai, Kodumudi and Musiri sites on the Kaveri River and eight sites on its tributaries (Figure 1). In order to estimate discharges of a given return period, a frequency distribution is compiled from the data series of annual maximum events. The probable design flood discharges have been calculated for return periods of 2-yr, 5-yr, 10-yr, 25-yr, 50-yr, 100-yr, 500-yr and 1000-yr by means of the GEVI probability distribution. The distribution has also been employed for approximation of the recurrence interval of average annual peak discharge ( ​\(\bar{Q}\)​ ), large flood (Qlf) and actually observed maximum annual peak discharge (Qmax). A visual check of the fit of the frequency distribution is probably the best way in determining how satisfactory an individual distribution fits the AMS dataset or which distribution fits “best”. Thus, flood frequency of all the sites of the Kaveri River and its tributaries are represented graphically (Figure 2, 3 and 4) which fairly represents the Kaveri Basin.

Figure 2. Magnitude–frequency curve for discharge gauging sites on the Kaveri River represented on the Gumbel Extreme Value Type-I (GEVI) probability graph paper.

Figure 3. Magnitude–frequency curve for discharge gauging sites on the tributaries of the Kaveri River represented on the Gumbel Extreme Value Type-I (GEVI) probability graph paper

Figure 4. Magnitude-frequency curve for discharge gauging sites on the tributaries of the Kaveri River represented on the Gumbel Extreme Value Type-I (GEVI) probability graph paper

1.1 Gumbel Extreme Value Type-I (GEVI) Probability Distribution

The GEVI (Gumbel, 1941, 1958) probability distribution has been applied as a universal model of extreme flood events principally in the perspective of regionalization procedures and has been recognized as a rational approach to predict the flood recurrence interval (Al-Mashidani et al., 1978; Hosking and Wallis, 1997). The GEVI probability distribution is expressed as;

​\(QT=\bar{Q}+[K(T)*\sigma Q]\)​                               (1)

​\(\bar{Q} ={ \sum{Q} \over n}\)​                                                           (2)

​\(\sigma Q = \sqrt { \sum{Q-\bar{Q} } \over n-1}\)​                                            (3)

where, QT = given return period’s discharge; QT  = average of annual peak discharge; K(T) = factor of frequency and is the return period’s (T) function. K(T) values are obtained from the book ‘Hydrology in Practice’ (Shaw, 1988); σQ = AMS’s standard deviation.

The return periods (T) were computed for average annual peak discharge ( K(T) ), large flood (Qlf) and peak on record (Qmax) for all sites by using the formula given below.

​\({1 \over T}=1-F(X)=1-exp[-e^ {-b(X-a)}] \)​      (4)

where, T = return period of a specified discharge; F(X) = probability density function (PDF) of an annual maximum series Q £ X; a and b = parameters allied to the moments of population of Q values. The parameters a and b are derived, as under, by equation (5) and (6).

​\(b= {\pi \over { \sigma Q \sqrt{6}}}\)​                                                        (5)

​\(b= \bar{Q} - {y \over {b}}\)​                                                       (6)

where, ​\(\bar{Q}\)​  = average annual peak discharge; sQ = annual peak discharge’s standard deviation; y = 0.5772.

1.2 Plotting Position and Frequency Curves

The plotting positions have been calculated using a variety of equations, nevertheless, Cunnane (1978) and Shaw (1988) argues that Gringorten’s approach is pertinent as it fits outliers more closely than other plotting positions. Consequently, by using the following equations (Gringorten, 1963), F(X) values were determined.

​\(P(X)=1-F(X)= {(r-0.44) \over (N+0.12)}\)​                           (7)

where, P(X) = the probability of exceedance; r = rank of flood magnitude; N = the number of observations of AMS.

In the Gringorten plotting positions, by judgment, a line can be constructed to fit the scatter. Notwithstanding, drawing the line mathematically is more appropriate. Creating confidence limits about the fitted line and its association between the AMS and the linearized probability variable is also essential, since the majority of AMS data are only available for a short period of time (Shaw, 1988). In the present work, the procedure given by Shaw (1988) has been adopted to fit the line mathematically and to construct the confidence limits. The plotted graphs are represented in Figure 2, 3 and 4.

4 . RESULTS AND DISCUSSION

A frequency distribution is amassed from a data series of AMS to estimate discharges of a given return period.  Peak flows have been estimated for different return periods such as 2-yr, 5-yr, 10-yr, 25-yr, 50-yr, 100-yr, 500-yr and 1000-yr by using GEVI probability distribution and the estimated discharges are given in Table 1. Accordingly, the highest estimated discharge (10408 m³s^-1) of 1000-yr return period was for the Musiri Site (Figure 1) on the Kaveri River in the Tamil Nadu state. Nevertheless, the peak discharge of 7690 m³s^-1 magnitude was observed on the Kaveri River at the Musiri Site in the year 2005 (Central Water Commission, 2020). The recurrence interval for this discharge is 135-yr (Table 2). Astonishingly, the estimated discharges of given return periods of the Kaveri River are remarkably low as compared to other Indian Peninsular Rivers (Hire, 2000; Hire and Patil, 2018; Patil, 2017; Pawar et al., 2020; Anwat et al., 2021).

Furthermore, an attempt has been made to find out the return periods of average annual peak discharge ( \(\bar{Q}\) ), large flood (Qlf) and actually observed maximum annual peak discharge (Qmax) by applying the GEVI distribution and result have been shown in Table 2. Unsurprisingly, as per GEVI distribution, the \(\bar{Q}\)  has the recurrence interval of 2.33-yr. The Qlf, which is well-defined as a flow that exceeds one standard deviation above the average annual peak discharge ( \(\bar{Q}\) ) (Hire, 2000), has the return period 6.93-yr. The analysis further confirms that the return period of the actually observed peak discharge (Qmax = 5571 m³s^-1) on the Amaravati River at the Nallamaranpatty is 636-yr which is high. It is interesting to note here that the Amaravati is a tributary of the Kaveri River having drainage area of only 9080 km². Nonetheless, the Qmax of the river at the Nallamaranpatty is much higher. However, the recurrence interval of the Kaveri River at the Musiri site, the lowermost site just before the commencement of the Kaveri Delta, is 135-yr. This site has the upstream drainage area of 66243 km². Moreover, some discharge gauging sites have much less return period of the Qmax (2325 m³s^-1) e.g. the Kabini River at T. Narasipur is merely 36-yr (Table 2). The observed annual peak discharges have been plotted against the return period or F(X) values (plotting positions) on the Gumbel graph paper, designed for the GEVI probability distribution. The magnitude-frequency graphs (Figure 2, 3 and 4) show that, the fitted lines are fairly near enough to the most of the data points and, consequently, can be reliably and expediently used to read the recurrence intervals for a given magnitude and vice versa. Remarkably, in plot of the GEVI distribution, the actually observed peak on record (Qmax) falls well in vicinity of the fitted line. This means the return period of Qmax(s) of all the sites projected by the GEVI probability distribution are likely to be quite reliable. Nevertheless, at three sites on the Kaveri River, viz. Urachikottai, Kodumudi and Musiri, the data points of lower magnitude floods are away from the fitted lines (Figure 2d, e, f). It may be attributed to controlled discharges as these sites are positioned downstream of the largest dam on the Kaveri River i.e. the Mettur Dam (Figure 1). Nonetheless, as stated earlier, the higher discharges are fairly close to the fitted lines (Figure 2d, e, f). Therefore, the GEVI probability distribution is an appropriate probability distribution for these sites too.

5 . CONCLUSIONS

Tables

Figures

Conflict of Interest

Acknowledgements

References