MODULE 22 TABLE OF CONTENTS
Lesson 1 – Resource Estimation Concepts ... 1
CONCEPTS ... 1
THE TRAINING PROJECT ... 2
NUMERIC EXCEPTIONS ... 2
Lesson 2 - Classical Statistics ... 4
CLASSICAL STATISTICAL ANALYSIS ... 4
Tables... 4
Lesson 3 – Generate Downhole Coordinates ... 11
WIREFRAMING ... 13
Lesson 4: Assay Data Flagging ... 15
FLAGGING/SELECTION ... 15
Flagging using Solid wireframes ... 15
Flagging using DTMs ... 15
Lesson 5 – Balancing Cut ... 18
Lesson 6 - Compositing ... 21
Lesson 7 - Geostatistics ... 25
THEORY ... 25
Variography ... 25
Anisotropy ... 27
Semi variogram formula ... 27
Semi variogram model formula ... 27
PRACTICE ... 30
Review ... 30
Nugget ... 30
Using Semi Variograms ... 32
Optimum Lag ... 33
Directional variogram, main: ... 39
Direction of maximum continuity, primary variogram; ... 40
Model the variograms:... 41
Indicator variograms: ... 44
Relative variograms: ... 48
Cross Validation: ... 51
Lesson 8 – Empty Cell Model ... 55
BLOCK MODEL CELL SIZE ... 55
Flagging... 55
Checklist ... 55
How to decide the block size ... 56
Subcelling ... 57
Lesson 9 – Modelling Principles ... 61
Declustering ... 61
Specific gravity and block size ... 62
Change of Support ... 62
Proportional effect ... 62
Interpolate parent blocks only ... 63
Multiple runs ... 63
Lesson 10 – Grade Interpolation ... 65
INTERPOLATION ... 65
How to decide search ellipse size ... 67
INVERSE DISTANCE WEIGHTING ... 68
Recommended Values ... 69
ORDINARY KRIGING ... 70
Ordinary kriging formula: ... 71
ORDINARY KRIGING, RELATIVE VARIOGRAMS ... 72
MULTIPLE INDICATOR KRIGING ... 73
KRIGING VARIATIONS... 78
Lesson 11 – Model validation ... 81
Global validation: ... 82
Declustered global estimate ... 82
Local validation: ... 83
Model validation: ... 84
Lesson 12 - Block Model Display ... 86
Lesson 13 – Resource Classification ... 88
Kriging variance: ... 89
Lesson 14 – Resource reporting ... 91
Lesson 15 – Cut-off grades and grade tonnage curves ... 95
Lesson 16 - Example NVG data Ordinary kriging start to end ... 98
Step 1: Classical statistics exhaustive population ... 98
Step2: Generate downhole coordinates ... 99
Step 3: Assign the wireframe to the assay file ... 100
Step 4: Classical statistics orezone ... 100
Step 5: Apply a balancing cut ... 101
Step 6: Composite the data to equal intervals ... 102
Step 7: Geostatistics ... 103
Step 8: Cross validation ... 109
Step 9: Build blank model ... 110
Step 10: Ordinary Kriging ... 111
Model report ... 114
Table of Exercises
Exercise 22.1 Classical Stats ... 6
Exercise 22.2 Generate Downhole Coordinates ... 11
Exercise 22.3 Flagging using solid wireframe ... 16
Exercise 22.4 Balancing Cut ... 18
Exercise 22.5 Compositing ... 21
Exercise 22.6 Nugget ... 30
Exercise 22.7 Omni Variogram ... 33
Exercise 22.8 Horizontal Fan Variogram ... 35
Exercise 22.9 Vertical Fan Variogram... 37
Exercise 22.12 Creating a Blank Block Model ... 58
Notes:
Lesson 1 – Resource
Estimation Concepts
After this lesson you will understand: • What are we attempting to model; • What decisions do we need to make;
• What are the most important decisions affecting the modelling; • How do we check how good our model is.
Concepts
Resource estimation can be conducted for 1D, 2D and 3D models. The resource sector generally requires 3D models except for gridding and other 2D techniques which use surface data to identify anomalous areas that are indicative of prospective subsurface mineralisation.
