Joint wave-equation traveltime inversion of direct and reflected waves for VSP data

Full-Waveform Inversion (FWI) of surface data builds the long- and short-wavelength components of the subsurface model using the low-wavenumber kernel of the diving wave and the high-wavenumber migration isochrones, respectively. FWI is prone to falling into local extremes when the penetration depth of the diving wave is insufficient. Wave-Equation Reflection Traveltime Inversion (RWETI) adopts the tomographic kernel of reflections to update the velocity model. RWETI cannot obtain satisfactory inversion results, especially when large errors are present in the velocity of the shallow part. For VSP survey, the direct wave is a transmitted wave with a large penetration depth, which can be used for building the background velocity model around the borehole. We develop a joint Wave-Equation Traveltime Inversion (WETI) method of direct and reflected waves for VSP data. We employ a global optimization parameter inversion method to implement the separation of down- and up-going waves and the separation of and P- and S-waves, construct a joint objective function based on the traveltime residuals of down-going direct and up-going reflected waves, derive the corresponding adjoint-state equation and gradient formulas, and introduce a hierarchical inversion workflow of background velocity and reflectivity model. The applications on synthetic and field VSP data reveal that joint WETI of direct and reflected waves can reconstruct kinematically accurate velocity model, which is a reasonable starting model for subsequent FWI. Additionally, joint WETI is more accurate than direct/reflected-wave-based WETI and less time-consuming than cascaded WETI of direct and reflected waves.