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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
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
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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