10. Probability Analysis for Estimation of Annual One Day

8/12/2019 10. Probability Analysis for Estimation of Annual One Day

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PROBABILITY ANALYSIS FOR ESTIMATION OF ANNUAL ONE DAY

MAXIMUM RAINFALL OF JHALARAPATAN AREA OF RAJASTHAN,INDIA

Bhim Singh*, Deepak Rajpurohit, Amol Vasishth and Jitendra Singh

College of Horticulture and Forestry, MPUAT Campus, Jhalarapatan, Jhalawar - 326 023 (Rajasthan), India.

Abstract

The daily rainfall data of 39 years (1973-2011) were analyzed to determine the annual one day maximum rainfall of Jhalarapatan

area of Rajasthan, India. The observed values were estimated by Weibull's plotting position and expected values were

estimated by four well known probability distribution functions viz., normal, log-normal, log-Pearson type-III and Gumbel.

The expected values were compared with the observed values and goodness of fit were determined by chi-square (2) test.

The results showed that the log-Pearson type-III distribution was the best fit probability distribution to forecast annual one

day maximum rainfall for different return periods. Based on the best fit probability distribution, the minimum rainfall of 44.74

mm in a day can be expected to occur with 99 per cent probability and one year return period and maximum of 252.98 mm

rainfall can be received with one per cent probability and 100 year return period. The results of this study would be useful for

agricultural scientists, decision makers, policy planners and researchers for agricultural development and constructions of

small soil and water conservation structures, irrigation and drainage systems in humid south-eastern plain of the Rajasthan,

India.

Key words :ADMR, return period, frequency, probability distribution.

Plant ArchivesVol. 12 No. 2, 2012 pp. 1093-1100 ISSN 0972-5210

Introduction

Rainfall is one of the most important natural input

resources to crop production and its occurrence and

distribution is erratic, temporal and spatial variations in

nature. Most of the hydrological events occurring as

natural phenomena are observed only once. One of the

important problem in hydrology deals with the interpreting

past records of hydrological event in terms of future

probabilities of occurrence. Analysis of rainfall and

determination of annual maximum daily rainfall would

enhance the management of water resources applications

as well as the effective utilization of water resources

(Subudhi, 2007). Probability and frequency analysis of

rainfall data enables us to determine the expected rainfall

at various chances (Bhakar et al., 2008). Such information

can also be used to prevent floods and droughts, and

applied to planning and designing of water resources

related to engineering such as reservoir design, flood

control work and soil and water conservation planning

(Agarwal et al., 1988 and Dabral et al., 2009). Though

the rainfall is erratic and varies with time and space, it is

commonly possible to predict return periods using various

probability distributions (Upadhaya and Singh, 1998).

Therefore, probability analysis of rainfall is necessary

for solving various water management problems and to

access the crop failure due to deficit or excess rainfall.

Scientific prediction of rains and crop planning done

analytically may prove a significant tool in the hands of

farmers for better economic returns (Bhakar et al.,2008).

Frequency analysis of rainfall data has been attempted

for different return period (Bhakar et al., 2006; Barkotullaet al., 2009; Nemichandrappa et al., 2010; Manikandan

et al., 2011 and Vivekanandan, 2012). Probability and

frequency analysis of rainfall data enables us to determine

the expected rainfall at various chances. The probability

distribution functions most commonly used to estimate

the rainfall frequency are normal, log-normal, log-Pearson

type-III and Gumbel distributions. Kumar (2000) and

Singh (2001) concluded that the log-normal distribution is

the best probability model for predicting annual maximum*Author for correspondence:E-mail: [email protected]


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1094 Bhim Singh et al.

daily rainfall for Ranichauri (Tehri-Garhwal) and Tandong

(Sikkim), respectively. Kumar et al. (2007) predicted

annual maximum rainfall and inferred that log-Pearson

type-III probability distribution function can be used to

design hydraulic and soil and water conservation

structures at Almora and similar places in Uttarakhand.

Subudhi (2007) found that normal distribution is the best

fit for predicting the annual maximum daily rainfall of

Chakapada block of Kandhamal district in Orissa. There

is no widely accepted procedure to forecast the one day

maximum rainfall (Barkotulla et al., 2009). In the present

study, an attempt was made to determine the statistical

parameters and annual one day maximum rainfall

(ADMR) at various probability levels using four

probability distribution functions, viz., normal, log-normal,

log-Pearson type-III and Gumbel distribution.

