DG-based joint transmission-reflection traveltime tomography and its application of borehole seismic data

The limitations of the coverage range and density of transmission wave often result in less-than-ideal results in traveltime tomography. By contrast, joint transmission-reflection traveltime tomography can not only recover deep structures that transmission tomography cannot detect, but also optimize its inversion results. In this article, we perform joint tomography on borehole seismic (vertical seismic profile, reverse vertical seismic profile, and crosswell seismic) data to obtain near-wellbore structures. In the first part, we solve the factored eikonal equation by the discontinuous Galerkin (DG) method to calculate the transmission/reflection traveltime. Owing to the large wavefront curvature near the source point, the traveltime errors generated by the numerical simulation will propagate from the source to all the calculation domains. According to the factorization principle, the equation solution is decomposed into two parts to solve the point-source singularity. To further improve the accuracy of solving traveltime, we use the DG method to solve the factored eikonal equation with additive factors (the factored DG method), obtaining second-order accuracy solution. The adjoint-state method is employed in the inversion section to calculate the gradient of the misfit function. We use the traveltime difference observed inside the model to define the misfit function, which is more suitable for borehole seismic and avoids the influence of surface normal vectors on gradients. Numerical tests applied on models indicate that the joint tomography method has the potential to accurately inverse the seismic structure information near the well and recover the deep underground structure.