Finite-frequency sensitivity kernels and hierarchical traveltime and waveform inversion of direct and reflected waves from vertical seismic profile data

Wave-equation-based traveltime or waveform inversion updates the velocity model along finite-frequency sensitivity kernels and exhibits higher accuracy than ray-based tomography. Owing to special geometry, the kernels for vertical seismic profile (VSP) data are different from those for surface seismic data. We have used the Born approximation to compute the traveltime and waveform kernels for direct and reflected waves from VSP data. Based on the property of sensitivity kernels for different information and arrivals, we develop a hierarchical inversion scheme: joint wave-equation traveltime inversion (JWETI) of direct and reflected waves, joint waveform inversion (JWI) of direct and reflected waves, and full-waveform inversion (FWI) of the reflected wave. The traveltime kernels, the waveform kernels, and the migration isochrones are used to retrieve the long-, intermediate-, and short-wavelength components of the velocity model, respectively. Numerical tests on synthetic and real walkway VSP data reveal that the JWETI stage obtains a plausible velocity macromodel, which can be treated as a starting model for subsequent JWI and FWI stages. The JWI stage is a transition between JWETI and FWI and helps to improve the structures of the deep layers of the model. The FWI stage offers some high-wavenumber contents to the inverted model. For the VSP survey, the transmitted direct waves can be used to reduce the dependence of FWI on the initial model. However, the conventional reflection-based wave-equation traveltime inversion and FWI still suffer from cycle skipping when the initial model is far away from the real model. In contrast, the hierarchical inversion scheme can yield reasonable inversion results around the borehole, even when large velocity errors are present in the starting model.