Materials and Methods

Jhalarapatan area is situated in the humid south-eastern plain zone-V of the Jhalawar district of Rajasthan

state at 24033' N latitude and 76010' E longitude with an

elevation of 426.72 m above mean sea level covering an

area of 1282 km2. The mean annual rainfall was 921.5

mm which distributed in 38 rainy days. Area receives 92

per cent of annual of the rainfall during south-west

monsoon i.e.from June to September. The study area

has the expansion of fertile plain having rich black-cotton

soil and is watered dominantly by Ahuand Kalisindh

rivers. The annual mean, maximum and minimum monthly

relative humidity of the region are 69%, 91% (August)

and 40% (April), respectively. The annual mean,maximum and minimum monthly mean daily temperatures

in the district are 27.5C, 48.8C (May) and 5.5C

(February), respectively.

Data collection and analysis

Daily rainfall data of Jhalarapatan raingauge station

has been used for the present investigation. Time series

rainfall records for the period of 39 years (1973 to 2011)

have been collected from Water Resource Department,

Government of Rajasthan, Sinchai Bhawan, Jaipur.

Annual maximum daily rainfall was sorted out from these

data (table 1) and using statistical techniques for dataanalysis. The statistical behavior of any hydrological series

can be described on the basis of certain parameters. The

commonly used procedures of statistical analysis as

followed by Gupta and Kapoor (2002) have been followed

herein. The computation of statistical parameters includes

mean, standard deviation, coefficient of variation and

coefficient of skewness were taken as measures of

variability of hydrological series. All the parameters have

been used to describe the variability of rainfall in the

present study.

Return period

Return period or recurrence interval is the average

interval of time within which any extreme event of given

magnitude will be equalled or exceeded at least once

(Patra, 2001). Return period was calculated by Weibull's

plotting position formula (Chow, 1964) by arranging oneday maximum daily rainfall in descending order giving

their respective rank as:

R

NT

1+= (1)

Where, N - the total number of years of record and

R- the rank of observed rainfall values arranged in

descending order.

Weibull's plotting position formula was used for

computation of observed ADMR amounts at the return

periods of 1.01, 1.05, 1.11, 1.25, 2, 4, 5, 10, 20 and 40

years.

Frequency analysis using frequency factors

Frequency or probability distribution helps to relate

the magnitude of extreme hydrologic events like floods,

droughts and severe storms with their number of

occurrences such that their chance of occurrence with

time can be predicted successfully. Observed values of

ADMR can be obtained statistically through the use of

the Chow's general frequency formula. The formula

expresses the frequency of occurrence of an event in

terms of a frequency factor, Kr, which depends upon the

distribution of particular event investigated. Chow (1951)

has shown that many frequency analyses can be reduced

to the form

)1( TVT KCXX += (2)

Where,Xris maximum value of event corresponding

to return period T; Xis mean of the annual maximum

series of the data of length N years, Cvis the coefficient

of variation and Kris the frequency factor which depends

upon the return period T and the assumed frequency

distribution. The expected value of annual maximum daily

rainfall for the same return periods were computed fordetermining the best probability distributions. Calculations

of frequency factor of the four distributions namely

normal, log-normal, log-Pearson type-III and Gumbel are

discussed as


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ProbabilityAnalysis forEstimationofAnnual OneDayMaximumRainfallofJhalarapatanAreaofRajasthan, India 1095

Normal distribution

For normal distribution, the frequency factor 'Kr' can

be expressed by following equation (Chow, 1988)

Kx

T

T=

(3)

This is the same as the standard normal variate z.

The value of z corresponding to an exceedence of

p(p = 1/T) can be calculated by finding the value of an

intermediate variable w:

where ,

wp=

ln1

2

12

( 50.00 < p ) (4)

Then calculating z using the equation (5)

z ww w

w w w=

+ ++ + +

2 515517 0 802853 0 010328

1 1432788 0189269 0 001308

2

2 3

. . .

. . .(5)

When, p > 0.5, 1-p is substituted for p in equation (4)

and the value of z is computed by equation (5) is given a

negative sign (Bhakar et al.,2006). The frequency factor

KTfor the normal distribution is equal to z, as mentioned

above.

