Laplace-domain waveform inversion versus refraction-traveltime tomography

SUMMARY Geophysicists and applied mathematicians have proposed a rich suite of long-wavelength velocity estimation algorithms to construct starting velocity models for subsequent pre-stack depth migration and inversion. Refraction-traveltime tomography derives subsurface velocity models from picked first-arrival traveltimes. In contrast, Laplace-domain waveform inversion recovers long-wavelength velocity structure using the weighted amplitudes of first and later arrivals. There are several implementations of first-arrival traveltime inversion, with most based on ray tracing, and some based on the damped monochromatic wave equation, which accurately represent simple and finite-frequency first arrivals. Computationally, Laplace-domain wavefield inversion is quite similar to refraction-traveltime tomography using damped monochromatic wavefield, but the objective functions used in inversion are radically different. As in classical ray trace-based traveltime inversion, the objective of refraction-traveltime tomography using damped monochromatic wavefield is to match the phase (traveltime) of the first arrival of each measured seismic trace. In contrast, the objective of Laplace-domain wavefield inversion is to match the weighted amplitudes of both first and later arrivals to the weighted amplitudes of the measured seismic trace. Principles of refraction-traveltime tomography were used to generate velocity models of the earth one century ago. Laplace-domain waveform inversion is a more recently introduced algorithm and has been less rigorously studied by the seismic research community, with many workers believing it be equivalent to finite-frequency first-arrival traveltime tomography. We show that Laplace-domain waveform inversion is both theoretically and empirically different from finite-frequency first-arrival traveltime tomography. Specifically, we examine the Jacobian (sensitivity) kernels used in the two inversion schemes to quantify the different sensitivities (and hence the inversion results) of the two methods. Analysing both surface responses and sensitivity results, we show that the Laplace-domain waveform inversion's sensitivity to later arrivals provides significantly improved resolution of deeper velocity structure than the first-arrival traveltime tomography. We demonstrate this capability using numerical inversion examples using a synthetic five-layer model and the synthetic BP benchmark model. Because of the similar algorithmic structure, Laplace-domain waveform inversion fits neatly as a starting velocity model pre-processing component of a larger (multi) frequency-domain wave equation inversion solution package.