Moveout analysis of wave-equation extended images

Conventional velocity analysis applied to images produced by wave-equation migration with a crosscorrelation imaging condition uses moveout information from space lags or focusing information from time lag. However, more robust velocity-estimation methods can be designed to simultaneously take advantage of the semblance and focusing information provided by migrated images. Such a velocity estimation requires characterization of the moveout surfaces defined jointly for space- and time-lags extended images. The analytic solutions to the moveout surfaces can be derived by solving the system of equations representing the shifted source and receiver wavefields. The superposition of the surfaces from many experiments (shots) is equivalent to the envelope for the family of the individual surface. The envelope forms a shape that can be characterized as a cone in the extended space of depth, space lag, and time lag. When imaged with the correct velocity, the apex of the cone is located at the correct reflection depth and at zero space and time lags. When imaged with the incorrect velocity, the apex of the cone shifts in the depth direction and along the time-lag axis. The characteristics of the cones are directly related to the quality of the velocity model. Thus, their analysis provides a rich source of information for velocity model-building. Synthetic examples verify the derived formulas characterizing the moveout surfaces. The analytic formulas match the numeric experiments well, demonstrating the accuracy of the formulas. Based on information provided by the extended imaging condition, future application for velocity updates can benefit from the robustness of the depth-focusing analysis and of the high resolution of the semblance analysis.