Regional Analysis of Seasonal maximum monthly rainfall Using L-moments: A case study for Brahmaputra Valley region of ,

Rubul Bora (  [email protected] ) Chandra Nath Bezboruah College https://orcid.org/0000-0002-3164-2255 Abhijit Bhuyn C.K.B. College Biju Kumar Dutta The Assam Kaziranga University

Research Article

Keywords: monthly maximum rainfall, India, Regional Rainfall Frequency

Posted Date: July 20th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-703101/v1

License:   This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Regional Analysis of Seasonal maximum monthly rainfall Using L- moments: A case study for Brahmaputra Valley region of Assam, India

RUBUL BORA, Department of Mathematics, The Assam Kaziranga University, , Assam. (India)

E-mail: [email protected] (ORCID: 0000-0002-3164-2255)

ABHIJIT BHUYAN, C.K.B. College, Teok, Jorhat, Assam (India)

[email protected]

BIJU KUMAR DUTTA, Department of Mathematics, The Assam Kaziranga University, Jorhat, Assam. (India)

[email protected]

Abstract measures. Also, the L-moment ratio diagram, heterogeneity measures, and Z-statistics measures are In this study, Regional rainfall frequency analysis has calculated for regional homogeneity testing. Finding out been carried out in Brahmaputra Valley. For this best fit probability distribution among the Pearson type purpose, daily rainfall data of 11stations are collected III (PE3), generalized extreme value(GEV), generalized for about 15 years from the Indian Meteorological Pareto (GPA), generalized normal(GNO), and the Department (IMD), Guwahati. The daily rainfall data generalized logistic(GLO) for Location, Scale and are then converted to seasonal maximum monthly Shape. Regional growth curves are considered for rainfall such as the Pre-monsoon, Monsoon, and Post- different return periods and develop the regional monsoon periods. Then based on L-moment, we find L- relationship. Monte Carlo simulations are discussed for coefficient variation, L-skewness, L-kurtosis, and large return periods. discordancy measure for regional homogeneity

1 Introduction region. Moreover, the most important reason for low rice production in the entire country is due to a lack of Rainfall is a peculiar phenomenon that varies in adequate and proper water supply in the lake (Panigrahi both space and time. Rainfall distribution is very uneven, and Panda, 2002). The rainfed food production system is and it does not only vary greatly from place to place but under stress as a result of changing rainfall patterns and a also varies year to year. Rainfall is a crucial and lack of adequate land and water resource utilization dominating component in any agricultural program’s (Khan and Hanjra, 2009). Crop planning for a region is development and implementation in any particular primarily influenced by several factors, including

1 irrigation process, drainage, irrigation system quality, soil distributions to express the parameters In1995, according characteristics, topography, and socioeconomic to Bobee and Rasmussen using regional data decreases conditions. However, rainfall magnitude and distribution sampling uncertainty by adding additional data and in space and time are the most important factors in reduces model uncertainty by providing for a better rainfed areas (Shetty et al., 2000). Again, the early end of distribution option. H.J.Fowler and C.G. Kilsby (2003) rains has an impact on production due to several moisture did their research work by using regional rainfall analysis stresses, especially when Kharif crops are in critical from 1961 to 2000 and observed that extreme stages of grain formation and growth (Dixit et al., 2005). precipitation has changed the nature of global climatic change in United Kingdom. Hossein et al. (2014) used L- It is important to manage agriculture on a moment for daily rainfall frequency analysis to develop scientific (technical) basis to make it even more of a rainfall estimation techniques in Iran. region's rainfall pattern in order to maintain crop production at a consistent level. This entails analyzing Frequency analysis gives data that can be used the series of dry and wet spells in a given area to take the to estimate how often a specific occurrence will occur. It necessary steps to prepare a crop plan in rainfed areas. is using probability distributions to collect information on Farmers can find that scientific forecasting of a wet and the volume and frequency of extreme events. (L.V. Noto dry spell analysis is beneficial to improving their fields and G. La Loggia, 2009). Deka et al. (2009) studied and cropping strength, thus improving their economic maximum rainfall daily for 42 years in the North-East condition. There are different methods for rainfall region taking 9 sites by using L-moment and LQ- frequency analysis such as daily, weekly, monthly, moment. Ishfaq et al. (2016) observed the annual daily seasonally annually, and consecutive days maximum maximum rainfall by using different moments in Pakistan rainfall. All are important to forecasting rainfall patterns and concluded that the fit probability distribution was and help the farmers to take better steps during selected by using Goodness of fit estimation. Md. agricultural plantation. It also helps to construct small Ashraful Alam et al. (2017) observed the best fit and big damps, bridges, drainage systems, and different probability distribution for monthly maximum rainfall by engineering projects relating to hydrology. For at-site using the L-moment method in Bangladesh from rainfall data analysis, we have required long-term data commonly use nine distributions. sets because the accuracy of the estimate is directly The probability distribution chosen is crucial proportional to the length of data. But in the case of since a bad choice could lead to significant bias and regional frequency analysis, the biggest advantage is seen inaccuracy in new design flood predictions, especially to be an increase in record length. Many studied have for longer return periods. This can lead to either an been done for regional flood and rainfall frequency underestimation or an overestimation, which can have analysis. Out of these, Hosking and Wallis (1997) major consequences in the operation of our real life. A, developed the L-moment which has a significant S. Rahman et al. (2013) gave stress for finding best- advantage over than ordinary moment that it combines a fitting probability distributions for flood in Australia in probability-weighted moment (PWM) and a linear their research work. In this research paper, we are trying function of the data set in a linear way. Greenwood et al. (1979) had explained the inverse form of the different

2 to find out the best fit probability distribution to get an 1 −2 1 accurate return period for seasonal rainfall analysis. 푘 −1 (푥−휉) 푘−1 (푥−휉) 푓(푥) = 훼 {1 − 푘 훼 } [1 + {1 − 푘 훼 }] (1)

