Triangulation Based Volume Calculation Andrew Y.T Kudowor and George Taylor Department of Geomatics University of Newcastle

Transcription

1 Triangulation Based Volume Calculation Andrew Y.T Kudowor and George Taylor Department of Geomatics University of Newcastle ABSTRACT Delaunay triangulation is a commonly used method for generating surface models which have many uses, including height interpolation and volumetric analysis. This method can be adapted to accurate volume estimation using borehole data. This paper compares volume estimation of three different techniques which use triangulated surfaces for volume calculation. Some of the problems in using Delaunay triangulation for volume calculation are identified and modifications are suggested to adapt it for more accurate volume estimation. A routine for automated volume estimation from borehole data, which incorporates some of these techniques, has been developed. 1. INTRODUCTION Results of volumetric analysis estimated from borehole data are a major input in many mining operations. Traditional programs have been based on the application of the end-area or prismoidal methods to some form of gridding technique. Triangulation methods have been chosen as a powerful alternative to these methods in most current programs for volume calculation. Techniques for volume calculation are utilised in different commercial programs with the user unaware of the underlying principles used. This can lead to inappropriate application of these techniques. The methods for the generation of triangulated surfaces are introduced, and the available techniques for manipulating these surfaces for accurate volume estimation are presented. 2. TRIANGULATION AND VOLUME CALCULATION. TRIANGULATION Triangulating a set of N points in the plane indicates joining them by non intersecting straight line segments so that every region internal to the convex hull is a triangle (Preparata et al. 1985). The basic concept of any triangulation procedure is to attempt to produce a unique set of triangles that are as equilateral as possible and with minimum side lengths. There are several methods available for triangulating the same set of points. Delaunay triangulation (DT) has been the dominating method for its ability to divide the surface into optimal triangles according to some useful criterion associated with it: EMPTY INTERNAL CIRCLE CRITERIA - Given a triangle T defined by points P1, P2, P3 of a DT of a set of points P, no other point of P is internal to the circle defined by P1, P2, P3. This property defines the concept of constructing a Delaunay triangulation. MAX-MIN CRITERIA:- The minimum of all the angles in the triangulation is maximised. That is given four points forming a quadrilateral, the diagonal which divides it into two triangles is optimal in the way that maximises the lesser of the internal angles, this guarantees the best possible triangle shape for the set of points. This is the Lawson criterion (Lawson 1972). These criteria has been utilised in several algorithms for Delaunay triangulation. Complete details of such algorithms are given by (Defloriani et al. 1992), (Watson 1992) and (Jian-Ming et al. 1990). The method implemented in the routine used here is similar to the incremental insertion algorithm, meaning that data points are added one at a time into an existing triangulation. It was proposed by Lee and Schachter (1980), who suggests an iterative technique to triangulate a set of points inside a rectangular region. This was improved by Macedonio et al. (1991) by triangulating a set of points inside any convex region. This algorithm was chosen because of the ease and simplicity of adapting its data structure to volumes. For further readings on these features consult Macedonio et al. (1991).

2 VOLUME CALCULATION To illustrate the volume calculation concept from triangulated surfaces, consider a set of points of borehole locations. These points can be divided into triangles by connecting the data points as the vertices of these triangles. The volume calculation is then reduced to finding the volume under each prism formed using formula [1]. volume = 1 / 2[ x1 ( y2 y3) + x2( y3 y1 ) + x3( y1 y2_ ]( z1 + z2 + z3) / 3 [1] Where, (x1, y1), (x2, y2) and (x3, y3) are the planar co-ordinates of the triangle vertices and z1, z2 and z3 are the corresponding elevation with respect to sea level or thickness between geological horizon. (Fig. 2b). The advantages of triangulation have been reported in the literature. These include: Deriving the triangles from the data points, that is nodes of the triangle honour the observed 3D locations, the resulting surface then include the observed points, hence eliminating the need for smoothing and interpolation. The observed density of information is maintained; Point x y z Ro small triangles occur where the point density is high and correspondingly large triangles where the density is low. It has the ability to represent discontinuity in the extent surface, and an irregular edge is enhanced. These are good features for volume applications. However, for accurate volume estimates there is a need to pay special attention to the manipulation of the z (height) values. In most DT algorithms, common Table1: x, y, and z point co-ordinates occurrences such as two points being coincident and three points of a potential triangle are co-linear are handled by rejecting (dropping) the points. This is an undesirable feature for DT application to volume estimation. The plausible alternative to be considered in volume applications is to shift the points co-ordinates to maintain its z value influence in the volume estimation. Figure1 illustrates the effect on volume estimates of rejecting and the extent of shifting the coordinates, due to the occurrences mentioned above. Let us assume that for handling only whole numbers, the point Ro coincides with point 1 (Table1) hence as indicated above has to be dropped in the course of the triangulation. Figure 1 shows the volume computed for the original position of Ro (a), when it was dropped (b) and when it was shifted with various approximations (c). This indicates that as far as the z value of the point remains significant, small shifts improves volume accuracy than dropping the point. See table1 for the co-ordinates of the points used for accessing the influence of point rejection and shifting on volume estimates. 2 a 3 2 b 3 c 2 3 Ro R s 1 4 Va= R e 1 4 Vb= Vc= Rs=120.01, Vc= Rs=120.10, 54,10 Vc= Rs=121.00, a.triangulated surface with original position of coincident point Ro with volume estimate b.triangulated surface with the coincident point dropped,and volume estimated c.triangulated surface with shifted coincident point with volumes for various approximations. Fig. 1. Effect of point shifting and dropping on volume estimates

