Teleseismic body-wave imaging techniques such as receiver function analysis can be notoriously difficult to employ on ocean-bottom seismic data due largely to multiple reverberations within the water and low-velocity sediments. In lieu of suppressing this coherently scattered noise in ocean-bottom receiver functions, these site effects can be modeled in conjunction with shear velocity information from seafloor compliance and surface wave dispersion measurements to discern crustal structure. A novel technique to estimate 1-D crustal shear-velocity profiles from these data using Monte Carlo sampling is presented here. We find that seafloor compliance inversions and P-S conversions observed in the receiver functions provide complimentary constraints on sediment velocity and thickness. Incoherent noise in receiver functions from the MOANA ocean bottom seismic experiment limit the accuracy of the practical analysis at crustal scales, but synthetic recovery tests and comparison with independent unconstrained nonlinear optimization results affirm the utility of this technique in principle.