Theory
Classical statistical analysis should be repeated using the same procedures described in the Lesson 2. However, this classical stats analysis will have the following differences:
• The analysis will be run for sample composites
• Only flagged composites will be used for the analysis, those assays inside the wireframe
• Stats will be run for each geological / lithological / structural / mineralogical domain separately
Final decisions will be made regarding the method of grade interpolation, variography, mixing of population and top cuts. Mixing of grade populations within each domain should be carefully considered using the same process described in Lesson 2. If it is not possible to separate grade populations using domaining, then the MIK grade interpolation method should be used.
Variography
Variography will be run for each element and domain separately. For every domain there are three variograms, each at right angles to each other. For example, if we have three elements and five domains, the task will be to generate 45 directional variograms. If MIK is applied, then the number of final variograms will be 450 (if 10 thresholds are used).
The first step would be to generate omni variograms. Omni variograms will indicate the general ranges and variances of grade populations and whether the chances of getting good directional variograms are good or bad. They also assist with the lag sizes. Variograms are to be generated using the process Stats | directional anisotropy of grade distribution.
A direction of maximum continuity should be identified from the horizontal variogram rosette. That will be the azimuth of the main axis. Then a rosette of vertical variograms should be generated with the azimuth of dipping equal to the azimuth of the longest continuity of horizontal variograms. A variogram with longest ranges will show the angle of dipping of the main axis of directional anisotropy.
Downhole variograms are to be used to model nugget effect.
Once the azimuth and dipping of the main axis of directional anisotropy is identified, three variograms are to be generated and modelled. The first variogram will be in the direction of the main axis, the second one –
Notes:
perpendicular to the first variogram, and the third one – perpendicular to the first two variograms.
If geology and mineralisation are well studied and interpreted, sometimes the main directions of directional anisotropy are obvious and the above steps could be simplified or skipped. The directional variograms are to be displayed and modelled in Stats | Semi Variograms. It would be a good practice to generate Direct, Log and Relative semi variograms to obtain the main features.
When experimental variograms are displayed on screen, they can be modelled using the Model menu. Select the variogram type (e.g. Model | New | Spherical). Then you will be prompted for the number of structures. Select the number of structures (for example 2). Then you will be able to model the nugget effect and sills of every structure using the mouse. When you specify the model parameters with the mouse, MICROMINE will display the modelled variogram parameters. Please note that Sill parameters there are actually C values. Sill values will have to be calculated (Sill = C partial sill + Nugget).
All modelled variogram parameters should be saved to a Form.
Variography can take from several days to several weeks depending on the number of elements, domains and selected interpolation method. If MIK is selected, the exercise could be very time consuming due to the large number of variograms to be modelled.
CHECKLIST:
• Note the “Sill” in MICROMINE is actually, the partial sill.
• Use downhole variograms (or vertical) to estimate nugget effect
• Make sure all 3 variograms have the same nugget, C value and total sill
Notes:
Anisotropy
There are two types of anisotropy:
1. Zonal. The sills and the ranges are different in the three directions. If this occurs two structures must be used with a second long range to get the sills back to the same level at a very large range, such as 10,000 metres
2. Geometric. The sills are the same but the ranges are different for the three directions
Semi variogram formula
The semi variogram and variogram is basically the same thing, technically they are different as the semi variogram is divided by two.
Algorithm as follows, semi variogram –
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2Once the semi variogram has been displayed then a model must be fitted to the gamma values.
Semi variogram model formula
The rule of thumb is that a spherical model is fitted to most gold data; in some circumstances an exponential model may be used.
3D modelling requires three variograms orthogonal to each other, the nuggets should be the same and the partial sills must be the same, the ranges can differ for each direction
The nugget is best determined from the downhole data as the data is the most closely spaced, the lag can be determined from the omnidirectional variogram which is an average of the lag spacing.
