Multiscale analysis of well- and seismic data

Acoustic/elastic surface seismic and borehole oriented remote sensing methods generally derive their existence from the presence of singularities, regions of rapid variation, in the medium properties that carry the waves. Coherent reflections emanate at these singularities and the primary aim of this paper is (i) to find a characterization for the singularity structure by means of scaling exponents and singularity spectra; (ii) to better understand how the singularity structure is being mapped from space (the medium) to space-time (the wavefield); (iii) to initiate a theoretical discussion on the implications of the multiscale analysis findings in relation to wave theory. Applying multiscale analyses to well- and seismic reflection data shows two things. Firstly the earth's subsurface behavior is much more intricate as assumed within the current piece-wise smooth medium representations. And, secondly, it reveals a direct relationship between the singularity structure of the medium, the well-log, and that of the wavefield, the reflection data. The medium's singularity structure shows that the singularities are not limited to jump discontinuities and that they lie dense. These important observations allude to the questions (i) how to improve the understanding of the medium's scale dependence in relation to the amplitudes of waves (ii) how to explain for the mapping of the singularity structure within the current wave theory.