New generalized phase shift approach to solve the Helmholtz acoustic wave equation

We have developed and given some proof of concept applications of a new method of solving the Helmholtz wave equation in order to facilitate the exploration of oil and gas. The approach is based on a new way to generalize the "one-way" wave equation, and to impose correct boundary conditions. The full two-way nature of the Helmholtz equation is considered, but converted into a pseudo "one-way" form with a generalized phase shift structure for propagation in the depth z. Two coupled first order partial differential equations in the depth variable z are obtained from the Helmholtz wave equation. Our approach makes use of very simple, standard ideas from differential equations and early ideas on the non-iterative solution of the Lippmann-Schwinger equation in quantum scattering. In addition, a judicious choice of operator splitting is introduced to ensure that only explicit solution techniques are required. This avoids the need for numerical matrix inversions. The initial conditions are more challenging due to the need to ensure that the solution satisfies proper boundary conditions associated with the waves traveling in two directions. This difficulty is resolved by solving the Lippmann-Schwinger integral equation in an explicit, non-iterative fashion. It is solved by essentially "factoring out" the physical boundary conditions, thereby converting the inhomogeneous Lippmann-Schwinger integral equation of the second kind into a Volterra integral equation of the second kind. Due to the special structure of the kernel, which is a consequence of the causal nature of the Green's function in the Lippmann-Schwinger equation, this turns out to be extremely efficient. The coupled first order differential equations will be solved using the "modified Cayley method" developed in Kouri's group some years ago. The Feshbach projection operator technique is used for constructing a solution that is stable with respect to "evanescent" or "non-propagating" waves. This method is tailored to the coupled first order differential equation formulation. Non-reflecting or absorbing boundary conditions are used to get rid of the reflections or wraparound from the artificial boundaries of the velocity grid.