Consider an acoustic half space $\{ (x,y,z):z > 0\} $ characterized by density $\rho (x,z)$ and sound speed $c(x,z)$, where $\rho $ and c are both close to constant. The problem of recovering $\rho $ and c using as data a line source impulse response function or a plane source impulse response function measured at the surface $\{ z = 0\} $ is studied. The spectral structure of the linear operators arising in this problem is studied in some detail both analytically and numerically. A specific numerical solution method is proposed and illustrated in an example with synthetic data.