Geometric simplicity as a migration criterion: An application of computational topology to seismic imaging
August Lau, Chuan Yin · 2009
With more sophisticated algorithms like reverse-time migration and waveform inversion, it requires a closer examination as to how we do preprocessing of the input seismic data. Too much signal processing could remove signal which used to be called “noise” since previous noise was an incomplete understanding of signal based on the wave equation. Too little preprocessing could cause migration artifacts when the migration operator is presented with data which does not fit the assumptions. We introduce a new criterion for migration based on geometric simplicity. Most migration optimization or tomography optimization methods are based on amplitude like RMS measurements or semblance or correlations. Geometric constraints are usually difficult to define for migration or inversion since geometry is in general a global concept. In this paper, we will use the Betti numbers from computational topology to describe geometric simplicity. The larger the Betti numbers are in an area, the more complex the seismic response would be for migration. The BO (Betti number of zero homology group) measures connectivity and B1 (Betti number of first homology group) measures one-dimensional holes and B2 (Betti number of second homology group) measures two-dimensional holes. This gives a geometric criterion for migration or inversion methods like tomography.