2-D and 3-D Trend Analysis - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 234

Chapter 7
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 describ-
ing 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 applica-
tions 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 indepen-
dent 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 station-
ary 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, geometri-
cally, 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 applica-
tion 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