Domains Interpolation in Ore Geology

November 2008Parkkinen GeoConsulting: Domains Interpolation 1

Domains Interpolationin Ore Geology

Keywords:Mineral deposits, 3D models, intersecting structure s, geostatistics, variogram analysis, variogram map s, ordinary kriging, indicator kriging, nearest neighbourhood, inverse distance, domains, categories, interpolatio n, block models, solids, isosurfaces, stereographic pr ojections, anisotropy, ellipsoids, mineral resource


Domains Interpolationin Ore Geology

Introduction

This novel (?) method combines the advantages of Ordinary Kriging and Indicator Kriging without some serious difficulties of conventional methods. In short DI handles datasets of domains (while Indicator Kriging handles cutoff grades), each of them separately but together with other data. This method suites especially well to the study of intersecting structures and to the study of categories.

Benefits:1. Nugget can be set to zero (like in IK).2. Distinct separation of domain values from other values helps geostatistical analysis (like in IK).3. Domains may overlap but they are not within each other (unlike in IK).4. There is plenty of flexibility in domain definition. This is also a method for the finding and defining of

domains, possibly not otherwise recognized.5. Several sets of domains can be compared, e.g. densities, grade distributions, lithology classes, ore

types, etc., while the quantification method is the same for all.

On the following pages I intend to describe the method in details by applying it to an artificial case calledExperiment. In each phase results of Domains Interpolation will be compared with conventional Ordinary Kriging and finally with several other methods. In this case conventional Ordinary Kriging means the treatment and modeling of all data as a single target without the division into domains. This singletarget I call All Domains .

All calculations and figures base on work with Gemcom Surpac Versions 6.03 and 6.1.


Procedures

1. Definition of DomainsA domain may be categorically defined like lithology class, rock type, or mineralization type. It may be numeric like grade interval, any quantification interval, or ratio interval. If numeric, then cutoffs may well be o verlapping.

2. Conventional geostatistical analysis of each doma in.In sequence, values in each domain are set to be 100 while all other values are set to be 0. Variogram analysis gives anisotropy parameters to conduct interpolation: azimuths and plunges of symmetry axes with sill values and respective ranges. (In the following exercise, however, original values 0, 100, 500, and 1000 will be used to make it easier to follow the procedure. In fact, it is all the same what values will be used.)

3. Block model creation and interpolationA distinct attribute is named for each domain. Interpolation (preferably Ordinary Kriging) is then run for each domain separately.

4. IsosurfacesIsosurfaces should help solid creation. If not available, then also block solids can be used similarly.The suitability of different isosurfaces should be tested. A result, e.g. isosurfaces >= 80 % and >=4 0 % were found good enough (experience helps!).

5. Solid constructionIso-surfaces (80 and 40) will show probable inner and outer limits for solid boundaries. Also original sample values or drill cores must be used. Now we have a spatial expression for each domain.

6. Constrained geostatistical analysis and constrain ed interpolationNew geostatistical analysis will be done inside solids of each domain and with composite values. Composites will then be used to do interpolation inside solids of each domain. If there is room between solids, also interpolation outside all solids should be done. In the best case, no empty room will be left if one of domains covers what other domains leave. If Domains are non-numerical categories no new analysis nor interpolation is needed.

7. Mineral resource estimates8. Conclusions

Appendix 1. Additional aids: symmetry visualizatio nAnisotropy ellipsoids and stereographic projections

Appendix 2. Description of alternate methods


1. Definition of Domains: Case Experiment

Data description, definition of domains

100 virtual drill holes in 10*10 unit grid and into the target, 100*100*100 unit rock cube. Holes are vertical and 100 units long. Sample length is 1 unit. Of samples, 1145 (each 1000 ppm) were classified to be ore, 2450 (each 500 ppm) were classified to be low grade marginal ore and 5870 (each 100 ppm) were assayed to be mineralized country rock and the rest (each 0 ppm) were classified to be barren country rock. Coordinate system and holes are shown in Figures 1-2. In Fig.2B. there is also a rough sketch outlining major structures to be further studied by geostatistical means.Domains selected to treatment: Domain1000 (red), Domain 500 (yellow), Domain100 (g reen).

Figure 1.Coordinate system and drill holes 1-100, surface le vel.

Figure 2 A-B.A. Drill holes inside 3D coordinate framework.B. Sketch outline of Domains (Domain1000 colored blu e).

A B


Summary

AreaX (easting) 1000 - 1100Y (northing) 10000 - 10100Z 0 - 100

DrillingHole-id 1-100Holes 99 Hole lengths 100 mSection intervals 10mHole intervals 10 mHoles azimuth/plunge 000/90

SamplesSample length 1 m

Figure 3 A/BBasic statistics of drill core samples.

