IJWREM: Volume 3, Number 1, January-June 2012, pp. 67-73

1,2 Indian Institute of Tropical Meteorology, Pune-411008* Correspondence Author: bdkul@tropmet.re.in

STATISTICAL ANALYSIS OF THE SPATIAL VARIABILITY OF VERYEXTREME RAINFALL IN THE KRISHNA RIVER BASIN

B.D. Kulkarni*1 and N.R. Deshpande2

Abstract: Annual extreme rainfall series of 1-3 day durations at stations located in and around the Krishna basinwere subjected to statistical analysis in order to estimate point Probable Maximum Precipitation (PMP) andmaximum rainfall of different return periods for the durations of 1-3 days. Daily rainfall data of 321 stationsranging from 1901-2000 (with varying data length) has been considered for the present study. Hershfield statisticaltechnique is the widest accepted one; the same has been applied in obtaining point PMP estimate of stations overthe Krishna basin, as this method is based on the actual observed rainfall data of long-period stations. Spatialpatterns of 1-3 day extreme rainfall over the basin have been prepared. Extreme Value Type-I (EV1) distributionhas been fitted to 1-3 day extreme rainfall series and various return period values were estimated. Using the samefit it was found that, PMP estimates for 1-3 day durations, have return period of the order of 1000-year. Extremerainfall features and estimates of point PMP and maximum rainfall for different return periods documented in thisstudy will be useful for designing and planning the water resources projects in the basin.

Keywords: Extreme Rainfall, Extreme Value Type-I Distribution, Return Period, PMP

1. INTRODUCTION

Worldwide existing approaches of estimating PMP primarily include hydrometeorological and statisticalapproaches. Statistical procedure for estimating PMP is normally used whenever sufficient rainfall data areavailable and is particularly useful for making quick estimates or where other meteorological data, such asdew point, wind records, etc. are lacking. The physical approach of estimation of PMP over a river basintakes into consideration the physiographic condition of terrain, catchment characteristics and the moistureholding capacity of the atmosphere during the movement of the particular weather system and is preferredto the statistical technique. A value approximate to the physical upper limit of precipitation can be foundfrom existing data set within large temporal and spatial domain. Extreme so found usually need to beproperly enlarged to get PMP. The techniques used to estimate PMP design rain depths depend on therainfall data available and may vary with catchment size and location, as well as with the meteorologicalconditions responsible for causing extreme rainfall over the catchments of interest.

Considering the importance of the subject, an attempt has been made in this study to carry out extremerainfall analysis of 321 stations in the Krishna basin by statistical methods to estimate point PMP andmaximum rainfall of different return periods which are often needed for proper planning, management anddesign of different types of water resources projects in a river basin.

2. PHYSIOGRAPHIC FEATURES OF THE KRISHNA BASIN

The Krishna river and its tributaries form an important integrated drainage system in the central portion of theIndian Peninsula. As per the Khosla (1949) classification, the entire basin has been divided into 5 sub-catchments

68 B.D. Kulkarni and N.R. Deshpande

(No. 306 to 310). The drainage area of the entire basin is about 2,58,948 km2 of which 26.8% lies in Maharashtra,43.8% in Karnataka and 29.4% in Andhra Pradesh. By considering the orography, geographic location andrainfall characteristics, the entire basin has been divided in to 5 sub-basins (see Figure 1). Details of sub-catchments and number of raingauge stations in each of the sub-basin are given in Table-1.

Table 1Sub-catchments of the Krishna River Basin

Sub-basin Name of sub/zone No of Stationsin each catchment

306 River Krishna up to its confluence with River Bhima, excluding river Bhima 60

307 River Bhima 62

308 Tungabhadra up to Tungabhadra dam site 49

309 Vedavati up to confluence with Tungabhadra 75

Figure 1: Location of Stations in and Around the Krishna Basin

3. DATA USED AND METHODOLOGY

Daily rainfall data of 321 stations for the period 1901-2000 (with varying data length) located inside theKrishna basin was considered for this study. Location of these rainfall stations and demarcation of differentsub-basins are shown in Figure 1. Description of different sub-basins, corresponding area and data availabilityof these sub-basins are given in Table 1. Rainfall features such as seasonal rainfall, annual rainfall; highestrainfall amounts, PMP and maximum rainfall of different return periods for 1-3 day durations were estimatedfor each of 321 stations considered in the study. These estimates were then interpolated at regular grids atthe resolution of 0.1° × 0.1° using inverse squared distance method. Spatial patterns were displayed on thebasin map using GrADS software (Version1.8)

