A model inversion framework is proposed for the recovery of the depth profile of a rough surface. A broadband sound source is placed above the surface of interest and the scattered sound pressure is measured at a microphone array. The problem is modelled analytically using the Kirchhoff approximation, which provides a computationally efficient forward model, with reasonable accuracy at high frequencies or in the far field. The inverse problem is formulated in a statistical sense within the Bayesian framework and sampled using a Markov chain Monte Carlo algorithm. In order to shorten the burn-in sampling phase, an initial solution obtained by deterministic optimisation is used. Special attention is devoted to modelling the smoothness of the surface using a prior probability distribution. The procedure is demonstrated experimentally on a surface with one-dimensional roughness.