1. Geo479/579: Geostatistics Ch13. Block Kriging

2. Block Estimate  Requirements An estimate of the average value of a variable within a prescribed local area  One method is to discritize the local area into many points and then average the individual point estimates to get the average over the area  This method is computationally expensive

3. Objective  See how the number of computations can be significantly reduced by constructing and solving only one kriging system for each block estimate  Block Kriging

4. Block Kriging  Block Kriging is similar to the point kriging  The mean value of a random function over a local area is simply the average (a linear combination) of all the point random variables contained in the local area  Where V A is a random variable corresponding to the mean value over an area A, and V j are random variables corresponding to point values within A Equation 13.1

5. Point Kriging  In point kriging, the covariance matrix D consists of random variables at the sample locations and the location of interest (12.14) (12.13)

6. Point Kriging  In point kriging, these are point-to-point covariances. For block kriging, these are point- to-block covariances (the block of interest)

7. Block Kriging  Point-to-block covariances required for Block Kriging

8. Block Kriging  The covariance between the random variable at the i th sample location and the random variable V A representing the average value over the area A is the same as the average of the point-to- point covariances between V i and the random variables at all the points within A

9. Block Kriging  The Block Kriging System  The average covariance between a particular sample location and all of the points within A Equation 13.3 Equation 13.4

10. Block Kriging  The Block Kriging Variance:  The value C is the average covariance between pairs of locations within A Equation 13.5 Equation 13.6

11. Ordinary Kriging Variance  Calculate the minimized error variance by using the resulting to plug into equation (12.8)

12. Block Estimates vs. the Averaging of Point Estimates The average of the four point estimates is the same as the direct block estimate The average of the point kriging weights for a sample is the same as the block kriging weight for the sample Figure 13.1

13. Varying the Grid of Point Locations within a Block  When using the Block Kriging approach - How to discretize the local area for block being estimated? The grid of discretizing points should be always regular The spacing between points may be larger in one direction than the other if the spatial continuity is anisotropic (Figure 13.2)

14. Discretizing Points  The shaded block is approximated by six points located on a 2X3 grid. The closer spacing of the points in a north-south direction reflects a belief that there is less continuity in this direction than in the east-west direction. Despite the differences in the east-west and north-south spacing, the regularity of the grid ensures that each discretizing point accounts for the same area, as shown by the dashed line Figure 13.2

15. Discretizing Points  Discretizing points 16, Estimates are similar  Sufficient discretizing points number 2D block: 4x4 = 16, 3D block: 4x4x4 = 64 Table 13.2

16. Block Kriging vs. Inverse Distance Squared Block Estimates  A plus symbol denotes a positive estimation error while a minus symbol denotes negative estimation error  The relative magnitude of the error corresponds to the degree of shading indicated by the grey scale at the top of the figure

17. Block Kriging vs. Inverse Distance Squared Block Estimates Figures 13.3, 13.4 Figure 13.4

18. Case Study  Comparison of summary statistics for Block Kriging and Inverse Distance Weighted  Inverse Distance Weighted has larger errors  For Inverse Distance Weighted, there are several large overestimation where relatively sparse sampling meets much denser sampling  Inverse Distance Weighted did not correctly handle the clustered samples, giving too much weight to the additional samples in the high-valued areas  Block Kriging showed some underestimation due to its smoothing effect and the positive skewness of the distribution of the true block values

19. Block Kriging vs. Point Kriging Figures 13.3, 13.5

20. Block Kriging vs. Inverse Distance Squared Block Estimates Table 13.3 Estimates Table 13.4 Errors

21. Block Kriging versus Point Kriging Table 13.5 Errors