The problem of contouring a faulted surface known at randomly spaced points is analyzed and different types of solutions are proposed. The data may in fact be from a field which satisfies an elliptic partial differential equation; if the equation is harmonic, the surface corresponds to the displacement of a membrane properly raised from a horizontal plane in correspondence to the data points and cut along the faults.
If the equation is biharmonic, the surface corresponds to the displacement of an elastic plate, properly riveted in correspondence to the data points and again cut along the faults.
A third method analyzed, that corresponds to a family of interpolation methods, is that of two–dimensional estimation. The technique used is that of modeling the autocovariance of the data as a function of the distance between the points only. The surface will depend upon the particular function chosen and it will tend to be peaked at the data points, if the function is peaked at the origin, and smoother if the autocovariance is smoother. When faults are present, the distance between two points is defined to be the length of the shortest linking path, not cutting a fault. In the latter case, it is shown that the set of functions eligible to be chosen as autocovariances is very limited.
The first method has the useful property that maxima and minima of the surface are data points. The second method generates smoother surfaces that sometimes may overshoot. Both methods are implemented by iteratively smoothing the interpolated lattice (except in the neighborhood of data points), and therefore are rather expensive in terms of computer time. The third method is not iterative and is less expensive; since the surfaces that it generates are noisy, it may be used to supply a tentative solution to be refined with an iterated smoothing.
These different techniques arc discussed in detail and some examples of their application are shown.