Important considerations for 3D modelling are the search ellipse, compositing, domaining, the interpolation method, dealing with erratic high grades, anisotropy, block sizes and validation
(1) The search ellipse includes sample grades relevant to the estimation of block grades and excludes redundant (not required) grades;
(2) The compositing ensures the grades used for estimation are weight averaged back to the same length so the estimation process is not biased.
(3) Domaining divides the deposit into separate areas such as lodes that have unique geological or grade characteristics that must be interpolated and modelled independently.
(4) The interpolation method is the method selected for modelling. This may be Inverse distance weighting (which does not require variography); ordinary kriging; median indicator kriging or multiple indicator kriging. Classical statistics, in particular the shape of the histogram, the shape of the probability plot and the coefficient of variation are useful to select the most appropriate interpolation method.
(5) Erratic high grades can be allowed for by applying balancing cuts to grades or by using nonlinear methods such as multiple indicator kriging. An allowance must be made for high grades so that they do not bias the entire model and affect large areas of the model to bias the model higher.
(6) Anisotropy is the preferred continuity of grade in one direction; isotropy means the grade is equal in all directions. All deposits should exhibit anisotropy and this reflects the nature of deposition and the style of mineralisation. Gold in particular is very changeable and more prone to continuity in one direction to another
(7) The size of the blocks required can be directed by the engineers who indicate the SMU (Smallest Mining Unit) or by the drillhole spacing, and by the
Notes: style of the deposit. The parent cells can be used for estimation and subcells
can improve the definition to provide an accurate volume.
(8) Validation can be both global and local. Global validation means the raw sample data and the wireframe envelope are compared to the block model tonnes and grade to ensure the model reflects the data that was used for the estimation.
The Training Project
A good example for resource estimation is an iron project because the data is more regular and can produce strong models. For the resource estimation we will use iron data –
Files used in this training project: Collar: IRON COLLAR.DAT Survey: IRON SURVEY.DAT Assay: IRON ASSAY.DAT
Assay: IRON ASSAY COMP2.DAT Wireframe: Iron.tdb
Numeric Exceptions
Always have the numeric exceptions ticked on for all three categories: Ignore characters, Ignore blanks and <x = 0.5x. <x = 0.5x means that any below detection value such as <0.01 g/t is actually taken by MICROMINE and halved in value to become 0.005 to be used as a real number in the interpolation process. There is no zero value in a geology database, ensure that below detection values are used and not zeros.
Notes:
Lesson Summary
This lesson has introduced some of the fundamental concepts involved in Resource Estimation. In the following lessons, we will put these concepts into use:
search ellipse compositing domaining
the interpolation method dealing with erratic high grades anisotropy
block sizes validation
numeric exception
Good Practice
Use Numeric Exceptions throughout.
Help Topics
For information on: See:
Notes:
Lesson 2 - Classical Statistics
Classical Statistical Analysis
The aim of the classical statistical analysis is:
• To check for the mixing of grade populations and the necessity of separation of grade populations if there are more than one population, • To derive the top cut grades for grade interpolation process,
• To determine the natural cut-off grades for interpretation of mineralisation, • To determine the distribution of grades,
• To assess the validity of Kriging interpolation process, and • To obtain the statistical parameters for grades.
Classical statistics are used to examine initially the global population and then the mineralisation population. The histogram, the cumulative frequency plot and the probability plot all reveal information about the distribution of population grades.
Tables
To obtain statistical parameters, run Stats | Descriptive | Normal/Ln for each element and for each potential domain separately. The statistical parameters are to be recorded in the output file, tabulated and included into the report.
Histogram
The histogram bin size is selected so the shape of the distribution is apparent, it must be small enough to show the shape and large enough to contain sufficient data. The histogram shows two populations, one at low background grades and another at higher grades; there is a discreet break between the two. The moments such as mean, medium etc are displayed on the right of the histogram. Probability plot
The change in the angle of the line on the probability plot helps to indicate the grade at which mineralisation grades occur as opposed to background grades. In other words this is the cut-off grade separating country rock from the ore zone. This grade is used for the grade to design the outlines on the drillholes for the ore zone interpretation. This grade is applied for the delineation of mineralisation polygons.
