INTERPOLATION
The block model is generated simultaneously with interpolation of grades. This means the following routine should be used:
1. Specify all input parameters, files, search ellipse and variogram parameters in the processes Modelling | 3D Block Estimate | IDW or Modelling | Kriging. Run the interpolation.
2. The generated block model should now be flagged for domains etc. using wireframes or outlines (as specified in chapter 8).
3。 All unflagged cells (above the surface, outside of wireframes etc.) should be filtered out (File | Filter | Subset).
4。 All the assigned cells should be checked for whether all cells are populated with grades or not (Stats or Min/Max). There should be no missing grades in the block model. would update the later ones (File | Merge | MM).
7. Repeat all steps from 1 to 6 for all domains and for all elements.
The interpolation process can take from several hours to several days.
CHECKLIST:
• Apply top cuts if necessary
• Interpolate grades using several methods for validation
• All cells in the Resource model should be informed with grades.
• If Kriging was used, run cross validation to check if the variogram is appropriate. Run the variogram model that produces a good estimate with the lowest kriging error.
• If MIK is used, make sure the search parameters are the same for all bins for a particular interpolation run
• The number of interpolation runs should be equal minimum to: No of elements x No of domains x No of interpolation volumes x No of interpolation methods. It is a good idea to save all these runs in a macro.
Notes:
Search ellipse
When defining the search ellipse in MICROMINE –
• Define the number of sectors, if there is a large amount of data, then eight or sixteen is an appropriate number of sectors to employ.
• Define the maximum and minimum number of points to be used in the search ellipse, a maximum of 6 means a total number of six per sector will be used, the six nearest samples to the point of estimation will be used for the estimate; the other points in the sector will not be used. With the minimum set to 2, if only one sample is found then the point of estimation will not be estimated. This is data declustering performed on the fly by MICROMINE in the estimation process.
• Define the attitude of the search ellipse, the azimuth, dip and plunge. This should be defined to include the relevant samples for estimation and to exclude the redundant points. Several different search ellipses of increasing sizes may be required to interpolate all of the blocks in the model. Often three runs may be needed to populate the model, the search ellipse will be increased in radius with each run, the blocks that have already been estimated will not be overwritten, only empty blocks will be populated by each new run.
Notes:
How to decide search ellipse size
The search ellipse radius is determined from the variogram parameters or from the sample spacing. For the iron example the radius of 250 metres above contains around seven drillholes in the ellipse each with two metre interval samples, so there are abundant samples for the first pass estimation because the drillholes are spaced around 100 metres apart.
Save the search ellipse forms. Load the search ellipse in 3D Viewer, bring up the grid and look from different angles with the ellipse transparency on to observe that the attitude of the search ellipse is the required design. This is an important validation step.
Notes:
Inverse Distance Weighting
Inverse distance weighting uses the inverse of the distance to the value of a selected power as the mechanism whereby the samples are preferentially weighted. For the simple example the unknown grade at the point of estimation is ?, results are tabulated
Using a power of 2 for the inverse distance weighted calculation the point of estimation equates to 0.234 kg/m3.
Where the algorithm result = point of estimation; d = distance; v = data value;
n = number of points to calculate cell node value; w = size of the power
Notes:
For the different powers, 1, 2 and 5, as the power is increased then the weighting on the nearest sample to the point of estimation increases, the higher the power then the greater this weighting to the nearest samples.
With an inverse power of 1 the grade weights are more evenly spread amongst the samples, based evenly on the distance from the point of estimation. As the power increases to a power of 5 then the samples closest to the point of estimation at 6 and 7 metres respectively receive nearly all the weighting. The grade estimate increases to 4.28 since the 2 nearest grades are also the highest grades; most of the other samples have very little influence on the grade
When using MICROMINE for inverse distance weighting for iron, interpolate both the cut and uncut fields in the composite file. For gold set a power of 2 or 3; 3 is most commonly used for gold. For iron a power of 2 is appropriate.
