1. Lecture (16) Introduction to Stochastic Hydrology

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

3. Monte-Carlo Sampling Uniform random number generator: Multiplicative Congruence Method developed by Lehmer [1951]. Ni 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, Ui is a pseudo-random number in the range {0,1}, and " MODULO" notation indicates that Ni is the remainder of the division of (A.Ni-1) by M.

4. Uniform Random Number Example

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

6. 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), Ui 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 Ui to be used. If m is 12, a normal distribution with tails truncated at six times standard deviation is produced

7. Example: Generate a Gaussian distribution using uniform random numbers. Random number generator gives numbers distributed uniformly in the interval [0,1] nm= 1/2 and s2 = 1/12 u Procedure: Take 12 numbers (r1, r2,……r12) from your computer’s random number generator (ran(iseed)). Add them together. Subtract 6 Þ Get a number that looks as if it is from a Gaussian pdf! A) 5000 random numbers B) 5000 pairs (r1 + r2) of random numbers D) 5000 12-plets (r1 + r2 +…r12) of random numbers. C) 5000 triplets (r1 + r2 + r3) of random numbers E) 5000 12-plets (r1 + r2 +…r12 - 6)of random numbers. E Gaussian m= 0 and s= 1 Thus the sum of 12 uniform random numbers minus 6 is distributed as if it came from a Gaussian pdf with m = 0 and s= 1. -6 0 +6 12 is close to ¥ !!

8. 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.

9. Terms