The quantile function is in the form 2 Study region and data collection Q(F) In this study, we collect the rainfall data for the 훼 (1−퐹) 푘 11 sites in the Brahmaputra valley of Assam. They are 푄(퐹) = 휉 + 푘 [1 − { 퐹 } ] (2) Beki, Dibrugarh, Goalpara, Golaghat, Guwahati, Jorhat, 3.1.2 Generalized extreme value distribution (GEV) Kokrajahar, North Lakhimpur, Sibsagar, Tezpur, and A three- parameter distribution for maximum is Kampur from IMD, Guwahati for about 15 years of which gives shape, scale and location parameters. GEV The rainfall data. We take the Brahmaputra valley region for probability distribution function and the quantile function our research because the maximum numbers of Tea of the are given by gardens are found in this area. This is the main pillar of GEV our economy. Also, maximum numbers of mini tea- 1 1 푘−1 garden are found in this region. The production of tea 1 (푥−휉) (푥−휉) 푘 훼 훼 훼 mainly depends on the amount of rainfall. So, the study 푓(푥) = [1 − 푘 ] 푒푥푝 [−{1 − 푘 } ] (3) of rainfall frequency is necessary for providing the 훼 푘 knowledge of the future rainfall patterns of these types of 푄(퐹) = 휉 + 푘 [1— (−푙푛퐹) ] (4)) 3.1.3 Generalized Pareto distribution (GPA) mini tea planters as they are not able to manage water in the dry season like a big tea garden. In precipitation analysis, the GPA distribution is applied. This is also three parameters distribution. The 3 Methods parameters can be estimated by using L-moments

3.1 Different probability distributions utilizing in our methods. The pdf and cdf are as follows: investigation

−1 Hosking and Wallis ( ) and Rao and 푓(푥) = 훼 exp[−(1 − 푘)푦] (5) Where Hamed used different probability1997 distributions −1 −푘 log {1 − 푘(푥 − 휉)/훼, 푘 ≠ 0 under . We used these methods in this (2000) 푓(푥) = { (푥 − 휉) ⁄ , 푘 = 0 훼 discussion. L − moment And we consider five distributions, for that the probability density functions and quantile 0 ≤ 푥 ≤ 훼⁄ 푖푓 푘 > 0, 0 ≤ 푥 <∝ 푖푓 푘 ≤ 0. 푘 functions are given below. (PDFs) 퐹(푥) = 1 − exp(−푦) 3.1.1 Generalized logistic distribution (GLO) 푘 훼{1−(1−퐹) } 휉 + 푘 , 푘 ≠ 0 A three-parameter distribution consisting of 푥(퐹) = 푓(푥) = { (6) 휉 − 훼 log (1 − 퐹) , 푘 = 0 location , scale , and shape is GLO. The 3.1.4 Generalized Normal (lognormal) (GNP and(휉) quantile functions(훼) for this are(푘) given by PDF 3

A three- parameter distribution is which where X is a real valued random variable with and are real. A special gives shape, scale and location parameters.GNO The probability distribution function and quantile function of distributioncase for the two function F p, r, sand are the are given by 1,0.s 1,r,0 generally used andmoments defined asM M GNO

(푥−휉) 1 1 (푥−휉) 2 1 exp [− log{1−푘 훼 }−2[−푘 log{1−푘 훼 }] 푠 1 훼푟 = 푀1,0.푟 = ∫0 푥(퐹)(1 − 퐹) 푑퐹 (12) 푓(푥) = 훼(2휋)2 (7) The quantile function is in the form 1 푟 훽where푟 = 푀 1,푟,0 = ∫ℝ0 푥(퐹)퐹 푑퐹is the,푟 =conventional 표,1.2,.. moment(13) of Q(F) order 푝, about 푟, 푠 ∈ the ,origin 푀푝,0,0 and is the inverse of 훼 −1 푄(퐹) = 휉 + 푘 {1 − 푒푥푝 (−푘휙 (퐹)} (8) calculatedp at the probability 푥(퐹). CDF 푥 3.1.5 Pearson type III (PEIII) F In 1993, Hosking and Wallis gave the general A three- parameter ( represent form L-moments in connection with PWM as location, scale and shape) distribution휉, 훼 is 푎푛푑 PE3 훽 which gives shape, scale and location parameters. The probability 푟 푟 ∗ 푟 ∗ distribution function and quantile function of the PE3 are 휆푟+1 = (−1) ∑푘=0 푝 푟,푘 훼 푘 = ∑ 푘=0 푝 푟,푘 훽 푘, 푟 = 0, 1 … . given by (14) where defined by Hosking and Wallis (1997) as ∗ 푝푟,푘 훼 −1 훼−1 (푥−휉) 푟−푘 푓(푥) = (훽 훤훼) (푥 − 휉) 푒푥푝 {− 훽 } (9) ∗ (−1) (푟+푘)! 푟,푘 2 2 3 푝 = (푘!) (푟−푘) ! ( 15 ) 2 훾 −1 훾 2 푄 (푥) = 휇 + 휎 [훾 {1 + 6 휙 (퐹) − 36} − 훾] (10) Therefore the first four L-moments, which are the linear combination of PWMs, are

2휎 4 1 where 휇 = 휉 + , 훼 = 2 푎푛푑 훽 = 휎 푚표푑(훾) 훾 훾 2 By putting in the above four 휆1 = 훽0 (16) 1 distribution functions, 퐹we = have 1 − 푇the Quantile Function for 휆2 = 2훽1 − 훽0 (17) return period years and is known as statistical ) distribution function. 푇 휆3 = 6훽2 − 6훽1 + 훽0 (18

