Geology Reference
In-Depth Information
3.4.3
Interpolation
All contouring methods require interpolation between control points in order to find the
structure contours. Discussed here are linear interpolation between nearest neighbor points
in a TIN network and interpolation to a grid.
3.4.3.1
Linear Interpolation
Linear interpolation is based on the assumption that slope between the data points is a
straight line. As a contouring technique it is also called mechanical contouring (Rettger
1929; Bishop 1960; Dennison 1968). This is a standard approach for producing topographic
maps where the high points, low points, and the locations of changes in slope are known,
allowing accurate linear interpolation between control points (Dennison 1968). This
method may produce unreasonable results in areas of sparse control (Dennison 1968; Tear-
pock 1992). The resulting map is good in regions of dense control and is the most con-
servative method in terms of not creating closed contours that represent local culmina-
tions or troughs. The method tends to de
-emphasize closed structures into noses, is good
for gently dipping structures with no prominent fold axes, and is often used in litigation,
arbitration and oil-field unitization (Tearpock 1992). This method is applied to the ex-
ample data in Fig. 3.9. The contours suggest an anticline with two separate culminations.
3.4.3.2
Interpolation to a Grid
Mapping by gridding requires interpolation between and extrapolation beyond the
control points to define values at the grid nodes prior to contouring. Gridding variables
always include the choice of the grid spacing and the interpolation technique. A typi-
cal characteristic of gridded data is that the original control points do not fall on the
contoured surface. The reason for this is that the control points are not used to make the
final map.
The simplest gridding technique is linear interpolation. The structure contour map in
Fig. 10a is the result of linear interpolation to the nodes of a 10
10 grid, then linear-inter-
polation contouring between the nodes. The best-controlled part of the structure resembles
the triangulated map of Fig. 3.9. The contouring algorithm for this technique forces all
contours to close within the map area which is not a geologically realistic assumption.
Another simple interpolation technique is the inverse distance method. In this method
the value at a grid node is the average of all points within a circle of selected radius around
the node, weighted according to distance, such that the farther-away points have less in-
fluence on the value. The weighting function is usually an exponential, such as one over
the distance squared (Bonham-Carter 1994). Figure 3.10b is a structure contour map
produced by inverse-distance interpolation. The contours are significantly more curved
than those produced by linear interpolation of the TIN (Fig. 3.9) or the linear interpolated
grid (Fig. 3.10a). The side view of the inverse-distance interpretation (Fig. 3.10c) shows
that the control points do not all lie on the interpolated surface.
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