Figures 1 A-B contain basic statistics of drill core samples.In fact the statistics stand for 1 m composite samples.In this case these are identical to original samples.In Fig. 2B note Variance! It will have an important rolein the following procedures.


2. Conventional Geostatistics

Figure 5 A-D. Down-hole variogramsA Domain 1000B Domain 500C Domain 100D Target

Figure 4 A-D. Omni-variogramsA Domain 1000B Domain 500C Domain 100D Target

First step is usually the modeling of down-hole and omnivariograms. Down-hole varios, if arranged according toazimuthal directions of drill holes, give information ofdeposit intersections. Moreover they give information onthe nugget , the constant portion of variance.

Omni variograms give preliminary information on overallranges and also on the nugget.

In Figures 4 A-D omni variograms of each domain and theentire Target tell that the average range of influence(in interpolations) is about or less than 40 m. Nugget seems to be about zero.

In Figures 5 A-D down-hole variograms confirm thatnugget is zero. They also imply a complex structure.In Fig. 4A the angles in the variogram may mean (in fact,we know it already) that there are several, perhapsthree different intersections, first about 30 m in thickness,second about 70 m and third about 100 m.

In this exercise Domains 100, 500, and 1000 are all treatedseparately and following Procedures, page 2.They will be combined later on. At the same time. conventional geostatistics and modeling of All Domainswith original domain values will be carried along asa reference procedure.


Variogram maps and directional variograms

Variogram maps and linked variograms able rapid survey of anisotropy in all azimuthal directions and plunges.Conventional techniques include start with horizontal maps followed by a systematic survey of vertical sectionsuntil a candidate for the best anisotropy direction is found. Then follows the search of other two directions to define the three-axial symmetry expression to be used in 3D interpolation. Variomaps contain general information of variance. In Figures 6A-D can be seen how the high grade and,respectively, high variance vertical and northerly plate (Domain 1000), its presence or its absence, dominates all maps. Best looking anisotropy and longest range do not consider the horizontally longest direction of the plate but they occur from corner to corner as illustrated by best anisotropy.

Figure 6 A-DVariogram maps and examples of directional variogra ms, variances and models on section 000/00 (horizont al). A. All Domains: best horizontal anisotropy 020/00. C . Domain 500, best horizontal anisotropy 165/00.B. Domain 1000: best horizontal anisotropy 015/00. D . Domain 100: best horizontal anisotropy 020/00.

A. All Domains

B. Domain 1000

C. Domain 500

D. Domain 100


Search for main anisotropy directions

Horizontal or conventional 2D variograms do not always imply best anisotropy directions. A systematic variogrammap display on vertical planes is therefore necessary. For this demonstration, for each domain, I studied verticalvariogram maps at 10 degree intervals (perpendicular 10 000, 010, 020,.170), and on these maps I stud ied variograms at 5 degree intervals (72 variograms in each map). When the best anisotropy direction (and plunge) foreach domain was defined, I used the Surpac option to search for the semi-major direction or axis. Af ter thatSurpac calculated the direction/plunge of the third or minor anisotropy axis. The results are in Figures 7A-D.

A. All Domains

B. Domain 1000

C. Domain 500

D. Domain 100

Figure 7 A/B C/DBest overall anisotropy directions of All domains, Domain 1000, Domain 500, and Domain 100.


Figure 8 A/B C/DKriging parameters forblock modelExperiment.mdl.

A. All Domains

B. Domain 1000 D. Domain 1000

C. Domain 500


3. Block model creation and interpolation

Because drilling grid is 10*10 units, proper block size would be 3*3*3 units. To find out the way interpolation agentsbehave, block model Experiment.mdl extends outside sample-space. There are also missing drill holes and missing samples. Block model attributes and dimensions are shown in Figure 9. Figure 10 gives an example of block model section: Y= 10060, All Domains kriging, also drill holes shown.

Figure 9Block model Experiment.mdl

Figure 10.All Domains block section Y = 10028.5.Colors correspond to domains 1000 (red), 500 (yello w-brown), and 100 (green),Note that blocks even next to drill core samples ha ve beendiluted!


Figure 11 A-B.A. All Domains block model, constrained by Y = 10030 (above) and grade > 100.B. Summary of Domain Interpolation block models cons trained similarly to A.

BA


4. Isosurfaces

Iso-surface connects block centres according to the cutoff grade given. In this project Isosurface >=50 envelopes space where the probability of sample values equal to or greater than domain value (cutoff grade) is >= 50 %. I defined two isosurfaces for each domain. Isosurface 80 outlines the core (measured) and isosurface 50 outlines the domain boundary (indicated). As seen in Figur es 11 and 12A, domain isosurfaces fit together rather well in spite of the differences in spatial grade distributions.

