In contrast with time-migration, depth migration requires the definition of an interval velocity model. A ray-tracing step links the interval velocity model and the focusing operators. Because of this indirect link, interval velocity can not be directly measured from the data, but is estimated through an inversion process that involves back-projecting travel-time information into the velocity model. This inversion process is underdetermined, and requires a fair amount of human interaction and interpretive judgment to constrain the model and converge to a useful solution. The interpretive process is crucial, but expensive, especially when compared with computational costs which keep failing. To reduce the need for interpretive judgment we split the velocity estimation problem into two steps. First, determine the kinematics of the best focusing operators. Second, estimate the interval velocity from the focusing operators. In this paper we present a methodology to address the first of these two tasks.
Berkhout 1996 recently proposed a method to estimate the best focusing operators in the prestack domain. However, he suggests an updating scheme that requires picking (manual or automatic) of timing discrepancies between images and operators on a trace-by-trace basis. We feel that this approach can be unreliable in complex regions and aim for a much sparser parameterization of the perturbations to the focusing operators. For this purpose, we use the concept developed by Audebert 1996 of homothetic scaling of the travel-time field. The optimal scaling can be reliably and efficiently measured from the data at each reflector location by scanning over appropriate ranges of the scaling factor (). In addition, by using focusing rather than coherency over offsets, there is a potential to do residual migration, making scanning computationally efficient Fomel (1996). Consistent with our previously stated goal of deferring the estimation of interval velocity as much as possible, we prefer to use the measured scaling factor to update the focusing operators, rather than the velocity model. A general theory needs to be developed that relates measured scaling factors to focusing operators. In this paper we show that for a simple case of a constant perturbations, the best focusing point towards the correct perturbation in the focusing operators.