True Amplitude One-Way solution in blocky media

The numerical solution of seismic waves propagation in heterogeneous media can in-volve different methods which are generally based on the solution of the second-order wave equation. Then the second-order wave equation can be solved completely by using a finite difference scheme and it is well-known that this results in a computa-tional burden, especially in case of three-dimensional problems. This surely explains that a lot of people prefer to solve an approximate problem which involves either a truncate expansion of the solution or an approximate wave equation arising from the factoring of the exact one [2]. In the simplest case of a homogeneous medium, the solution can be obtained from the inversion of a system of one-way wave equations based on the decomposition of the wave into a down-going part and an up-going one which reproduces the physical phenomenon very faithfully. In 1996, M. De Hoop [1] developed a new theory based on the micro-local analysis of the wave equation in a complex medium. By using the theory of pseudo-differential operators, one can derive a system of one-way equations even if the medium is heterogeneous. As far as the numerical solution is concerned, it is associated to the inversion of an approximate system which takes more or less the characteristics of the medium into acount. In this work, we intend to show that the amplitude of the numerical solution can be improved by adding a transmission term in the system. This term can be included in the system from two ways and we show that one of them is optimal