We introduce and demonstrate the applicability of a parameter identification algorithm based on the Hilbert transform to nonlinear ocean mooring systems. The mooring dynamical system consists of a submerged small body and includes a geometrically nonlinear restoring force and a nonlinear dissipation function incorporating both viscous and structural damping. By combining a recently developed methodology with a generalized averaging procedure, parameter estimation from the slowly varying envelope dynamics is enabled. System backbone curves obtained from data generated by numerical simulation are compared to those obtained analytically and are found to be accurate. An example large-scale experiment is also considered.