The Double Focal Transformation and its Application to Seismic Data Reconstruction

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 has different properties than the seismic signal and can be penalised. However, in cases that subsurface information is available, this information cannot be used in most reconstruction methods. Therefore, the focal transform was derived as a way to include knowledge about the subsurface within the data reconstruction algorithm and, thereby, increase its potential reconstruction capabilities. The basic principle of the focal transformation is to focus seismic energy along source and receiver coordinates simultaneously. The seismic data are represented by a number of spikes in the focal domain 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 data, from which the propagation operators serve as the basis functions of this data decomposition method. Results on 2D synthetic and field data illustrate the method's virtues.