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lecture9 - CWR 6536 Stochastic Subsurface Hydrology Optimal...

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CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using Kriging
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Purpose of Kriging To estimate regional distribution of a spatially variable parameter To estimate accuracy of regional distribution Need scattered point measurements of the variable of interest Need knowledge of the spatial correlation structure
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Purpose of Kriging Estimate value at an unmeasured point Use estimated values to: produce map of variable use as input parameter for deterministic or stochastic groundwater flow/transport model
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Desirable Characteristics of Estimated Values Linear, i.e. weighted linear combination of observed values: Unbiased, i.e. Efficient, i.e. minimum estimation variance for given number of observed points is minimized Kriging is sometimes referred to as a BLUE estimate ) ( ) ( ˆ 1 0 i N i x T x T = λ [ ] [ ] ) ( ) ( ˆ 0 0 x T E x T E = ( 29 - = - 2 1 0 2 0 0 ) ( ) ( ) ( ˆ ) ( i N i x T x T E x T x T E λ
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Simple 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’). Assume random field of interest is T(x). Define a zero mean random field Y(x) as Y(x)=T(x)-m(x). Since the expected value of T(x) is m, Y(x) is a zero mean random variable with covariance, PTT(x,x’), and variance, σ Y2(x)= σ T2(x).
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Simple Kriging Define kriging estimate as: Check for bias: Choose λ i so that estimation variance is minimized: ) ( ) ( ) ( where ) ( ) ( ) ( ˆ 1 0 0 i i i i N i i x -m x T x Y x Y x m x T = + = = λ [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( ˆ 0 1 0 1 0 0 x m x Y E x m x Y x m E x T E i N i i i N i i = + = + = = = λ λ
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