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Unformatted text preview: CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters using Kriging 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 i N i x T x T ∑ = λ [ ] [ ] ) ( ) ( ˆ x T E x T E = ( 29 ∑ =  2 1 2 ) ( ) ( ) ( ˆ ) ( i N i x T x T E x T x T E λ 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...
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 Spring '11
 Graham

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