relative coordinate kriging

8/13/2019 Relative Coordinate Kriging

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M E T E C H U S E R S C O N F E R E N C E 2 2 - 2 6 O C T O B E R 2 0 0 1

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RELATIVE COORDINATE KRIGING

Stephen Mundell, Geologist, Mount Isa Mines Limited

INTRODUCTION

Understanding the application of grade estimation techniques is hard enough without the added difficulty ofdealing with complex geology. Factors that cause variations in the orientation and placement of mineralisationsuch as folding and faulting need to be taken into account when resource modelling. Both folding and faulting onmany scales are common occurrences in the George Fisher Ag-Pb-Zn Deposit, affecting the spatial distributionof mineralisation and consequently, grade estimation techniques.

The Medsystem kriging interpolation technique (M624V1) allows the use of a fixed orientation search ellipsoid foreach kriging run. The estimation derived using the technique (without surface tracking software) on structurallycomplex orebodies may be valid for global estimates, however local (eg stope scale) errors may occur owing to

variations in the orientation of mineralisation.

In-house software incorporating Medsystem procedures and programs has been developed by MIM to providean estimation technique that is better adapted to the complex geology of the deposit. It is the purpose of thispaper to outline and evaluate the methods used to account for structural deformation when estimating gradedistribution at George Fisher.

GEOLOGY

Mineralisation

The George Fisher Deposit, locatedin NW Queensland (Figure 1), is astratiform Ag-Pb-Zn deposit hostedby the Urquhart Shale Formation ofthe Mount Isa Group. The UrquhartShale Formation is a dolomiticsiltstone/shale unit that containsvariable amounts of fine-grainedpyrite and other sulfides (eg galena,sphalerite, pyrrhotite, andchalcopyrite). Bedding strikesapproximately 10-15 west ofmagnetic north and dips variablybetween 30-85 to the west

(averaging approximately 55). Thedeposit is bound to the south by apost mineralisation fault and isstructurally open to the north.

Mineralisation is bedding-parallel(apart from minor cross-cuttingveins) in pyritic shales, occurringmainly as centimetre to metre widebands of sulfide separated by barren siltstones and shales (Figure 2). Ore-waste boundaries are always verysharp and can be defined by a single bedding plane. There is no mineralisation halo that crosses the ore-wasteboundary. Typical sulfide mineralogy within the orebodies is galena + sphalerite + pyrite + silver sulfides pyrrhotite. In a broad sense, the mineralisation is continuous along bedding with individual sulfide bands beingable to be followed for tens to hundreds of metres, despite open folding/flexuring of the strata. Native silver is

present throughout the deposit, however local concentrations are recognised. The phase is not continuous,rather occurs as small, thin wafers on fault/fracture planes.

George Fisher

Figure 1:Location map of the George Fisher Mine


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Stratigraphic Domains

The stratigraphic positions of domains were decided by analysing the entire deposit and identifying sections ofthe stratigraphic column where mineralisation was not present. These unmineralised domains, which consistedmainly of siltstone, divided the mineralised domainsthat contain the orebodies (see Figure 2).

The domain boundaries are truly stratigraphic over the entire deposit. The advantage of stratigraphic domainingis to constrain mineralisation to particular strata and stop unmineralised packages interfering with interpolations

of mineralisation.

Figure 3:Fault plan of central 12/L George Fisher showing faults and fault blocks. Northern fault N76 is not shown. North is attop.

S73EFault

Block

S73WFault

Block

M72Fault

Block

L70FaultBlock

K68FaultBlock

G69Fault

Block

I74Fault

Block

K68 Fault

G69 Fault

L70 Fault

M72 Fault

S73 Fault

I74 Fault

R75 Fault


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Figure 4:12L Plan showing examples of mineralised domains offset by faults. Unmineralised domains separate the

mineralised domains.0BM0 B Mineralised Domain (B Orebody) Yellow

0CM0 C Mineralised Domain (C Orebody) Pink

0CDM CD Mineralised Domain (CD Lens) Grey

0DM0 D Mineralised Domain (D Orebody) Turquoise

0GU0 G Upper Domain (G Orebody) Blue


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RELATIVE COORDINATES EASTING-OFFSET METHOD

Two important geological properties of the deposit are:

1. The position and stratification of mineralised strata is dependent on the bounding stratigraphic surfaces (iewhere folding and faulting are present, the mineralised strata follow bedding), and

2. Mineralisation is continuous in the plane of bedding.

These properties are the basis to the relative coordinate system used at George Fisher to remove somestructural deformation effects from the data.