Figure 12.Section Y = 10028.5, drill holes, and isosurfaces 5 0 and80 of domains 1000, 500, and 100.Drill holes: Isosurfaces:

Domain 1000, 80 / 50

Domain 500, 80 / 50

Domain 100, 80 / 50


Figure 13 A-B.A. Isosurface 50 % of Domains 100, 500 and 1000.B. Isosurface 80 % of Domains 100. 500 and 1000.Note how 50 % surfaces fit well together .

A. Isosurface 50 %.B. Isosurface 80 %.


Figure 14 A-D.Section Y = 10028.5.Isosurfaces 50 % and 80 % of all Domains.Respective block models.A. Block model of All Domains, coloring

according to cutoff grades 100, 500,and 1000.

B. Block model of Domain 1000, coloringto illustrate probabilities 80 % and 50 %.

C. Block model of Domain 500, coloringas above.

D. Block model of Domain 100, coloringas above.

Note that(1) Interpolation of All Domains, which meansall grade values together, produces rather mixed results inside deposit domains.(2) Surpac kriging procedure tends toproduce values higher than values of originaldata, e.g. values up to 1026 when maximumvalue should be 1000. This is a problem!!See also Figure 10.

14 A: 14 B, C, D:

A

B

C

D

Isosurface- and block models compared


5. Solid construction

Principles:- Solids separately for each Domain.- Solids base on sectional polygons.- Sections created in 3 directions.- Section intervals vary depending on the

complexity of structures.- Polygons fixed to drill core sample

borders where possible.

Practice:I created solids only for Domains 500 and 1000.XZ-, YZ- and XY- directions were used forpolygon outlining while XY (elevation) wasbest for Domain 1000 and YZ (easting) wasfound best for Domain 500. XY-based outliningwas not sufficient because polygons could notbe fixed to drill core samples. So the finaloutlining was done in XZ-sections.

Solid Domain 100 was created by subtractingDomains 500 and 1000 from a cube coveringall Domains (Surpac outersecting). Non-sampled holes I left inside Domain 100 withoutoutlining.

Figure 15 A/B C/DA and B illustrate preliminary polygon outlining fo r Domains 1000 and 500.Respective solids with same polygon strings are in C and D.

A

B

C

D


A

B

C

D

Figure 16 A/B C/DA: All Domains, C: Vertical section Y = 10040. B: Domains 1000 and 500. D: Horizontal section Z = - 60.

A Surpac feature is the difficulty to handle longstraight lines. Therefore straight lines had to becurved and therefore many surfaces are a bitwrinkled. Another type of wrinkling can be seenin some close or tight places where two ormore Domains meet.

Boundaries of neighbouring solids tend tooverlap or there may be openings betweenthem. In principle the first alternative is easierto handle by Surpacs outersect solidsprocedure. The same procedure I used to outlineDomain 100 (+ 0) solid. Unfortunately it doesnot function well, and solids easily remain false.


6. Constrained geostatistical analysis and constrain ed interpolation

Domains modeling only gives main structures, main bodies. In order to survey the bodies or solids a new geostatisticalanalysis should be done and, consequently, a new interpolation in each solid. This particular case, however, is one thatcan be compared to the analysis of category domains, and no further analysis is needed. In fact, because all valueswithin a Domain solid are equal, there is no variance to analyze.

Constrained interpolation inside Domain solids can be now made by using any interpolation method. My choice in this case is isotropic Nearest Neighbourhood because sample distribution is very even. According to variogram analysis(Figures 4-5) we can set a general range value of 30 m for each Domain.

A weak point of this demonstration is the treatment of negative values and zero values. Zero values derive from zero assay values, whereas negative values derive from the Surpac feature of forcing negative value -99 to blocks thatare beyond the reach of interpolation range. In this demonstration there were 139 zero assay values (Fig.17), and in themodeling they were taken into account. However, in reporting mineral resources both negative and zero values will beleft out to not complicate explanations.

Figure 17.Composite samples distribution in Domain solids sho ws that solidmodeling succeeded rather well. Only one Domain500 sample strayedto Domain100. I did not correct this error we will see the consequencelater on.


Figure 18 A/B C/D E/FA-D: Final Domains Interpolation Block Model. E-F: All Domains Block Model.A. All blocks seen from the south-east. Note the blu e blocks in line! E. Cutoff grade 500, from the SE.B. Model constrained by X => 1025, Y=> 10020, from t he south-west. F. Constrained by X => 1025,C. Domains 500 and 1000 from the south-east. and Y => 10020.D. Domains 500 and 1000 from the north-east.


