Constrained Traveltime Inversion Using Reflected Waves of Surface Shot Gathers

We formulated an approach to estimate parameters of a layered earth model such as interval velocity, reflector depth, and dip by inverting reflected traveltime with constraints on dipping directions and depth to reflectors. We applied the “shooting method” (Lines, Bourgeois and Covey, 1984) to calculate modeled reflection time. We derived an explicit expression of the Jacobian matrix. The efficiency of computation is achieved by inverting model parameters layer by layer from shallow to deep, which is equivalent to solving a diagonal matrix. We used the LevenbergMarquardt (L-M) method (Marquardt, 1963) and the singular value decomposition (SVD) technique (Lanczos, 1961; Golub and Reinsch, 1970) to solve the least-squares problem. Two kinds of criteria, RMS error and the maximum deviation, were introduced to keep track of the convergence of iterations to reduce the possibility of rejecting an acceptable modified model. Two errorfree synthetic examples, performed on a PC computer, showed the approach was stable and fast.