Fitting the variogram model is done interactively, the Noel Cressie statistic can show the quality of the fit using a least squares regression, however the best guide is a visual fit of the line to the gamma values.
The smaller the Noel Cressie statistic then the better the variogram model fit, theoretically. However in practice the first and second gamma values greatly influence this result. Use the visual fit in conjunction with the test button.
Ordinary kriging requires the data to resemble a normal population. If there are mixed populations (this is apparent on the histogram and probability plot) then a method such as multiple indicator kriging must be used.
The variograms are saved together in a form that retains the model parameters and the attitude; azimuth and plunge of the variogram. The weighting is then performed automatically within the ordinary kriging routine.
Notes:
Search ellipsoid parameters
The parameters entered here define a search ellipse used to select samples for modelling. That is, the samples that will be used to calculate the estimated value.
Radius - Enter the primary radius of the search ellipse. This value is a length or distance that becomes the base value by which the three factors below are multiplied to determine the dimensions of the search ellipse.
Azim (deg)
Enter the Azimuth (bearing in degrees) of the long axis of the search ellipse. This has a range of values 0 - 360 measured clockwise from north = zero. It corresponds to geological strike, or the trend of the long axis of a plunging body.
Azim factor
Enter the factor for the length of the long axis of the ellipse. This will be multiplied by the Radius to determine the actual length of the Azimuth axis.
The Azimuth factor is generally the longest dimension of the search ellipse.
Commonly the Radius is set equal to the along-strike search, typically 1.25 to 1.5 times the average section spacing, and the azimuth factor set to 1. The other two factors would then be defined as decimal values between zero and one.
Plunge (deg)
The plunge is the downward inclination of the orebody along the strike. It must be positive; plunge values are always in the range 0 - 90.
For example, a tabular structure with Azimuth 30 degrees and a Dip of 60 degrees to the South-East will have Azim = 30 and Dip = -60. If the plane
Notes:
contains a tubular or elliptical structure that plunges downward at 30 degrees to the north-east then the value required is Plunge = 30.
If the same tabular structure has a tubular or elliptical structure that plunges downward at 30 degrees to the south-west then the value required is still Plunge
= 30 but the Azim must be 210 and the Dip required is 60.
Thick factor
This describes the thickness component of the search ellipsoid. Enter a factor for the length of the search axis perpendicular to the plane of the Azimuth and Dip values. This value is multiplied by the Radius value to determine the actual length of the Thickness search axis. The thickness factor usually describes the short axis of the search ellipsoid.
Dip (+/- deg)
Dip is an angle, with range -90 to 90 measured from the horizontal, perpendicular to the azimuth axis. It corresponds to geological dip. The convention used throughout MICROMINE is that clockwise rotation, looking in the Azimuth direction, has negative dip values and counterclockwise rotation has positive dips. Thus a bed striking at zero degrees and dipping 60 degrees east will have a Dip angle of -60.
Dip factor
Enter a factor for the length of the dip axis of the ellipse. This value is multiplied by the Radius value to determine the actual length of the Dip axis search. This is the down dip search dimension of the search ellipsoid.
Notes:
Practice
Iron deposit (3d): The iron deposit will be modelled using ordinary kriging and median indicator kriging. The iron deposit is a good example because it has an excellent linear population and produced strong variogram models with geometric anisotropy. assigned to the assay file. This ensures we know which grades are the relevant mineralised grades and which grades fall outside the wireframe and are redundant. This file has already been produced and is IRON ASSAY.DAT; the code field should be WFCODE with the code iron. A 2 metre composite file was also created called IRON ASSAY COMP.DAT.
Nugget
The IRON ASSAY.DAT file is now used to calculate the variograms and so, to create the variogram models. Do not use the composite file at this stage because it may inadvertently display zonal anisotropy because compositing smooths the data in the file and will change the variance to a greater degree in one direction than another. The composite file is only used for the interpolation.
