Integral formulation for migration in two and three dimensions

Abstract Computer migration of seismic data emerged in the late 1960s as a natural outgrowth of manual migration techniques based on wavefront charts and diffraction curves. Summation (integration) along a diffraction hyperbola was recognized as a way to automate the familiar point-to-point coordinate transformation performed by interpreters in mapping reflections from the x, t (traveltime) domain into the x, z (depth domain).We will discuss the mathematical formulation of migration as a solution to the scalar wave equation in which surface seismic observations are the known boundary values. Solution of this boundary value problem follows standard techniques, and the migrated image is expressed as a surface integral over the known seismic observations when areal or 3-D coverage exists. If only 2-D seismic coverage is available, wave equation migration is still possible by assuming the subsurface and hence surface recorded data do not vary perpendicular to the seismic profile. With this assumption, the surface integral reduces to a line integral over the seismic section, suitably modified to account for the implicit broadside integral. Neither the 2-D or 3-D integral migration algorithms require any approximation to the scalar wave equation. The only limitations imposed are those of space and time sampling, and accurate knowledge of the velocity field.Migration can also be viewed as a downward continuation operation which transforms surface recorded data to a deeper hypothetical recording surface. This transformation is convolutional in nature and the transfer functions in both two and three dimensions are developed and discussed in terms of their characteristic properties. Simple analytic and computer model data are migrated to illustrate the basic properties of migration and the fidelity of the integral method. Finally, applications of these algorithms to field data in both two and three dimensions are presented and discussed in terms of their impact on the seismic image.