The purpose of this paper is to provide a novel model to characterize the nonlinear restoring force of a marine vertical riser by using position-and-velocity-dependent polynomials. This model permits the obtaining of a specific state space representation via the Lienard transformation for the design of state observers that identify the structural parameters of vertical risers. The main results presented here are: (i) an approximation of the nonlinear restoring force by means of polynomials and its incorporation into a distributed parameter (DP) model, (ii) the transformation of the DP model into a Lienard system and (iii) an analysis of its observability and identifiability properties.