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Monte Carlo Simulation

Monte Carlo Simulation. CWR 6536 Stochastic Subsurface Hydrology. Steps in Monte Carlo Simulation. Create input sample space with known distribution, e.g. ensemble of all possible combinations of v, D, q, m values

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Monte Carlo Simulation

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  1. Monte Carlo Simulation CWR 6536 Stochastic Subsurface Hydrology

  2. Steps in Monte Carlo Simulation • Create input sample space with known distribution, e.g. ensemble of all possible combinations of v, D, q, m values • Run each realization of v, D, q, m values through model to produce output sample space • Repeat experiment many times to get accurate representation of input sample space and accurate statistics of output sample space • Calculate statistics of output sample space, i.e. pdf, mean, variance, etc. as a function of location

  3. Steps in Monte Carlo Simulation

  4. Primary questions to ask • How to generate input samples? • are random inputs correlated with each other • are random inputs correlated in space or time • How many replicates are required to get reliable output statistics? • test input statistics to be sure they are generated correctly • test convergence of output statistics to constant values • calculate approximate number of replicates needed as get an idea of magnitude of mean and variance of the output.

  5. Generating random variables of arbitrary distribution • Generate uniform distribution of random numbers between 0 and 1 (yi) • yi can be considered the cdf of a random variable xi with the arbitrary distribution G(x)

  6. Example: Exponential Distribution • Now x is a random variable with cdf G(x) =1-e-ax and pdf ae-ax • Thus can use uniform distribution random number generator to generate random variable of any distribution

  7. Generating random fields/processes(See Deutsch & Journel, 1998; Goovaerts, 1997) • Spatially distributed random fields • Categorical indicator simulation to honor specific geometrical patterns (i.e. layering) • Sequential Gaussian simulation, LU decomposition, and/or Turning Bands generator to simulate distribution of continuous properties within geometry • Can generate conditional or unconditional simulations; must specify mean & spatial covariance structure, as well as data to be used for conditioning) • Temporally correlated random processes: • Markov process generators (specify m,s2, transition probabilities)

  8. How many replicates are sufficient? • Test input statistics for convergence to known moments • Test output statistics for convergence to constant values • Use confidence intervals to estimate number replicates required to give desired accuracy once have estimate of mean and variance of output

  9. 95% Confidence Intervals • Consider the moment estimator • If is normally distributed, a=1.97 std err( )

  10. 95% Confidence Intervals • Suppose want 95% confidence intervals to be +/- 10%

  11. 95% Confidence Intervals • For the mean • For the std dev

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