Seismic first-arrival tomography with functional description of traveltimes

Abstract

A two-dimensional smoothness-constrained least-squares inversion scheme was applied to seismic first-arrival time data. The inversion scheme was based on the functional description of traveltimes; thus, it did not require a step of ray tracing, and traveltimes were obtained by a finite-difference eikonal solver. The Laplacian difference of cell slownesses was used for the smoothness constraint. Model velocities were obtained by matrix inversion including the QR decomposition and iterative LSQR method. The Jacobian matrix of the partial derivatives was constructed by a finite-difference approximation based on the perturbation of the cell slowness. Since the construction of the Jacobian matrix was the most time-consuming step of the inversion scheme, Broyden's update was used for this matrix, and it was replaced by its numerical approximation obtained from Broyden's method after the third iteration and for all subsequent iterations to expedite the inversion process. This significantly improved the computational performance of the scheme by reducing the computer time for the calculation of the Jacobian matrix. The algorithm was tested by using a number of synthetic and field data sets. The test studies included both surface seismic refraction and crosshole seismic data acquisition configurations. Also image appraisal analyses were performed for the solutions obtained from both surface and crosshole field data sets by calculating model covariance and model resolution matrices. The presented algorithm yielded satisfactory results during the test studies. The stability and fast convergence rate were the main characteristics of the algorithm.