Statistical Analysis of the Spatial Variability of Very Extreme Rainfall in the Krishna River Basin 69

4. METEOROLOGICAL SITUATIONS RESPONSIBLE FOR CAUSING HEAVY RAINFALL

The Krishna basin is bounded on the west by mountains of the Western Ghats which exerts a profoundinfluence on rainfall over the basin. During the southwest monsoon season (June to September), the monsoonwinds from the Arabian Sea are intercepted by mountain ranges of the Western Ghats producing veryheavy rainfall over the mountainous part of the basin than over the plateau to the east. The heavy rainfallreceived by the basin during the monsoon months is mostly caused by the following meteorological situations:

1. Monsoon depressions / cyclonic storms after originating in the head Bay of Bengal and crossing theeastern coast of the country move in west to northwesterly direction. These disturbances causeheavy to very heavy falls of rain along and near their tracks. Heavy rainfall is generally found tooccur in the southwest quadrant of these disturbances.

2. Westward or west-northwestwards movement of low pressure areas and upper air cycloniccirculations particularly between 700 hpa and 500 hpa.

3. Position and strength of the axis of the seasonal monsoon trough, especially when it is south of itsnormal position, north peninsular region receives heavy to very heavy rainfall and

4. North-south oriented trough in the lower /middle troposphere.

However, the conditions in the interior of the basin are somewhat unfavorable for heavy rainfall exceptin association with the depressions from the Bay of Bengal.

5. ANALYSIS OF HIGHEST RAINFALL

Magnitudes of highest rainfall are the basic tools in the estimation of Probable Maximum Precipitation(PMP) or extreme rainfall which is possible at a given point for a given duration.

According to Hershfield (1962) the magnitude of point PMP at an individual station should normallynot exceed three times the highest observed rainfall from a long period of rainfall data. Dhar et al (1981)have, however, shown that at some of the stations over India, the highest ever recorded 1-day rainfalls werenearly equal to their respective estimates of point PMP. As an example, Bhagamandala station (KarnatakaState) recorded the highest 1-day rainfall of 84 cm on 25 Jul., 1924 and it was found that its estimate ofpoint PMP was near about this magnitude. Similarly, Beed, a station in Marathawada region of Maharashtrastate has recorded 35 cm rainfall on 24th July, 1989 which was found to be almost equal to the estimate ofphysical upper limit of rainfall (i.e. PMP) for this station (Kulkarni et. al,1991).

In view of the importance of highest rainfall for different durations and their usefulness in many typesof hydrological analyses, highest ever observed 1, 2 and 3-day rainfalls for each of 321 stations werecatalogued. The generalized maps of highest ever recorded 1, 2 and 3-day rainfalls have been preparedand the same are shown in Figure 2 Highest ever recorded rainfalls over each of the 5 sub-catchments of theKrishna basin for 321 stations and for 1, 2 and 3-day durations with their date of occurrences are given inTable 2.

Figure 2: Spatial Pattern of 1, 2 and 3-day Extreme Rainfall (cm) Over the Krishna Basin

70 B.D. Kulkarni and N.R. Deshpande

5.1 Estimation of PMP by Statistical Method

Statistical method used for estimating PMP at a station or over an area is based upon the assumption thatinformation regarding extreme rainfall is contained in the long period rainfall records of that station / area.PMP is defined as theoretically the greatest depth of precipitation for a given duration that is physicallypossible over a given size storm area at a particular geographical location during a certain time of the year(WMO, 1986). Statistical procedure for estimating PMP is normally used whenever sufficient rainfall dataare available and is particularly useful for making quick estimates or where other meteorological data, suchas dew point, wind records, etc. are lacking.

