2-D and 3-D Trend Analysis

Abstract

One of the early applications of the general linear model is trend surface analysis (Krumbein and Graybill, An introduction to statistical models in geology. McGraw-Hill, New York, 1965). In the late 1960s, this technique was competing with universal kriging originally developed by Huijbregts and Matheron (Can Inst Min Metall 12:159–169, 1971). To-day, both techniques remain in use for describing spatial trends or “drifts” in variables with a mean that changes systematically in two- or three-dimensional space. Simple moving averaging as practiced by Krige (Two-dimensional weighted moving average trend surfaces for ore valuation. In: Proceedings of the symposium on mathematical statistics and computer applications in ore valuation, Johannesburg, pp 13–38, 1966) or inverse distance weighting methods can be equally effective when there are many observations.

Trend surface analysis was one of the first computer-based methods widely applied in geophysics, stratigraphy and physical geography in the 1960s. Initially, it was assumed that the residuals from a best-fitting trend surface should be independent and normally distributed but Watson (J Int Assoc Math Geol 3:215–226, 1971) clarified that polynomial trend surfaces are unbiased if the residuals satisfy a stationary random process model. Examples of 2-D trend surface analysis include variations in mineral composition in the Mount Albert Peridotite Intrusion, eastern Quebec. A 3-D extension of the method applied to specific gravity data shows that, geometrically, serpentinization of this peridotite body occurred along a northward dipping inverted pyramid. 2-D and 3-D polynomial trends of copper in the Whalesback Deposit, Newfoundland, illustrate how numbers of degrees of freedom are affected by autocorrelation of residuals in statistical significance tests. A useful approach to regional variability of variables subject to both deterministic regional trends and local variability that can be characterized by stationary variability of residuals is to extract the trend by polynomial-fitting and to subject the residuals from the trend to ordinary kriging using 2-D autocorrelation functions. This approach is illustrated by application to (1) depths of the top of the Arbuckle Formation in Kansas, (2) the bottom of the Matinenda Formation in the Elliott Lake area, central Ontario, and (3) variability of sulphur in coal, Harbour seam, Lingan Mine, Cape Breton, Nova Scotia. Use of double Fourier series expansions instead of polynomials to describe regional trends is illustrated on copper in exploratory drill-holes originally drilled from the surface into the Whalesback deposit. Use of 2-D harmonic analysis is also illustrated by application to density of gold and copper deposits in the Abitibi area on the Canadian Shield, east-central Ontario. An advantage of using double Fourier series instead of ordinary polynomials in trend surface analysis is that many geological features are to some extent characterized by similarity of patterns along equidistant straight lines. Periodicities of this type are accentuated by harmonic analysis.