Exercise 22.6 Nugget
1. Open the semi variogram form by selecting Stats | Semi variograms.
Notes:
2. The Semi Variogram Type will be Downhole in this instance.
3. Select the IRON ASSAY.DAT as the Raw Data file. We initially have to create a semi variogram file from a raw data file. Later these semi variogram files can be re-used by clicking the second option.
4. Apply a filter to the Raw Data file in order to only use data inside the wireframe. Make sure you save this filter using Forms | Save as.
5. Select Show Variance under Data Values and complete the form as shown below:
6. Under Semi Variograms, select Show semi variograms and Write semi variograms to file. The name of the File will be Vario DH of Type DATA.
7. You will of course have set Numeric Exceptions and then saved the form using the Forms button (third from left on the Toolbar) as Downhole Nugget.
8. This will display a Downhole variogram to determine the nugget size.
9. This gives a Nugget Coefficient established at about 50.
Notes:
Using Semi Variograms
When you create a semi variogram, you will notice that the menu at the top of the screen changes. The two menu items of particular interest are Display and Variogram.
The menu items with icons are available from the Semi Variogram Toolbar. The ones that are probably used the most under Display menu are:
Form – you should always save your forms so that you can easily reproduce a result.
Dump – creates a screen shot of the active window that you can paste into a report that you might be generating as you work.
Zoom + Area – allows you to zoom into the area of interest. The relevant scale will also be shown automatically.
Display Mode – takes you to the variogram parameters page where you can make changes or check entry details.
Show Together – if you have more than one set of parameters, all of them can be shown on the screen at the same time. This is an alternative to leafing throught them by using the Page Up and Page Down keys on your keyboard.
When using Show Together, you might want to use Display Mode first and change the Display Mode of some of the less likely candidates to None. This will simplify the display and let you focus on the more likely candidates.
All of the items under the Variogram menu are of use:
Previous – lets you leaf back through individual variograms. This can be done more easily by using the Page Up key on the keyboard.
Next – lets you leaf forward through individual variograms. This can be done more easily by using the Page Down key on the keyboard.
Model – lets you model a curve through the points to represent a best fit representation of an ideal curve.
Notes:
Optimum Lag
Determine the optimum lag size by using an omni directional variogram with various lag sizes. The omni directional variogram displays the average of lags.
We will use the composited data found in the IRON ASSAY COMP.DAT file.
For this iron example, lags of 50 to 60 metres produce well behaved variogram results. This distance can now be used to find the direction of maximum continuity.
Exercise 22.7 Omni Variogram
1. Open the semi variogram form by selecting Stats | Semi variograms.
2. The Semi Variogram Type will be Omnidirectional.
3. Select the IRON ASSAY COMP.DAT as the Raw Data File. Keep the filter WFcode = iron and select TFECUT65 as the Semi variogram field.
Notes:
4. Select Show Variance under Data Values, click the Omnidirectional Semi Variograms button and complete the form as shown below:
5. Save the above form as Optimal Lag – Omni Comp.
6. Under Semi Variograms, select Show semi variograms and Write semi variograms to file. The name of the File will be Var Omni Comp of Type DATA.
7. Save the main form also as Optimal Lag – Omni Comp.
8. These will display Semi Variograms. To leaf through the displays, use the Page Up and Page Down buttons on your keyboard. It is advisable to zoom in to the left-hand side using the magnifying glass with the square inside it from the toolbar
9. The screenshot below shows the orange 60_50 values. This gives a reasonable fit.
Notes:
10. We should now repeat the search with a tighter range of say 35 to 75, or even tighter.
We will use a value of 50° for our lag or interval for now. The next step is to discover the direction of maximum continuity. This will have the longest total range. Set the variogram fan for 30 degree increments for 180 degrees, it is not necessary to do 360 degrees as one half is the mirror of the other. Set the tolerance to 15 degrees so they do not overlap and apply a conical search. The geology is often a very good guide to the direction of maximum continuity.
