A Comparison of Two Travel-time Tomography Schemes for Crosshole Radar Data: Eikonal-equation-based Inversion Versus Ray-based Inversion

Abstract

Some test studies were performed for comparison of two travel-time inversion schemes for tomographic evaluation of crosshole ground-penetrating radar (GPR) data. The first scheme was a linearized inversion based on Tikhonov regularization (Method 1). In this scheme, ray tracing was not a part of the inversion algorithm and the Jacobian matrix was calculated by numerical differentiation. Travel-time calculations were performed by a finite-difference eikonal equation solver. Model velocity fields were updated by matrix inversion techniques using iterative conjugate gradient solvers. The inversion process was stabilized by a smoothness-constrained regularization. The second scheme was based on a ray tracing algorithm (Method 2) and velocities were updated by a simultaneous iterative reconstruction technique (SIRT) using both straight- and curved-ray approximations. The test studies included synthetic travel-time data sets generated from the models with various velocity distributions. Broyden’s update was implemented within Method 1 to expedite the calculation of the Jacobian matrix, and this greatly improved the computational performance. In the tests, the effect of the regularization parameter on the models from Method 1 was examined. Also, how the straight-ray and curved-ray assumptions affected the solutions from Method 2 was illustrated. The effect of the initial velocity distribution on the resulting tomograms was exemplified by the solutions from both Method 1 and Method 2. The velocity tomograms from Method 1 were characterized by smaller travel-time residuals, Euclidean distances and lower errors in the velocity of cells. Also, the convergence rates of the solutions from Method 1 were faster than those from Method 2. Method 1 better imaged the zones with the high velocity contrast than Method 2, and both methods produced similar velocity distributions within the zones with low velocity contrast. Overall, Method 1 yielded better solutions compared to Method 2, and the curved-ray inversion generated relatively better results than the straight-ray inversion.