Automatic Kriging And Contouring In The Presence Of Trends (Universal Kriging Made Simple)

Abstract

Abstract The main purpose of this paper is to present a new automatic interpolation and contouring technique which automatically recognizes the spatial structure of the dam to be contoured even when, trends are present and data are not regularly spaced. A brief relief of contouring problems is presented the idea of spatial structure and its characterization by a generalized covariance is then introduced and the POLYPAC program is described. Applications in various areas (oil shales, sulphide mines, coal field, cement industry, geophysical data) are illustrated and a detailed appendix covers the mathematical theory of the intrinsic random functions of order k, which is the basis for the program. References to other similar programs are included. Conceptual Background Introduction Attempting to survey everything that has been written on the subject of automatic contouring is a challenging task as ideas are generated in various fields with little communication between each other, such as coal reserves, ore reserve estimation or geophysical mapping, not to mention geography and meteorology. Also, the concern of contouring packages for high cosmetic quality like G.P.C.P., CSPL. Stampede, etc... tends to push people away from the fundamentals of the problem and its conceptualization, Most of the time the interpolation method used in a program is unknown to the map user, or he is left with the choice of up to 32 different interpolation subroutines in the same package, with no further hint at which method is adapted to his problem. It is our opinion that most mapping problems can be encased in the frame of a unifying theory which will allow the generation of the most adequate map for a given purpose. This theory has been known as kriging or simply geostatistics and was developed by Matheron, The full name of the most complete version of it is "theory of intrinsic random functions of order k", and all its intricacies have now been turned into another black box, which produces the best theoretical maps within the framework of distance weighted and trend-surface interpolation. It is one of the purposes of this paper to show which concepts are needed to understand why a given interpolation algorithm is good in certain cases and unacceptable in others. These concepts of spatial structure, drift, trend, high-frequency noise, estimation variance have been known for a long time, but we must admit that map users should be presented with something else than a truckload of equations. The contouring problem is easily split in two; the first problem is that of the generation of II. regular grid of points, the second being that of aesthetically contouring this fine grid. So far only the first problem retained the attention of geostatisticians, with few exceptions. Broken contours are not acceptable even if the theory behind the grid generation technique is faultless (Olea, 1972). There now exists a program matching the most sophisticated theory and superb contours.