Histograms, Log Histograms and (Log) Probability Plots should be generated for each element and potential domain using Stats | Distribution process. Use filters to separate domains if possible. All graphs should be plotted, studied and
Notes: included into the report. The potential mixing of grade populations, grade
cut-offs for interpretation and top-cuts for grade interpolations should be determined from the histograms and cumulative frequency plots. One of the most important questions is to identify the number of grade populations. The number of populations can be estimated using the Stats | Distribution, (select Probability Plot and Natural Log options). When the probability plot is displayed, run Model | Decompose from the top menu to obtain the statistical parameters of grade populations. This can subsequently be displayed on the drill traces and checked against the geological model and structures.
Obtain the grade value that separates background grades from mineralization grades. This is the inflection point on the probability plot. Alternatively this figure may be stated by the Government. This grade value is the grade at which all mineralization polygons will be interpreted in section.
This Stage can take from several hours to several days depending on number of elements and number of domains. Domains are separated data for different block models, such as different lodes, or weathering or very high grade, high grade and low grade areas.
Please note, it is a good practice to calculate a coefficient of variation that might indicate a potential quality of variograms (COV = St. Dev. / Mean) and the nature of interpolation method required. For a COV of 1 or less than 1.5 IDW may be appropriate, above 1.5 a kriging method should be applied.
Interpretation
When interpreting the orebodies and wireframing the ore zones and mineralisation, a decision needs to be made whether a hard boundary or soft boundary is used. A typical hard boundary is where mineralisation is truncated by a fault; a typical soft boundary may be where a stockwork of veins extends from the main mineralisation zone into the country rock. A wireframe can be used for the hard boundary, however grades outside the wireframe or indeed no wireframe may be used when modelling the soft boundary.
CHECKLIST
• Classical stats tables are generated for all elements and domains • COV values are calculated
• Histograms, log histograms and probability plots are generated for all elements and domains
• Tables and graphs are studied and modelling methodology is determined • Use numeric exceptions if there are any character values in numeric fields • The distributions are described
Notes:
Exercise 22.1 Classical Stats
1. Select Stats | Descriptive | Normal/Ln. 2. Fill in the dialog box:
Prompt Setting
Input File IRON ASSAY
Type DATA
Fields | Field Name (1) Fields | Field Name (2)
T FE SIO2
Output File DESCRIPT IRON
As discussed in Lesson 1, all Numeric Exceptions should be selected.
3. Click Run. You should see a window displaying the Normal and Logarithmic descriptive stats showing T Fe. Click Next to review the SIO2 stats. 4. Click Close to exit. You can right-click on the Output File: DESCRIP IRON.REP to view the file. Copy and paste to you Resource document and then Close.
5. Select Stats | Distribution with:
Prompt Setting
File IRON ASSAY
Type DATA
Graph field T FE
Graph type HISTOGRAM
Values used NORMAL
6. Numeric Exceptions are all set. 7. Graph Limits are set to:
Prompt Setting
Graph min 0
Bin Size 1
Graph max 71
Notes: 8. Graph Options are:
9. Click OK and you should see the histogram graph below:
The histogram of the exhaustive (total) iron population shows a very strong normal distribution suitable for Inverse Distance Weighting or Ordinary Kriging interpolation, which will be explored in later lessons.
Notes: 11. From the menu, select Mode to change the Graph type to CUM
FREQUENCY to display the following cumulative frequency graph:
The cumulative frequency curve shows the frequency of grades at varying grade cut-offs.
12. Change the Graph type again to PROBABLITY PLOT for this graph:
With the probability plot set Normal, the major inflection point around a grade of 10% indicates the change from background to the mineralised grades.
Notes: 13. Change the Values used to NATURAL LOG.
When the probability plot is set to natural log and the population plots as a straight line then the distribution is normal, this is the case for the iron population.
Notes:
Lesson Summary
This lesson has introduced the concepts of: • Descriptive stats
• Histograms
• Cumulative frequency graphs • Probability plots
• Natural Log scales
To create these outputs and graphs, we used Stats | Descriptive and Stats | Distribution.
Good Practice
Keep saving your outputs into a document or folder so that you can build a resource estimation report as you work through the process. This will be easier while you have the data fresh in your mind.
Label everything carefully so that it will be easy to understand what it is in a months time or to a third party reader.
Help Topics
For information on: See:
Descriptive stats Descriptive stats
Notes:
Lesson 3 – Generate Downhole
Coordinates
Prior to interpolation, the assay data file should be desurveyed (i.e. 3D co-ordinates should be calculated for the centroid of each sample interval) using Dhole | Generate | Downhole Coordinates.