3 3. TECHNIQUES FOR MANIPULATING TRIANGULATED SURFACES FOR VOLUMES Having a set of N irregular distributed points P(xi,yi) as borehole locations with values of Zi as elevation or thickness of the layer between geological horizons, Delaunay triangulation can be used for volume reconstruction to estimate the volume between a surface and a datum or between two surfaces. To illustrate the volume estimation between triangulated surfaces, assume two triangulated surfaces using Delaunay criteria. The following available techniques can be used to manipulate the triangulated surface for volumes: Simple prisms With the nodes of the triangles being the elevation or thickness points, the simple prism method takes a triangle from one triangulated surface model, finds the elevation difference and calculates the volume for the vertical prism formed (Fig. 2b). Fig. 2 a. Triangulated surface for generating simple prisms. b. Simple vertical prism. c. Isopachytes. d. Complex vertical prisms This method is accurately utilised when the boundaries of the triangulated surfaces are identical and the difference between volume from each triangulated surface to the same datum taken as the net volume. Cut and fill estimates cannot be identified. Isopachytes Where the boundaries of the triangulated surface are not identical, the isopachyte method can give accurate estimates when surface elevation does not change rapidly. This method involves determining the difference between the elevation values at every point location in the two surfaces, and generate a triangulated surface from the resultant points. The estimated cut and fill volume between the two surfaces is the volume between the generated surface and the zero datum (Fig 2c.) Complex Prisms This method gives more precise results (see investigation results) and is generally accepted to be highly accurate relative to the data points. The method takes into consideration the effect of every point in both surfaces (Fig.2d). Complex prisms involve projecting a triangle from one surface, and each triangle it overlaps is taken and the intersection polygon calculated. The polygon is then broken down into triangles to form vertical prisms for which the volume is individually calculated. However, this method is a trade off between accuracy and time. Though it is generally accepted as the ultimate method for volume estimate, very few programs have implemented it, because of the amount of processing required to find overlapping triangles and the cost of computer resource required, see Hall and Watts (1991) for further reading.