Hershfield (1961) for the first time used the statistical method for estimation of PMP for USA. InCanada (Bruce and Clark, 1966) has shown that the PMP estimates obtained by this approach are closelycomparable to those obtained by the conventional moisture maximization and storm transposition methods.According to Wiesner (1970) this method has an advantage of taking into account the actual rainfall data,expressing it in terms of statistical parameters and is easy to use. The World Meteorological Organization(WMO) in their various manuals and technical publications (WMO, 1986) has also recommended thismethod for estimation of PMP for those river basins whose daily rainfall data are available for a long-period of time.

A brief description of Hershfield technique for estimating point PMP for station is as follows:

XPMP = Xn Sn Km (1)

Where, XPMP = PMP estimate for a station, Xn = mean of the annual maximum series, Sn = standarddeviation of the annual maximum series, and Km = frequency factor which depends upon the availability ofdata period and the return period.

According to Hershfield (1961), values of the frequency factor ‘Km’ is obtained by using the followingequation :

Km = 1( max ) / 1X Xn Sn (2)

Where, Xmax = largest value of the annual maximum series, 1Xn mean of the annual maximumseries omitting the largest value from the series, and S

n–1 = standard deviation of the annual maximum

series omitting the largest value from the series. Km values from all stations are then plotted against the

Xn values respectively and a smooth envelope curve is drawn. K envelope value is picked up from thecurve for each station’s. PMP for each station is then calculated using equation (1) by replacing Km with Kenvelope value. Details of the Hershfield technique are available in various publications (WorldMeteorological Organization (WMO), 1986, Dhar & Kulkarni 1975). Here Km Vs Xn were plotted separatelyfor each sub-basin. PMP estimates thus obtained by applying the Hershfield technique for each 321 stations

Table 2Ranges of Highest Observed Rainfall (cm) in Different Sub-basins. Stations which has Recorded Highest Value has

been shown in the Bracket

Sub-basin Rang of Highest observed rainfall (cm)

1-day 2-day 3-day

306 9.3-44.0 (Mahableshwar) 13.7-76.2(Mahableshwar) 17.4-106.2(Mahableshwar)

307 12.0-51.6 (Khandala) 15.3-88.2 (Lonavala) 17.2-110.1 (Lonavala)

308 10.7-54.9 (Agumbe) 13.6-884 (Agumbe) 17.2-113.6 (Hirebyle)

309 7.9-26.1(Sindhanoor) 10.3-34.8 (Halebid) 13.0-49.3 (Kanakatte)

310 12.3-50.9 (T. Kothapalem) 16.1-66.9(T. Kothapalem) 16.3-82.2 (T. Kothapalem)

Statistical Analysis of the Spatial Variability of Very Extreme Rainfall in the Krishna River Basin 71

for 1, 2 and 3-day durations are then plotted over base map of the study basin and generalized PMP mapshave been prepared. Figure 3 (a, b & c) shows the generalized point PMP maps by statistical method fordurations of 1, 2 &3-day over the basin. It is seen from this figures that point PMP estimates for stationsover the Krishna basin broadly range from about 18 to 79 cm for 1-day, 27 to 132 cm for 2-day and 31 to172 cm for 3-day durations.

Figure 3: Spatial Pattern of 1, 2 and 3-day Extreme Rainfall (cm) Over the Krishna Basin

1-Day 2-Day 3-Day

5.2 Return Period Analysis

The most important factors to be taken into consideration in the designing of engineering structures aresafety, economy and efficiency. Thus, structures have to be designed to withstand the pressure to whichthey may be subjected during their estimated economic life, which may vary from less than 10 to greaterthan 1000 years or so. Thus, design of water control structures focuses on the prediction of runoff duringmajor hydrologic events. This requires estimates of the frequencies of occurrence of rainfall of a givenduration and intensity, for analysis of the potential costs and benefits of building adequate controls. Theseestimates are called return periods. Thus the Recurrence Interval or Return Period (T) is the frequencywhich is expected, on an average, a given rainfall event to re-occur. The standard method of estimating theprobability of occurrence of maximum rainfall for hydrologic design is to fit a known probability distributionto the maximum annual rainfall series and estimating its parameters by statistical procedures. Varioustheoretical and empirical distributions have been proposed by various hydrologists, which are generallyapplicable to the annual maximum rainfall series. A comprehensive study of various distributions wasmade by Hershfield and Kohlar (1960) who found that the Extreme Value Type-1 (EV1) or Gumbeldistribution is the most suitable one. The same has been used in this study.

