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# lecture12 - CWR 6536 Stochastic Subsurface Hydrology...

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CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters

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Block Kriging Kriging systems discussed to date use point measurements to estimate point values of the random field at unmeasured locations…. called point, or punctual, kriging Block kriging uses point measurements to estimate average values of the random field over a prescribed area. Useful if trying to estimate parameters for discretized models
Block Kriging Block Kriging equations are similar to point kriging equations, e.g. for simple kriging: Where is the covariance between a particular sample location and all the points within A0 = ) , ( ~ ) , ( ~ ) , ( ~ ) , ( ~ ) ( ) ( ) ( ) ( 0 3 0 2 0 1 0 0 0 3 0 2 0 1 3 2 1 3 3 3 2 3 1 3 2 3 2 2 2 1 2 1 3 1 2 1 1 1 N N N N N N N N N N x A P x A P x A P x A P A A A A ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P ) ,x (x P YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY YY λ = 0 ) , ( 1 ) , ( ~ 0 0 A i YY i x d x x P A x A P YY

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Block Kriging Once λ i are determined construct estimate from And estimation variance from Where ) , ( ~ ) ( ) ( ) ( ˆ 0 1 0 0 2 0 2 Y i N i i x A P A A A YY Y - = = λ σ ) ( ) ( ) ( ) ( ˆ 0 1 0 0 i N i x Y A A m A Y + = y d x d y x P A A A P A A YY A YY Y = = 0 0 ) , ( 1 ) , ( ~ ~ ) ( 2 0 0 0 0 2
Block Kriging Integrals can either be determined analytically from analytical point covariance functions or estimated by a discrete sum: If estimating average covariances numerically a balance must be struck between too few discretizing points which may not accurately estimate the average and too many discretizing points which can be computationally expensive It is important to use the same number of discretization points for the calculation of the point-block covariances and the block-block covariance = N j i j YY i x x P N x A P YY 1 0 ) , ( 1 ) , ( ~ = = = N k k j YY N j x x P N A A P A YY Y 1 1 2 0 0 0 2 ) , ( 1 ) , ( ~ ~ ) ( σ

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Block Kriging Each block kriging weight can be shown to be the average of the point kriging weights throughout the area A Thus the block kriging system yields an estimate
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lecture12 - CWR 6536 Stochastic Subsurface Hydrology...

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