All geological domains and mineralised envelopes (or seams) can be interpreted interactively on screen using Strings or Outlines in VizEx. These Strings or Outlines are used to generate solid wireframes; Strings are generally used to create surfaces or DTM’s such as weathering surfaces.
When all strings or outlines have been generated, they should be loaded into the 3D Viewer and checked for potential errors (missing interpreted sections or drillholes etc).
Interpretation can take from 1 to 2 days, to several weeks depending on the complexity of deposit, necessity to interpret geology, domains and multiple elements.
Internal dilution can also be automatically integrated into the interpretation process. MICROMINE can use Dhole | Compositing | Grade to set a trigger grade at which compositing occurs in conjunction with various settings for internal dilution, minimum grade, minimum and maximum waste intervals. Once the settings are enabled then a new grade composite file incorporating the internal dilution can be produced. The new file is then set up in section and the interpretation is performed on this file instead of the assay file.
Exercise 22.2 Generate Downhole Coordinates
1. From the menu, select Dhole | Generate | Downhole Coordinates and fill in as below:
Notes: 2. The Collar/survey File More button needs to be filled in as follows:
3. You will need to write in new field names for the new Interval file fields (East, North and RL):
4. Close the Collar/Survey Setup form and Run.
5. Examine the modified Interval File by right-clicking on it.
If you are using the Drillhole Database option on the 3D Coordinates form, you can just tick the Create new Coordinate fields tickbox and the new fields in the Interval file will be generated in accordance with you Form Options under Options | Forms.
Notes:
Wireframing
The geological model is an attempt to model by wireframing the mineralisation in the ground, there are no economic decisions at this stage, it is a mineralisation model.
Wireframing was covered in a previous section of the training.
At the completion of the wireframing, assign the wireframe to the assay file to allow the wireframed mineralised data to be segregated from the unmineralised data outside of the wireframe. The finished wireframe is Iron.tdb. A tdb file is a database file and is short for triangle database. Use Iron.tdb for all work.
Notes:
Lesson Summary
This lesson has introduced the concepts of • Wireframing
Good Practice
Checklist
• All sections are interpreted for mineralised zones and/or geology; • All Strings/Outlines are snapped to drillhole intervals;
• Interpretation is visually checked in 3D.
Help Topics
Notes:
Lesson 4: Assay Data Flagging
FLAGGING/SELECTION
When all the wireframes are generated, they should be used to code the assay database in order to select the part of the database to be used for geostatistics and grade interpolation. If the wireframing stage was not required, the interpreted outlines can also be used to flag the Assay database. Prior to flagging, additional fields should be generated in the files where the flags will be recorded. The Assay file should be de-surveyed. By this method the assay file will be coded for those intervals inside the wireframe and those outside of the wireframe, and then a filter can be applied to consider grades only in the wireframe.
Flagging using Solid wireframes
If solid wireframes are used to flag assays (or points), this should be done using Modelling | Assign |Wireframes. There is no necessity to use the sub blocking option; because we are flagging assay data and sub blocking factors are not required. If there are several overlapping solid wireframes, the flagging process should be run several times using the solids in the required order or recording flags into different fields.
Flagging using DTMs
DTMs can be used to flag any point data (e.g. samples) for their location below, above and outside of the DTM using process Strings | DTM | Assign. It is always a good practice to avoid absent values in numeric fields if possible. Therefore, it is recommended to use all three options in the process and to generate flag values for the point data below, above and outside of the DTM (e.g. generate values 1, 2 and 3). Make sure the DTM covers all samples. A grid file can be generated and converted to a DTM to get sufficient coverage.
Notes:
Exercise 22.3 Flagging using solid wireframe
1. Select Modelling | Assign |Wireframes and fill in as below:
2. We will not use a Block Model here so select Point Data.
3. The Input file will be IRON ASSAY.DAT. Once it is selected, you will need to right-click on the filename and select Modify. You will need to insert a new Field after SiO2. The Field Name is WFcode, Type C (Character) and Width 5. You do not need to give decimals as this will default to 0.
