lecture19and20 - Approximate Analytical Solutions to the...

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Unformatted text preview: Approximate Analytical Solutions to the Groundwater Flow Problem CWR 6536 Stochastic Subsurface Hydrology System of Approximate Moment Eqns to order ε 2 • Use φ (x), as best estimate of φ (x) • Use σ φ 2 =P φφ (x,x) as measure of uncertainty • Use P φφ (x,x) and P f φ (x,x) for cokriging to optimally estimate f or φ based on field observations ) ( ) ' , ( ) ' , ( ) ( ) ' , ( ) ' ( ) ' , ( ) ' , ( ) ' ( ) ' , ( ) ( ) ( ) ( 2 ' ' ' ' 2 ' 2 1 1 1 1 1 1 1 x x x P x x P x F x x P x x x P x x P x F x x P x x F x f x ff x f x x f x φ φ φ φ φ φ φ φ φ φ φ ∇ ⋅ ∇ + ∇ ⋅ ∇ + ∇ = ∇ ⋅ ∇ + ∇ ⋅ ∇ + ∇ = ∇ ⋅ ∇ + ∇ = Fourier Transform Techniques • Require an infinite domain • Require coefficients in pdes for P f φ 1 and P φ 1 φ 1 to be constant • Require input covariance function to be stationary. • Convert pdes for covariance functions P f φ 1 and P φ 1 φ 1 into algebraic expressions for S f φ 1 and S φ 1 φ 1 . Spectral Relationships • • • Assume a form for S ff , inverse Fourier transform to get P f φ 1 and P φ 1 φ 1 ; Multiply by ε 2 = σ lnK 2 to get P f φ and P φφ 2 1 ) ( ) ( 1 k k JS ik k S ff f - = φ 4 2 2 1 2 1 ) ( ) ( ) ( 1 1 1 k k S J k k k JS ik k S ff f = = φ φ φ 3-D Results • Assume 3-D exponential input covariance for P ff. • 3-D Head-LnK Conductivity Cross-Covariance: +--- =-- λ ξ λ ξ φ λ ξ ξ λ ξ λ ξ λ χ σ χ ξ e e J P K f 1 2 2 cos ) , ( 3 3 3 3 2 ln λ ξ σ ξ- = e P K ff 2 ln ) ( 3-D Results • 3-D Head Covariance • 3-D Head Variance: 3 2 2 2 ln 2 λ σ σ J k h = ( 29 ( 29...
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This note was uploaded on 03/27/2012 for the course CWR 6536 taught by Professor Graham during the Spring '11 term at University of Florida.

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lecture19and20 - Approximate Analytical Solutions to the...

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