May 2012

Uniform Conditioning

Estimating recoverable reserves in the face of sparse data can present unique challenges. Often, drillhole data is not available at a distribution representative of the scale of mining – but we are forced to use this data to estimate at the small scale. In these types of scenarios the geostatistical technique Uniform Conditioning (UC) can prove a powerful tool for grade estimation and model validation.

What is it?

UC is a non-linear estimation technique which determines the grade distribution above cutoffs inside a larger block. The principle of the tool is that data exists to accurately estimate grade at a larger “panel” scale. But you cannot reliably estimate the selective mining unit (SMU) sized blocks in the deposit (Figure 1). For instance, at the beginning of a project when only sparse exploration data is available, but you would like to have some knowledge of the behavior of the deposit at the mining scale.

Figure 1: Schematic of the terms used in Uniform Conditioning (UC).

How it works

The basis of UC is to use a robust estimation of the panel grade (via kriging) to condition the results at the SMU scale. This is where the name of UC originates - the estimate of the recoverable reserves is conditioned to one variable – the panel grade.

After estimating the panel grade, the crux of the UC calculation is related to change of support. The sample sizes of the composites, panels, and SMUs are vastly different from each other and cannot be compared directly. A change of support function is needed to convert the data to a normal score distribution for analysis.

The Uniform Conditioning MSBasis procedure (psuc.dat) computes the normal scores transformation (Gaussian anamorphosis) of the input data. If the data is already transformed, this option can be disabled. The anamorphosis function defines how the real grade units correspond to the normal score units. In the UC procedure this is accomplished through the use of Hermite polynomials. You can choose to specify the polynomial coefficients used via an input file, or have the procedure calculate them on the fly. The number of terms in the polynomial expansion can range up to 100 – the more terms used, the better the polynomial fitting will be, but the more speed is sacrificed. The detailed mathematical calculations will not be presented here, but more information can be found in the geostatistical literature related to volume-variance calculations and non-linear estimation methods.

What are the assumptions and limitations?

UC assumes the relationship between the SMU grades and the panel grades is bivariate normal. Therefore, if we know the normal score panel grade, we know the mean and variance normal score SMU distribution. UC also assumes the change of support model for the SMU’s can be extended to the panels.

One limitation of UC is panels with the same estimate will have the same grade and proportion curves, regardless of the surrounding data. This reinforces the need for robust panel estimates – as they are critical to the UC results.

Some may view the fact that UC does not provide information regarding where the high or low grade SMU’s are within the panel as a limitation. However, this is the underlying premise of UC – panel grades can be predicated reliably, but SMU grades cannot.

Running UC in MineSight – MSBasis procedure psuc.dat

Input – MineSight drillhole composite file or ASCII data (x, y, z coordinates and grade).

Output – Reports the grade, proportion, and metal content at each specified cutoff for the block. Results can be output to the 3D block model or to ASCII.

General preparation workflow:

• Decide the SMU and panel sizes. The SMU size will be based on the deposit type and mining equipment limitations. The panel size should be based on drillhole spacing – procedure p52201.dat can provide insight for this spacing analysis.
• Decide the grade cutoffs of interest for analysis. These could be chosen based on economic or mining parameters (i.e. low, medium, and high grade cutoff values). Up to 10 cutoffs can be used.
• Initialize a project with the block model size equal to the panel size. Add the necessary items to the block model for storage of the UC output. For each cutoff, three model items must be specified (i.e. if five cutoffs are used, 15 items are needed).
  • Items to store the grade above cutoff
  • Items to store the proportion above cutoff (0-1 range)
  • Items to store the metal content above cutoff
• Estimate the panel/block size grades (it is common to use ordinary kriging – procedure p62401.dat or pintrp.dat) – run the kriging with the block discretization set at the SMU resolution within the panel. This value will be used to condition the UC output.
• Calculate the variance at SMU support and panel support (via running procedure psblkv.dat twice at the two different sizes).

Now you are ready to run UC via procedure psuc.dat.

Example – Uniform Conditioning for a single panel –

Panel size is 100 x 100 x 100 based on drillhole spacing.

SMU size is 20 x 20 x 20 based on our mining equipment and deposit type.

After running ordinary kriging (discretizing at 5 x 5 x 5), we obtain grade estimates for copper at the panel scale:

Figure 2: Panels/blocks colored by CUKRG (estimate of the panel from ordinary kriging) and query window showing UC results.

After determining the variances at the SMU and panel scale (0.1979 and 0.1258 respectively), we run procedure psuc.dat to run Uniform Conditioning for five cutoffs. The cutoff setup is:

Cutoff start = 0.0

Increment = 0.1

Results of UC for highlighted block (upper left hand panel in Figure 2):

Based on the conditional kriged grade estimate for the panel (0.186) – the above table appears reasonable. A graph of the grade/proportion curve also highlights the results:

Figure 3: Grade/Proportion Curve for example panel.

The illustration below represents possible SMU distributions of grade inside the block based on the UC results. It should be noted that because we don’t know the spatial distribution of the SMU, either scenario shown is equally likely.

Figure 4: Two possible scenarios of grade distribution for the example block at the SMU scale.

Reserves reporting after running UC can be accomplished via the procedure smures.dat. After setting up the smu table file (with a real or “dummy” zone item), you can select the number of UC cutoffs per block and the respective grade items for analysis. Grades and proportions above cutoff from UC can also be converted to discrete grades and percentages using the calculation script cp-ModelCalcTool.pyz. These discrete grades and percentages can then be used in a multiple ore% setup with MineSight Interactive Planner (MSIP). The August 2011 Newsletter – Managing Multiple Indicator Kriging Models contains more details on analyzing the results from UC (along with the syntax for the model calculation script). The article is geared toward MIK reporting – but similar setups can be used for UC output as well.