4. The Wiretrame will be selected from the IRON.TDB for the Type and the orebody Name is also iron.
5. Under Attributes to Assign you can tick Clear target field and Overwrite target field. Click the More button.and fill in (by double-clicking) as below:
6. Create a Report file name, eg WFassign. 7. Click Run.
8. Right-click the Report file to show 3933 records were updated.
Notes:
Lesson Summary
This lesson has introduced the concepts of
Good Practice
• The flagged samples are displayed in 3D and checked visually • It is a good practice not to have absent values (blanks) for flags • DTMs cover all samples involved in the resource estimation
Help Topics
For information on: See:
Notes:
Lesson 5 – Balancing Cut
The balancing cut is used for block models produced using a linear estimation method such as Inverse distance weighting or ordinary kriging. A balancing cut is the use of a more conservative grade instead of a few higher grades in the interpolation process. A balancing cut is determined by using a Cumulative Frequency plot for the mineralized grades only, mineralised grades are those inside the wireframe. At the grade where 97.5 percent of the grades occur, read from the cumulative frequency curve, is the grade to be used for the balancing cut.
Exercise 22.4 Balancing Cut
1. Select Stats | Distribution with:Prompt Setting
File ASSAY IRON
Type DATA
Filter Selected
WFCODE = 1 numeric
Graph field T FE
Graph type HISTOGRAM
Values used NORMAL
2. Save your Filter as Mineralisation.
3. In the plotted example the grade at which 97.5 percent of the population occurs is 65.
Notes: 4. Select Fields | Calculate with the following settings:
Prompt Setting
File IRON ASSAY
Type DATA
Filter Selected
Mineralisation 5. Right-click on IRON ASSAY and select Modify. 6. Insert a new row before T Fe:
Prompt Setting
Field TFe Cut 65
Type N
Width 5
Decimals 2
7. Close and Save.
8. Fill the table on the right hand side:
Prompt Setting
Input T Fe
Function Cut highs to
Input 65
Result TFe Cut 65
9. Click Run.
10. Right-click IRON ASSAY and you can check that rows 11 and 12 have been reduced down to 65.
The cut grades will be used for the interpolation process and for reports on the resource and the reserve.
The 97.5 percent figure is a western standard discovered from the modelling of many projects, generally the grade at 97.5 percent is sufficient to reduce the influence of the high grades in the estimation avoiding sections of the model biasing the overall result too high. A balancing cut is required because if another hole was drilled down beside the high grade samples it is quite likely that a much lower sample grade will be returned, this is the nugget effect, and as such the model must be more conservative for very high grades.
Notes:
Lesson Summary
This lesson covered the concepts of :
• Cumulative Frequencies for just the minerealisation. • Applying a cut to the data
Good Practice
Help Topics
For information on: See:
Notes:
Lesson 6 - Compositing
When deciding upon interval length, if the composite length is not obvious, a histogram should be produced for the Assay file, it should be calculated in File | Fields | Calculate by subtracting FROM from TO. Then run the process Stats | Distribution for the INTERVAL field, study the obtained histogram and make a decision on the composite length. Produce the balancing cut prior to compositing.
Samples can be composite using Dhole | Compositing | Downhole process. Set up sample composite length equal to average sample interval length. Avoid mixing of samples from different populations or geological domains when composites are generated. It can be achieved by applying filters and then appending the result files together.
Generally wireframes will be assigned to the assay file before compositing, if so then use the wireframe assign field as the Constant field in the compositing routine.
CHECKLIST:
• Run stats and make sure there are no strange composite interval lengths
• Use weighted average method for compositing • Run compositing separately for each domain
• Set up minimum composite length (usually equal to the half of the composite length)
• Use numeric exceptions if there are any character values in numeric fields
Exercise 22.5 Compositing
1. We already have interval lengths in the field INT. Determine the most frequent sample interval to composite to for the iron data. Plot the interval size on the histogram as follows –
Notes: 2. The image below should be produced.
Clearly a two metre interval should be used for the composite length, by using two metres most assays will remain unaltered for the estimation but ultimately all will be of equal length.
3. Select Dhole | Compositing | Downhole and fill in as below (Output File is IRON ASSAY COMP1):
There will be errors reported but they are only where there are breaks in mineralisation within a given drillhole. The first item in the report file rep will be in hole CK2 at a depth of 326.3m. This is the start of the second mineralisation in that drillhole.