3.2 Moments of Distribution 휆 4 = 20훽 3 − 30 훽 2 + 12 훽 1 − 훽 0 (19)

Probability Weighted Moments and L-Moments Also, we have

Probability Weighted Moments was

1 given by Greenwood et al. first time(PWMs) as ∗ 휆푟 = ∫0 푥퐹(푥)푃푟−1(퐹)푑퐹, 푟 = 1,2,… (20) (1979) where is the shifted Legendre ∗ 푟 ∗ 푘 th 푝 푟 푠 polynomial.푃푟 (퐹 ) = ∑푘=0 푃푟,푘퐹 r 푀푝,푟,푠 = 퐸[푋 {퐹(푥)} {1 − 퐹(푥)} ] = 1 푝 푟 푠 ∫0 [푥(퐹)] 퐹 (1 − 퐹) 푑퐹 (11) 4

The L-moments ratio (LMRs) is given by Hosking and Wallis (1993) developed two tools Hosking and Wallis (1997) as: for Discordancy measure . It is a measure of

dissimilarity. The difference (D betweeni) a site's L-moment Coefficient of L-Variation, ratios and the average L-moment ratios of similar sites is 휆2 Coefficient of L-skewness 휏 = 휆1 (21) used to calculate the similar sites. This data can also be 휆3 used to find out incorrect data sets. The discordance 휏3 = λ휆2 (22) Coefficient of L-Kurtosis, τ λ 4 . assessment calculates the distance between the 4 2 = (23) giveni site and the group's middle. τ τ and τ are denoted in short by and (D )

, 3 4 퐿퐶푣, 퐿퐶푠 퐿퐶푘 Hosking and Wallis (1993, 1997) established the The sample estimation of L-moments can be formula for discordance measure by taking the sites in defined as: a group as follows NS λ ^ r ∗ Let be a vector containing the r+1 k=0 r,k k = ∑ p b (24) (푖) (푖) (푖) With LMRs, 푢푖 = [ˆ휏 ˆ휏 3 ˆ휏 4 for] the site , then the group (푖) (푖) (푖) average ˆ휏for NS , ˆ휏 sites 3 , is ˆ휏 given 4 equation (푖)

−1 푛 (푗−1)(푗−2)…(푗−푟) 푏푟 = 푛 ∑푗=푟+1 (푛−1)(푛−2)…(푛−푟) 푥푗 (25) 27 where , for is the ordered sample and n is 1 푁푆 푢̅= 푁푆 ∑푖=1 푢푖 ( ) the sample 푥푗 size.푗 =The 1, 2,sample … 푛 estimates The sample Covariance matrix is given by the equation of and are unbiased. 28 푁푆 푇 훽푟 휆푟 푆 = ∑푖=1(푢푖 − 푢̅)(푢푖 − 푢̅) ( ) Again, another equivalent statistical tool was The discordancy measure is given for the site given by Hosking and Wallis (1997) for the L-moments 푖 by the equation ratio which can be used to judge the consistency with the data sample. These are 29 1 푇 −1 λˆ λˆ λˆ 푖 푖 푖 ˆτ ˆτ ˆτ 퐷 = 3 푁푆 (푢 − 푢̅) 푆 (푢 − 푢̅) ( ) λˆ λˆ λˆ 2 3 4 3 4 = 1 , = 2 = 2 (26) If is high, a site is declared to be unusual. 3.3 Regional rainfall frequency analysis Hosking and퐷 푖Wallis (1997) calculated푖 a critical value for At our study site, we use the steps outlined by the discordancy statistic against the number of sites in

Hosking and Wallis (1997) for regional rainfall an area. If is greater than퐷푖 the critical value, the site is frequency analysis. Step-by-step instructions are declared to퐷 be푖 discordant. 푖 provided in the subsections below. Following are the tabulated form for discordancy 3.3.1 Screening of data by using the Discordancy measures for seasonal rainfall of gauging sites. measure 11

5

In this analysis we use the free package of the LMOMENTS contains Fortran-77 source code for these routines on request from hoskingOwat son. ibm. com, or from the Statlib software repository, http : //lib . stat . emu. edu/ general/lmoments

Table-1: Name of sites, Number of records, 1st L-moment, L-CV, L-skewness, L-kurtosis and discordancy measures for maximum monthly rainfall of the Pre-Monsoon period of the Brahmaputra Valley

Name of Site Number of 1st L- L-CV L- L- discordancy record moment Skewness kurtosis Beki 15 351.78 0.2031 -0.0717 0.3080 2.23 Dibrugarh 15 348.45 0.1856 -0.0466 0.0984 0.67 Goalpara 15 411.04 0.1618 0.0667 0.1964 0.13 Golaghat 15 231.77 0.1477 0.0001 0.1309 1.07 Guwahati 15 265.40 0.1933 0.0378 0.1753 0.12 Jorhat 23 284.17 0.1800 0.1239 0.1118 0.23 Kokrajhar 15 466.74 0.1588 0.1103 0.1304 0.39 North Lakhimpur 15 382.32 0.1376 0.1914 0.2412 0.90 Sibsagar 15 265.16 0.2353 0.2623 0.0347 2.77 * Tezpur 15 275.41 0.1515 0.2288 0.2926 1.42 Kampur(Nagaon) 14 158.69 0.2169 -0.0986 0.1156 1.06

Table-2: Name of sites, Number of records, 1st L-moment, L-CV, L-skewness, L-kurtosis and discordancy measures for maximum monthly rainfall of the Monsoon period of the Brahmaputra Valley

