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lecture10 - 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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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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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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Ordinary Kriging Ordinary kriging is optimal estimation of a random field, e.g. T(x), with an unknown mean, but a known semivariogram γ TT(x,x’). Note that simultaneous estimation of the mean and covariance from field data gives a biased estimate of the covariance If statistics must be estimated from field data it is often preferable to estimate the variogram which does not require simultaneous estimation of the mean
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Ordinary Kriging Define kriging estimate as: To ensure unbiasedness: Therefore must choose λ i so that: this is called the unbiasedness constraint ) ( ) ( ˆ 1 0 i N i i x T x T = = λ [ ] [ ] [ ] ) ( ) ( ) ( ) ( ) ( ˆ 0 1 1 1 0 x m x m x T E x T E x
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