D. F. Watson

Triangle based interpolation

Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slant-top proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C1surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C1,interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slant-top interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.

Matheronian geostatistics—Quo vadis?

Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.

Measures of variability for geological data

Diverse global and local measures of variability appear in the geological literature and, along with methods used to apply them, have been subject to some debate. Measures of variability for three data types—replicate, locational, and compositional—are considered; the source and nature of the variability determine the appropriate type of measure. To illustrate the effects of these measures and expose their inadequacy when improperly applied, the variability of a three-column data set is interpreted under three different suppositions. Geologists need to be aware of the confusion and misleading results that can develop from the use of variance as a measure of variability for locational or compositional data.

A method for assessing local variation among scattered measurements

Automatic triangulation of scattered locations permits analysis of local variation in a dependent variable through calculation of a roughness index. This is approached by treating triangles of the triangulation (including the dependent variable) as vectorial structures, and accumulating at each data point the vector sum of the cluster of triangles surrounding it. The roughness index is defined as the complement of the ratio of the area of a triangle cluster to the area of component triangles as projected onto a gradient plane defined by their vector sum. The roughness index provides a measure of consistency of data values relative to surrounding observations and can be interpreted as a local index of reliability of interpolation.

Algebraic dispersion fields on ternary diagrams

Previously published dispersion fields on ternary diagrams have been constructed variously, and their derivations have not been well-specified. Here an explanation of their bases is provided through an algebraic method for calculating two related forms, designated thesilhouette dispersion field and thegirth dispersion field. Such dispersion estimates can be made more precise by specifying the percentage of samples that fall within the field. Because such fields represent a mechanistic rather than a probabilistic approach, their use in comparison of sample sets must be viewed with caution.

Neighborhood discontinuities in bivariate interpolation of scattered observations

Because of the need for computational efficiency, bivariate interpolation methods applied to scattered observations often involve two stages. Initially the variable is estimated at regular grid nodes using a running subset of data (usually of fixed number). This, however, will produce discontinuities in the interpolated surface. Thus a second stage, curvilinear interpolation technique, is applied to estimated values to smooth out the effect of discontinuities. Such problems can be overcome efficiently in processing large data sets by interpolating over natural neighbor subsets. Interpolation procedures that generate discontinuities in the interpolated surface are inappropriate for geological applications, where dislocations due to structural complications may be present.

A derivation of the Isted formula for average mineral grade of a triangular prism

The Isted formula, which provides an exact calculation of the spatial average of mineral grade in a triangular prism under the assumption of linear change, is derived, and its limitations outlined. This formula may be used to estimate the mean of any trivariate function whenever variation in the vertical direction can be considered negligible.