Nonlinear ocean wave modeling: Beyond the Volterra Series Expansion and Fokker-Planck Equation

Non-Gaussian sea surfaces are treated using the Markov State Model (MSM). A standard method is given for extending the Non-Linear Dynamical (NLD) equation associated with the Fokker-Planck Equation (FPE), which is the basis of the MSM. This is necessary for processes which are sampled in space and time, as indicated by the Kramers-Moyal Expansion. It is shown how the Volterra Series Expansion (VSE) can be derived from the extended NLD equation, and how the VSE can be orthogonalized. The orthoganalized form is useful for obtaining the VSE from the input-output statistical coherence measures, where the Wiener process basis functions are regarded as the input. A simple example of a sampled sea surface is given to illustrate this approach.