Wave equation depth migration—a new method of solution

We present a wave propagation method rigorous in one-way and two-way wave theory for complex velocity varying media with new solutions. In the horizontal wavenumber domain, the first-order differential system that governs acoustic wave propagation can be written in terms of field vectors that are coupled in the wavenumber variables through convolutions between the medium and the fields. The differential system can be uncoupled by introducing a reference system with reference velocity equal to the reciprocal of the rms slowness. The uncoupled system of equations has propagator solutions that are coupled in the wavenumber variables. These solutions can be decoupled by introducing simple approximations. This scheme can be exploited for wave equation depth migration. It then is convenient to introduce new field variables that relate to upgoing and downgoing waves in the reference medium. One-way and various two-way wave equations for the laterally varying medium then can be derived by introducing the down-up wave interaction (DUWI) model. The differential equation for the downgoing (incident) field is derived in the zero-order DUWI model, which neglects the interactions with the upgoing field, resulting in a pure one-way wave equation for the downgoing field. Similarly, the zero-order DUWI model yields a one-way wave equation for the upgoing field. In the first-order DUWI model, the downgoing field from the zero-order DUWI model is used as a source for the upgoing field. This solution gives a quasi two-way wave equation which may be used to migrate overturning waves. Noteworthy, the differential equations we derive have analytical solutions for migration in the wavenumber domain. Simple approximations lead to numerically fast migration schemes that can be implemented in a manner like the split-step Fourier migration schemes.