In this paper, a computationally-efficient scheme is developed to assimilate radar images into a pseudo-spectral wave evolution model. The nonlinear wave model uses a fourth-order Runge-Kutta scheme to integrate in time a coupled set of equations for the evolution of the free surface elevation and velocity potential at the free surface. An asymptotic expansion of the model variables in terms of wave steepness parameter is used to close the system of equations and determine the vertical velocity at the free surface from the velocity potential at the free surface. A variational data assimilation scheme is then developed to find an optimal initial wave field that minimizes a cost function defined as the squared difference between model predictions and radar observations over an assimilation interval. The conjugate gradient method is used for the minimization scheme with the adjoint technique used to calculate the gradient of the cost function with respect to the initial condition. Numerical experiments have been conducted with one dimensional pseudo-observations that have been generated by the forward model by adding uncorrelated noise, as well as synthetic radar data to validate the proposed assimilation scheme.

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