The easting-offset method was designed to transform the data into an optimal form for the standard Medsystemkriging program (M624V1). Ordinary kriging uses a planar search strategy, however, as described above, themineralisation at George Fisher is commonly non-planar. In this untransformed form, the mineralisation iscontinuous along bedding, however it is discontinuous in the frame of reference of a planar search strategy. Theaim of the software is to make the data continuous in the frame of reference of the search by transforming it into aplanar form.

The theory behind the method is that points, whether samples or block centroids with the same stratigraphicposition in a domain, have an equal perpendicular offset from the hanging wall or footwall of the domain (Figure5a). If all points in the same stratigraphic position are plotted such that the measured offsets are transferred to anoffset from a planar surface, the points will be aligned in a plane parallel to the planar surface, producing anunfoldeddata set in a new coordinate system (Figure 5b). Perpendicular offsets are used because they track thegeology closer where the bedding orientation varies. Figure 6 shows an example.

The software used to measure the offset does so in the east orientation of the block model (ie 90 0 0). To measurea perpendicular easting offset in the 90 0 0 orientation from a surface, the surface must be first rotated such that itis vertical and striking north-south (ie 0 0 90). Given that the domains at George Fisher strike approximately north-south, a dipping surface can be rotated about the north axis (0 0 0) to satisfy the requirements. Figures 7 and 8show the rotation

Figure 5:a) All points shown are in the same stratigraphic position within a domain. The perpendicular offsets of each fromthe hanging-wall surface are equal. b) After the offsets are transferred to the plane of dominant dip, all points are in the oneplane. This is the optimum form for the Medsystem kriging technique. Since the process to transform the data is known, anyinterpolation in the relative coordinates can be transformed back into real space.

Using the method, the following are true in transformed coordinates:

Blocks with the same offsets lie in the same plane. Samples with the same offset are in the same plane Blocks and samples with the same offset lie in the same plane.

Hanging-wallsurface

Perpendicularoffsets fromhanging-wallsurface, all ofequal length

Stratigraphicplane defined by

offsets

Transfer offsets to plane ofdominant dip of originalsurface

All samples in newcoordinate system are inthe one plane (ie unfoldeddata set).

a) b)


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Thus, if kriging is performed in the transformed coordinates with a planar search oriented 0 0 90, the samples willbe aligned such that large-scale structural effects are removed.

Relative Coordinate Kriging Procedure

Step 1: Surface creation

Surfaces defining the domain boundaries and fault surfaces are created by triangulation using points fromdiamond drillholes, mapping and interpretation. The surfaces are used for:

1. Creating domain and fault block solids for coding the block model and drillholes.

2. Calculating offsets (domain surfaces only).

Two surfaces (hanging-wall and footwall) are used to define a domain. Currently, internal surfaces cannot be incorporated in the procedure.

Step 2: Coding

The block model and composites are coded for fault blockand domain. The same solids are used for

both the block model and the composites to ensure parity.

Step 3: Define domain to estimate

Define the fault blockand the stratigraphic domainto be kriged. Also define the rotationrequired to rotatethe stratigraphic domain to vertical. Only one domain can be kriged at a time owing to the differentvariography of each domain.

