Inversion of CMP gathers for P- and S-velocity

A method for inverting common-midpoint gather data for the parameters P-velocity, S-velocity, density, quality factor, and source waveform is described. The method uses an iterative nonlinear least-squares formulation for overdeter-mined problems. Therefore, an efficient modeling theory is required to enable the required iterations of parameters to be carried out within a reannable amount of time. The iterations proceed until a solution is located that minimizes the square error between the modeled and real data. The practical limitations are the ability of the modeling theory to simulate the real data, and the sizes of the data and parameter spaces which determine matrix dimensions and hence speed of the algorithm. A simple ray tracing scheme for horizontal layers is used as the modeling theory, and an inversion for P- and S-velocities in a 16-layer model is performed for synthetic data generated by the same modeling theory. In this case, the inversion scheme is like a form of velocity analysis that refines P-velocity by allowing for nonhyperbolic moveout caused by ray bending, and also computes 5-velocity using the amplitude and phase variation with offset. The P- and S-velocities resulting from the inversion are in reasonably close agreement with the true values used to generate the synthetic data.