A Computer Method for Four-Variable Trend Analysis Illustrated by a Study of Oil-Gravity Variations in Southeastern Kansas
Abstract
A method for fitting four-variable trend hypersurfaces by least squares has been programmed for the IBM 7090 computer. The program fits first-, second-, and abbreviated third-degree hypersurfaces to irregularly spaced data. The program automatically contours the intersection of each hypersurface with a block whose top, bottom, and four sides represent planes located in three-dimensional space. This permits the four-variable or four-dimensional hypersurfaces to be visualized. The program also automatically plots original data and residual values in a series of horizontal slice maps. The theory and operation of the program are discussed and illustrated in detail.
The program has been used to interpret variations in crude oil gravity from place to place and in different Paleozoic stratigraphic horizons in southeastern Kansas. Hypersurfaces were fitted to API oil gravity as a function of geographic location and depth below the surface. The four variables involved are (1) API gravity, (2) well depth, (3) north-south geographic coordinates, and (4) east-west geographic coordinates.
The trend hypersurfaces, distribution of residual values, and other considerations suggest that oil-gravity variations in southeastern Kansas have been affected by both well depth and environment of deposition. The tendency for API gravity to increase with depth is complicated by regional effects that may reflect differences in environment of deposition. The result is an overall increase in API gravities in a west-northwest direction. Of interest is a tendency for residual API gravity "highs" and "lows" to be clustered in certain geographic areas even though oils from different stratigraphic zones are involved. This, in turn, suggests that the depositional environment may have affected oil gravities in a given locality much the same way from one geologic period to the next.
The computer program described in this report may have a number of geological applications, and can be used readily by anyone having access to an IBM 7090 or 7094 computer.
