Multi-dimensional Fast Marching approach to wave propagation in heterogenous multipath environment

Sound propagation is described by the wave equation. If in an homogenous free field its resolution is straightforward, any variation from this hypothesis makes the wave equation solution not analytically tractable especially in a shallow water environment. The direct resolution of the wave equation requires in most cases numerical methods such as FDTD (Finite Difference Time Domain) or PSTD (Pseudo Spectral Time Domain). A direct approach however is often extremely computationally expensive (the spatial and temporal discretisation has to be small, of the order of λ/10, for stability criterion) and approximations are necessary for practical reasons. For low frequencies applications, mathematical models for shallow water propagation include Normal Mode Model or Parabolic Equation Model. For higher frequencies (above 1kHz), the most popular method for wave propagation in shallow water is based on Ray theory and geometrical acoustics. Thanks to the infinite frequency assumption, the wave equation simplifies to the eikonal equation which propagates the wavefront of the acoustic pulse. The ray trajectories are computed as perpendicular to the wavefront. In the ideal case of a constant sound velocity profile and perfectly flat interfaces for the surface and the seafloor, an elegant solution is derived from the Mirror theorem: source images are easily geometrically computed by successive symmetries of the source itself. In a second step folding the straight paths linking all the source images to a target computes the multipath. Unfortunately this method fails for non flat seabeds, non constant depth or non constant velocity profile. In this paper we propose an extension to the Mirror theorem to take into account any interface geometry or sound velocity variation (horizontally or vertically) by solving the eikonal equation using the Fast Marching algorithm. We will show that multipath can then be solved by wrapping the wavefront propagation at each interface.