Impedance inversion from transmission data for the wave equation

We study the problem of determining an unknown impedance eta(x) in the wave equation eta(x)u(tt) - (eta(x)u(x))(x) = 0 using transmission data u(X, t) for X less than or equal to t less than or equal to 3X. Here u(x, t) is an impulse response function, satisfying u(x, t) = 0 for t less than or equal to 0 and -u(x)(0, t) = delta(t), the Dirac delta function. We prove a uniqueness result, and discuss a numerical solution method. A central ingredient in the analysis is that we can relate the transmission inverse problem to an inverse problem for reflection data (X = 0) whose theory is quite well understood.