Abstract
In oil exploration, changes in soil depth or thickness of a rock are indicators for the presence of the so called "seismic horizons" that identify the possible presence of oil. The data concerned are obtained by making measurements at randomly distributed sparse points. It is of interest to reconstruct the full model of the surface of the rock or the terrain, from the knowledge of the few sparse data points. This reconstruction cannot be achieved by using ordinary interpolation methods, as these methods assume that the reconstructed surface is smooth. Instead, a fractal prior model for the terrain has to be assumed. A constraint fractal formation then follows, with the constraints being the data points available. The dimension of the fractal used is inferred from the data points that are available, on the basis of the assumption that a fractal model applies and from the fact that a fractal exhibits the same properties at all scales. Several tools for the creation of artificial fractals of varying degrees of roughness are used to give a wide range of data for the reconstruction experiments. A tool to measure the fractal dimension of a surface, or a set of sparse data points, is an important part of the reconstruction process. Several methods of fractal dimension measurement are developed and thoroughly tested with many different surfaces. The reliability of the dimension calculation and how this changes with different levels of sparsity is investigated. Both tools are then modified to enable the production and measurement of anisotropic fractals - fractals with different levels of roughness in different directions. These sorts of fractal surfaces have received little or no attention in the literature and fractal reconstructions using prior knowledge of the anisotropy have not been done before. Several different versions of the fractal reconstruction method are developed and the control of the dimension of the reconstructed surface is carefully investigated. Example reconstructions are then presented, using both artificial and real fractals. The subsampling of the data is performed both at random and in regular patterns and the reconstruction is forced to extrapolate from as well as interpolate between the data points. Finally the reconstruction method is modified to incorporate knowledge of any anisotropy in the fractal surface. The method is tested on both real and artificial data and shows significant advantages over the regular isotropic reconstruction.