Interpolate only the grades in the wireframe and define the blocks from the file to update the block model. The composite file must be used; if grade intervals are not of equal length then the model will be biased and will be a less accurate estimate.
Notes:
Exercise 22.13 Inverse Distance Weighting
Ordinary kriging
Kriging is an interpolation method, which uses the measured anisotropy of the deposit to preferentially weight the samples to varying extents in the three defining directions within the deposit. Anisotropy may or may not be present dependent upon the nature of the deposit. Anisotropy is the uneven distribution of grade within the deposit. If the deposit is isotropic and the variogram range does not change with direction then an omnidirectional variogram may be fitted.
The omnidirectional variogram will have a tolerance of 90 to look in all directions and will weight the samples as an average of all variogram models. The weighting mechanism is determined by the variograms that are modelled. The variogram model is then applied to the kriging algorithm to estimate block values. The variogram models are a geostatistical measure of variation in grade with distance along a spatially defined direction. Three variogram models will be
Notes:
produced for 3d modelling in x,y,z and two in x and y for 2d modelling. Kriging is dependent upon being able to model variogram models, variography will confirm or disprove the geologist’s intuition and assumptions relating to the deposit.
Ordinary kriging formula:
matrix. A second matrix is set up which calculates the mean squared differences between the data points and the cell nodes. This is accomplished using gamma values from the variogram models. This matrix is the M2 matrix. The K matrix is divided by the M2 matrix to obtain the kriging weight, which is λ. Any left overs are accounted for by the Lagrange coefficient,µ. The Lagrange parameter is a condition in the equation that requires that the total kriging weights sum to one.The ordinary kriging dialog boxes are the same as the Inverse distance weighting with the exception that the routine uses not only distance but also the variogram models in the three orthogonal directions to weight the estimation. The three variogram models are setup as saved forms in the semi variogram parameters box. The longest range is the main direction, followed by the intermediate and the third is the shortest direction. Some rules apply to the form for saving the variograms –
• The nugget must be the same for all three variograms
• The partial sills must be the same for all variograms
• The three variograms must be orthogonal to each other, note that if the main direction is 180 degrees with zero dip, then the second direction must be 270, not 90 degrees, the angle must be bigger.
• If these parameters are wrong then the kriging variogram form will not be saved, this is a validation step in MICROMINE.
Block kriging can be used which will then enable the discretisation, this means that several points can be estimated into the block and are then averaged for the block estimate.
Iron example;
Notes:
Run the routine to generate the block model output file.
Ordinary kriging, relative variograms
Once the relative variograms are saved together in a form then the ordinary kriging routine can be run. There is no need for a back transformation for the grade estimate, this is performed automatically; the kriging variance and kriging standard error however are rendered useless and should not be used.
Notes:
Run the routine and report the tonnes and grade.
Multiple Indicator Kriging
MIK takes account of different anisotropy at different grade levels. MIK is particularly useful where there are mixed populations present as it is a non linear method. It is also better at handling the higher grade values and avoids the need to cut top grades.
MIK uses different grade levels by asking for a cut-off value above which all raw data values are transformed by an indicator to a value of one. All values below this cut-off are assigned an indicator of zero. This is an indicator transformation.
The various grade bins are selected by using a cumulative distribution frequency curve to group grades into percentile bins. The indicators are then modelled with semi variograms and these variograms are applied to each cut-off.
By applying a more suitable variogram model to the various grade levels the anisotropy is correctly honoured. This type of modelling method is attempting to model higher grades that are mixed in with the main population and cannot be domained out to be modelled independently.
Micromine has an MIK option. The indicator variography and the indicator kriging both exist with multiple cut-offs with associated multiple variogram models. Up to ten cut-offs are available, so that deciles can be defined.
Notes:
Example:
Apply a cut-off; model the variography, run the model with the indicator cut-off and associated variogram model. Do this for each cut-off; you will then have five kriged models.
Indicator bins, Au, more importantly the range will vary for each indicator.
