APPLICATION OF STATISTICAL METHODS TO HOROLOGICAL PROBLEMS OF UTTAR PRADESH.

Abstract

New techniques have been employed in seeking answers to a few complex hydrological problems pertaining mostly to the state of Uttar Pradesh. The applications have been made to calculation of floods, rainfall and droughts. The applied techniques and the derived models are based on quite new approaches. These comprise mostly the technique of modelling in the comparatively new field of stochastic hydrology wherein there has been very little contribution of a similar nature in India. Other fields which have not been taken up on a wide scale in hydrology consist of simulation and sequential generation. The thesis makes practical appli cation in this field on the basis of Indian data. In the estimation of storage from reservoirs, numerical solutions have not been coming forth on the basis of the queueing theory, Contribution has been made in this field from two approaches including that based on the theory of queues. The hydrologic problems handled consist of flood flows, rainfall, droughts, reservoir storage. In the field of flood flows, usual techniques employed by hydrologists and engineers which are mostly empirical in character and do not take proper cognizance of catchment or basin characteristics 2 have been discussed. It has been indicated that mathematical modelling, as applied to the hydrological problem of flood flows, based on stochastic processes, is a more rational app roach. Stochastic probability models have been deduced for estimating the design flood with three measures of risk, viz. recurrence interval distribution, encounter probability and expected recurrence interval, for twenty river sites in India. Parameters involved in the estimating equation for design flood, which are not based on empirical co-efficients, are deduced from the historical data, on account of which greater rationality has been added in the estimation. The series of flood peaks which are considered on a yearly basis have been found to obey the Poisson distribu tion. The probability mass function of the number of exceedances or hazard events ( i.e. flood peaks ) has also been found to follow the Poisson process. The encounter probabi lity, the expected recurrence interval and the distribution of the hazard event magnitudes all follow the exponential func tion. A theoretical basis has thus been provided for the esti mation of the expected recurrence interval for various levels of probability. This has been done for the first time in India. In the next problem of rainfall, which is the princi pal phenomenon governing floods, basic studies have been carried out with respect to the series of yearly rainfall maxima for New Delhi, Lucknow and Allahabad for durations been of 5 and 60 minutes. They have/treated as stochastic pheno mena obeying the Poisson distribution. Stochastic models have been derived which may be used for the estimation of the magnitude of yearly rainfall maxima for any expected recurrence interval. This is a completely new approach in the field of rainfall studies. The estimation derived there from is considered to be more rational in comparison with the usual empirical methods, as the involved parameters are deri ved from the recorded data, whose characteristic features are thereby likely to be reflected more realistically in the derived models. The degree of the level of risk with respect to the expected recurrence interval has to be decided in accordance with the nature of the structure for whose flood estimation the expected yearly rainfall maxima is required. Stochastic modelling of rain storms thus provided an aid in the estimation of flood peaks whose utility in serving timely warnings and consequential indirect flood protection is well known. The next problem taken up has been that of simula tion and sequential generation of rainfall data, in which field the applications of this nature are quite uncommon. 3 4 The role of the technique is brought out in getting over the lacuna due to smallness of data. This has been done by incr easing the size of the data series. In small-sized data, it is not possible to get an idea of all possible combinations of sequences which are likely to become more manifest when the size of the data is increased artificially. The approach is based on Monte Carlo techniques which have been applied with respect to the available data pertaining to ten six-hourly rain storms recorded at Lucknow from 1956 to 1965. By the application of the sequential generation technique to the series of first-hourly rainfalls, whose size of the recorded data is only 10, an artificial series of 100 pseudorandom numbers has been generated. On the basis of a stochastic pro bability model, which has been worked out for the estimation of the second to the sixth hourly rainfalls on the basis of the first hourly rainfall, it has been possible to generate the series of rainfalls for subsequent hours. The derived model has been found to follow a non-stationary Markov process. It is felt that a new vista has thus been opened out for making fuller use of usually available small-sized data in a compara tively new hydrologic field. The problem taken up next pertains to that of drou ghts whose significance arises from the enormity of losses caused by droughts in the sector of agriculture which plays a vital role in national economy. The utility of the estima tion of expected droughts with varying recurrence intervals in minimising the economic losses due to droughts by providing an aid in planning for taking protective measures has been explained. The basic character of the series of droughts and their fundamental differences from those of floods have been elucidated. Taking the restrictions in the possible distribu tions of droughts into consideration, studies have been confi ned to four types of possible standard distributions. It has been found, after applying a number of tests, that the most fitting distribution is that of Pearson Type III. The involved comparative studies with possible distributions in the deriva tion of the appropriate distribution for the series of droughts has not been carried out for Indian data so far. Prom the available historical data for droughts of the Ganga and the Yamuna rivers, it has been found that both these rivers have recorded droughts corresponding to an expected recurrence inter val of 200 years. The last prohlem taken up is that of storage from reservoirs wherein the objective is to estimate the safe yield or storage capacity of a reservoir with a specific level of probability of failure ( i.e. non-filling or even emptiness of 5 6 the reservoir due to failure of rainfall or drought). Whereas theoretical approaches have been tried for the problem earlier, numerical solutions and applications have been quite rare. No work of this nature had been carried with Indian data. This has been done for the Matatila reservoir on the basis of the available data, for the period 1962-68, with respect to inflows and outflows. The storage has been estimated with a level of probability of 2 per cent. A review has been given regarding the past methods employed for the estimation of reservoir storage whose main lacunae have been non-consideration of sam pling errors, statistical homogeneity and independence and unrealistic assumption of a constant outflow apart from non availability of numerical solutions. Two methods have been considered in some detail for seeking numerical solutions for the estimation of reservoir storage. This has been done for the first time tm India. The series of inflows has been found to obey a lognormal distribution. An estimation has been made both regarding the draft or the outflow and the storage, emp loying an equation in which the involved parameters are the yield and storage ratios and the constants are deducible from the probability density function of a lognormal distribution. The estimates for both the draft and thestorage are likely to be rational and realistic. The second method used for the estimation of reser voir storage is based on the queueing theory whose approach and fundamental characteristics have been explained. Reser voir storage has been estimated from the service function following a queueing process. The involved parameters are deduced from the available recorded data. The estimate provided for the reservoir storage based on the queueing theory compares favourably with that obtained by the first method, based on Monte Carlo techniques.