Where the last interval of the mineralisation is less than 2 metres, the same TO value is used and the actual interval is written in.
Notes: 4. Intervals (INT) in uncomposited IRON ASSAY.DAT file vary. The image
below has less relevant fields hidden.
5. Recalculate your intervals by right-clicking on IRON ASSAY COMP1 and selecting Edit, then select Calculations from the toolbar.
Prompt Setting
Input TO
Function Minus
Input From
Result INT
Clear result field Selected Overwrite result field Selected
6. The composited IRON ASSAY COMP1.DAT file has 2 metres intervals and values for the mineralisation only (again some fields are hidden). This file is then used for all further interpolation.
Notes:
Lesson Summary
This lesson has introduced the concepts of • Compositing the mineralisation
Good Practice
Help Topics
Notes:
Lesson 7 - Geostatistics
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 | Semi Variograms.
The second step would be to identify the main axis of directional anisotropy if any. If omni variograms are reasonable, a rosette of horizontal directional variograms should be generated. Ideally, a variogram map should also be generated which will clearly show the minimum and maximum ranges and 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
• Handle zonal anisotropy (if any) by adding another structure to variograms Variogram properties Nugget co gamma Sill co + c Range m
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 –
( )
( ) (
( ))
2 12
1
∑
=−
=
N h i i iy
x
h
N
h
γ
Once 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.
Files:
Collar: IRON COLLAR.DAT Survey: IRON SURVEY.DAT Assay: IRON ASSAY.DAT Assay: IRON ASSAY COMP.DAT Wireframe: IRON.TDB
Review
After the 3d coordinates were created for the assay file, the wireframe was 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
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.
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.
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:
Notes:
The display of the Omni directional variograms;
Directional variogram, main:
Step 4:Finding the principle direction;
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.
Notes:
Principal direction: 142 degrees azimuth, fitted model;
Notes:
Intermediate: Directional variogram 232 degrees azimuth, fitted model;
Notes:
3rd direction: 180 degrees azimuth, 90 degrees dip, fitted model;
These are the required variogram models for Ordinary kriging or to establish the search ellipse dimensions for Inverse distance weighting.
Indicator variograms:
Only use indicator variograms if you intend to interpolate using a non linear model method such as median or multiple indicator kriging.
For median indicator kriging determine the median of the grades inside the wireframe by plotting the cumulative frequency curve of the data. The also find the grade ranges at 10 percent, 20 percent etc up to 90 percent. These grade cut-offs will be used for the bins in the median indicator kriging routine and the 50th percentile or median will also be used at the cut-off grade in the indicator
Notes:
Grade thresholds for the bins:
% of data Fe % 10 26.3 20 29 30 33.8 40 37.6 50 42 60 46.8 70 51.8 80 56.9 90 61.1
Notes:
The median occurs at a cut-off grade of 42% Fe; use this cut-off grade for the indicator variograms. Use the same procedure as for semi variograms to find the direction of maximum continuity, the intermediate variogram and the third direction. Note the nugget and partial sill must still be the same for all three indicator variograms.
Notes:
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.
Notes:
Relative variograms:
If the model area exhibits proportional effect, where the mean and variance change in proportion to each other across the model area then a relative variogram must be used to ensure the sills of the lags are all at the same level so the variograms appear sensible and will allow the fitting of a variogram model. A test for proportional effect can be conducted by using modelling | 3d block estimate | statistical.
Display the result in Stats | scattergrams | simple linear, plotting the mean on one axis versus the standard deviation on the other axis.
If the result plots as close to a straight line then a proportional effect is present, if the cloud is wide as is the case with the iron example then no proportional effect exists and relative variograms are not required.
Notes:
To produce an ordinary kriged model using relative variograms the variograms will be modelled using the relative gamma values. When the normal direct variograms are modelled the relative gamma value is also stored inside the file. Use the relative gamma values for the 142, 232 and vertical normal direct variogram output files by selecting display relative variogram from file. Then fit a model and save the form for all three relative variograms.
Notes:
Cross Validation:
Cross validation is conducted by removing a raw data value and using the surrounding raw data values to estimate the removed value. The value is then compared to the estimate and is repeated throughout the dataset. The total average estimates are compared to the actual estimates; if the variogram model is robust the figures should be very close.
Notes:
The average error statistic should be close to zero and the standard deviation of the error statistic close to one.