4 4. INVESTIGATION METHODOLOGY AND RESULTS To investigate volume estimation using the methods described above, a reference model in the form of trapezoidal prism (Fig.3) of area 1200 units 2 was built, of which the top and bottom consists of nine points. The actual volume was determined to a zero datum by decomposition to be 11,000 units 3, units 3 and units 3 for the top, bottom and between the two surfaces respectively. Fig.3 Reference Model Additional points for both the top and bottom of the model were created (not interpolated) on the surface, which consists of 9/9, 15/15, 20/20, 24/24, 35/35, 43/42, 49/48, 55/54, 61/60, 69/68, 81/81 points. Each of the surface models was triangulated and subsequently, volumes were estimated from the planar triangulated surfaces for each point density using the three techniques. The percentage error of each point density for each surface model was calculated for each technique used (Table 2) and a bar chart of point density against percentage error was drawn (Fig.4) from which conclusions were made. RESULTS The Simple Prisms and Isopachytes techniques, exhibited approximately the same influence on the accuracy of volume estimates for each point density (PD), for both Top and Bottom surface models (Fig 4a and b). Though the percentage error was small, especially for the top model, the obvious expectation is that an increase in PD which increases the number of triangles hence closer approximation of the true surface, should yield reduction in the % error of estimation or increase accuracy. The two techniques which were used in estimating volume to a zero datum, gave a more accurate result for the relatively rugged top surface model, with increase in PD limiting the error to less than 0.8%. However there were undulating effect which is more of an overestimation. The Bottom model on the other hand exhibited some undulation with greater underestimation between PD 9 and 35, where no points are dropped during the triangulation stage. Beyond PD 35 the estimates appears consistently overestimated below an error of 1%, where points are dropped for each point densification during triangulation. For volume estimates between the two surface models using Isopachytes and complex prisms, the two techniques differ in the resultant volume estimates (Fig 4c). Relative to the actual difference of the volume estimates between the top and bottom models, the complex prism method gives a more precise estimate, as it consistently gives estimates equal to the actual difference. However, as it stands, the two estimates show similar undulating effect with the Isopachytes technique giving more accurate estimates at PD 15, 35, 49, 61 and 69, and the Complex Prism at PD 20, 24, 43, 55, and 81 with difference between the errors insignificantly small.

5 Table2: Percentage error in volume estimates Top Volume Point Density % Error of Estimation Simple Prism Isopachytes Bottom Volume Point Density % Error of Estimation Simple Prism Isopachytes Top-Bottom Volume Point Density % Error of Estimation Isopachytes Complex prism

6 a % Error of Top Volume Estimation Program A (Simple prism) % error of estimation Program B (Isopachytes) % error of estimation % Error No of Points -0.6 b. % Error % Error of Bottom Volume Estimation No of Points Program A (Simple prism) % Error of estimation Program B (Isopachytes) % Error of Estimation c. % Error % Error of Top-Bottom Volume Estimation No of points Program A (Isopachytes) % Error of estimation Program B (Complex prism) % Error of Estimation Fig.4 Percentage error in volume estimation a. of Top surface. b. of bottom surface. c. between the two surfaces.

7 CONCLUSION This paper described the various techniques for manipulating triangulated surfaces for volumes. For accurate volume estimates, an illustration of some of the drawbacks for adapting DT to volume application which includes point rejection as a result of common occurrences such as coincident points and three points of a potential triangle being co-linear are presented, alongside modification such as point shifting in DT for adaptation to accurate volume estimates. As points are shifted by approximation to the next decimal place, volume estimates approaches the true value than when they are dropped (Fig. 1). The difference techniques for triangulation based volume estimation were investigated and the results discussed. The complex prism method has proven the most precise of the three for volume calculation between surfaces. Relative to the actual difference of the volume estimates between the top and bottom models, the complex prism method gives a more precise estimates, as it consistently gives estimates equal to the actual difference. For accuracy of volume estimates between surfaces the isopachyte method has proven competitive. the two programs show similar undulating effect with program A (Isopachytes) giving more accurate estimates at PD 15, 35, 49, 61 and 69, and program B (complex prisms) at PD 20, 24, 43, 55, and 81 with difference between the errors insignificantly small. REFERENCES De Floriani L., and Puppo E., An on-line algorithm for Delaunay triangulation. Graphic models and image processing, Vol.54, No.3, pp Hall & Watts Systems, Volume Techniques. Documentation, Department of Geomatics, University of Newcastle. Jian-Ming Z., Ke-ran S., Ke-ding Z., and Qiong-hua Z., Computing constraint triangulation and Delaunay triangulation: A new algorithm. IEEE transactions on magnetics, Vol. 26, No.2. Joseph O Rourke Computational Geometry in C. Cambridge University press, p346. Lee D.T., and Schacter B.T., Two algorithms for constructing a Delaunay triangulation. International journal of computer and information science, Vol. 9, No. 3, pp Macedonio G., and Pareschi M.T., An algorithm for triangulation of arbitrarily distributed points: Applications to volume estimate and terrain fitting. Computers and geosciences, Vol.17, No.7, pp Preparata F.P., and Shamos M.I., Computational Geometry - An introduction. Springer-verlag New York Inc., p391 Watson D.F., Contouring: A guide to the analysis and display of spatial data. Pergamon Press, pp321.