Log-normal distributionFor log-normal distribution, it is assumed that Y= ln

X is normally distributed [the value of variate 'X' (rainfall)

is replaced by its natural logarithm]. The expected value

of rainfall 'XT', at return period T, can be obtained from

the relation

XT= exp(Y

T) (6)

)KC(YY TVYT += 1 (7)

Where, ' 'Y is the mean and 'Cvy' is the coefficient

of variation of Y.

andy

yT

T

yK

= (8)

The value of frequency factor 'KT' can be computed

using equation (5) or found from the standard normal

distribution table.

Log-Pearson type-III

In log-Pearson type-III distribution, the value of

variate 'X' (rainfall) is transformed to logarithm (base

10). The expected value of rainfall 'XT' can be obtained

by the following formulae

XT

= Antilog X (9)

and Log X = M + KTS (10)

where, 'M' is the mean of logarithmic values of

observed rainfall and 'S' is the standard deviation of these

values. Frequency factor KTis taken from Benson (1968)

corresponding to coefficient of skewness (Cs) of

transformed variate as

+

= 11

66

23

SS

S

T

CCz

CK (11)

Gumbel distributionIn Gumbel distribution, the expected rainfall 'X

T' is

computed by the following formula

)1( TVT KCXX += (12)

Where, X is mean of the observed rainfall, CVis the

coefficient of variation; KT- frequency factor which is

calculated by the formula given by Gumbel (1958) as

Distribution Probability density function Range Equation for the parameters in terms

of the sample moment

Normal

2

2

1

2

1)(

=

x

exf < x < = =x Sx

,

Log-normal (y = lnx)

2

2

1

21)(

= yyy

x

exf

0


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+=

1lnln5772.0

6

T

TKT

(13)

Testing the goodness of fit of probability

distribution

The expected values of maximum rainfall were

calculated by four well known probability distributions,viz., normal, log-normal, log-Pearson type-III and Gumbel

distribution at different selected probabilities i.e.99, 95,

90, 80, 50, 25, 20, 10, 5, 2.5, 2, 1 and 0.5 per cent levels.

Among these four distributions, the best fit distributions

decided by chi-square test for goodness of fit to observed

values. The chi-square test statistic is given by the

equation (14)

=

=

k

i i

ii

E

EO

1

2 )( (14)

Where, Oi is the observed rainfall and E

iis the

expected rainfall and will have chi-square distribution with

(N- k -1) degree of freedom (d.f.). The best probability

distribution function was determined by comparing Chi-

square values obtained from each distribution and

selecting the function that gives smallest chi-square value

(Agrawal et al.1988). If Cal. Tab2 2

for (N- k -1)d.f. then

the difference between observed and expected values is

considered to be significant.

Regression model

Regression models were developed for estimating

the ADMR to return periods in the present study and

found the coefficient of determination (R2).

Results and Discussion

One day maximum daily rainfall corresponding date

for the period of 39 years (1973 to 2011) is presented in

table 1. The maximum (231.6 mm) and minimum (45.6

mm) annual one day maximum rainfall was recorded

during the year 2000 (21-July) and 2008 (6-Aug),

respectively. This indicates that the mostly fluctuations

was observed during the decade 2000-11. The average

for these 39 years rainfall was found to be 111.84 mm. It

was also observed that 17 years (43.6%) received oneday maximum daily rainfall above the average (fig. 1),

however, no general trend in rainfall occurrence was

observed during the study period. The distribution of one

day maximum rainfall received during different months

in a year is presented in fig. 2. From the figure, it can be

seen that July received the highest amount of one day

maximum rainfall (46%) followed by August (38%) and

September (13%). This is due to fact that the study area

received most of its rain from southwest monsoon (92

per cent of annual rainfall). The average, standard

deviation, coefficient of variation and skewness of ADMR

for 39 years and their respective logarithmictransformation is given in table 2. These statistical

parameters can be used to find the estimated one day

maximum rainfall from different probability distribution

functions.

The ADMR for the period of 39 years was plotted

against return period in years which was calculated from

Weibull's method and presented in fig. 3. Observed rainfall

were found for return periods of 1.01, 1.05, 1.11, 1.25, 2,

4, 5, 10, 20 and 40 year and presented in table 3. The

expected ADMR for different probability distributions

such as normal, log-normal, log-Pearson type-III andGumbel were calculated and presented in table 3 for

different return periods. The expected ADMR for

different probabilities are graphically represented in fig.

