Introduction

It is well known that, even if correct velocity information is provided, any conventional time-migration algorithm outputs erroneous images of subsurface structures whenever overburden velocities vary laterally. Depth migration has been demonstrated to be superior to time migration in handling general velocity variations. However, even today, time migration is still widely used in industry. Black and Brzostowski (1992) gave two reasons to explain the continued wide-spread use of time migration. Their first reason concerns computational costs. They estimated that depth migration usually requires an order of magnitude more computer time than time migration. With today's computer power, such a cost ratio can become significant when data collected in a moderate-sized 3-D survey are to be imaged. The second reason is related to the sensitivity of a migration process to velocity errors. Numerical experiments show that time migration is less sensitive to velocity errors than depth migration and is thus useful in starting an iterative algorithm of migration-velocity analysis. I believe that another reason that may extend the use of time migration in the future is the possibility of developing efficient algorithms with which a time-migrated image can be converted into the corresponding depth-migrated image. In this paper, I refer to this type of algorithms as time-to-depth conversion.

The simplest algorithm for time-to-depth conversion is to stretch the vertical axis of a time-migrated image in accordance with the relation between the vertical traveltime and depth. Obviously, such an algorithm is valid only for layered media. Assuming that time migration focuses the energy diffracted from a scatterer to a point, Hubral (1977) introduced the concept of the image ray that defines a time-to-depth conversion valid for arbitrary variations of overburden velocities. The cascade of time migration and image-ray corrections was later proposed (Larner et. al., 1981) as an approximation to true depth migration. Experiments with field data show that the image-ray corrections are often important in the presence of lateral velocity variations. Black and Brzostowski (1992) recently studied the errors of time migration and pointed out that the image-ray method is invalid for reflectors of large dips. They suggested that image-ray corrections should be supplemented by additional spatial corrections.

One of the most important applications of the image ray method is in the process of migration-velocity analysis that usually requires an efficient prestack migration algorithm. However, up till now, the concept of image ray is limited to poststack imaging and the computation of image rays is done through ray tracing that is inefficient. In this paper, I present my studies in two aspects of the image-ray method. Assuming that the validity condition of the image ray method holds, I generalize the image-ray concept to prestack imaging. I derive the mapping functions of the image-ray corrections for both profile imaging and constant offset imaging. I also describe how to compute these mapping functions without actually tracing image rays. For the sake of simplicity, I address the problem only in two-dimensional cases although the concepts and algorithms readily generalize to three dimensions.