The problem of inferring side information regarding an underwater acoustic (UWA) channel from the environmental signature imprinted on waveforms by the propagation medium, particularly the transmitter/receiver configuration and relative speed vector, is addressed in this paper. The main contribution of this work is to develop techniques to automatically deal with wavefront classification, i.e., to assign the appropriate number of surface and bottom bounces to a propagation path detected in the set of channel responses observed at a receiver array, considering possible omission or duplication of some paths. This is a key step to develop automatic and robust source localization and velocity estimation algorithms, which are also overviewed here. Our inference algorithms directly operate on linear time-varying UWA channel responses, represented as 3-dimensional delay-Doppler-depth functions (DDDF). At each depth a sparse delay-Doppler spread function (DDSF), analogous to the impulse response of time-invariant systems, is estimated using Basis-Pursuit (BP) methods. The ensemble is processed with energy-based methods to detect the spatio-temporal signatures of wavefronts, which are then matched to candidate environmental configurations by a parametric propagation model. The proposed approaches have been assessed using simulation and empirical data collected during the CALCOM'10 sea trial for various communication ranges from 300 m to 2 km and varying relative speed within the limit of 6 knots.