Multifractional splines: from seismic singularities to geological transitions

A matching pursuit technique in conjunction with an imaging method is used to obtain quantitative information on geological records from seismic data. The technique is based on a greedy, non-linear search algorithm decomposing data into atoms. These atoms are drawn from a redundant dictionary of seismic waveforms. Fractional splines are used to define this dictionary, whose elements are not only designed to match the observed waveforms but also to span the appropriate family of geological patterns. Consequently, the atom’s parameterization provides localized scale, order and direction information that reveals the stratigraphy and the type of geological transitions. Besides a localized scaling characterization, the atomic decomposition allows for an accurate denoised reconstruction of data with only a small number of atoms. Application of this approach to angles gathers allows us to track geological singularities from seismic data. Our characterization bridges the gap between the analysis of the main features within geologic processes, i.e. the geologic patterns, and the interpretation of their associated seismic response. A case study of Valhall data is presented.