Step 4: Surface rotation and gridding

Stratigraphic surfaces are rotated to vertical and gridded to a 5m by 5m grid in long-section. The griddingand rotation steps are performed for a number of reasons:

The gridding gives numerous regularly distributed points that can be measured quickly for accuratecalculation of offsets. The points created by the geologist during triangulation are spaced too widely

for the procedures to measure perpendicular offsets.

a) b)

Figure 6:a) Offsets at acute angle to surface: Shows surface (solid line) and predicted stratigraphic position with sameoffset as the red point. All offsets are of the same length in the same orientation. Note how the solid and dashed lineconverge at the area of variable orientation. b)Offsets perpendicular to dominant surface dip:Note how the stratigraphicposition of points with equal offset is equidistant from the surface surface. Although some offsets are not exactlyperpendicular, sensitivity tests show that it is not significant provided the discrepancy is less than 15.


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The rotation ensures that measurements of easting-offset are in the same frame of reference as the blockmodel and perpendicular to the surface (ie the easting-offset is truly in the 90 0 0 MEDS orientation).

Gridding smooths out bumps and overturns in wireframes (Figure 7).

Step 5: Calculation of block relative coordinates

The block centroids are rotated around the same point and by the same amount as the surfaces.Distances are measured from the rotated block centroids to the gridded hanging-wall and footwall, togive the hanging-wall distance (HWDIS) and footwall distance (FWDIS) respectively (Figure 8). TheHWDISvalue is negative as east is positive.

There is now a question of which surface should be used to align the blocks (ie to use as zero offset).Three are available the hanging-wall, the footwall and the mid-wall (Figure 8). The mid-wall is amathematical construction defined by the plane midway between the hanging-wall and footwall. If ahanging-wall or footwall is used for alignment, blocks on the hanging-wall or footwall would line uprespectively, however if the domain fluctuates in width, blocks toward the other surface would not match.The mid-wall, on the other hand can partially account for fluctuations in the width of a domain, and henceis used for alignment. The topic of alignment and variations in width will be discussed further.

The mid-wall offset distance is calculated by the equation (HWDIS + FWDIS)/2and is stored (MWDIS)(Figure 8). The transformed coordinates are then calculated. TRANX(Northing) remains the same as allrotations are about the northing axis. TRANY(Easting) is equal to the mid-walldistance (ie the easting-offset). TRANZ(RL) remains the RL of the block after the rotation.

Surface withoverturn.

Section lookingnorth.

Surface rotated tovertical and

gridded to 5m X5m

Resultant surfacecreated from gridpoints. Note removalof overturn.

Figure 7: Diagram showing how gridding can remove bumps and overturns in surfaces. Surfaces are gridded in long-section view. Sectional view looking north.


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Figure 8:Diagram of pre- and post-rotation surfaces showing the various distances measured and calculated by the software.

Step 6: Composite relative coordinates calculated

The composites are rotated around the same point and by the same amount as the surfaces and blocks.The same distances are measured, calculated and stored as for block centroids in Step 5. Distances arefrom the centroid of the composite. If we view the composite file using the transformed coordinates, wesee that the samples are now aligned in a planar form (ie the orebody has been effectively unfolded)(Figure 10shows an example of aligned transformed drillholes).

Step 7: Kriging

Grades are kriged into blocks using the relative coordinates calculated in Steps 5 and 6. The actualprocedure of kriging grades uses the Medsystem program M624V1. The only difference betweenstandard ordinary kriging and relative coordinate kriging is that the latter reads the new transformedcoordinates of the composites and blocks. As with the standard procedure, it allows for definition ofvariables such as search parameters, variogram parameter files, multiple passes of different searchparameters and search ellipsoids and sample weighting. The orientation of the search ellipsoid is always0 0 90 (MEDS Angles) because the transformation rotates the samples to vertical, striking north-south.Batch processes can be set up to run the model for a number of domains so there is no user inputrequired.

Requirements of the Easting-offset technique

Domain of a consistent width. As with any technique, if the orebody doesnt vary in width, there is greaterlikelihood for samples at the same stratigraphic position to be matched.

Dip ranges less than 30. Sensitivity tests show that the interpolation is not significantly affected if thedomain dip ranges less than 15.

