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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
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
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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