Lecture (16) Introduction to Stochastic Hydrology

Stochastic Approach stochasticA “stochastic” modeling approach can be used to calculate the probability of a future value lying between two specified limits.

Monte-Carlo Sampling Uniform random number generator: Multiplicative Congruence Method developed by Lehmer [1951]. N i is a pseudo-random integer, i is subscript of successive pseudo-random integers produced, i-1 is the immediately preceding integer, M is a large integer used as the modulus, A and B are integer constants used to govern the relationship in company with M, U i is a pseudo-random number in the range {0,1}, and " MODULO" notation indicates that N i is the remainder of the division of (A.N i-1 ) by M.

Uniform Random Number Example

Generation of a Random Variable from any Distribution  Inverse of Distribution Function.  Transformation Method.  Acceptance-Rejection Method.

Transformation Method (1) Random number generator for normal distribution (from central limit theory):" Observations which are the sum of many independently operating processes tend to be normally distributed as the number of effects becomes large" with mean (μ=0) and unit standard deviation (σ=1), U i is the i-th element of a sequence of random numbers from a uniform distribution in the range {0,1}, and m is the number of U i to be used. If m is 12, a normal distribution with tails truncated at six times standard deviation is produced

0-6+6 Thus the sum of 12 uniform random numbers minus 6 is distributed as if it came from a Gaussian pdf with  = 0 and  = 1. E A) 5000 random numbers B) 5000 pairs (r 1 + r 2 ) of random numbers C) 5000 triplets (r 1 + r 2 + r 3 ) of random numbers D) plets (r 1 + r 2 +…r 12 ) of random numbers. E) plets (r 1 + r 2 +…r ) of random numbers. Gaussian  = 0 and  = 1 12 is close to  Example: Generate a Gaussian distribution using uniform random numbers. Random number generator gives numbers distributed uniformly in the interval [0,1] n  = 1/2 and  2 = 1/12 u Procedure: Take 12 numbers (r 1, r 2,……r 12 ) from your computer’s random number generator (ran(iseed)). Add them together. Subtract 6  Get a number that looks as if it is from a Gaussian pdf!

Exercise For project no. 1 generate a time series of the rainfall depth from a normal distribution using the sample mean and sample variance calculated in Ex 1. for the next 12 months. Assume independency between the values.

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