lecture11 - CWR 6536 Stochastic Subsurface Hydrology...

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CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using Kriging
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Types of Kriging Simple kriging is optimal estimation of a random field, e.g. T(x), with a known mean, m(x), and a known covariance PTT(x,x’). Ordinary kriging is optimal estimation of a random field, e.g. T(x), with an unknown constant or linearly trending mean, but a known semivariogram γ TT(x,x’). Universal kriging is optimal estimation of a random field, e.g. T(x), with an unknown polynomial trending mean, but a known semivariogram γ TT(x,x’).
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Universal Kriging Assume random field has the form: Define kriging estimate as: To ensure unbiasedness: Therefore must choose λ i so that: ) ( ) ( ˆ 1 0 i N i i x T x T = = λ [ ] [ ] 2 0 0 0 1 1 1 0 ) ( ) ( ) ( ) ( ˆ cx by ax m x m x T E x T E x T E N i i i i N i i i N i i + + + = = = = = = = λ λ λ = = = = = = = = N i i i N i i i N i i i N i i x x y y x x 1 2 0 2 1 0 1 0 1 1 λ λ λ λ ... ) ( where (x) ) ( ) ( 2 + + + + = + = cx by ax m x m x m x T ξ
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Universal Kriging Proceed as before minimizing the estimation variance subject to these 4 constraints, i.e. Use the method of Lagrange multipliers which adjoins the constraints to the objective function. = = = = + ∑ ∑ = = = = = = = N i i i N i i i N i i i N i i j i N i i N i N j j i j i x x y y x x x x x x x x x - TT TT 1 2 0 2 1 0 1 0 1 1 0 1 1 0 0 1 subject to ) , ( ) ( 2 ) , ( ) ( ) ( Minimize λ λ λ λ γ λ γ λ λ
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Universal Kriging
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