This paper develops an inversion method for the seabed transition layer at the water-sediment interface, often found in muddy sediments, which provides density and sound-speed profiles that were previously not resolvable. The resolution improvements are achieved by introducing a parametrization that captures general depth-dependent gradients in geoacoustic parameters with a small number of parameters. In particular, the gradients are represented by a sum of Bernstein basis functions, weighted by unknown coefficients. Compared to previous forms found in the literature, the Bernstein-based parametrization can represent a wider range of depth-dependent geoacoustic profiles using fewer parameters which leads to reduced uncertainty and improved resolution. In addition, the Bernstein basis is the most stable polynomial representation in that small perturbations to the unknown coefficients result in small, localized perturbations to the geoacoustic profile, thereby providing an efficient exploration of the parameter space using Markov-chain methods in nonlinear inversion. Geoacoustic profiles at four mud sites on the Malta Plateau are studied with the proposed approach. Results show exceptional resolution of density profiles, estimated with low uncertainty and clear sensitivity to sediment features of centimeter scale.