Variography must be completed using the transformed data set. In some deposits there will be greatadvantage in completing variography with a transformed data set. Since the samples are aligned aftertransformation, there is a greater chance that a variogram search will find more pairs, particularly within thin,folded domains. Sometimes the expected advantages are not seen owing to the statistical properties of thedeposit. For example, take a deformed deposit that has high, short-range variations in grades. If thefolding/flexuring is on a large scale compared to the ranges required for the variogram to reach greater than70% of the sill, little advantage will be noticed. The search will find enough samples to define the variogrambefore the effects of deformation cause the samples to fall outside the planar variogram search. Despite thelack of enhancement of variography, there is good cause to krige using the easting-offset method to obtain thegeometric advantages, as discussed later.

Point kriging must be used over block kriging . Through the transformation, a block will become distortedowing to the changing orientations of the orebody surface. Also, after the rotation, the orientation of the block

Hanging-wall

Footwall

Mid-wall

Block centroid tobe transformed

HWDIS

FWDIS

MWDIS =

(HWDIS+FWDIS)/2

East (+ve)

Rotation

RotatedHanging-wall

RotatedFootwall

RotatedMid-wall


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with respect to the orebody (and hence search) is different from the pre-rotation relationship. Thus the normalmethod used by software for discretisation will not work correctly. A correction has to be applied to allow blockkriging.

Domain must strike north-south. All rotations are performed around an axis oriented north-south. Errorsmay occur if the strike of the domain deviates from this orientation since offsets will not be perpendicular to theorebody surfaces. Sensitivity tests show that errors are not significant if the difference in strike is less than000 15.

What are the anticipated results?

BLUE Best Linear Unbiased Estimator. Geological zonation (across strike) within domains will be modelled. One of the deficiencies of ordinary

kriging techniques that do not take into account structural effects is an unrealistic representation of gradezonation within the domain. Rather than having zones continuous and parallel to bedding (as predicted by thegeology), zebra striping is the result (Figure 9). The amount of zebra striping tends to increase withincreased sample separation. It is anticipated that by applying the easting offset method, the striping effect willbe eliminated, providing a geologically realistic representation of grade continuity and zonation withindomains.

Figure 9: An example of zebra striping. The domains hanging wall and footwall are shown in orange. The green line is afault. Note how the grade zonation (yellow) does not follow the domain boundaries.

Accurate local scale estimations . By providing a realistic representation of grade continuity, local (stopescale) estimates will be more accurate and reliable. The striping induced by fixed orientation searches caninadvertently cause local high or low-grade estimations.

Optimisation of search strategy because the samples are aligned, allowing confident use of a planarellipsoid and search strategy. If it is known that the mineralisation is truly stratigraphic (ie parallel to beddinglike at George Fisher), it is ideal to match samples at the same stratigraphic position between drillholes.

Unfolding the domain allows:

1. Variography searches to find the maximum number of samples from the correct stratigraphiclocation.

2. Application of a planar search ellipsoid to the transformed coordinates knowing that it will includesamples from the same stratigraphic location.

RELATIVE COORDINATE KRIGING RESULTS AND COMPARISONS

Comparisons between grade estimations of a stope from the block model and the actual mined grades aredifficult owing to the complexity of the mining operations. Numerous sources are being produced at any one timewith all ore reporting to a common location. Thus comparisons between estimated and actual are restricted todirect observations of the geology (underground exposure and drillholes). In the following sections the results ofthe modelling process are compared with real geology and a model created using the standard Medsystemprocedure.


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Note that all figures on the following pages are the product of real data.

Comparison between the results of relative coordinate kriging and real geology

In some areas, the relative coordinate kriging technique honours the geology very closely, however in other areas,there appears to be little correlation. The reasons for the lack of honouring of drillholes are due primarily to rapid

changes in orebody width and complex internal deformation of the orebody. Where blocks do not honour drillholesclosely, there are extreme structural effects present.

