  We study an inverse source problem for the acoustic wave equation in a random
waveguide. The goal is to estimate the source of waves from measurements of the
acoustic pressure at a remote array of sensors. The waveguide effect is due to
boundaries that trap the waves and guide them in a preferred (range) direction,
the waveguide axis, along which the medium is unbounded. The random waveguide
is a model of perturbed ideal waveguides which have flat boundaries and are
filled with known media that do not change with range. The perturbation
consists of fluctuations of the boundary and of the wave speed due to numerous
small inhomogeneities in the medium. The fluctuations are uncertain in
applications, which is why we model them with random processes, and they cause
significant cumulative scattering at long ranges from the source. The
scattering effect manifests mathematically as an exponential decay of the
expectation of the acoustic pressure, the coherent part of the wave. The
incoherent wave is modeled by the random fluctuations of the acoustic pressure,
which dominate the expectation at long ranges from the source. We use the
existing theory of wave propagation in random waveguides to analyze the inverse
problem of estimating the source from incoherent wave recordings at remote
arrays. We show how to obtain from the incoherent measurements high fidelity
estimates of the time resolved energy carried by the waveguide modes, and study
the invertibility of the system of transport equations that model energy
propagation in order to estimate the source.
