Geological modelling. Gridded Models

Transcription

1 Geological modelling This white paper discusses the meaning and generation of a geological model. The emphasis is on gridded seam models which are commonly used in coal. Mining companies make investment and production decisions based on their understanding of the deposit geology. This geology is usually represented by a model. In the distant past the model was a set of hand drawn sections and plans. Today computers allow us to build those models electronically, and view them in 3D, section or plan. The mine planning software industry generally uses two types of models: 1. Block models. These represent the deposit as a series of cubes of fixed or variable size. Block models are ideal for massive deposits such as copper, gold or iron ore. The block is located in space with an XYZ coordinate system and attributes of the block store mineral information (e.g. percent Fe, grams Au). 2. Gridded models. These models represent the deposit as a series of layers. Gridded models are ideal for layered deposits such as coal, bauxite, laterites and phosphate. This paper is primarily focused on the gridded model although some comments are common to both gridded and block models. Gridded Models Typically exploration drilling provides data values Z, at irregularly locations in X and Y space. The process of gridding uses these irregularly spaced data values to create or estimate grid values on a regular spacing. The imaginary grid is usually square but can be rectangular. The size of the grid cells should reflect the data distribution. In Figure 1 the data is spaced at 90 metres and 130 metres and a grid cell size of 25m has been selected. Typically grid cells should be ¼ to 1 / 5 of the data spacing. Figure 1: Typical grid spacing. 130m 90m Page 1 of 13 November 2005

2 The estimation of the Z grid values from irregular data values forms the body of this paper. Modelling algorithms Minex provides three algorithms for estimating the grid values from the data values. These are: 1. Distance weighting 2. Kriging (although not discussed in this version of paper) and 3. General or growth modelling. Distance weighting methods assume an interpolated grid point value will be most influenced by the nearest data point and more distant data points will have less influence on determining the interpolated grid value. The most common distance weighting method is inverse distance squared. ID 2 assigns values to a mesh node based on the inverse distance squared of the data point to the mesh node. The power factor can be varied by the user but is frequently squared (Figure 1). Figure 1: Inverse Distance method. λ = i N i = 1 1 d α i 1 d α i One feature of inverse distance weighting is that the value of a grid point is bounded by the values of the data used to estimate that point. Thus if the seam ash is between 10 and 20 the grid values using inverse distance can only fall between 10 and 20. This effect can be viewed positively as it is conservative, but it can also be considered as a negative as it won t highlight data trends. In Figure 2, five data values 0,100,1000,100,100 (A,B,C,D,E) have been used to create a grid at 25m centres using inverse distance squared. Three points are significant: 1. Unless a grid cell falls exactly on a data point then the model has difficultly honouring the data. This is apparent at point C. Page 2 of 13 November 2005

3 2. Between data points (e.g. B and C) the weights assigned to the data values cause the resultant grid down below 100. Between points B and C this averaging is caused by the downward influence of point A and D. 3. If inverse distance is used to generate a model outside the data then the model will trend to the average of the data values. Figure 2: Inverse distance weighting models tend to the average. B C E F A D SMG Growth Method The growth method is a two stage process. Stage 1 surrounds the data with four grid mesh points. In stage 2 the data is discarded and the grid mesh grows outwards to fill in the remaining grid mesh cells. The growth method is based on a technique used by IBM in the STAMPEDE 1 planning software and described by Batcha and Reese. They describe stage 1 as follows: to determine the mesh point values of a square containing one or more data points, the centroid of these data points is first determined. If only one data point falls within a square, the data point is the centroid). Next a plane is established passing through the centroid. To establish this plane, values of surrounding data points are taken into consideration. The technique involves searching out the nearest data point falling within each of eight equal sectors around the centroid. These data points are then used in a least squares fit to determine a plane passing exactly through the centroid. The weighting of the selected data point values is such that the closer the data point to the centroid, the greater the weight in the least squares fit. The value of the plane at each of the four mesh points of the grid square is taken to be the value at that particular mesh point. (1) STAMPEDE Surface Techniques, Annotation and Mapping Programs for Exploration Development and Engineering Page 3 of 13 November 2005

4 Figure 3 shows a set of data points in red with the first pass nodes coloured in blue. Figure 3: First pass of the growth method surrounds the data with grid nodes. Stage 1 uses a sextodecimo (16 sectors) search to estimate the stage 1 nodes. In Figure 4 assuming one point per sector the top right and lower left points (marked x) are excluded when determining the plane for the point at the sector centre. Figure 4: Sextodecimo sector search to determine stage 1 nodes, data points ticked are used if search is limited to one point per sector. x R x Page 4 of 13 November 2005

