Factorized One-Way Wave Equations

Abstract

The method used to factorize the longitudinal wave equation has been known for many decades. Using this knowledge, the classical 2nd-order partial differential Equation (PDE) established by Cauchy has been split into two 1st-order PDEs, in alignment with D’Alemberts’s theory, to create forward- and backward-traveling wave results. Therefore, the Cauchy equation has to be regarded as a two-way wave equation, whose inherent directional ambiguity leads to irregular phantom effects in the numerical finite element (FE) and finite difference (FD) calculations. For seismic applications, a huge number of methods have been developed to reduce these disturbances, but none of these attempts have prevailed to date. However, a priori factorization of the longitudinal wave equation for inhomogeneous media eliminates the above-mentioned ambiguity, and the resulting one-way equations provide the definition of the wave propagation direction by the geometric position of the transmitter and receiver.