Name of Site Number of 1st L- L-CV L- L- discordancy record moment Skewness kurtosis Beki 15 716.50 0.1893 0.3148 0.2113 0.40 Dibrugarh 15 542.65 0.0968 0.0622 0.1920 1.15 Goalpara 15 606.60 0.1890 0.3332 0.2164 0.40 Golaghat 15 358.70 0.1606 0.1557 0.2384 1.42 Guwahati 15 386.98 0.1803 0.4147 0.3746 1.59 Jorhat 23 396.17 0.1277 0.1835 0.3506 1.09 Kokrajhar 15 904.37 0.1762 0.2939 0.0624 1.26 North Lakhimpur 15 839.07 0.1158 0.0051 0.0819 1.20 Sibsagar 15 397.45 0.1542 0.1802 0.0761 0.39 Tezpur 15 391.08 0.1287 0.2003 0.1797 0.85

6

Kampur(Nagaon) 14 339.28 0.1954 0.2321 0.0507 1.25

Table-3: Name of sites, Number of records, 1st L-moment, L-CV, L-skewness, L-kurtosis and discordancy measures for maximum monthly rainfall of the Post-Monsoon period of the Brahmaputra Valley

Name of Site Number of 1st L- L-CV L- L- discordancy record moment Skewness kurtosis Beki 15 106.49 0.3861 0.0464 -0.0114 0.42 Dibrugarh 15 108.25 0.3271 0.1871 0.0012 1.16 Goalpara 15 126.46 0.3801 0.2470 0.3180 1.26 Golaghat 15 91.87 0.3272 -0.0318 -0.0465 0.66 Guwahati 15 108.58 0.2941 -0.0109 0.2493 1.13 Jorhat 23 108.25 0.3868 0.3015 0.0962 0.74 Kokrajhar 15 166.03 0.4495 0.1454 0.0801 1.29 North Lakhimpur 15 144.28 0.3872 0.2159 0.2006 0.42 Sibsagar 15 90.62 0.3243 0.2814 0.1589 1.02 Tezpur 15 85.29 0.2434 -0.0901 0.1710 1.59 Kampur(Nagaon) 14 85.16 0.3686 -0.1209 -0.0766 1.30

Hosking and Wallis (1997) gave the critical value for for discordancy measure corresponding to the number of study sites considered. 퐷푖

Table-4: Critical values for

퐷푖

No. of 5 6 7 8 9 10 11 12 13 14 15 sites ≥ Critical 1.333 1.648 1.917 2.140 2.329 2.491 2.623 2.757 2.869 2.971 3.00 value The bold numbers represent the critical value for our study area

L-moments ratios for 11 sites

7

-0.10-0.050.000.050.100.150.200.250.30 -0.10-0.050.000.050.100.150.200.250.30 0.35 Pre-Monsoon 0.24 0.24 Pre-Monsoon 0.30 Sibsagar 0.22 0.22 0.25

0.20 0.20 0.20

L-CV 0.18 0.18 0.15

L-kurtosis

0.16 0.16 0.10

0.14 0.14 0.05 Sibsagar

0.12 0.12 0.00 -0.10-0.050.000.050.100.150.200.250.30 -0.10-0.050.000.050.100.150.200.250.30 L-Skewness L-skewness

Fig 1 Fig 2

. .

Result and discussion 3.3.2 Heterogeneity measure means identification of homogeneous region From Table-1 to Table-3, it is clear that there is A heterogeneity measure is the second statistic, only one site Sibsagar whose value is 2 77 which is which is used to estimate the level of heterogeneity in a more than its critical value of 2푖 623 for 11 study sites. 퐷 . grouping of sites and whether they can be called We have the interval range for as 0 39 1 59 . homogeneous. Hosking and Wallis (1997) suggested this for maximum monthly rainfall of푖 the Monsoon푖 season 퐷 . ≤ 퐷 ≤ . test. The heterogeneity measure , for a group of sites and 0 42 1 59 for maximum monthly rainfall of that would be expected in a homogeneous 퐻푖 area. and the Post-Monsoon. ≤ 퐷푖 ≤ season.. But from Table-4, the critical are the three levels of variability. The heterogeneity1 2 value of for in this region is 2.623. So, discordancy 푉 , 푉 test푉3 is then formulated as measure shows퐷푖 that there is only one site Sibsagar which is discordant for maximum monthly rainfall of the Pre- 30 monsoon season in our study area. But from Fig. 1 and 푉푖−휇푉푖 푖 퐻 = 휎푉푖 , 푖 = 1, 2, 3. ( ) Fig. 2, we have observed that the L-CV is higher and L- And and are given by Kurtosis is lower for Sibsagar than the other site of the 푉1, 푉2 푉3 region. But, again if we used the regional formula for 31 푁푆 (푖) 푅 2 푁푆 estimating the rainfall for different return periods it will 푉 1 = ∑푖=1 푁푖( ˆ휏 − ˆ휏 ) / ∑푖=1 푁 푖 ( ) not give an absurd result for the Sibsagar site and also the 2 푉 = region is found to be homogeneous including the site 푁푆 (푖) 푅 2 (푖) 푅 2 푁푆 ∑푖=1 푁푖 √ {(ˆ휏 − ˆ휏 ) + ( ˆ휏 3 32− ˆ휏 3 ) } / ∑ 푖=1 푁 푖 Sibsagar. Though it is found to be discordant we can ( ) include the site for our study. 푁푆 (푖) 푅 2 (푖) 푅 2 푉3 = ∑ 푖=1 푁 푖 √ {( ˆ휏 3 − ˆ휏 3 ) + ( ˆ휏 4 − ˆ휏 4 ) } / 33 푁푆 ∑푖=1 푁푖 ( ) 8 where represents the number of site, is the record Table-5: Heterogeneity Measures of our study area for

푖 Pre-monsoon, Monsoon, and Post-monsoon rainfall length at푁푆 each site and , and 푁are the regional (푅) 푅 푅 average L-moments, canˆ휏 be obtainedˆ휏3 ˆ휏using4 the following formulation Heterogeneity Values

