# Probability analysis for consecutive-day maximum rainfall for Tiruchirapalli City (south India, Asia)

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## Abstract

In the design of irrigation and other hydraulic structures, evaluating the magnitude of extreme rainfall for a specific probability of occurrence is of much importance. The capacity of such structures is usually designed to cater to the probability of occurrence of extreme rainfall during its lifetime. In this study, an extreme value analysis of rainfall for Tiruchirapalli City in Tamil Nadu was carried out using 100 years of rainfall data. Statistical methods were used in the analysis. The best-fit probability distribution was evaluated for 1, 2, 3, 4 and 5 days of continuous maximum rainfall. The goodness of fit was evaluated using Chi-square test. The results of the goodness-of-fit tests indicate that log-Pearson type III method is the overall best-fit probability distribution for 1-day maximum rainfall and consecutive 2-, 3-, 4-, 5- and 6-day maximum rainfall series of Tiruchirapalli. To be reliable, the forecasted maximum rainfalls for the selected return periods are evaluated in comparison with the results of the plotting position.

## Keywords

Rainfall Return period Probability distribution Chi-square test## Introduction

Several applications in water resources engineering require appropriate estimate of rainfall depth and its return period from available historic data. Estimation of flood in watersheds, water balance studies, water management studies, rainwater harvesting, detention and retention pond design, evapotranspiration estimation, irrigation planning, etc. are some of the examples where rainfall provides a vital input to design and modeling. Planning and development of water resources at the local or regional level require comprehensive and reliable information of hydrological data of the area under investigation. Prober database is needed to assess the water availability of a region, the absence of which can lead to erroneous planning and design. Long period data can provide reliable water resource assessment. The degree of uncertainty increases if the data length is short. Mathew and Vivekanandan (2009) examined the effect of data length on water resource assessment. The results of the study indicate that the lower the data length, the higher is the likelihood of overestimating water resource availability in the regions. Rainfall at a particular place is also known to be influenced by the results of its local/regional atmospheric and geomorphologic environments.

An important aspect in hydrology is to interpret the future probabilities of occurrence from past records of hydrologic events. Vivekanandan and Mathew (2010) chosen probabilistic modeling to fit six different distributions for annual d-day maximum rainfall of different values of d such as 1, 2 and 3 days for the Devgadhbaria region of Gujarat (India). Chi square and Kolmogorov–Smirnov tests are used to judge the applicability of the distributions for modeling of the recorded rainfall data. The standard procedure for estimating the frequency of occurrence of hydrological event is frequency analysis. The objective of frequency analysis of hydrologic data is to relate the magnitude of extreme events to their frequency of occurrence using probability distributions.

The study of extreme rainfall events involves the selection of a sequence of the maximum observations from the respective data series. Goswami et al. (2006) examined the trend of daily high (*R* > 100 mm) and highest (*R* > 150 mm) rainfall events over a relatively large region covering 1803 stations for the period 1951–2000. The finding of the study shows that there is a 10 % increase per decade in the level of heavy rainfall event since the early 1950s and more than two times increase in very heavy events. Khan et al. (2007) investigated spatial and temporal variability of daily and weekly precipitation extremes in South America. They have proposed a new measure called the precipitation extremes volatility index to measure the variability of extremes. An analysis of their study indicates the increasing trend of daily maximum rain in the Amazon basin. Guhathakurta et al. (2010) carried out the frequency analysis of rain days, heavy rainfall days and also 1-day extreme rainfall, to observe the impact of climate changes on extreme weather events and flood risks in India. The report shows that the frequency of heavy rainfall events is decreasing in major parts of central and north India, while such events are increasing in Peninsular India and also east and north-east India. The present study aims to evaluate the rainfall magnitude for different return periods and also to ascertain the type of probability distribution that best fits the rainfall.

## Study rainfall station and data

Tiruchirappalli City, known also as Trichy, is an urbanized watershed in the Cauvery River basin. The terrain of the city is flat. The city lies at an altitude of 78 m above sea level and is traversed by the rivers Cauvery and Coleroon. The Coleroon river forms the northern boundary of Trichy. There are few hills located within the city, with the prominent among them being Golden Rock, Rock Fort and the one in Thiruverumbur. During heavy downpour times, the low-lying and improper drainage areas of the city are often subjected to inundation. This happens due not only to blockage of drains, but also to undersized stormwater drains. The study on temporal distribution of rainfall will provide useful information to the city planner. The present study uses 100-year continuous daily rainfall data, obtained from the Indian Meteorological Department located at Pune.

