Multiple Indicator Kriging (MIK) is often used to estimate the grades of panels, which can be localised into smaller blocks, for deposits with wide spaced drilling and/or skewed sample grade distribution.
This article discusses a possible enhancement to the standard MIK method, in which local average grade values are estimated for each indicator bin for each panel, instead of using constant grade values (based on the whole sample database) for each indicator bin.
A comparison between the standard method and the proposed method shows the proposed method can give a better local estimate of the panel, and therefore selective mining unit (SMU) grades.
MIK was developed by Andre Journel in 1983. It is a non-linear grade estimation method, generally used to model deposits with skewed grade distributions and low continuity. MIK is commonly used to:
- Estimate panel grades (Figure 1) for non-gaussian and skewed distributions.
- Estimate grades recoverable from SMU size blocks using change of support and, optionally, localized multiple indicator kriging (LMIK).
- Provide resource SMU grade distributions.
An indicator is used to estimate the percentage of a block above given cut-off grades. The general MIK process is as follows:
- The percentage of the block above each grade cut-off is estimated using 0’s (below cut-off grade) and 1 (above cut-off grade).
- Using these percentages, the percentage of each bin (percentage between cut-offs) is calculated.
- The bin percentage is multiplied by the mean grade of all samples, with grades between the cut-offs used for that bin.
- The panel grade or E-type estimate is the sum of these percentages multiplied by the average sample grade.
- Change of support is then used to adjust the grade distribution to enable the estimation of SMU-sized block grades.
- LMIK may be used to distribute metal within SMU sized blocks within the panel.
Figure 1 Panel and SMU block sizes
Advantages of MIK
MIK provides some significant advantages over linear estimation methods, including:
- It is less prone to conditional bias and over-smoothing of grades.
- It reduces the need for cutting extreme sample grades, by explicitly modelling the continuity of grades across a range of cut-offs. Consequently, grades above the highest selected cut-off can be restricted by interpolation over short distances.
- It allows estimation of recoverable resources, inclusive of ore loss and dilution, over a range of selective mining unit sizes.
- It does not depend on prior assumptions about the shape of the grade distribution.
A proposed enhancement of the method
MIK, as originally implemented, used the average grade of the whole dataset for each of the grade bins when calculating the panel grade distribution. This was computationally efficient, but did not adequately reflect local variations in grade distribution.
A suggested solution to the issue of non-local average bin grades, is to calculate the average grade for each indicator bin, based on the neighbouring samples around each panel. Surrounding samples are used to estimate the sample average bin grades, rather than applying the deposit average. The search ellipse used to select these sample would be based on the anisotropic variogram ranges used in the indicator estimate but increased to obtain sufficient number of samples. Using this method ensures the panel grade distribution compensates for local trends in grades. Also, the discrete high-grade and low-grade zones, common within deposits with very skewed grades, will have appropriately estimated panel grades. The estimation of the grade for SMU-size blocks will then also be more consistent, being based on a distribution estimated using the local grades.
By using Datamine software to apply this method, only a small increase in time is required. In addition, using a Datamine macro ensures the steps are carried out consistently.
AMC is not aware of any other application of a local average bin grade instead of deposit average bin grades.
Comparing the existing and proposed method
A comparison of MIK estimates using global and local bin average grades was carried out on a deposit with a highly skewed grade distribution (Figure 2).
Figure 2 Log probability plot of sample grades
The deposit was drilled on a 50 m x 40 m spacing. The parameters used in the panel grade estimation were:
- Eight grade bins based on percentiles of the grade distribution.
- Panels 12.5 m x 12.5 m x 10 m in size.
- Estimation search ellipse for indicator and average grade estimates: 60 m x 80 m x 100 m
- Minimum of 4 samples and a maximum of 16 samples used to calculate the local grade average and indicator percentage of each bin
Table 1 lists the grade cut-offs used for the grades bins and the average bin grade based on the global sample data.
Table 1 Bin cut-off grades and global average grades
A comparison between the panel grades (E_type) at different cut-offs is shown in Figure 3.
This figure shows the tonnes are very similar. However, the AMC model has higher grade at higher cut-offs.
Figure 3 Grade-Tonnage Curve – Original versus AMC
Figure 4 shows a scatter diagram between the panel grade estimates using the original grades compared to AMC grades.
This plot shows that the grades trend around the 45 degree line (red dotted line), with the AMC estimate showing higher grades. The correlation co-efficient is 0.91.
Figure 4 Scatter diagram comparing original panel grades to AMC Grades
Note dotted line is the 45 degree line
Figure 5 is a scatter diagram for bin 4 grades (cut-off 0.09) with a correlation coefficient of 0.9. It clearly shows using the total sample data average grade for the top bin limits the local block grade estimate.
Figure 5 Scatter diagram for above bin 4 comparing original panel grades to AMC Grades
Note dotted line is the 45 degree line
The log probability plot (Figure 6) shows a similar trend, with both estimates having a similar mean grade, original 0.191 versus AMC 0.196. The AMC estimate has a slightly higher variance original 1.70 versus AMC 1.78.
Figure 6 Log probability plot
Note: the blue line indicates original grades and the red line indicates AMC grades
Table 2 compares the bin global averages between the original model and AMC model.
Table 2 Comparison between bin global average grades<Round figures in table to 2dp>
|Bin||Probability %||Cut-off||Global Average||AMC Average||AMC Median|
Figure 7 shows the grade of the original panels compared to their elevation, coloured by the AMC panel grade. This shows the AMC grade is generally similar to the original grades but better represents the local sample grades.
Figure 7 Original grade versus RL
AMC plans to continue this study to review the impact on SMU grades after the application of change of support and LMIK.
This initial study shows that the use of local average bin grades, instead of constant bin grades, can give a better estimate of the local grade distribution. The increase in local panel grades is particularly evident above the higher cut-off grades.
AMC’s suggested alternative to the standard MIK method, by using local bin grades, maintains the suitability of MIK to model skewed distributions with low grade continuity. The benefit of this approach is that it potentially improves the estimate of the local panel and SMU grades.