Measures Pre- Post- Monsoon 34 푁푆 (푖) monsoon monsoon 푅 ∑푖=1 푁푖 ˆ휏 푁푆 H ˆ휏 = ∑푖=1 푁푖 ( ) 1 0 70 0 04 0 56

푁푆 (푖) 35 H2 푅 ∑푖=1 푁푖 ˆ휏3 −0.54 −0.80 −1 84. 3 푁푆 ˆ휏 = ∑푖=1 푁푖 ( ) H 3 1 02 0 56 2 19 푁푆 (푖) 36 − . − . . 푅 ∑푖=1 푁푖 ˆ휏4 4 푁푆 ˆ휏 = ∑푖=1 푁푖 ( ) − . − . . Result and discussion

Also, and represent the standard From the above table, we observe that the value deviation of where푉푖 푉푖 . i 휇 휎 of 0 04 and 0 56 for maximum monthly

푉 푖 = 1,2,3 rainfall1 in the Pre-Monsoon, Monsoon, and Post- To assess the heterogeneity measures, a Kappa 퐻 = −0.70, − . − . Monsoon period respectively which are less than . So, distribution (Hosking, 1988) is appropriate for grouping 1 the heterogeneity measures of our study site can be average L-moment 1, , . From this Kappa 푅 푅 푅 considered as a homogeneous region. Though the site distribution, a large ˆ휏 number , ˆ휏3 ˆ of휏4 regions are Sibsagar is found to be discordant in Table-1, we simulated. The data is presumed to be free 푁 of푠푖푚 cross- consider the site for regional frequency analysis then the correlation and serial correlation, and the regions are region is found to be homogeneous including this site in assumed to be homogeneous. The sites' record lengths are Table-5. believed to be the same as their real-world counterparts.

are determined for each simulated area. 3.3.3 Z-Statistics Criteria and L-moment ratio diagram The푉푖(푖 = heterogeneity 1, 2, 3) can be calculated by using the for Goodness of fit equation 3 3 30 . If퐻 H푖 i is big enough, an area is said to be heterogeneous. ( . . ) An area is appropriate homogenous, Two methods for determining the best-fit according to Hosking and Wallis (1997) if , probabilistic model were developed by Hosking and probably heterogeneous if , and certainly 퐻푖 < 1 Wallis (1997). The Z-Test and the L-moments ratio heterogeneous if . 푖 1 ≤ 퐻 < 2 diagram are the two procedures used for this goal 퐻푖 ≥ 2 According to Hosking and Wallis 1993 . (a) Z- Test Criteria: statistics and based on the measures and( do), This method compares the simulated L-skewness not have 퐻 enough2 퐻 capacity3 to tell the difference푉2 between푉3 homogeneous and heterogeneous areas, whereas and L-kurtosis values of the fitted distribution to the regional average L-skewness and L-kurtosis values of based on does far better. As a result, the statistic,퐻1 secondary data. The main advantage of this test is that it which is based푉1 on , is suggested as a primary퐻1 predictor of heterogeneity. 푉1

9 works directly to the regional average and it was given by Hosking and Wallis (1997. The formula is given below Result and discussion From Table-6, it has been observed that for the 37 Pre-monsoon rainfall period all the four distributions i.e. 퐷푖푠푡 퐷푖푠푡 푅 GLO, GEV, GNO, and PE3 have Z-statistics values less 푍 = (휏4 − ˆ휏4 + 퐵4)/휎4 ( ) where is the regional average of and are than the critical value1 64. But out of these four 푅 the biasˆ휏 and4 standard deviation of ˆ휏respectively.4 퐵4, 휎4 Z probability distributions, the. GLO has the smallest - ˆ휏4 And these are defined as statistics value 0 11 therefore it is the best fitting distribution for the( .maximum), monthly rainfall of the Pre- 38 monsoon period in our study area. In the same way, for 1 푁푠푖푚 푚 푅 퐵4 = 푁푠푖푚 ∑푚=1 ( 휏4 − ˆ휏4 ) ( ) the Monsoon period rainfall, the best fit probability distribution is also GLO with the least critical value of 0 00 among the distributions GLO, GEV, and GNO. And 1 푁푠푖푚 푚 푅 2 2 4 4 푠푖푚 4 휎 = √[(푁 푠푖푚−1) {∑ 푚=1 ( 휏 4 − ˆ휏 ) − 푁 퐵 }] (39) for. the Post-monsoon period maximum monthly rainfall, the best fit probability distribution is PE3 which has the represents the number of simulated data set and m least critical value of 0 87 among the three distributions

푁is푠푖푚 the simulated region caused by Kappa GEV, GNO, and PE3. Here, we observe that GPA 푡ℎ . distribution.푚 distribution has a critical value of more than the Z- statistics value of 1.64 for all the seasonal rainfall thought Table-6: The statistics measures of the study area for the year. seasonal maximum|푍| rainfall

(b) L-moment ratio diagram:

Hosking (1991) defined the L-moment ratio -statistics values diagram as a curve that illustrates the theoretical Distributions Pre- | | Post- 푍 Monsoon relationships between L-skewness and L-kurtosis of monsoon monsoon various candidate distributions. When summarising GLO 0.11 0.00 2.68 regional results, L-moments ratio diagrams have been GEV 1.46 0.95 1.04 presented as a way for discriminating between candidate GNO 1.25 1.22 1.11 distributions. PE3 1.35 1.76 0.87 GPA 4.53 3.15 2.31

10

0.4 0.4 Pre-Monsoon Rainfall Monsoon Rainfall GLO GLO GEV GEV GNO GNO 0.3 PE3 PE3 GPA 0.3 GPA

0.2

Regional Average 0.2 Regional Average

L-Kurtosis L-Kurtosis

0.1 0.1

0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 L-Skewness L-Skewness

Fig. 3 Fig 4

.