## Analysis methods

### Probabilistic methods

Probability distributions are widely used in understanding the rainfall pattern and computation of probabilities. In the present study, the probability of exceedance of rainfall *T* = *m*/(*N* + 1), where *m* is the order or rank and *N* is the total number of events. It was computed using the Weibull’s plotting position formula and applied to the observed rainfall data. The continuous probability distribution log normal (LN), Pearson type III (P III), log-Pearson type III (LP III) and extreme value type I (EV I) were used to evaluate suitable probability functions.

### Log-normal distribution

*X*

_{av}is the mean values,

*k*is the frequency factor and

*σ*is the standard deviation and

*N*is the sample size.

The value of *k* is determined considering the coefficient of skewness as zero.

### Pearson type III distribution

*k*is obtained from the theoretical table available for the Pearson type III distribution with skew coefficient (Patra 2001).

### Log-Pearson distribution

This method is extensively used in hydrologic frequency analysis. The traditional fitting procedure consists of transformation of natural data into logarithms and fitting logarithmic data to a Pearson type III distribution by the method of moments.

### Extreme value distribution

### Chow method

*k*) for log-normal distribution and presented a theoretical table. The value of

*k*can be obtained using the skewness coefficient (

*C*

_{s}) and coefficient of variation (

*C*

_{v}). In log-normal distribution, these two parameters are related as

*C*

_{s}value is adopted in case of Chow method, whereas for the log-normal approach

*C*

_{s}value is considered as zero.

### Chi-square test

*O*

_{ i }and expected number of occurrence

*E*

_{ i }can be developed as:

This statistical test judges whether or not a particular distribution adequately describes a set of observations by making a comparison between the actual number of observations and the expected number of observations. The *χ* ^{2} distribution has *v* (=*N* – *h* − 1) degrees of freedom, in which *N* is the total number of sample data and *h* is the number of parameters used in filling the proposed distribution. The value of *χ* ^{2} for various degrees of freedom against distribution percentages is available in the *χ* ^{2} table. In hydrology, 95 % level of confidence is considered as the typical value (Patra 2001). The *χ* ^{2} value corresponding to 98 (i.e., *v* = 100 − 1 − 1) is 124. In general, the probability distribution that provides the least Chi-square value is considered as a best-fitting probability distribution in the recorded range for the given data.

## Analysis of data

Maximum rainfall for 1, 2, 3, 4 and 5 consecutive days using the plotting position method (Suribabu et al. 2015)

Return period (year) | One-day rainfall (mm) | Two-day rainfall (mm) | Three-day rainfall (mm) | Four-day rainfall (mm) | Five-day rainfall (mm) |
---|---|---|---|---|---|

10 | 152.4 | 174.8 | 185.7 | 199.4 | 214.4 |

50 | 297.96 | 333 | 336.2 | 328.7 | 345.5 |

100 | 318.9 | 366.2 | 368.5 | 383.5 | 403.3 |

| 145.56 | 158.2 | 150.5 | 129.3 | 131.1 |

| 20.94 | 33.2 | 32.2 | 54.8 | 57.8 |

For design of drainage system for any urban area, a vital but tricky consideration is the return period of the “extreme” rainfall events. Generally, a best value will lie between overestimating and underestimating the risks involved and a major deciding factor is the cost. When design is done based on the 10-year return period, the risk involved will be more, whereas if 100-year return period is considered the risk probability will be less. Hence, the selection of appropriate design value is becoming crucial. It is very important that the selection of probability distribution for a particular data set should not provide an underestimated design value. The data corresponding to the 50-year return period can be used in the study area as it falls within the underestimating and overestimating design value. In particular, the design estimate presented here would be beneficial and valuable guidelines during the construction of new drainage systems and rehabilitation of existing drains in the study area, as poor drainage has been identified as one of the major factors causing flooding in the area.

## Conclusions

Extreme rainfall events for the study area is found out through five different methods and compared with plotting the position method which uses present data by ranking the events as per Weibull’s method. The analysis of results indicate that the there is a significant difference in rainfall amount between the 10-year and 50-year return periods for 1- to 5-day consecutives rainfall. The probability analysis performed in this study depicts that the difference in rainfall amount for 1- to 5-consecutive day rainfall estimates between the 50- and 100-year return periods is found to be insignificant. Hence, the hydraulic design based on the 50-year return periods holds good even for the 100-year return period rainfall for the study area. In comparison to the estimates based on five probability distribution functions, LP III distribution shows the least value of Chi-square value for the return period up to 100 years and can be adopted for estimation of rainfall amounts at various probability levels.

## References

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## Copyright information

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