The double focal transformation and its application to data reconstruction

Many seismic data processing and imaging processes require densely and regularly sampled data, whereas the actual measurements are mostly irregularly and sparsely sampled. Therefore, seismic data reconstruction methods are utilised as a pre-processing step. Within the class of transformation-based reconstruction techniques, observed seismic data is decomposed into certain basis functions, such as plane waves, parabolas or curvelets. In the corresponding model space the aliasing noise is assumed to have different properties than the seismic signal and can be suppressed. However, in many cases subsurface information is available that cannot be used in these traditional reconstruction methods. Therefore, the double focal transformation was derived as a way to incorporate knowledge about the subsurface in the reconstruction algorithm. The basic principle of the double focal transformation is to focus seismic energy by a back-propagation of the seismic data at the source and receiver side to certain depth levels. As a result, the seismic data are represented by a limited number of samples in the focal domain in a localised area, whereas aliasing noise spreads out. By imposing a sparse solution in the focal domain, aliasing noise is suppressed and data reconstruction beyond aliasing is achieved. To facilitate the process, only a few effective depth levels need to be included, preferably along the major boundaries in the subsurface. Propagation operators from these boundaries to the surface (focal operators) serve as the basis functions of this data decomposition method. Including more depth levels allows a sparser data representation, and hence, increases the reconstruction capability. The more precise the subsurface information is known, the more accurate these propagators can be computed. However, very precise operators are not necessary for a good reconstruction result, because in the reconstruction step (the inverse focal transformation) the effect of these operators is again removed. The calculation of the double focal transformation requires a non-linear inversion process, where the samples in the focal domain are estimated such that they - after inverse transformation - match the input data at the measurement locations. Because the inversion process is under-determined, an extra constraint on the focal domain is applied, for which the minimum L1 norm is chosen. This forces the distribution in the focal domain to be sparse and - thereby - suppresses the aliasing noise. For the inversion a so-called spgl1 solver has been used that is guaranteed to converge to the desired minimum of the defined objective function. It utilises a steepest decent type iterative process, called Spectral Projected Gradient. Seismic data reconstruction via the double focal transform method appears to be robust against inaccuracies in the focal operators up to roughly ten percent velocity error. Furthermore, the method was extended to the full 3D case, where each focal transform sub-domain in principle contains a 5D data space. In addition to the basic focal transformation, the method can be combined with other transforms in order to increase data compression. As an example, the double focal transformation can be combined with the linear Radon transformation, such that the seismic data can be represented sparser and fewer focal operators are necessary. Satisfactory results of focal domain data reconstruction beyond aliasing on 2D and 3D synthetic and 2D field data illustrate the method’s virtues