Currently used one-way wave equations in depth fail at horizontal propagation. One-way wave equations in time do not have that shortcoming; they are omni-directional in space. In these equations, spatial derivatives appear in a pseudo-differential operator-the square root of the Laplacian. With an appropriate definition of this operator, we have proved via ray theory that the solutions of one-way wave equations in time asymptotically approximate the solutions to the two-way wave equation to leading order for forward or reverse time propagation. For us "true-amplitude" is meant in this ray-theoretic (asymptotic) sense. The inverse series in powers of iw in the frequency domain becomes a series in progressing waves in the time domain. The propagation of the leading order progressing wave is governed by the eikonal equation for the two-way wave equation and the slowly varying amplitude of this leading order progressing wave satisfies the same transport equation as for the two-way wave equation. This theory provides a solid theoretical base for the Explicit Marching algorithm for solving reverse time migration and anticipates an inversion-a true-amplitude reverse time migration. We present the correct initial value problems for forward and reverse time Green''s functions. These Green''s functions are the analytic extensions of the Green''s functions for the two-way wave equation with their imaginary parts being the Hilbert transforms of those real (two-way) Green''s functions. The Kirchhoff approximation of asymptotic ray theory in frequency domain applies to progressing waves in time domain, except that the incident wave must also be the analytic extension of the data for the two-way wave equation. A Green''s identity relating solutions of oneway wave equations and their adjoint is derived. This allows us to develop Kirchhoff integral representations from propagation of surface data into the Earth and the propagation of reflection data to the upper surface. Those identities plus identification of the adjoint operator for the forward modeling operator lead to migration and inversion formulas using our analytic Green''s functions. Observed data at the upper surface must be extended to analytic data in order to apply the inversion theory.
True-amplitude one-way wave equations in depth (Zhang, 1993; Zhang et al. 2003) yield high quality solutions for forward modeling problems and inversion. A shortcoming of these equations is that they yield poor results near and fail at horizontal propagation. In theory, the separation into equations for up-going and down-going wave equations has a pathology in the neighborhood of horizontal propagation where the terminology, upward and downward, loses meaning.
Zhang et al. (2007) derived a new equations from the twoway wave equation. They involve the first derivative term in time and so-called analytic Green''s functions. These analytic Greens functions are complex-valued with the imaginary part being the Hilbert transform in time of the real-valued Green''s function of the two-way wave equation. These equations can handle horizontal propagation and turning waves. Further, they provide a new way of doing reverse-time migration, called "Explicit Marching" (EM) method. Unlike the conventional finite-difference methods, EM does not suffer from stability and numerical dispersion problems.