7. Mineral Resource Estimates

This is the final stage of present demonstration: mineral resources by several different methods. In Figure 19 the results are from left to right:Domains Interpolation as illustrated in this demonstration,All Domains Ordinary Kriging as illustrated in this demonstration,Inverse Distance 2 as described in Appendix 2.Indicator Kriging as described in Appendix 2.Nearest Neighbourhood as described in Appendix 2.Nearest Neighbourhood isotropic as described in Appendix 2.inverse Distance 2 isotropic as described in Appendix 2.

Some observations:- Conventional methods, except Nearest Neighbourhood, smoothes grade values towards low value end.- Conventional methods, except isotropic ones, over-estimate over-all average grade.- Conventional methods, except Nearest Neighbourhood, under-estimate high grade volumes.- Conventional methods, except Nearest Neighbourhood, do not preserve nor imply grade distribution structures correctly.

Nearest Neighbourhood method comes closest to Domain Interpolation. This is apparently because sample geometry is even and simple.

Figure 19. Resource estimates at different grade ranges (or cu toff grades 0; 99; 499; 999) with volumes (density = 1) and average grades.


Conclusions:

Whenever domains can be defined, they should be used. Worst results were given by isotropic Inverse Distanceand Indicator Kriging.

Discussion:

There are two ways to correct my results and to further develop ideas presented in this demonstration:1. All processes and procedures can be repeated. My database is free for distribution. Just contact me.2. I intend to reduce drill holes and to reduce samples to find out how this method works with less and less

information and less and less explicit geometry.


Appendix 1. Additional Aids

Anisotropy ellipsoids

Anisotropy ellipsoids visualize the same parameters that were used for the kriging of Domains.As illustrated in Figures 1 A-C, anisotropy axes show the longest ranges obtainable, that is from corner to corner.

Figure 1 A-C.Anisotopy ellipsoid of All Domains (blue) is shown together with ellipsoids of Domain 1000 (red), Domai n 500 (yellow), and Domain 100(green). In same figures Domain bodies are illustra ted by respective block models. Scale is the same f or bodies and ellipsoids.

A B C


Figure 18 A-CAnisotropy ellipsoids like in Fig. xA-B but all in one and projected against XY-, XZ- and YZ planes. No te that XY-projection (A)is what might be a result of conventional 2D geosta tistical analysis if major axes are sub-horizontal. However, Domain 500 symmetry(yellow) would probably be lost without 3D domain g eostatistics.

A. XY-plane B. XZ-plane C. YZ-plane


Stereographic Diagrams

Stereographic diagrams are a very handy means to compare between different symmetry features. The mostimportant feature is the axial plane including major and semi-major axes. If the ranges of two principal anisotropydirections (or lengths of main axes) are close to each other and substantially longer than the shortest axis, thenwe can conclude that we are dealing with tabular bodies like all bodies (Domains) in this project. In fact, Domain500 should have exactly same orientation with Domain 100, but the diagram shows some difference. It derives fromminor differences like from Domain 100 being thicker than Domain 500.

Figure zPrincipal anisotropy axes on a stereographic projec tion with explanations.All Domains axial symmetry (blue stars) apparently represents a summary or integration of all symmetry inside our virtualdeposit and its surroundings. However, as this geos tatistical analysis has shown, this is very far fro m being true. In contrary,All Domains axial symmetry is misleading. It gives a nearly correct approximate of one component, name ly of Domain 1000, butit misses the other two ones.


Appendix 2. Description of Alternate Methods

All Domains, that is no domain boundaries set, were interpolated using isotropic Inverse Distance and isotropicNearest Neighborhood methods and 30 m range for search sphere as indicated by geostatistical analysis.For anisotropic Inverse Distance and anisotropic Nearest Neighborhood parameters defined for All Domains Ordinary Kriging were used (Fig. 1A, Fig 2A). For All Domains Indicator Kriging parameters were define byspecial geostatistical analytics and by using cutoff grades of 99, 499 and 999 (Fig. 1B).

Figure 1 A-BA. Ordinary Kriging parameters for All Domains.B. Indicator Kriging parameters for All Domains, c utoff grades set to emulate domain grades.

A

B


Figure 2 A / B.A. Anisotropic Inverse Distance and anisotropic Near est

Neighbourhood parameters for All Domains.B. Isotropic Inverse Distance and isotropic Nearest

Neighbourhood parameters for All Domains.

Figure 3.Anisotropy ellipsoids of Indicator Kriging.Cutoff grades: 99 green-blue, 499 yellow, 999 red.

A

B

Domains Interpolation in Ore Geology

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