5 The MINEX growth algorithm has been implemented with a limit of points per sector (defaults to 3). This avoids unnecessary searching to find points. In forming the stage 1 nodes search or scan distance is critical. In Figure 4 the scan distance of R collects points from all sectors to estimate the plane around the centre of the search. If the scan were reduced to ½R then determining a value at the top of Figure 4 would not use points at the bottom of Figure 4. Scan distance needs to balance the conflicting aims of infilling the model but not mixing populations. If the data at the top and base of Figure 4 is a single population then the scan distance should be adequate to bridge this gap. The second pass of the growth process grows away from these initial four mesh points and progressively fills in the remaining model. Jones using oil exploration terminology describes the second stage as: After the nodes around all wells are calculated the wells are removed from further consideration. The program then makes a series of passes over the grid. At each pass it calculates values for any grid nodes that have not been assigned a value and that are adjacent to an assigned node. In other words, each iteration enlarges the calculated region around the original well locations. The stage 2 process is also described by Crain as follows: The second stage fills in the remaining mesh points. These are calculated by taking the average of two planes (a secant plane and a tangent plane ) calculated at each grid intersection. The secant plane is calculated in a manner analogous to the first stage. The tangent plane is produced by a first order finite difference equation. This generation proceeds outwards from the known mesh values of stage one. The MINEX implementation is very similar although the secant plane is usually described as interpolation and the tangent plane as extrapolation. In Figure 5 the secant plane is generated for point X by using the nodes that match the searching parameters (coloured green in Figure 5). A hardwired control also affects the secant plane. Nodes must exist in at least three of the sectors to generate the secant plane. As with stage 1 the weights of the nodes is based on distance to build a least squares plane. However if there are 5 or less points available to estimate the secant plane then an inverse distance weighted average estimate is used in lieu of a least squares estimate. Page 5 of 13 November 2005

6 Figure 5: Generation of secant planes for point X during stage 2 is controlled by green nodes (assuming for clarity one point per sector) X As the secants planes bridge across the void between stage 1 nodes, they do not reflect the gradient or trend of the data. This gradient or trend is supplied by the tangent planes. The tangent plane is determined by only using the local nodes. In Figure 6 the yellow nodes determine the tangent plane. A first order partial differentiation equation is used to determine the tangent value. This is determined from triangles formed at the surrounding 4 mesh points at the node to be estimated. In Figure 6 two triangles will used to calculate a tangent value at node X. If no triangles can be formed due to a lack of node values then only the secant value is used as the estimated value. Generally however two estimates of node X are available a secant value and a tangent value. In Batcha and Reese s implementation X is assigned the average of the tangent and secant planes. The MINEX implementation allows the user to control the weight assigned to the extrapolation (tangent) estimate. By default the weighting is 2/3 interpolation or secant and 1/3 extrapolation or tangent. In Minex the user can disable the extrapolation estimate; this will tend to flatten the model between data points. With extrapolation on the user may override the default weight ratio of 2/3:1/3 although the ratio can only increase to 0.5:0.5 (secant : tangent). Page 6 of 13 November 2005

7 Figure 6: Yellow points are used in determining tangent planes for node X. X Unlike inverse distance methods the growth method can generate values that exceed the data values. This occurs for two reasons: 1. In stage 1 the plane around each data point is built from the surrounding data and can thus generate a node greater than or less than the data values. 2. The tangent plane or extrapolation component of stage 2 as it grows form a local trend This point is illustrated in Figure 6. Page 7 of 13 November 2005

8 Figure 6: Growth Method used to model data from Figure 2. ID 2 Growth The growth method is often explained in MINEX marketing using a simplified section view. These sections are presented in Appendix A. References BATCHA, J.B. and REESE J.R Surface determination and automatic contouring for mineral exploration extraction and processing. Colorado School of Minex Quarterly, 59: CRAIN I.K Computer interpolation and contouring of two dimensional data a review. Geoexploration 8: JONES T.A., HAMILTON D.E. and JOHNSON C.R. Contouring Geologic Surfaces with the computer Van Nostrand Reinhold Company. Page 8 of 13 November 2005

9 Appendix A: Growth Method Figures A1 to A9 present the growth method in a cross sectional view. Figure A1: Four drill holes with the seam data as shown. Step 1 the data is surrounded with the grid mesh or nodes. A B C D Borehole Figure A2: Regional trends are created around B using A and C and around C using B and D. A B C D Borehole Trend around B Trend around C Page 9 of 13 November 2005

10 Figure A3: Planes are moved so they are matched to data points B and C. A B C D Borehole Adjusted trend around B Adjusted trend around C Figure A4: Mesh node values are estimated around the data points. A B C D Borehole Computed Value Page 10 of 13 November 2005

11 Figure A5: Data is discarded before stage 2 is commenced. Computed Value Figure A6: Missing nodes are estimated from first stage nodes using Extrapolation and Interpolation. Computed Value Interpolated Extrapolated Page 11 of 13 November 2005

12 Figure A7: New point is estimated by averaging the extrapolation and interpolation New Point Computed Value Interpolated Extrapolated Figure A8: The node from Figure A7 is used as data to estimate the next node using the extrapolation and interpolation process. New Point Computed Value Interpolated Extrapolated Page 12 of 13 November 2005

13 Figure A9: Final surface showing smooth connections between mesh points. Computed Value Modelled Surface Page 13 of 13 November 2005