0.2 g/t everything above 0.2 has a primary variogram attitude of 260 degrees 0.5 g/t everything above 0.5 has a primary variogram attitude of 260 degrees 0.9 g/t everything above 0.9 has a primary variogram attitude of 265 degrees 1.5 g/t everything above 1.5 has a primary variogram attitude of 275 degrees 6.5 g/t everything above 6.5 has a primary variogram attitude of 290 degrees Do for the three directions and save the formset, repeat for each grade cut-off;
alternatively do once at the 50th percentile, this is called median indicator kriging and is much faster as only three variograms and one variogram formset is needed. Also the variograms at the 50th percentile are always the easiest to do.
Create a blank block. Setup the cut-offs and enter the corresponding variogram formset. Then run to generate an MIK model that applies different weights from the different variograms at various grade levels. An e type estimate for each block will be produced. MIK is more likely to be employed in resource estimation than grade control; however some of the large nickel operations do use this method in addition to others. Production pressures restrict its use.
How the estimate is produced: then an estimate can be obtained.
If the grade thresholds are as follows –
1, 2, 3, 5, and 10 and the probabilities are 1, 0.82, 0.61, 0.46 and 0.12
You need to obtain the mean of each bin; ex the mean of the raw data below 1 g/t is 0.5
Notes:
The mean of the data between 1 and 2 g/t is 1.45 g/t. MICROMINE calculates the mean value automatically and then produces the e type estimate.
For the block estimate, the weighted equation becomes
(0.18 x 1.45) + (0.18 x 1.45) + (0.18 x 1.45) + (0.18 x 1.45) + (0.18 x 1.45) = 5.42 g/t.
This is a single block estimate, called an E type estimate. This is then repeated to estimate all of the blocks in the blank model.
MICROMINE example; Median indicator kriging;
Median indicator kriging uses one Indicator variogram modelled for the median at the 50th percentile defaulted to all of the grade thresholds, instead of different variograms for each threshold. Median indicator kriging is not as accurate but it is faster and is still a non linear technique that can deal with any high grades and dual populations.
Once the three indicator variograms are modelled then go to Modelling | 3d block estimate | multiple indicator kriging and save the indicator variogram form, this should be used in the cut-off box. All of the remaining dialogs are the normal setup as for inverse distance weighting, interpolate the Composite file but interpolate the TFE field, do not model the cut field.
The cut-off box allows a form to be saved that should include all of the cut-offs for the grade bins and the indicator variogram form containing the 3 orthogonal indicator variograms; in this case the form is as saved as IND and is defaulted to
Notes:
each grade cut-off. The grade estimate is the method to calculate the bin grade to be used for the weighted estimate, if mean is selected it is the average of grades between 0 and 10%, 10 and 20% etc; if median is selected then it is the median grade for grades between 0 and 10%, 10 and 20% etc.
The deciles do not have to be used for all bins, the top bin may be at 95% or some other figure that in the opinion of the modeller appropriately deals with the influence of a few high grades.
The IND formset includes the 3 saved indicator variograms for the median.
Notes:
The first indicator variogram is the direction of maximum continuity.
The second indicator variogram is the intermediate variogram with the second longest range.
Notes:
The third indicator variogram is the variogram with the shortest range.
Again use block kriging and interpolate the parent cells only. Run the MIK routine.
Kriging variations
Notes:
The kriging calculations are not straightforward and only the advent of fast processing of millions of multiple equations by computers has allowed their application as an alternative to simpler methods. Numerous variations exist upon the listed algorithm with ordinary, universal, disjunctive, indicator kriging etc also available.
Note that –
Simple kriging differs from ordinary kriging in that simple kriging interpolates from a constant or known mean whilst ordinary kriging applies a local mean which varies across the sample area. MICROMINE supports ordinary kriging.
Cokriging requires an inverse relationship of one element to another in the model area such as nickel and magnesium.
Disjunctive kriging is another name for co indicator kriging.
Notes:
Lesson Summary
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