4. From the figure, it can be observed that the estimated

annual ADMR for different probability distributions are

following the same trend of observed rainfall. All four

probability distribution functions were compared by chi-

square test of goodness of fit and then selecting the

function that gave the smallest chi-square value

Table 1 : One day maximum daily rainfall for the period of 1973

to 2011.

S. Rainfall S. RainfallYear Date Year Date

no. (mm) no. (mm)

1. 1973 23-Jul 128.0 21. 1993 17-Jul 80.2

2. 1974 20-Aug 140.0 22. 1994 1-Jul 105.2

3. 1975 2-Sep 75.3 23. 1995 3-Sep 68.44. 1976 8-Jul 64.2 24. 1996 21-Aug 101.4

5. 1977 22-Jul 83.0 25. 1997 7-Aug 85.4

6. 1978 4-Jul 122.6 26. 1998 12-Jul 72.6

7. 1979 7-Aug 76.2 27. 1999 30-Jul 164.2

8. 1980 22-Jun 76.4 28. 2000 21-Jul 231.6

9. 1981 21-Jul 62.8 29. 2001 2-Jul 186.0

10. 1982 17-Aug 125.4 30. 2002 8-Aug 54.2

11. 1983 1-Sep 122.0 31. 2003 16-Sep 83.4

12. 1984 10-Aug 147.0 32. 2004 24-Aug 136.6

13. 1985 9-Aug 173.0 33. 2005 5-Jul 133.014. 1986 27-Jul 230.4 34. 2006 10-Aug 178.8

15. 1987 25-Aug 198.0 35. 2007 8-Jul 67.0

16. 1988 5-Aug 100.6 36. 2008 6-Aug 45.6

17. 1989 6-Aug 80.6 37. 2009 22-Jul 152.0

18. 1990 3-Jul 69.4 38. 2010 5-Sep 51.0

19. 1991 22-Jul 75.2 39. 2011 20-Jul 91.0

20. 1992 16-Aug 124.2


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Table 2 :Computation of statistical parameters of annual one day maximum rainfall.

Statistical parameter Formula Computed value Logarithmic transformation

Average X =

=N

i

iX1N

1X 111.84 2.01

Standard deviation () =

=n

i

i XX

N 1

2)(

1

1 49.035 0.185

Coefficient of variation (CV)

Mean

DeviationStandard=VC 0.438 0.092

Coefficient of skewness (Ck)

=

=

=

N

i

i

k

XXN

M

NN

MNC

1

3

3

3

3

2

)(1

)2)(1( 0.870 0.175

Table 3 :Observed and expected one day maximum rainfall at different probability levels.

S. Probability Return Period Observed Rainfall Expected Rainfall (mm)

no. (%) (years) (mm)

1. 99 1.01 44.4 11.97 43.01 44.74 31.42

2. 95 1.05 51.0 35.55 52.77 53.75 47.85

3. 90 1.11 62.8 48.99 59.31 59.82 57.91

4. 80 1.25 69.4 70.58 71.52 71.30 71.60

5. 50 2 100.6 111.84 102.32 101.05 103.79

6. 25 4 140.0 144.54 135.88 134.89 137.40

7. 20 5 152.0 152.48 145.57 144.93 147.11

8. 10 10 186.0 172.70 173.48 174.53 175.79

9. 5 20 230.4 188.14 198.36 201.75 203.30

10. 2.5 40 231.6 200.08 220.02 226.07 230.28

11. 2 50 203.31 226.27 233.20 238.90

12. 1 100 211.72 243.40 252.98 265.59

13. 0.5 200 218.07 257.19 269.15 292.17

Normal Log Normal Log Pearson Gumbel

Table 4 :Chi-square values at different probability levels for different distributions.

