  We propose and study several inverse problems associated with the nonlinear
progressive waves that arise in infrasonic inversions. The nonlinear
progressive equation (NPE) is of a quasilinear form $\partial_t^2 u=\Delta f(x,
u)$ with $f(x, u)=c_1(x) u+c_2(x) u^n$, $n\geq 2$, and can be derived from the
hyperbolic system of conservation laws associated with the Euler equations. We
establish unique identifiability results in determining $f(x, u)$ as well as
the associated initial data by the boundary measurement. Our analysis relies on
high-order linearisation and construction of proper Gaussian beam solutions for
the underlying wave equations. In addition to its theoretical interest, we
connect our study to applications of practical importance in infrasound
waveform inversion.