The results of the iron estimation by cross validation were 8.0575 for the standard error and -0.005125 for the error statistic. The standard error is a little high and could be improved but the error statistic is close to zero and is a good result.
Notes: Actual versus estimated values can be plotted on a scattergram to see how well
the kriging process reproduces the sample data. Actual value versus the error statistic demonstrates the conditional bias
The Means are very close so the global cross validation is good, the precision is 19%, and the result was influenced by some low grades that did not produce a low estimate because of the amount of data found by the search ellipse. The cross validation is reasonable for the direct variograms to be used for ordinary kriging.
Notes:
Lesson Summary
This lesson has introduced the concepts of
Good Practice
KeepHelp Topics
For information on: See:
Notes:
Lesson 8 – Empty Cell Model
Grades can also be interpolated into an empty cell block model generated (or imported) earlier.
Block Model Cell Size
Model cell size should be selected depending on the following parameters: Drilling density
Variability of grade Smallest mining unit SMU Final model size
The cell size should be sized to be small enough to produce a grade map for grade distribution and big enough that it reflects available data.
When block model cell size is selected and the extent of model is calculated, an empty cell block model should be generated using Modelling| 3D Block | blank block model. The block sizes should be saved in a form and restrict to wireframe option should be used to save only the blocks in the mineralisation.
Flagging
If wireframe solids and DTMs are modelled, then they should be used to flag the block model. Basically, flagging should be carried out the same way as described in the section 6. The only difference would be if sub blocking is required.
When the block model is flagged for all possible domains / zones / ore bodies etc, all other cells (unflagged) should be deleted from the model to reduce the size of the file and number of records (File | Filter | Subset). That will also help to control the interpolation process.
The process should not take more than several hours.
Checklist
The model should not have too many cells. An average block model has several hundreds of thousands cells.
Generate sub blocks to represent volume more accurately.
Generate as many new fields for flags, as there are wireframe/string/outline types that are used for flagging. It is easy to combine them later, if necessary.
Notes:
How to decide the block size
The block size of 10 metres east, 20 metres north and 5 metres in rl is displayed at the centre of the search ellipse. This block size for iron estimation is appropriate given the sample spacing of 100 metres. The blocks must reflect grade distribution, showing a local map of grade occurrence, so the block cannot be too big as the grade change will not be shown and cannot be too small because the file will be unnecessarily large and the grade estimate will become less reliable.
The block size for an iron deposit will be bigger than the block size for more densely spaced shear hosted gold deposits or VMS hosted base metals because the samples are more closely spaced and the geology is far more variable.
Notes: If the sample spacing for a gold deposit was 25m between sections, 10m
between the drillholes along the section and 1m sample intervals then an appropriate block size would be 5m by 2m by 2.5m in rl.
Subcelling
Subcelling is the creation of smaller blocks on the edge of the wireframe when the parent cell is not fully inside the wireframe. The numbers entered into the sub block boxes are how much the parent cell is divided by to define the subcell in metres. If the parent cell is 10 metres in east and the sub blocks east is entered as 5 then the subcells will be 2 metres in the easterly direction. A sub block factor is different, the cells are not subblocked, rather a number between zero and one is defined for the percentage of the block inside the wireframe.
Notes:
Exercise 22.12 Creating a Blank Block Model
1. We know we are going to be asked for the extents of the Iron wireframe, so firstly open that in Vizex in Plan view. Jot down values to encompass the wireframe in Eastings and Northings. You should have jotted down something like: 19000mE-19900mE and 35000mN-36500mN.
2. Switch to Looking North view and note the RLs as well. You should have something like -650mRL-0mRL.
3. Select Modelling | 3D Block Estimate | Blank Block Model and fill in as below. The Output entries are all typed in as we are creating a new file.
4. Under Restrictions | Wireframes click on the More button. Fill in the details as shown below. Again, the Block Factor Field is going to be created and the Sub-block values represent the number of times you want the block sub-divided not the size to which you want it sub-divided.
Notes: 6. Click on the Block Definitions button. This is where we use the values that
we jotted down earlier.
7. Enter the values but notice that you are asked for the Block Centre. To accommodate this and keep our blocks on round number co-ordinates, add half the relevant block size (Spacing) to each Origin Block Centre and subtract half the relevant block size from each End Block Centre.