Problems of changing domain width

It has been discussed previously that as part of the transformation, samples are aligned according to an offsetfrom a calculated plane (the domain mid-wall). If the software is presented with a number of complete (iestructurally unaffected) domain intersections of the same length, the transformed intersections will be alignedsuch that the middle of the intersections are in the one plane (oriented 0 0 90) (Figure 10).

Figure 10: Diagram showing a real non-planar data set (left) and the equivalent transformed data set (right). Drillholes havebeen numbered ( indicating the transformed DDH). The interpretation of geology is shown in purple. The transformed datashows that the effect of the large-scale folding has been removed. NOTE: Through the transformation, the drillholes aremirrored. This has no effect on the interpolation.

If planes of the same orientation are imposed on different parts of the intersections, each plane will match upsamples from the same stratigraphic location between drillholes. If the orebody changes in width (because offaulting or internal folding) the intersection will still be aligned according to mid-wall offset, however the 0 0 90planes will not match up samples from the same location (Figure 11). If the hanging-wall most or footwall most

sample on the longest intersection is chosen, the 0 0 90 plane will not match up with any sample on the shorterintersection.


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Figure 11: Diagram showing a domain that changes width. Real non-planar data set is on the left and the equivalenttransformed data set is on the right. Drillholes have been numbered ( indicating transformed DDH). The interpretation ofgeology is shown in purple. Vertical lines (dashed green) have been placed at the endpoints of the longest intersection in thetransformed data set to indicate the orientation of the search ellipsoid. As shown, stratigraphically similar samples do notmatch with the vertical lines because of the change in width. Refer to text and compare with Figure 10.

The problems of internal deformation

If the situation presented in Figure 12a is considered:

The diagram shows 4 drillholes of approximately the same length. The interpreted geology between thedrillholes suggests that the area has undergone deformation

Figure 12b shows the transformed drillholes together with an interpretation of the geology. The transformationhas transferred the drillhole intersections into single plane. In doing so, the effects of internal deformation arenot also regularised. If the aligning of samples is investigated, it is seen that the yellow samples do not lineup between drillholes 2, 3 and 4. Subsequently, when kriging, samples from different strata may beconsidered by the software as being from the same stratigraphic position when they are not. The effect ofthis on the block model is a smoothing effect (Figure 12c). Trying to counter this problem is almostimpossible.


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a) b)

Figure 15 a) Fixed orientation Search Ellipsoid. b) Easting offset sample selection. Note how selection is in the plane of thedomain boundaries.

In some cases it can be seen that the grade of blocks kriged using the standard procedure honours the drillholesbetter than blocks estimated using the relative coordinate technique. These situations mainly occur in areas wherethe dip is significantly different from the average dip where the orebody width is changing. The reason for theimproved honouring of drillholes is that the difference between domain dip and the search ellipsoid is so great thatfew other drillholes are selected. The relative coordinate technique will allow more samples to be selected,however since the domain width is changing, there is a high likelihood that samples from the same stratigraphicposition will not be matched between drillholes, as described previously.

With all factors considered, the easting-offset method produced a better, more geologically correct result than thestandard procedure. With the accurate tracking of across strike geological zonation displayed, geologists are ableto evaluate lenses of ore on the hanging wall or footwall of a domain. Evaluation is carried out to see whether a

lens is economical as an individual source, or if it should be included (or bulked in) with the rest of the domain.This procedure would not be able to be completed as successfully with the standard procedures because of thezebra striping illustrated in Figure 9.

CONCLUSIONS

The easting-offset method provides a valid method for the resource estimation of complex, although reasonablyunderstood orebodies. The results obtained at the George Fisher Deposit have allowed application of advancedtechniques of resource definition in compiling the Mineral Resource. More importantly, the complex programs andprocedures required by the technique have been simplified so they can be used by the mine geologist as part oftheir daily routine.

The development of the software has demonstrated the strong link between geology and geostatistics. Continual

advances in the understanding of the ore deposit geology have driven the adaptation of the technique to suit thestyle of mineralisation.

relative coordinate kriging

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