0.4 Post-Monsoon Rainfall GLO GEV GNO 0.3 PE3 GPA

0.2

L-Kurtosis

0.1 Regional Average

0.0 0.00.10.20.30.40.5 L-Skewness

Fig 5

.

Table-7: Regional average L-moments:

L-moments Pre-monsoon Monsoon Post-monsoon (Regional) L-CV 0 1791 0 1543 0 3537

L-Skewness 0.0765 0.2144 0.1168 L-kurtosis 0.1646 0.1934 0.1044 . . . 11

Result and discussion 3.3.4 Development of regional parameters and Fig. 3, Fig. 4, and Fig. 5 represent the graph of quantiles the L-skewness and L-kurtosis of regional average for the A suitable probability distribution is needed for fitting of five distributions (GLO, GEV, GNO, PE3, and regional frequency analysis after the homogeneity GPA) on the same graph during the Pre-Monsoon, analysis of the study area has been completed. The goal Monsoon, and Post-Monsoon periods of maximum is to create a probability distribution that is not only good monthly rainfall. It is clear from the graph that the for regional frequency analysis, but also for estimating regional average (Table-7) values for the Pre-Monsoon, the regional growth curve's quantiles. GLO, GEV, GPA, Monsoon, and Post-Monsoon periods; L-skewness and L- GNO, and PE3 are among some of the candidate kurtosis are oriented toward the GLO, GLO, and PE3 probability distributions for the regional frequency distributions, respectively. Therefore, the L-moment ratio analysis of our study region. diagram also shows the best fit probability distribution for these seasons of our study region.

Table-8: Regional Parameters of the Pre-monsoon, Monsoon and the Post-monsoon maximum monthly rainfall:

Dist. Pre-monsoon Monsoon Post-monsoon Location Scale Shape Location Scale Shape Location Scale Shape GLO 0.978 0.177 -0.076 0.947 0.143 -0.214 - - - GEV 0.870 0.292 0.151 0.865 0.208 -0.068 0.726 0.549 0.084 GNO 0.975 0.314 -0.157 0.941 0.252 -0.444 0.926 0.612 -0.240 PE3 1.000 0.320 0.468 - - - 1.000 0.637 0.713

Regional estimates of quantiles for different return periods Table-9: Pre-monsoon rainfall

Return Period Distribution 1 2 5 10 20 50 75 100 500 1000 GLO 0.619 0.978 1.237 1.402 1.563 1.781 1.888 1.954 2.388 2.591 GEV 0.611 0.974 1.262 1.427 1.568 1.730 1.798 1.837 2.046 2.121 GNO 0.611 0.975 1.258 1.421 1.565 1.736 1.812 1.857 2.118 2.224 PE3 0.610 0.975 1.259 1.422 1.565 1.733 1.807 1.851 2.102 2.203

12

Table-10: Monsoon rainfall

Return Period Distribution 1 2 5 10 20 50 75 100 500 1000 GLO 0.696 0.947 1.177 1.348 1.533 1.815 1.966 2.065 2.805 3.210 GEV 0.696 0.942 1.194 1.371 1.550 1.795 1.915 1.989 2.475 2.700 GNO 0.695 0.941 1.198 1.376 1.551 1.786 1.898 1.967 2.409 2.610

Table-11: Post-monsoon rainfall

Distribution Return Period 1 2 5 10 20 50 75 100 500 1000 GEV 0.252 0.924 1.499 1.850 2.167 2.550 2.718 2.817 3.379 3.597 GNO 0.250 0.926 1.496 1.844 2.160 2.550 2.727 2.832 3.463 3.729 PE3 0.248 0.925 1.502 1.849 2.160 2.537 2.705 2.804 3.386 3.624

Result and discussion 3.3.5 Development of regional relationship for Seasonal rainfall Again from Table-8, we have found that for the

Pre-monsoon season maximum monthly rainfall, GLO is Following are the regional rainfall frequency the best fit probability distribution among the four relationship for the 11 gauging stations of the distributions GLO, GEV, GNO, and PE3. For the Brahmaputra Valley for the best fit probability Monsoon season also GLO is the best fit probability distributionsGLO GLO, and PE3 together with the distribution among the three distributions GLO, GEV, regional parameters given in Table-8. and GNO. Finally, among GEV, GNO, and PE3; PE3 is , the best fit probability distribution for the Post-monsoon Pre-monsoon rainfall: season maximum monthly rainfall to describe the three 1st L parameters Location, Scale, and Shape. −0.076 푗 1 moment푄푇 = [−1.of site351j + 2.329 × ( 푇−1 ) ] × − 40 Table-9 to Table-11; gives us the different return periods of best fit distributions GLO, GLO, and ( ) Monsoon rainfall: PE3 for the Pre-monsoon, Monsoon, and Post-monsoon L rainfall. It can be extended to more return periods. −0.214 푗 1 푠푡 moment푄푇 = [0.of279site+j 0. 668 × ( 푇−1 ) ] × 1 − 41

13 ( )

Post-monsoon rainfall: The formula for repetitions, the Bias and relative RMSE are No explicit formula of quantile function for PE3 푀 distribution exists. Therefore, for estimating quantiles for [푚] 42 each site of the region we can multiply the regional 1 푀 {ˆ푄 푖 (퐹)−푄푖(퐹)} 푖( ) 푀 ∑푚=1 푄푖(퐹) growth curve given in Table-11 for PE3 distribution by 퐵 퐹 = ( ) the 1st L- moments of the site can be used. 1 43 [푚] ⁄2 where represents the quantile estimates for the period 1 푀 {ˆ푄 푖 (퐹)−푄푖(퐹)} 2 푖 푚=1 푖( ) 푗 푅 (퐹) = [ 푀 ∑ { 푄 퐹 } ] ( ) T of the푄 푇site j. The estimated quantiles' regional average relative bias, 3.3.6 Monte Carlo Simulations for estimating rainfall absolute relative bias, and relative RMSE are calculated quantiles by using the followings.

