1. CWR 6536 Stochastic Subsurface Hydrology Optimal Estimation of Hydrologic Parameters

2. 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

3. 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 A 0

4. Block Kriging Once  i are determined construct estimate from And estimation variance from Where

5. 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

6. 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 identical to the average of the point estimates over area A at much computational savings Same concepts can be applied to ordinary kriging, universal kriging, indicator kriging, or cokriging

7. Block Kriging Similarly it is possible to use measurements that are obtained as averages over a certain area or volume. In this case the kriging equations will be This makes it possible to simultaneously use data collected by different methods with different support volumes

8. Indicator Kriging Recall that it is often useful to define indicator transforms of continuous random variables: where various levels of threshold values x k can be used For these indicator transform variables the mean, covariance and semi-variogram can be determined. The mean value is equivalent to the probability that the random variable has a value less than or equal to the threshold value. i.e.

9. Indicator Kriging Thus indicator kriging can be used to optimally estimate the probability of exceeding a sequence of thresholds throughout the region of interest. The collection of probabilities at a particular point provides an estimate of the conditional cumulative distribution function at that point Indicator means, variograms and covariances functions must be estimated separately for each threshold value ( 5 to 15 threshold values typically sufficient) Either simple kriging or ordinary kriging (and co-kriging) algorithms can be used depending on the information available

10. Co-Kriging In cases where more than one hydrogeochemical variable is sampled at a site, the estimate of any one variable can be improved by incorporating measurements of other correlated random variables into the estimation equation For example if transmissivity and head measurements are available at a particular site can obtain a better estimate of transmissivity by using both head and transmissivity measurements Co-kriging is most useful if the variable you would like to estimate under-sampled compared to another correlated variable

11. Simple Co-Kriging Simple co-kriging provides the best linear unbiased estimate of a random field, e.g. T(x), with a known mean, m T (x), covariance P TT (x,x’), and cross-covariance P TH (x,x’) with another variable H(x) with known mean m H (x), covariance P HH (x,x’).

12. Simple Co-Kriging Define simple co-kriging estimate as: Check for bias: Proceed as before minimizing the estimation variance by taking derivatives with respect to i and i and setting equal to zero. This results in a linear system of N T +N H equations with N T +N H unknowns:

13. Simple Co-Kriging Co-Kriging equations: Or in matrix notation: Once  i and   are determined construct estimate and estimation variance from:

14. Properties of Co-Kriging Process If T and H are not very correlated weights associated with H observations will be much smaller than weights associated with T observtions. Cokriged estimate will look a lot like kriged estimate, cokriging variance will be approximately equal to kriging variance If T and H are highly correlated and are both measured at all locations will not get too much improvement in estimate of T over kriged estimate Get most improvement if T and H are highly correlated and T is undersampled compared to H. Simple co-kriging most applicable if have derived required means, covariances and cross-covariances from physical model

15. Ordinary Co-Kriging Ordinary co-kriging provides the best linear unbiased estimate of a random field, e.g. T(x), with an unknown constant or linearly trending mean, and a known stationary covariance/ variogram and cross-covariance/cross-variogram structure Used when estimate covariances/variograms and cross-covariances/variograms from field data

16. Ordinary Co-Kriging Define ordinary co-kriging estimate as: Check for bias: Therefore must choose i and l so that: Proceed as before adjoining the constraints to the estimation variance equation, then minimizing by taking derivatives with respect to i and l and setting equal to zero.

17. Ordinary Co-Kriging This results in a linear system of N T +N H +2 equations with N T +N H +2 unknowns: Or in matrix notation:

18. Ordinary Co-Kriging Once  i and   are determined construct estimate and estimation variance from: It is possible that if the two fields are not strongly correlated the increase in uncertainty due to filtering out the mean of the second random field may completely wipe out decrease in uncertainty gained from measurements of that random variable

19. Properties of Ordinary Co-Kriging Process Ordinary Co-kriging applicable if can estimate covariances/variograms from field data. To determine experimental cross variogram must have pairs of both observations at all locations. This is not required to determine experimental cross-covariance. Cokriging matrix must be conditionally positive definite. Therefore not any function is acceptable to model the cross-variogram. Criteria for acceptability not well established for experimental cross-variograms