This paper proposes MpIC, an on-manifold derivation of the probabilistic Iterative Correspondence (pIC) algorithm, which is a stochastic version of the original Iterative Closest Point. It is developed in the context of autonomous underwater karst exploration based on acoustic sonars. First, a derivation of pIC based on the Lie group structure of S E ( 3 ) is developed. The closed-form expression of the covariance modeling the estimated rigid transformation is also provided. In a second part, its application to 3D scan matching between acoustic sonar measurements is proposed. It is a prolongation of previous work on elevation angle estimation from wide-beam acoustic sonar. While the pIC approach proposed is intended to be a key component in a Simultaneous Localization and Mapping framework, this paper focuses on assessing its viability on a unitary basis. As ground truth data in karst aquifer are difficult to obtain, quantitative experiments are carried out on a simulated karst environment and show improvement compared to previous state-of-the-art approach. The algorithm is also evaluated on a real underwater cave dataset demonstrating its practical applicability.