S. no. Probability Return Period (years) Normal Log-normal Log-Pearson Gumbel

1. 99 1.01 87.85 0.04 0.00 5.36

2. 95 1.05 6.72 0.06 0.14 0.21

3. 90 1.11 3.89 0.21 0.15 0.41

4. 80 1.25 0.02 0.06 0.05 0.07

5. 50 2 1.13 0.03 0.00 0.10

6. 25 4 0.14 0.12 0.19 0.05

7. 20 5 0.00 0.28 0.34 0.16

8. 10 10 1.02 0.90 0.75 0.59

9. 5 20 9.49 5.18 4.07 3.61

10. 2.5 40 4.96 0.61 0.14 0.01

Cal

2 115.23 7.50 5.84 10.57


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-100

-50

0

50

100

150

1973

1975

1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

1997

1999

2001

2003

2005

2007

2009

2011

year

Rain

fall(mm)

Fig. 1 :Deviation from the average of one day maximum annual rainfall during 1973-2011.

September

13%

August

38%

July

46%

June

3%

Fig. 2 :Distribution of one day maximum annual rainfall in ayear.

determined the best probability distribution function. The

chi-square values (table 4) for normal, log-normal, log-

Pearson type-III and Gumbel distributions were 115.23,

7.50, 5.84 and 10.57, respectively. Log-Pearson type-III

distribution gave the lowest calculated chi-square value

among the four probability distributions. Hence, log-

Pearson type-III has been found the best probability

distribution for predicting ADMR for Jhalarapatan area

of Rajasthan. According to this distribution, in a day the

minimum rainfall of 44.74 mm rainfall can be expectedto occur with 99 per cent probability and one year return

period and maximum of 252.98 mm rainfall can be

received with one per cent probability and 100 year return

period. A maximum of 101.05 mm rainfall is expected to

occur at every 2 years which is approaching average

ADMR. It is generally recommended that 2 to 100 years

is sufficient return period for soil and water conservation

measures, construction of dams, irrigation and drainage

works (Bhakar et al., 2006). Regression models were

developed from the observed ADMR against different

return period by using Weibull's method. The trend

analysis (fig. 3.) for prediction of one day maximumrainfall for different return period was carried out and it

is found that the exponential trend line gives better

coefficient of determination (R2) = 0.9782 and the

equation is: Y = 48.905 e0.0369Xwhere, Y-ADMR, mm

and X-Return period, Year.

Conclusion

The mean value of ADMR was found to be 111.84

mm with standard deviation and coefficient of variation

of 49.035 and 0.438, respectively. The coefficient of

skewness was observed to be 0.870. July month received

the highest amount of one day maximum rainfall (46%)followed by August (38%) and September (13%). The

frequency analysis of ADMR for identifying the best fit

probability distribution can be studied for four probability

distributions such as normal, log-normal, log-Pearson type-

III and Gumbel by using Chi-square goodness of fit test.

It was observed that all the three probability distribution

functions fitted significantly except normal distribution.

Log-Pearson type-III distribution was found to be the

best fitted to ADMR data by Chi-square test for goodness

of fit. A maximum of 101.05 mm rainfall is expected to

occur at every 2 years and 50 per cent probability which

is approaching mean ADMR. For a recurrence interval

of 100 years and one per cent probability, the annual one

day maximum rainfall is 252.98 mm. Regression model

for ADMR was developed by using Weibull's method to

predict the rainfall for different return period. The

coefficient of determination (R2) is 0.9782. This study

gives an idea about the prediction of ADMR rainfall to

design the small and medium hydraulic and soil and water

conservation structures, irrigation, drainage works,


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Y = 48.905e0.0369 XR2= 0.9782

0

50

100

150

200

250

1.0

26

1.0

53

1.0

81

1.1

11

1.1

43

1.1

76

1.2

12

1.2

5

1.2

90

1.3

33

1.3

79

1.4

29

1.4

81

1.5

38

1.6

1.6

67

1.7

39

1.8

18

1.9

052

2.1

05

2.2

22

2.3

53

2.5

2.6

67

2.8

57

3.0

77

3.3

33

3.6

364

4.4

445

5.7

14

6.6

678

10

13.3

33

20

40

Return period, Year

Rain

fall,m

0.5 1 2 2.5 5 10 20 25 50 80 90 95 99

Probability, %

0

50

100

150

200

250

300

350

Rainfall,mm

Observed Normal Log Normal Log Pearson Gumbel

Fig. 3 :Annual one day maximum rainfall vs return period by Weibull's method.

Fig. 4 :Estimated annual one day maximum rainfall at different probability levels.

vegetative waterways and field diversions. This study

also helps in developing cropping plan and estimating

design flow rate for maximizing crop production.

References

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10. Probability Analysis for Estimation of Annual One Day

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