## Section: New Results

### Approximation of one-way equations

Participants : Hélène Barucq, Julien Diaz, Taous-Meriem Laleg-Kirati.

Seismic migration techniques used in petroleum field are based on the resolution of the wave equation. We are interested in this study in a one way formulation of this equation. The numerical resolution of this problem is difficult and requires the approximation of a Fourier Integral Operator (FIO). Computing FIO is very heavy (long time computation and big storage space). An algorithm based on a Fourier transform representation of FIO was proposed in [82] where the symbol of the FIO was approximated by separation variables functions. Although this approach reduces the computational cost, it does not give good results in heterogenous media. The objective of this study consists in developing a fast and precise algorithm for FIO computation. So first we have been interested in the application of *curvelets* , which are recent multiscale tools for data representation, analysis, and synthesis. The curvelets suggest a new form of multiscale analysis combining ideas of geometry and multiscale analysis [49] , [50] . As shown in [48] , curvelets provides a parsimonious representation of FIO but unfortunately they are not as good numerically as described by the theory.
Then many other algorithms for FIO computation have been developed for example in [58] where a method based on the discrete symbol calculus was introduced. We have chosen to apply an algorithm proposed in [57] . Unlike the curvelets which works in the phase space, the product of the frequency and spatial spaces, this algorithm decomposes the FIO in the frequency domain. The FIO kernel is decomposed into two terms: a diffeomorphism which can be computed with a nonuniform Fast Fourier Transform FFT and a residual factor computed with a numerical separation of the spatial and frequency variables. The computational cost and the storage space in this case are sensitively reduced. A review article is under preparation.