The Mode button applies the value in the first row to all other valid rows. The display modes you can choose from are:
None: The data for that azimuth will not be displayed. Useful when you want to switch a direction off temporarily to simplify the display.
Line: Data for the azimuth will be plotted as a simple line graph. You can enter a symbol number when LINE is selected. The corresponding symbol will appear at each interval distance. Its size will vary proportionally to the number of pairs in that interval.
Graph: The data will be displayed as a graph with two lines. The area between the lines can be hatched. To generate the lines, alternate values from interval one to the maximum calculated distance interval are connected.
The intervening values are then connected back to the first interval value.
This displays the difference between values in adjacent intervals (but loses information on the number of pairs in each interval).
Symbol: The interval semi-variogram value for the azimuth will appear as a symbol. The symbol size is relative to the number of pairs in the interval.
Pairs: The Pairs display option displays a fixed size symbol with the number of pairs written beside the symbol.
Exercise 22.8 Horizontal Fan Variogram
1. We will use the Semi Variograms form again. The Semi Variogram Type will be Directional.
Notes:
2. Change the Semi Variograms File to Var Horiz Comp. The rest of the form stays unchanged.
3. You will notice that the button under Data Values has changed to Semi Variograms Directions. Click this and complete the form as shown below:
4. Save the above form as Horizontal Fan.
5. Notice that we are using the Display Mode of Lines. Lines are easier to read than Pairs but there is much more information in Pairs.
6. Save the main form also as Horizontal Fan.
7. Again leaf through the displays using the Page Up and Page Down buttons on your keyboard. It is advisable to zoom in to the left-hand side using the magnifying glass with the square inside it from the toolbar.
8. Your graphs should show that the best fit will be between 35 and 55 degrees for the Azimuth. The screenshot below comes from a second run from 35° to 55°. Blue 39° was slightly better than Pink 41°. We will use 40°.
Notes:
Exercise 22.9 Vertical Fan Variogram
1. Open the semi variogram form by selecting Stats | Semi variograms.
2. The Semi Variogram Type will again be Directional.
3. Change the Semi Variograms File to Var Vert Comp. The rest of the form stays unchanged.
4. Click the Semi Variograms Directions button and complete the form as shown below:
5. Save the above form as Vertical Fan.
6. Save the main form also as Vertical Fan.
7. Click Run to display the variograms as shown below.
Notes:
8. Again leaf through the displays using the Page Up and Page Down buttons on your keyboard. It is still advisable to zoom in to the left-hand side using the magnifying glass with the square inside it from the toolbar.
9. Your graphs should show that the best fit will be -6°.
Omni directional variogram to determine optimum lag:
The setup for defining the Omni directional variogram;
Notes:
The display of the Omni directional variograms;
Directional variogram, main:
Step 4:
Finding the principle direction;
Defining the settings to narrow down the principle direction variogram;
Notes:
Display of the variograms for the principle direction;
Direction of maximum continuity, primary variogram;
Step 5: Directions of maximum continuity; once the approximate direction of maximum continuity is known from step 4, then the lag can be experimented with and then the exact direction of maximum continuity in terms of azimuth and plunge can be investigated and modelled.
Notes:
Clearly the lag of 50 or 60 metres and a zero degree plunge produce the best behaved semi variograms. The variograms have a good regular pattern, sill out close to the variance and do not have a saw toothed appearance.
Model the variograms:
Step 6: The azimuth is 142 degrees, the lag 50 and the plunge zero. Then fit a spherical model to the gamma values on the variogram. Retain the same nugget as the downhole, vertical variogram with the same partial sills, one and two retained for all three orthogonal variograms.
Principal direction: 142 degrees azimuth;
Notes:
Principal direction: 142 degrees azimuth, fitted model;
Intermediate: Directional variogram 232 degrees azimuth;
Notes:
Intermediate: Directional variogram 232 degrees azimuth, fitted model;
Intermediate: Directional variogram 232 degrees azimuth, fitted model;