Monte Carlo simulations are used to assess the 44 accuracy of projected rainfall quantiles (Hosking and 푅 1 푁 퐵 (퐹) = 푁 ∑푖=1 퐵푖(퐹) (45) Wallis, 1997). For each homogeneous area, a region with 푅 1 푁 푖=1 푖 the same number of locations, record time at each station, 퐴 (퐹) = 푁 ∑ |퐵 (퐹)| (46) 푅 1 푁 heterogeneity, and regional average L-moment ratios as 푅 (퐹) = 푁 ∑푖=1 푅푖(퐹) ( ) The growth curve for the site is denoted by q F and previous data is simulated. For large M simulated i defined as locations, this technique was done M times. The bias, 푖 ( )

root mean square error (RMSE), and 90 percent error bounds were evaluated after errors in the simulated 푄푖(퐹) = 휇푖푞푖(퐹) growth curve and quantiles were obtained for every Amina Shahzadi et al. (2013) used Monte Carlo simulation. simulation technique for finding large return period for quantile estimation in Pakistan by using regional L- moments.

Table-12: Simulation result for the Monsoon season of the regional growth curve

Dist. Measures Return Period 1 2 5 10 20 50 100 500 1000 Abs. Bias 0.027 0.005 0.010 0.017 0.022 0.028 0.032 0.038 0.041 Bias 0.001 0.005 0.001 -0.003 -0.007 -0.011 -0.014 -0.021 -0.023 GNO RMSE 0.043 0.017 0.016 0.029 0.044 0.064 0.079 0.113 0.128 L. Bound 0.662 0.916 1.174 1.329 1.469 1.646 1.776 2.073 2.201 U. Bound 0.733 0.963 1.221 1.432 1.655 1.970 2.226 2.890 3.207 GLO Abs. Bias 0.026 0.007 0.009 0.016 0.022 0.030 0.037 0.051 0.057

14

Bias -0.000 0.007 0.002 -0.003 -0.009 -0.019 -0.026 -0.044 -0.051 RMSE 0.046 0.019 0.016 0.029 0.047 0.074 0.097 0.158 0.187 L. Bound 0.660 0.917 1.151 1.298 1.447 1.657 1.831 2.308 2.551 U. Bound 0.742 0.969 1.198 1.403 1.646 2.043 2.421 3.656 4.406 Abs. Bias 0.026 0.005 0.010 0.017 0.022 0.029 0.033 0.042 0.045 Bias 0.000 0.005 0.001 -0.003 -0.007 -0.014 -0.018 -0.028 -0.032 GEV RMSE 0.043 0.016 0.016 0.028 0.044 0.068 0.088 0.140 0.164 L. Bound 0.663 0.917 1.167 1.325 1.469 1.649 1.778 2.059 2.171 U. Bound 0.736 0.962 1.216 1.424 1.655 1.998 2.291 3.110 3.530

Table-13: Simulation result for the Pre-Monsoon season of the regional growth curve

Dist. Measures Return Period 1 2 5 10 20 50 100 500 1000 Abs. Bias 0.038 0.002 0.012 0.018 0.022 0.025 0.027 0.032 0.033 Bias 0.001 0.001 -0.000 -0.001 -0.001 -0.000 0.001 0.003 0.004 GNO RMSE 0.079 0.019 0.025 0.038 0.052 0.070 0.084 0.114 0.127 L. Bound 0.553 0.947 1.213 1.343 1.446 1.556 1.627 1.763 1.812 U. Bound 0.690 1.007 1.301 1.497 1.680 1.909 2.079 2.464 2.627 Abs. Bias 0.037 0.002 0.011 0.017 0.022 0.026 0.029 0.035 0.037 Bias 0.002 0.002 -0.000 -0.002 -0.002 -0.003 -0.003 0.001 0.004 GLO RMSE 0.086 0.021 0.027 0.040 0.056 0.080 0.100 0.157 0.187 L. Bound 0.555 0.946 1.186 1.320 1.437 1.575 1.669 1.869 1.947 U. Bound 0.714 1.011 1.283 1.482 1.689 1.986 2.235 2.921 3.272 Abs. Bias 0.038 0.002 0.012 0.018 0.022 0.025 0.027 0.031 0.033 Bias 0.001 0.001 -0.000 -0.001 -0.001 0.000 0.002 0.007 0.011 GEV RMSE 0.078 0.018 0.025 0.037 0.051 0.071 0.087 0.126 0.143 L. Bound 0.554 0.947 1.216 1.352 1.452 1.547 1.598 1.661 1.672 U. Bound 0.690 1.003 1.305 1.501 1.682 1.906 2.066 2.406 2.539 Abs. Bias 0.038 0.002 0.012 0.018 0.022 0.025 0.027 0.031 0.033 Bias 0.001 0.001 0.000 -0.000 -0.000 -0.001 -0.001 -0.001 -0.001 PE3 RMSE 0.078 0.019 0.025 0.039 0.052 0.068 0.079 0.102 0.110 L. Bound 0.553 0.948 1.214 1.342 1.446 1.560 1.637 1.794 1.855 U. Bound 0.688 1.007 1.301 1.498 1.679 1.903 2.066 2.428 2.579

15

Table-14: Simulation result for the Post-Monsoon season of the regional growth curve

