lecture9 - CWR 6536 Stochastic Subsurface Hydrology Optimal...

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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 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 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 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). Simple Kriging • Define kriging estimate as: • Check for bias: • Choose λ i so that estimation variance is minimized: ) ( ) ( ) ( where ) ( ) ( ) ( ˆ 1 i i i i N i i x-m x T x Y x Y x m x T = + = ∑ = λ [ ] [ ] ) ( ) ( ) ( ) ( )...
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This note was uploaded on 03/27/2012 for the course CWR 6536 taught by Professor Graham during the Spring '11 term at University of Florida.

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lecture9 - CWR 6536 Stochastic Subsurface Hydrology Optimal...

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