Block kriging estimation

1: Generate and store the point-to-point covariance look-up table. If the FFT- integration hybrid block covariance calculation method is used, compute and store the block-to-point covariance map(s)

2: for Each location u do

3: Search the conditioning data consisting of the closest original point data and block data

4: if the number of data (point or block) is large enough then

5: Compute or retrieve the needed local block-to-block, block-to-point, point-to-block and point-to-point covariance

6: Build and solve the mixed-scale kriging system 7: Compute kriging mean and variance for location u 8: else

9: Set node as uninformed and issue warning message 10: end if

11: end for

Parameters description

The BKRIG algorithm is activated from Estimation | bkrig in the algorithm panel. The main BKRIG interface contains three pages: “General”, “Data” and “Vari- ogram” (see Fig. 7.10). The text inside “[ ]” is the corresponding keyword in the

BKRIGparameter file.

1. Grid Name [Grid Name] Name of the estimation grid.

2. Property Name Prefix [Property Name] Prefix for the estimation output. 3. Kriging Type [Kriging Type] Select the type of Kriging system to be

solved at each node: Simple Kriging (SK) or Ordinary Kriging (OK). 4. SK mean [SK mean] Mean of the attribute. Only required if Kriging Type

[Kriging Type]is set to Simple Kriging (SK).

5. Block Covariance Computation Approach [Block Cov Approach] Select the method of computing block covariance: FFT with Covariance-Table or Integration with Covariance-Table.

6. Check block data reproduction [Check Block Reproduction] If checked, the estimated block average values are calculated and the relative errors compared to the input block data are computed for each realization. The results are shown on the Commands Panel, which is activated from

23 20 21 22 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9

Figure 7.10 User interface for BKRIG

7. Hard Data | Object [Hard Data.grid] Name of the grid containing the con- ditioning point data. If no point grid is selected, the estimation is performed conditioned only to block data.

8. Hard Data | Property [Hard Data.property] Property for the point data. Only required if a grid has been selected in Hard Data | Object [Hard Data.grid].

9. Assign hard data to simulation grid [Assign Hard Data] If selected, the hard data are relocated onto the estimation grid.

10. Min conditioning point data [Min Conditioning Data Point] Minimum number of point data to be retained in the search neighborhood.

11. Max conditioning point data [Max Conditioning Data Point] Maximum number of point data to be retained in the search neighborhood.

12. Search Ellipsoid for Point Support Data [Search Ellipsoid Point] Parametrization of the search ellipsoid for point support data, see Section6.4. 13. Min conditioning block data [Min Conditioning Data Block] Minimum

7.4 BKRIG: block kriging estimation 129

14. Max conditioning block data [Max Conditioning Data Block] Maximum number of block data to be retained in the search neighborhood.

15. Block Data From Select where the block data are to be found. There are two options: From File [Block From File]and From Point Set Object [Block From Pset].

16. Block Data From File [Block Data File] Only activated if From File [Block From File]is selected in Block Data From. The directory address of the block data file should be specified. The block data file format is shown in Fig.7.9. If no block data file is entered, the estimation is performed using only point data.

17. Block Data From Point Set Objects Only activated if From Point Set Object [Block From Pset]is selected in Block Data From.

18. Number of blocks [Number of Blocks] Number of blocks entered from point-set objects.

19. Input block average values [Block Average Values] Enter the input block average value for each block. The sequence of block data should be the same as that of the corresponding input point-set objects. The number of block average values should be the same as the Number of blocks [Number of Blocks]. 20. Consider block data error [Consider Block Error] If checked, the block

errors are considered.

21. Input block data errors [Block Error Variance] Only activated if Con- sider block data error [Consider Block Error] is checked. Enter the block error variance for each block. The sequence of block data should be the same as that of the corresponding input point-set objects. The number of error variances should be the same as the Number of blocks [Number of Blocks].

22. Point set objects [Block Grid i] Enter the point-set block objects. This allows users to conveniently use the pre-loaded point-set objects as the condi- tioning blocks. No property is required to be associated with the input point-set grids. The maximum number of grids entered in this way is 50. They have to be loaded from a file if there are more than 50 blocks.

23. Variogram parameters for simulation [Variogram Cov] Parametrization of the point-support variogram, see Section6.5.

Examples

BKRIGis run for two 2D synthetic cases corresponding to tomography and down- scaling applications. The reference model and input data for these two cases are given in Fig.7.11.

The reference field for the tomography case is discretized into 40 ×40 grid cells, each cell of dimension 0.025 × 0.025. The reference background field has been

(a) Reference (tomography) 4.5 4 3.5 2.5 3 2 4.5 4 3.5 2.5 3 2 4.5 4 3.5 2.5 3 2 4.5 5 4 3.5 2.5 3 2 4.5 5 4 3.5 2.5 3 2 4.5 5 4 3.5 2.5 3 2 3 (d) Reference (downscaling)

(b) Point data (tomography)

(e) Point data (downscaling)

(c) Block data (tomography)

(f) Block data (downscaling)

Figure 7.11 The reference field, the point and block data for the tomography and the downscaling examples

(a) Estimation (tomography) 4 5 0.5 0.4 0.3 0.2 0.1 0 4.5 4 3.5 3 2.5 2 3.5 3 2.5 0.8 0.65 0.5 0.35 0.2 (c) Estimation (downscaling) (b) Variance (tomography) (d) Variance (downscaling)

7.4 BKRIG: block kriging estimation 131

generated using sequential Gaussian simulation (Section 3.8.1) with the normal score variogram model:

γ (hx,hy)= 0.1 + 0.9Sph ⎛ ⎝ B8_hx 0.5 92 + 8 _hy 0.25 92⎞ ⎠ . (7.11) A high value heterogeneous area is added in the center of the simulated field, see Fig. 7.11a. The two columns of values at the left and right hand sides of the refer- ence model are retained as conditioning point data, see Fig. 7.11b. The block data are the 18 ray data, see their geometry in Fig. 7.11c. Each of the block datum value is obtained by averaging point values over the ray path.

The reference field for the downscaling case study is discretized into 40 × 50 cells, each cell of dimension 0.025 × 0.02. The background property is generated the same way as in the tomography case. Two high value heterogeneities are added into that background field. Figure 7.11d gives the reference model. Again, the two columns of values at the left and right hand sides of the reference model are retained as conditioning point data, see Fig.7.11e. The block data are 10 coarse grid data covering the entire field, see Fig.7.11f. Each of the block datum values is obtained by averaging point values over the block.

Variogram model (Eq. (7.11)) is used for the BKRIG examples. The input SK mean for the tomography case is 3.0 and that for the downscaling case is 2.7. The minimum and maximum number of conditioning data are 0 and 12, respectively.

Figure 7.12a and Fig. 7.12c show the smooth kriging estimation maps using both point and block data. The general patterns of the reference models are well reflected, for example the locations of the high value heterogeneities. Figure 7.12b gives the kriging variance for the tomography case. Low variances are found in areas close to conditioning data. Figure 7.12d gives the kriging variance for the downscaling case. In the middle area, there is less data constraint thus the variances are high. The block data in both cases are well reproduced; the average absolute errors are 1.8% and 0.5%, respectively.

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