Geo479/579: Geostatistics Ch17. Cokriging

1. Geo479/579: GeostatisticsCh17. Cokriging

2. Introduction Data sets often contain more than one variable of interest These variables are usually spatially cross-correlated

4. The Cokriging System A method for estimation that minimizes the variance of the estimation error by exploiting the cross-correlation between several variables Cross-correlated information contained in the secondary variable should help reduce the variance of the estimation errors

5. The Cokriging System When is the secondary variable useful in estimates? Primary variable of interest is under sampled then the only information we have is the cross correlated information

6. The Cokriging System The cokriging estimate is a linear combination of both primary and secondary data values This is the Equation used in Ordinary Kriging pg 279

7. The Cokriging System The development of the cokriging system is identical to the development of ordinary kriging system Estimation Error R can be defined as This is a modification of the error estimation in Ordinary Kriging( pg 279)

8. The Cokriging System Using matrix notation we can write w = { a1, a2, a3,…an, b1, b2, b3,…bm} Z = { U1, U2…..Ui, V1,….Vj}

9. The Cokriging System Using Equation 9.14 (p216), 12.6 (p283), 16.3 (p372) we can write

10. This is similar to Chapter 16

11. The Cokriging System 1) Unbiasedness condition • Note: Other nonbias conditions are also possible

12. It is similar to unbiasedness in Ordinary Kriging (p281) • We set error at as 0:

13. The Cokriging System 2) Minimizing error variance Lagrangean Relaxation:

14. The Cokriging System Lagrangean Relaxation: Original Lagrange parameter: (12.9)

15. The Cokriging System Equating n+m+2 partial derivatives of Var{R} to zero, we get the following system of equations

16. This is similar to minimizing the varianves of error in Ordinary Kriging • The set of weights that minimize the error variance under the unbiasedness condition satisfies the following n+1 equations - ordinary kriging system: (12.11) (12.12)

17. Minimization of the Error Variance • The ordinary kriging system expressed in matrix (12.13) (12.14)

18. The Cokriging System Positive definiteness must hold for the set of auto- and cross-variograms (Eq16.44, p391).

19. The Cokriging System If the primary and secondary variables both exist at all data locations and the auto- and cross-variograms are proportional to the same basic model then the cokriging estimates will be identical to the ordinary kriging estimates

20. A Cokriging Example

24. A Case Study • Compares cokriging and ordinary kriging • Undersampled variable U is estimated using 275 U & 470 V sample data for cokriging and only the 275 U data for ordinary kriging

27. Ordinary kriging 275 U values Using eq 17.11 for the variogram model

28. Cokriging 275 U and 470 V values Using eq 17.11 for the variogram model Two non-bias conditions 1) uses the initial conditions 2) uses only one nonbias condition

29. In the alternate unbiased condition, the unknown U value is now estimated as a weighted linear combination of nearby V values adjusted by a constant so that their mean is equal to the mean of the U values

30. Negative estimates occur because of the nonbias condition

35. Cokriging with two nonbias conditions is less than satisfactory A physical process with both negative and positive weighting scheme is difficult to imagine Cokriging with one nonbias condition considerably improved the spread of errors and bias Though we had to calculate global means of U and V

36. If the spatial continuity is modeled using semivariograms then they can be converted to covariance values for cokriging matrices by following equation:

37. The Cokriging System If we want an estimate over a local area A, there are two options: 1) Average of point estimations within A

38. The Cokriging System 2) Replace all the covariance terms and in point cokriging system, with average covariance values and

39. The Cokriging System With the unbiasedness conditions, we can calculate error variance as follows