Solving the problem of hydrometeorological data analysis by random process

The paper reviews the hydrometeorological data analysis (precipitation, flow, water level, etc.) to evaluate and predict mutations such as flood, drought and saline extremes to reduce the impact of climate change on the economy and life. The main method for solving the problem posed is the max-domain of attraction of extreme distributions with the Gumbel copula of random variables related to hydrometeorological data. Results presented in this paper are reviewed and verified through data supplied by hydrometeorological stations at the Tan Chau and Chau Doc districts, An Giang province from 1990 to present.


INTRODUCTION
robability and statistic theories have been widely applied in hydrometeorological analysis.In this review, we introduce the integration of random (or stochastic) process methods to evaluate, predict and analyze the extreme distributions in hydrometeorology.In other words, this review will be solved problems associated with flood peak, drought and saline extremes, contributing the significant economy and life.
The mathematical methods are mainly used in The copula method offers a efficiency for describing the dependence among multiple hydrologic variables.In the last section ofthe research, the data of maximum rainfall and water levels are treated in Tan Chau district, An Giang province, located in the Mekong River Delta of Vietnam.

Extreme Value distributions
Let X1, X2, …, Xn be a sequence of independent random variable.When we consider the max value of this sequence, If random sequence, ; 1,2, , ; i X i n independent and they have the same distribution X F x , we get: The symbol of density function equivalence We know that Fisher-Tippett ( [5]) and Gnedenko ([6]) has proved H will be one of three Extreme Value distributions: Gumbel, Fréchet, Weibull.
We given Generalized Extreme Value -GEV as: In there: We make the symbol Extreme Value distribution by GEV ( , , μ σ ξ ).So that Extreme Value distributions has three parameters.Specify, we classify GEV by shape parameter ξ as: The expectation: The variance of Gumbel distribution: is gammafunction.Characteristic of Gumbel distribution ξ 0 .
The density function (PDF) of Gumbel distribution will be: The Expectation of Gumbel distribution: 0, 577216 EX μ σ there, 0, 577216 Euler constant.The Variance of Gumbel distribution: In the next section of this review, application III will be presented the statistic methods Maximum Likelihood Estimation (MLE).
The method of moment is to find out the initial parameter 0 0 ; μ σ in Expectation of Gumbel Distribution, then MLE is used to estimate statistical parameters ; μ σ .The Newton-Raphson algorithm can be required to solve the set of equations.

The copula and distributions in the multivariate maximal domain of attraction.
Copula function is an efficient method for describing the dependence among multiple hydrologic variables.It offers a flexible way to construct a joint distribution independent from the marginal distributions.The main advantage of this approach is that construction of a joint distribution through a copula is independent of the marginal distributions of the individual variables.
Derived from the following definition of maximum suction region of multidimensional distribution: Let , , There: : φ t inf y φ y t .
To get attractive formulae, it is necessary to assume all marginal of F are the same so we proceed under the assumption: We now report the main result concerning risk aggregation: Suppose (1) holds where all marginal of F x are equal and all marginal of G x are Gumbel and (2), (3) hold.
Formulae for aggregation of risks may be readily obtained when F does not possess asymptotic independence.

Extreme Value distributions in the
hydrological models.Theory used in this application is shown in 2.1, and two specific assignments are solved as follows: Assignment 1: Find maximum Gumbel distribution function for maximum rainfall in Tan Chau, An Giang based on data at hydrological stations in Tan Chau, An Giang, Vietnam.
Assignment 2: Find maximum Gumbel distribution function for maximum water level in Tien river based on data at hydrological station in Tan Chau, An Giang, Vietnam.
Data supplied by Southern Hydrographical Meteorological Station at the Tan Chau and Chau Doc districts, An Giang province from 1990 to present.
The problems are solved specifically as follows: The expectation of Gumbel distribution: 0, 577216 EX μ σ there, 0, 577216 Euler constant.The variance of Gumbel distribution: By moment method we will find the statistical estimation, 2 2 0, 577216 0, 4501 0, 7797 6  there: The next section we will use maximum likelihood method for parameter estimation in distribution maximum to illustrate the use of actual data through Newton -Raphson algorithm.
First of all, to building the likelihood function, we have: , , , , , , We select μ and σ which satisfactory the system of equations: 0 0 μ σ Using the Newton -Raphson algorithm: 1 .
This iteration is repeated until the following inequality is satisfied: The problems for the maximum distribution of water lever (or rainfall)in Tan Chau station, if select k = 10 -4 .

WATER LEVEL PROBLEM
Based on data: 0 0 μ 379.9874; σ 50.00598 calculated by moment method in the previous method of maximum likelihood and Newton -Raphson algorithm, we get the results in repeated step as table below.
The extreme distribution function in this problem will have: