lecture15

# lecture15 - Stochastic Modeling Approximate Analytical...

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Stochastic Modeling Approximate Analytical Solutions CWR 6536 Stochastic Subsurface Hydrology

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Stochastic model predictions can be obtained in several ways: Exact analytical solutions Monte Carlo techniques Approximate analytical solutions Approximate numerical solutions
Recall the problem we are considering: 3-D steady-state contaminant distribution from a continuous point source in a uniform flow field Governing p.d.e. 0 at t m m , , as 0 B.C. 0 ) 0 , , , ( I.C. 0 2 2 2 = = = = = = - + + = z y x z y x c t z y x c x c v z c D y c D x c D zz yy xx

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Steady State Solution zz xx yy xx xx zz yy D D z D D y x B D v B x D D B m v g z y x c 2 2 2 where 2 ) ( exp 4 ) ( ) , , ( + + =  - = = πθ
Would like to determine: Mean concentration: E[c]=E[g(v)] Concentration Variance: σ c 2 =E[{g(v)-E[g(v)] 2 } 2 ] How can we determine these quantities if we do not have full pdf of v or c?

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Taylor Series Expansion Expand non-linear function g(v) in a Taylor series around Definition: Taylor series of function f(x) about a: Therefore: v ( 29 ! ) ( ) ( 0 n a x x x f x f n a x n n n - = = = ( 29 ( 29 ( 29 ... 6 ) ( 2 ) ( ) ( ) ( ) ( 3 3 3 2 2 2 + - + - + - + = v v v v g v v v v g v v v v g v g v g c
Evaluate Mean Concentration Take the expected value of the equation: To first order To second order [ ] [ ] ( 29 [ ] ( 29

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lecture15 - Stochastic Modeling Approximate Analytical...

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