I have constructed integral offset continuation operators by posing and solving an initial value problem for the offset continuation equation (1). For the special cases of continuation to zero offset (DMO) and continuation from zero offset (inverse DMO) the OC operators are related to the known forms of DMO operators: Hale's Fourier DMO, Born DMO, and Liner's ``exact log DMO.'' The discovery of these relations sheds additional light on the problem of amplitude preservation in DMO.
The wave-type process, described by equation (1), contains two different branches, associated with continuation to either larger or smaller offsets. In order to separate the desired (one-way) direction of continuation, one needs to use the first-order derivative of the recorded wavefield with respect to the offset. This requirement is eliminated in the case of inverse DMO, where the offset derivative vanishes to zero according to the reciprocity principle. I propose inverting the amplitude-preserving inverse DMO as a way to create the true-amplitude DMO operator.
In Part Three of this paper, I plan to describe a method of eliminating the first-derivative requirement in the general case of offset continuation. This method will allow me to proceed from the theory of amplitude preserving offset continuation to synthetic tests and real data applications.