  The recent discovery of inertial waves on the surface of the Sun offers new
possibilities to learn about the solar interior. These waves are long-lived
with a period on the order of the Sun rotation period ($\sim$27 days) and are
sensitive to parameters deep inside the Sun. They are excited by turbulent
convection, leading to a passive imaging problem. In this work, we present the
forward and inverse problem of reconstructing viscosity and differential
rotation on the Sun from cross-covariance observations of these inertial waves.