8. Select Forms | Save As and save the form for later use as Iron OBM. 9. Close the Block Defintions form.
10. Select Forms | Save As and save the form as Iron OBM.
11. Click the Run button. This may take a couple of minutes and progress is shown in the bottom left of the screen.
12. Right-click on Iron OBM in the Output File box and firstly select Min/Max. You will notice various details including the creation of about 110,000 records and that the data does not start until 19025mE.
13. Right-click on Iron OBM a second time and select view. Notice that the BF field stores a value between 0 and 1 representing the number of virtual sub-blocks inside the wireframe as a fraction.
Notes:
Lesson Summary
This lesson has introduced the concepts of
Good Practice
KeepHelp Topics
For information on: See:
Notes:
Lesson 9 – Modelling Principles
Declustering
If samples are clustered then the samples must be declustered to allow a fair estimation of the unknown value in the search ellipse. Declustering is required to minimise interpolation bias from high-density assay areas, which often occur in high-grade zones. If data is not declustered the clustered data has an undue overwhelming influence in the grade interpolation on the surrounding area. If a large number of raw data values are picked up by the search ellipse from one area then these points will preferentially ensure that this area weights the interpolation of the point of estimation more so than the scattered data points. Sectors or cells can be employed to decluster the data, a maximum and minimum number of points can be stipulated for each sector.
MICROMINE currently employs a sector method in some model modules to subdivide the search ellipse and allow the thinning of the number of points to be interpolated by specifying the maximum number of points allowed within the sector.
Notes:
Specific gravity and block size
Block size is determined by drill spacing and the smallest mining unit required for the resource, the SMU. The SMU is often suggested by the mining engineer, even at the stage of resource estimation. Specific gravity can be defaulted or interpolated for the blocks.
Change of Support
Discretisation is where ordinary kriging in this case estimates point grades inside the block which are then averaged to produce the block grade. Block kriging was designed to combat the change of support, where the grade of a truck load of ore is more even and reliable compared to the grade of a far smaller often more variable sample. Block kriging is now not considered the best method of dealing with the change of support but there are few practical alternatives.
Proportional effect
When the local variability of data changes across the model area this is known as heteroscedasticity, the proportional effect is a form of this. For the positively skewed distributions the local variance increases with the local mean. The proportional effect is detected from a scatterplot of the local mean versus the variance-calculated from moving window statistics. The proportional effect can be calculated in Micromine by using the Modelling | 3d | statistical, defining a block size and writing a file containing the local mean and variance.
Proportional effect will render the sample semi variogram uninterpretable. Clustering combined with the proportional effect results in the high clustered values contributing to the lower lags. The corresponding lag mean is large and because of the proportional effect the lag variance is also large. As distance (h) increases the data that contributes to the lag becomes more representative, the
Notes: lag mean and variance decrease. The trend results in the lag variance results in
overestimation of the semi variogram value at short range and also the relative nugget effect. An inaccurate variogram model generates inaccurate weighting. A relative variogram is required as opposed to a traditional variogram. The relative variogram is standardised by the gamma value divided by the lag variance.
Interpolate parent blocks only
If parent cells only are interpolated then it means that if the parent block was divided into subcells because it was on the wireframe border, then the subcells will be assigned the grade estimate for the parent cell and they will not be independently estimated.
Multiple runs
Often when the blank model is created and interpolated into the first search ellipse size is not sufficiently large enough to populate all blocks. When the name of the block model file is the same for ‘Define blocks from file’ and the output block model file then the grades will be written into the blank model. Note that the input field name, width, type and decimals must be the same in the input file as the define blocks from file data file. If they are different then a result will not be written. After the first run with the first search ellipse then increase the search ellipse size for run 2 and possibly run 3 until all blocks been populated with an estimated value.
Notes:
Lesson Summary
This lesson has introduced the concepts of
Good Practice
KeepHelp Topics
For information on: See:
Notes:
Lesson 10 – Grade
Interpolation
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.
5. If not all cells are all populated, then repeat the steps from 1 to 4 with altered (increased) search parameters until all cells are informed with grades. Generate a Run Number field in the output model file. You will have several saved search ellipses that increase in size, each one represents a grade interpolation run.
6. Add all the generated models together in such a way, that earlier models 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.
