Rainfall frequency analysis of Brahmaputra valley of Assam, India

Introduction

Rainfall is an atypical event that is both spatial and temporal. The distribution of rainfall is highly uneven and differs significantly not only across locations but also from one year to another. In any agricultural program, rainfall is crucial to the creation and execution of any given agricultural arrangement. Furthermore, the factor that causes the decrease in rice production throughout India is most significant because of the inadequate and improper supply of water in the lake (Panigrahi and Panda, 2006) [6] . There is stress on the rainfed food production system due to varying rainfall predictions and inadequate use of land and water utilization (Khan and Hanjra, 2009) [20] . Nevertheless, the most significant aspects in rainfed regions are the magnitude and spatial and temporal distribution of rainfall (Shetty et al. 2000) [4] . Once again, the premature cessation of rains affects production by several moisture stressors, particularly as Kharif crops are growing and producing grains (Dixit et al., 2005) [2] . Bora et al. 2022, 2021 [17,18] focused on the rainfall frequency for crop planning. There are various modes of rainfall frequency based on day, week, month, season, annual, and successive days of extensive downpour. Each of them is significant in predicting rainfall and assists farmers in making superior decisions in planting harvests. It is also beneficial in the design and building of both big and small bridges, dams, drainage networks, and other tasks involving hydrology. Numerous studies have been carried out on the analysis of regional flood and rain frequency. "Among them, Hosking and Wallis (1990 & 1997) [10][11][12][13] created the L-Moment, which has a major advantage over the Standard Moment, which is a product of a Probability-Weighted Moment (PWM) and a linear use of the data set". The inverse form of the various distributions had been explained by Greenwood et al. 1979 [9] to provide the variables. In1995, when Bobee and Rasmussen, 1995 [5] used regional samples, less sampling unreliability was created by assembling data and less model unpredictability by allowing a superior distributional choice". In their research work, H. J. Fowler and C. G. Kilsby, 2003 [7] resorted to using the regional rainfall analysis of the period from 1961 to 2000 and perceived that the significant rainfall has influenced the character of worldwide environmental transformation in the UK. In Iran, Hossein et al. [8] studied the L-Moment to determine the frequency of day-today rainfall for the development of techniques regarding rainfall estimation. Frequency anatomization provides information that is used in the determination of the likelihood of a certain event taking place. It utilizes distributions of probability to accumulate data based on the frequency and magnitude of significant events (L.V. Noto and G. La Loggia, 2009) [14] . Deka et al. 2019 [19] determined the highest daily rainfall of forty-two years in the nine states of North-East India through L-Moment and LQ-Moment. Ahmad et al. 2016 [1] measured the maximum rainfall per day by employing various consequences in Pakistan and then decided that the probability that was fit was chosen by employing the goodness-of-fit assessment. Md. Ashraful Alam et al. 2018 [15] concluded that the L-Moment approach provides the most accurate distribution of probability for maximum rainfall in Bangladesh per-month amongst nine frequently utilized distributions. Bora et al. 2021 [16] analyzed successive days of maximum precipitation employing the L-Moment technique. The selection of the probability distribution is vital because a faulty selection may cause significant bias and inaccuracy in updated design flood prediction, mainly for extended return periods. This may lead to either an underestimation or an overestimation, which could have substantial significance for the daily operations. Again, A. S. Rahman et al. 2013 [3] provided a method to identify the best-fitting probability distributions of floods in Australia. The best-fit probability distribution in this research paper will help us to obtain an appropriate return period to analyze seasonal rainfall.

Study region and data collection

We have made use of rainfall records of a total of 11 sites in the Brahmaputra valley of Assam in this study, viz. Beki, Dibrugarh, Goalpara, Golaghat, Guwahati, Jorhat, Kokrajahar, North Lakhimpur, Sibsagar, Tezpur, and Kampur, which are based on IMD, Guwahati, and contain about 15 years of precipitation measurements.

Methodology 3.1 L-Moments

L-Moments has been described by Hosking and Wallis [12] as-

From Eq.( 1), the initial four L-Moments will be as follows -

According to Hosking and Wallis, 1997 [13] , the L-Moments Ratio (LMRs) is -

Results and Discussion

The models of the frequency of regional rainfall of all three seasons are constructed, and their validity is ensured by the Jacknife resampling technique, also called leave-one-out (LOO). The sections below contain all the required information.

