Bremmer Series Decomposition of Solutions to the Lossless Wave Equation in Layered Media

In this paper we prove the validity of the following decomposition of the solutions to the lossless wave equation in layered media: The complete output from a K-layered media system, which is comprised of the superposition of primaries, secondaries, tertiaries, etc., can be obtained from a single state space model of order 2K-the complete model-or from an infinite number of models, each of order 2K, the output of the first of which is just the primaries, the output of the second of which is just the secondaries, etc. This decomposition of the solution to the lossless wave equation into physically meaningful constituents (i.e., primaries, secondaries, etc.) is called a Bremmer Series decomposition, after Bremmer, who, in 1951, established a similar decomposition. In many geophysical situations, where reflection coefficients are quite small, the decomposition can be truncated after secondaries or tertiaries; hence, it also represents a way to approximate the solution to the wave equation.