Simultaneous Inversion of Velocity and Density Profiles

~~~~~~Summary ~difference between the total field P and the incident field Po. Go is the Green's function associated with a point source in a homo-The multidimensional inverse scattering problem for an acous-geneous medium. U,(.) a [c 2 /c 2 (X)]-1 and U,(z) a ln[p(_)/po] tic medium is considered within the homogeneous background can be termed the "velocity scattering potential" and the "den-Born approximation. The objective is to reconstruct simulta-sity scattering potential", respectively. co is the propagation ve-neously the velocity and density profiles of the medium. The locity, and po is the density of the background medium. We medium is probed by wide-band plane-wave sources, and the time assume that c(x) and p(_) do not deviate significantly from the traces observed at the receivers are appropriately filtered to ob-background values of co and po; consequently the Uc(_) and U,(x_) tain generalized projections of the velocity and density scattering values are small with respect to 1. We also assume that U¢,(x) potentials, which are related to the velocity and density varia-and Up(x) have the bounded support V, which is disconnected tions in the medium. The generalized projections are weighted from the receiver array. integrals of the scattering potentials; in the two-dimensional ge-From (2), it can be deduced that the scattering pattern due to ometry the weighting functions are concentrated along parabo-U(x.) is that of a monopole, whereas the scattering pattern due las. The reconstruction problem for the generalized projections is to Up(z) is that of the sum of a monopole and a dipole. Therefore, formulated in a way similar to the problem of x-ray, or straight-the scattering due to density perturbations is most prominent for line tomography. The solution is expressed as a backprojection reflected waves, and the least prominent for reflected ones. operation followed by a two dimensional space-invariant filter-The incident wave is given as P,(z',w) = e', where. = ing operation. In the Fourier domain, the resulting image is a (cos , sin 8) is the unit vector which indicates the angle of in-linear combination of the velocity and density scattering poten-cidence of the plane-wave source. For the 2-D geometry, the tials, where the coefficients depend on the angle of incidence of Green's function is given as Go(z, ', w) =)(k-z'l)/4, the probing wave. Therefore, two or more different angles of where H(1)(indicates es the Hankel function of order zero and incidence are necessary to solve for the velocity and …