Preliminary study about variable density acoustic wave equation.
Speed of sound
In a dispersive medium sound speed is a function of sound frequency, through the dispersion relation. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity, while the energy of the disturbance propagates at the group velocity.
Simultaneous linearized inversion of velocity and density profiles for multidimensional acoustic media
The multidimensional inverse scattering problem for an acoustic medium is considered within the homogeneous background Born approximation. The medium is probed by wide-band plane wave sources, and the scattered field is observed along straight-line receiver arrays. The objective is to reconstruct simultaneously the velocity and density profiles of the medium. The time traces observed at the receivers are appropriately filtered to obtain generalized projections of the velocity and density scattering potentials, which are related to the velocity and density perturbations of the medium with respect to their nominal values. The generalized projections are weighted integrals of the scattering potentials; in two dimensions the weighting functions are concentrated along parabolas, in three dimensions they are concentrated over circular paraboloids. The reconstruction problem for the generalized projections is formulated in a way similar to the problem of x-ray, or straight-line tomography. The solution is expressed as a back-projection operation followed by a two or three dimensional space-invariant filtering operation. In the Fourier domain, the resulting image is a linear combination of the velocity and density scattering potentials, where the coefficients depend on the angle of incidence of the probing wave. Therefore, two or more different angles of incidence are necessary to reconstruct the velocity and density scattering potentials separately.
Velocity and density of a two‐dimensional acoustic medium from point source surface data
An inverse acoustic scattering theory and algorithm is presented for the reconstruction of a two‐dimensional inhomogeneous acoustic medium from surface measurements. The measurements of the surface pressure due to a harmonically oscillating surface point source at two arbitrary frequencies allows the separate reconstruction of the density and velocity of the subsurface. This is a first step towards solving the inverse problem of exploration geophysics.