5.2.1 Extreme Value Type-I (EV1) Distribution or Gumbel Distribution

If X follows Extreme value Type-I distribution then its Cumulative Distribution Function is given by,

F(x) = exp{ [ ( ) / ]}x u (3)

, , and 0x u

Here ‘u’ and ‘ ’ are the location and scale parameters of the distribution which are estimated from theobserved annual extreme rainfall series using the method of least squares. Kolmogorov-Smirnov test isused for testing the Gumbel fit for 1-3 day extreme rainfall series at individual station. Figure 4(a, b, c)shows the goodness of fit testing. It has been observed that nearly 80% stations follow Gumbel fit at 5%level of significance. Therefore further it is used for the estimation of return period values.

72 B.D. Kulkarni and N.R. Deshpande

5.2.2 Estimation of Return Period Values

Inverting the above equation for F(x),

XT = u + . (ln . ln (T/(T – 1))) (4)

Where T = 1/(1-F) is the return period and XT is corresponding return period value. Here ‘ln’ denotesthe natural logarithm. Equation (4) along with the estimates parameters, has been used in this study for thecomputation of return period values corresponding to 10, 25, 50 100, 500 and 1000 years for durations 1,2 & 3-day for all 321 stations located inside the Krishna basin. Spatial pattern maps for each of the abovereturn periods for the durations of 1-3 days have been prepared which will be useful in designing andplanning the hydraulic structures and also for proper management of water resources and flood controlstudies within the basin. Generalized maps for 10, 100 and 500 –year return periods for 1-3 day durationsare only shown in Figure 5 (a, b, & c) to keep the paper concise. From these Figures; it is seen that 10-year1-day extreme rainfall values ranges from 10 to 40 cm, 100 -years 1-day extreme rainfall is of the order of10 to 60 cm, 500-year 1-day extreme rainfall is of the order of 20 to 70 cm respectively.

Figure 4: Gumbel Fit for 1-3 Day Duration Extreme Rainfall Series at Each Station

Figure 5(a): Spatial Pattern of Estimated 1, 2 and 3-day Rainfall (cm) for 10-yr Return Period

Figure 5(b): Spatial Pattern of Estimated 1, 2 and 3-day Rainfall (cm) for 100-yr Return Period

Statistical Analysis of the Spatial Variability of Very Extreme Rainfall in the Krishna River Basin 73

Figure 5 (c): Spatial Pattern of Estimated 1, 2 and 3-day Rainfall (cm) for 500-yr Return Period

The equation for XT is further used to compute the return periods of PMP for 1-3 day durations. It was foundthat, for most of the stations in the Krishna basin, 1-3 day PMP have return periods more than 1000 years, whichsatisfy the minimum requirement suggested by the Central Water Commission, India (CWC, 1972).

6. CONCLUSIONS

Highest ever recorded rainfall over the basin was found to range 8 to 51cm in 1-day, 10 to 88 cm in 2-dayand 13 to 114 cm in 3-day durations respectively. Over the entire basin, point PMP estimates were found torange from about 18 to 79 cm for 1-day, 27 to 132 cm for 2-day and 31 to 172 cm in 3-day durations.Extreme rainfall, point PMP and maximum rainfall amounts for different return periods over the Krishnabasin presented in this paper will be useful in planning, designing and management of different types ofhydraulic structures for optimum utilization of water resources in the basin.

Acknowledgements

Authors are highly grateful to Prof. B.N. Goswami, Director, IITM, and Pune for his constant encouragementand support for carrying out this study. Authors are also thankful to Dr. N. Singh, Scientist-F and Head ofthe Climatology and Hydrometeorology Division, for his encouragement and support. Thanks are also dueto Director.

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