Dist. Measures Return Period 1 2 5 10 20 50 100 500 1000 Abs. Bias 0.124 0.004 0.014 0.019 0.022 0.025 0.027 0.031 0.033 Bias -0.016 0.003 0.002 0.002 0.004 0.006 0.008 0.016 0.020 GNO RMSE 0.457 0.055 0.049 0.073 0.099 0.133 0.158 0.216 0.241 L. Bound -0.153 0.857 1.378 1.636 1.830 2.045 2.179 2.436 2.529 U. Bound 0.000 1.026 1.606 2.055 2.497 3.082 3.531 4.613 5.102 Abs. Bias 0.126 0.004 0.014 0.019 0.022 0.025 0.027 0.029 0.030 Bias -0.013 0.002 0.003 0.004 0.005 0.005 0.006 0.007 0.007 PE3 RMSE 0.449 0.056 0.048 0.075 0.100 0.127 0.145 0.180 0.193 L. Bound -0.151 0.857 1.386 1.633 1.829 2.058 2.213 2.531 2.658 U. Bound 0.000 1.030 1.610 2.062 2.495 3.050 3.463 4.411 4.818 Abs. Bias 0.122 0.004 0.014 0.019 0.022 0.025 0.027 0.034 0.040 Bias -0.020 0.003 0.002 0.002 0.002 0.005 0.010 0.025 0.034 GEV RMSE 0.455 0.052 0.051 0.071 0.097 0.136 0.168 0.253 0.296 L. Bound -0.155 0.856 1.378 1.651 1.848 2.037 2.131 2.237 2.249 U. Bound 0.000 1.017 1.612 2.057 2.505 3.102 3.565 4.652 5.133

Regional Growth Curves for the three different seasons

2.5 Pre-Monsoon Monsoon GLO GLO 3.0 GEV GEV GNO GNO 2.0 PE3 2.5

2.0

1.5

Growth Curve Growth Growth Curve Growth 1.5

1.0 1.0

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 Gumbel Reduce variate-log(-log(F)) Gumbel Reduce variate-log(-log(F))

Fig. 6 Fig. 7

16

GEV 3.5 Post-Monsoon GNO PE3 3.0

2.5

2.0

Growth Curve Growth 1.5

1.0

0.5

0123456 Gumbel Reduce variate-log(-log(F))

Fig. 8 Result and discussion From Table-12, we observed that for return We choose GNO, GLO, and GEV as suitable periods 1, 2, 5, 10, and 20 absolute bias for GNO is distributions for the monsoon season. For this region, the slightly high than GLO and GEV while for high return regional average correlation is 0.27. Hosking and Wallis periods 50, 100, 500, and 1000 it is low. But in practice, (1997) have developed the algorithm for the simulation absolute bias and bias are not useful for quantities. So, procedure (in Table-6.1) which is used in our discussion. we observe the simulation results for GNO in the case of In this region there are 11 sites with record length and RMSE. It is low for the large return period 50,100, 500, has been simulated for M= 10,000 times and a regional and 1000 and relatively high for low return periods 1, 2, L-moment algorithm is used to fit the distributions to the 5, 10, and 20 as compared to the distributions GLO and data generated for the estimation of quantiles and a GEV. Again, error bounds (LEB, UEB) for GNO regional estimates. The L-Cv varies from 0.0968 to regional quantiles at large return periods (50, 100, 500, 0.1954, L-skewness= 0.2144, and L-kurtosis = 0.1934. and 1000) are smaller than the error bounds of GLO and GEV. So, we can conclude that GNO is the best for the The absolute bias, bias, and RMSE are estimation of quantitle in the monsoon period for large calculated for regional growth curves. For the pre- return periods. GEV is best for the return periods 1, 2, 5, monsoon and post-monsoon seasons, the regional 10, and 20 on the basis of relatively low values of average correlations are 0.50 and 0.67 respectively. We absolute bias, RMES, and smaller difference of error consider the distributions GNO, GLO, GEV, and PE3 for bounds of the regional quantiles. the pre-monsoon. Also for the post-monsoon we take GEV, GNO, and PE3. Accuracy measures are given in For the pre-monsoon season, the simulation Table-12 to Table-14. results are given in Table-13. It shows that for large return periods 50, 100, 500, and 1000, PE3 is best for less

values of absolute bias and smaller error bounds while for return periods1, 2, 5, 10, and 20, the absolute bias is

17 the same for all the distributions. So, PE3 distribution is steps in this process. Discordancy measures are taken the best choice for quantile estimation for the pre- with 1st L-moment and found that among 11 sites only monsoon season. In Fig. 6, GNO, GLO, GEV, and PE3 the Sibsagar site is discordant. But due to higher L-Cv are close at higher nonexceedance probabilities. and lower L-Kurtosis, the site can be considered as a homogeneous one under the regional formula. Also, Again, Table-14 shows that for large return when we apply regional heterogeneity testing, then all the periods 50, 100, 500, and 1000, PE3 is the best fit sites become homogeneous. Z- Statistics method shows distribution due to the lower RMSE and smaller error that GLO, GLO, and PE3 are the best fitting distribution bounds while for return periods 1, 2, 5, 10, and 20, the with the least critical value comparing with the standard absolute bias is same for all the three distributions. So, critical value 1 64 . The same result is found in the case PE3 distribution is the best fitting for quantile estimations of the L-moment ratio diagram. And then we find the in the post- monsoon period. At increasing . regional parameters for the Pre-monsoon, Monsoon, and nonexceedance probability, all of the distributions are Post-monsoon periods. Regional estimates of quantiles close, as seen in Figure 8. Thus for large return periods, for different periods are calculated. Finally, we develop a PE3 is best for the quantile estimation for the pre- regional relationship for the season’s maximum monthly monsoon and post-monsoon period. For the rainfall based on 1st L-moment by taking the best-fitting monsoon period, GNO is the best one. probability distributions. For large return periods, PE3 is the best for the quantile estimation for the pre-monsoon Acknowledgment and post-monsoon period, and for the monsoon period, We are acknowledged by the Indian GNO is the best one. Meteorological Department (IMD), Guwahati (Assam) for providing the rainfall data quickly for analysis. References 4 Conclusions Ahmad I, Abbas A, Saghir A, and Fawad M (2016) Finding Probability Distributions for Annual Daily

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