First-arrival traveltime tomography using second generation wavelets

Wavelet decomposition of the slowness model allows a multiscale description of the seismic first-arrival time tomography. We propose the introduction of the so-called second generation wavelets that could be used for any mesh structure and do not require a number of samples, such as the power of two in each direction for fast wavelet transform. A linearized procedure for inverting delayed travel-times considering either slowness coefficients or wavelet coefficients has been set up with an efficient ray tracing at each iteration of the inversion procedure. Wavelet decomposition over constant patches (Haar wavelet) or over linear patches (Battle-Lemarie wavelet) of coefficients at different scales are inverted as unknowns of the tomographic linearized system. Reconstruction of these coefficients depends dynamically on the local resolution when considering dense ray coverage. On simple synthetic examples, it has been found necessary to perform a local resolution analysis for specifying wavelet coefficients to be inverted. This resolution analysis could be performed for an initial smooth reconstructed medium and by designing a bit mask operator it allows fine scales to be inverted in specific areas of the model where the resolution is high while not being inverted in other areas where the resolution is poor: the wavelet decomposition will ease the multiscale reconstruction. A few synthetic examples, such as crosshole tomography or surface-surface tomography illustrate the multiscale feature of wavelet tomography. The second generation wavelet approach seems to be a flexible and rather promising tool for controlling the resolution variation of seismic first-arrival tomography.