Lecture (16)Lecture (16)

Introduction to Stochastic Introduction to Stochastic Hydrology Hydrology

Stochastic Approach • A “stochasticstochastic” modeling approach can be used

to calculate the probability of a future value lying between two specified limits.

Monte-Carlo SamplingMonte-Carlo Sampling

Uniform random number generator:Multiplicative Congruence Method developed by Lehmer [1951].

/MN = U

MMODULO B NA. = N

or

B , MNA. = MODULO N

ii

i-i

i-i

)()(

)(

1

1

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.

Uniform Random Number ExampleUniform Random Number Example

1.0,5.0,3.0,9.0

.......5,3,9,1,5,3,9:

)3(710

737319*8

)9(010

9911*8

)1(410

414115*8

)5(210

252513*8

)3(710

737319*8

9)(

)10()18(

0

1

sequence

remainder

remainder

remainder

remainder

remainder

seedN

MODULO N = N i-i

Generation of a Random Variable Generation of a Random Variable from any Distributionfrom any Distribution

Inverse of Distribution Function.

Transformation Method.

Acceptance-Rejection Method.

Transformation Method (1)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"

12

21

m/

- m/Uε =

m

i=i

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.

612

1

- Uε = i

i

If m is 12, a normal distribution with tails truncated at six times standard deviation is produced

σ + εμα = αα

0-6 +6Thus the sum of 12 uniform random numbers minus 6 is distributed as if it camefrom a Gaussian pdf with = 0 and = 1.

E

A) 5000 random numbers

B) 5000 pairs (r1 + r2)of random numbers

C) 5000 triplets (r1 + r2 + r3)of random numbers

D) 5000 12-plets (r1 + r2 +…r12) of random numbers. E) 5000 12-plets

(r1 + r2 +…r12 - 6) of random numbers.

Gaussian = 0 and = 1

P a Y Y2 ...Yn n n

b

P a ri 121

2i1

12

112

12 b

P 6 ri 6i1

12 6

12

e 12y2

6

6 dy

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 = 1/12u Procedure: Take 12 numbers (r1, r2,……r12) 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 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.

Terms