Introducing wavelets in traveltime tomography: Application to foothills
Résumé
First-arrival traveltime tomography (FATT) applied to wide aperture seismic data has proved to be an interesting tool to investigate the earth structures. The resolution of FATT is controlled both by the theoretical resolution imposed by the first-arrival traveltimes, and the experimental device as well as the structure itself, which may lead to uneven ray coverage. The theoretical resolution power of FATT is limited by the size of the first Fresnel zone for each ray. In the area of limited ray coverage, there may be inconsistencies between the limited resolution power of FATT and the numerical sensitivity kernel of FATT corresponding to rays. Indeed, one raypath induces high frequency information along the ray, even at locations where resolution of FATT is expected to be really low. This can lead to models that have a high degree of data fitting, but which are far from reality or that contain artefacts. A classical way to account for the limited resolution of FATT is to augment the tomographic system with smoothing constraints. However, this kind of smoothing is difficult to tune and generally lacks adaptivity. Frequently, when the smoothing is tuned to remove the footprints of raypaths in poorly illuminated areas, short-scale features that could be resolved according to the resolution of FATT are smoothed accordingly, leading to a loss of information in the well-resolved areas.The resolution of FATT generally varies locally. For example, considering surface acquisition geometries, the shallow structures will be sampled both by short- and long-offset rays running across the heterogeneities with different angles while the deeper structures will be sampled by long-offset rays only, hence a much narrower range of angles. This makes the resolution of FATT decrease fast with respect to depth. Two other factors creating non uniform ray coverage are the presence of low velocity anomalies which create shadow zones in the ray coverage and non uniform source and/or receiver arrays especially in the case of passive tomography.To account for this varying resolution power, adaptive parametrizations based on non structured grids (Bohm et al. 2000) or multigrid (Zhou 2001) have been developed. The size of the elementary cells are adapted to the local ray coverage such that the amount of control to each model parameter tends to be uniform. Here, we proposed an alternative adaptive parameterization based on the wavelet transform (Mallat 1999). We discussed in an EAGE abstract (Delost et al. 2006) the possibilities offered by wavelet parametrization with a canonical synthetic example. We here present a comparison between classical smoothing regularizations and wavelet parametrization in an application to complex foothills data provided by TOTAL.