Many different seismic attributes have been proposed so far in the literature to “mark” the presence of faults in a seismic cube. From these attributes, the challenge is now to extract, automatically, a fault network where each fault is singled out as a subset of points. For this purpose, we propose to use a method based on a cascade of two Hough transforms. The basic idea of this algorithm is that the intersection of a fault by a series of (x,z) cross sections is (approximately) a family of straight-lines. Each of these straight-lines is transformed into a point in a first parametric space thanks to a first Hough transform. For each fault, the set of points so obtained constitutes (approximately) a new straight-line in the parametric space which is then transformed into a point of a second parametric space thanks to a new Hough transform. Reverse transformations allow then to rebuild each fault as a set of points.
Seismic attributes such as, for example, variance, semblance, or coherency are commonly used to mark the presence of faults in a seismic cube. The presence of noise, however, has a strong tendency to blur the fault zones: this is why extracting automatically each fault as a set of points is still an active and challenging field of research (e.g., see [4],[5]). In this article we propose a new approach allowing all the faults to be extracted automatically in one go from a 3D seismic cube of attributes in such a way that each fault is represented by a distinct subset of points. After extraction, each subset of points can be approximated by a surface patch to provide a 3D geometric representation of the associated fault.
The Hough transform has been introduced by Paul Hough in 1959 to automatically detect trajectories of particles in bubble chambers (see [1] and [2]). It is a very efficient method used in image processing for detecting a particular shape in a 2D binary image. To detect parametric curves (lines, circle, parabolas, …), the basic idea is to transform each of them into easily recognizable patterns (ideally a point) in a parametric space. For example, in the frame of this article, the Hough transform is used to extract straight lines from a noisy 2D binary image itself obtained by thresholding seismic attributes (coherency, semblance,…) on a (x,z) seismic cross-section. As figure 1 shows, consider a point (xk,zk) corresponding, for example, to a pixel marked as “faulted” in a 2D cross section. If we choose a finite number of angles {?1,?2,...} for lines going through (xk,zk) we can compute their corresponding distances {r1,r2,...} to the origin. If this operation is repeated for all the pixels marked as “faulted” in the 2D (x,z) cross section, then, as figure 2 shows, it is possible to count in a 2D (r,?) grid the number of occurences of each pair (ri,?i). The 2D space corresponding to the 2D array so obtained is called the “parametric space” of the Hough transform.
