The Helmholtz-Kirchoff 2.5D integral theorem for sign-bit data

The Helmholtz-Kirchoff integral theorem is the basis for Kirchoff migration of seismic data. Essentially, Kirchoff migration is a solution to the wave equation in the presence of obstacles. In practice, prestack migration of typical seismic data sets can be very computer intensive. Consequently, it is beneficial to consider more efficient migration algorithms. The theory presented here offers, potentially, a much more compact Kirchoff migration scheme by reducing the seismic data into its residual sign bits, prior to migration. Sign-bit processing in the seismic industry has historically been limited to stacking, but the theory presented in this paper extends sign-bit data to wave equation processing. Included in this paper is a refinement of prior treatments of sign-bit processing based on statistical analysis. In other words, this treatment is purely mathematical, while earlier treatments tend to be partially intuitive and partially mathematical. This paper is the theoretical component of a dual presentation, in which the second paper will include computational examples.