Data screening through Discordancy Measure

Hosking and Wallis formula of discordancy measure [12,13] took the sites of a group as follows-

Let 𝑢 𝑖 = [ˆ𝜏 (𝑖) ˆ𝜏 3 (𝑖) ˆ𝜏 4 (𝑖) ] be a vector encompassing the LMRs, ˆ𝜏 (𝑖) , ˆ𝜏 3 (𝑖) , ˆ𝜏 4 (𝑖) for the (𝑖) site, then the group average for NS sites will be the given equation-

The sample equation for the Covariance Matrix is-

The discordancy measure for the site (𝑖) is shown by the equation-

As suggested by Hosking and Wallis [13] , if 𝐷 𝑖 is higher than the critical value, then the site 𝑖 is considered to be discordant. Based on Table 1 to Table 3, it is evident that the values of 𝐷 𝑖 are lower than the critical value (Hosking and Wallis, 1997) [13] in all sites and in all three seasons, except one site, Sibsagar, specifically in the pre-monsoon season. Thus, the discordancy measure indicates that only Sibsagar is discordant in this regard. However, as Figures 1 and2 have shown, the L-CV is bigger, in contrast to the smaller L-Kurtosis in Sibsagar. As such, we have found once again that with the help of heterogeneity measures (Table 5), our study region is homogeneous, including this site. Consequently, we apply all 11 sites to the development of a regional rainfall frequency model in our region.

Heterogeneity measures

The three levels of variability of Hosking and Wallis, 1997 [13] propose the heterogeneity measure of a group of sites that would be supposed to be homogeneous. The heterogeneity test is then expressed in the form of the following-

For 𝑉 1 𝑉 2 and 𝑉 3 , We use the following equations

where 𝑁𝑆 indicates the site's number, 𝑁 𝑖 indicates each side's record length and ˆ𝜏 (𝑅) , ˆ𝜏3 𝑅 and ˆ𝜏4 𝑅 is the regional average L -Moments, which can be achieved through the below given formulations.

Also, 𝜇 𝑉 𝑖 and 𝜎 𝑉 𝑖 gives the Standard Deviation of 𝑉 𝑖 for 𝑖 = 1,2,3.

According to Hosking and Wallis [13] , an area is considered properly homogeneous if 𝐻 𝑖 < 1, likely heterogeneous if 1 ≤ 𝐻 𝑖 < 2, and clearly heterogeneous if 𝐻 𝑖 ≥ 2.

Goodness of Fit-Tests:

Two goodness-of-fit tests, specifically the Z-Statistic criterion and the L-Moment Ratio Diagram (LMRD) proposed by Hosking and Wallis [13] , are employed to bring out the most suitable probability distributions for the three seasons of our research region.

Z-Test Criteria:

The primary benefit of this test is its direct application to the regional average, as presented by Hosking and Wallis [13] . The equation for this is forwarded below-

where ˆ𝜏4 𝑅 indicates the Regional Average of ˆ𝜏4 and 𝐵 4 , 𝜎 4 states the bias and standard deviation of ˆ𝜏4 respectively. And these can be presented as

𝑁 𝑠𝑖𝑚 denotes the simulated data quantity sets and 𝑚 is the 𝑚 𝑡ℎ region of simulation resulting from the Kappa Distribution. Table 5 indicates that during the pre-monsoon period, GLO (0.11), for the monsoon period, GLO (0.03), and for the postmonsoon period, PE3 (0.87) are the most effective distributions for our study region, relative to the critical value of 1.64.

LMRD: Hosking, 1991 [11] defined the LMRD as a curve that exemplifies the L-Skewness and L-Kurtosis of several distributions of candidates that have theoretical links." When summarising regional results, LMRDs have been considered as a method for identifying amongst candidate distributions. Again, from LMRD (Figures 3 to 5), it has been observed that the GLO distribution is found to be most suitable during the Pre-Monsoon and the Monsoon seasons, while for the postmonsoon period, the PE3 distribution is the best. Table 6 gives the regional average of L-moments. 4.4 Development of regional relationship: Here, we construct the models of the frequency of rainfall in the region of our study in all three seasons. To this end, the regional parameters have been computed in all the candidate distributions of each of the three seasons, and they are provided in Table 7. Regional growth factors of each of the candidate distributions in the various return periods are calculated as well, in each of the three seasons, and presented in Tables 7 to 9. The regional rainfall frequency for each of the three seasons is developed for our study region and is given below.

For Pre-Monsoon rainfall: 𝑄 𝑇 𝑗 stands for the quantile determines for the period 𝑇 of site 𝑗.

For Post-Monsoon rainfall, the best-fitting distribution is the PE3 distribution. As no explicit formula of the quantile function exists, so to estimate the quantiles for each study site of the region, we can proliferate the regional growth factors given in Table 9 for the PE3 distribution, by the1st L-Moments of the site.

Validity of the Model

In this part, the validity of the formulated models for different seasons within our study areas is verified through the LOOCV method. To this end, we shall omit one side and use the regional flood frequency procedure for the other ten locations. The regional factors are determined of identified best fitting distributions of 2 to 1000 years return period. This process is done on all 11 sites, and the Root Mean Square Error (RMSE) and the BIAS of each of the return periods are determined and are provided in Tables 10 to 13.

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