Inverse scattering in a stratified medium

Abstract

In this analysis the following problem is considered: A point harmonic source at frequencies ω1 and ω2 is located in the region z<0, where the assumption is made that both the sound speed and density profiles are known. For z>0 the sound speed and density profiles are unknown. These two profiles are recovered from a measurement of the pressure field for all r at some fixed depth z<0 at the two frequencies. Trace formula methods are used. The following assumptions are needed: First, c(z) and ρ(z) approach c1, ρ1 as z→−∞, and c2, ρ2 as z→+∞, and although c2, ρ2 are not known, it must be known that c2>c1. Second, the angular frequency of the source must be such that no trapped modes are excited. While the sound speed c(z) can be complex, the limiting values c1 and c2 must be real. If c(z) also depends on frequency and the form of the dependence is known, say, c(z)=cR(z) +iωλ(z)cI(z), then cR(z), cI(z), and λ(z) can be recovered. Explicit expressions are obtained in the realistic case when cI(z) is not large. If a density is known and cI(z) =0, then only a measurement at one frequency is required. Two numerical examples are given; in both examples ρ(z)≡1 and c(z) is real. The first example is for a monotically increasing profile and the second has a low velocity zone. Typical sediment parameters are used.