We invert reflection coefficient measurements of muddy sediment layers along a 12-km seabed transect on the Malta Plateau in the Mediterranean Sea using 1711 source transmissions recorded on a 32-element linear hydrophone array with both source and array towed by an autonomous underwater vehicle. Trans-dimensional Bayesian inference using reversible jump Markov chain Monte Carlo sampling is applied to obtain posterior probability densities of the number of homogeneous sediment layers, their depths, and their geoacoustic parameters. The forward sediment acoustics model is based on the grain-shearing model which obeys physical causality and provides correlation between important geoacoustic properties. Each dataset was treated as one-dimensional seabed structure inversion carried out on high performance clusters, and inversion results for multiple data sets were combined to yield a two-dimensional subsurface profile including full uncertainty analysis. Comparisons of inversion results to piston and gravity core estimates show agreement in both geoacoustic parameter values and depths of discontinuities. In the range-dependent model constructed from inverting the entire data set, dipping and terminating layers are observed along the track with high vertical resolution on the order of 10 cm.