To solve this least-squares problem, the linear approximation of the cost function must be determined. The four-dimensional variational (4DVar) method estimates the physical state by minimizing the squared error between the value estimated by the physical model and the observed data as a cost function. Observed ocean data are often spatially sparse, and the unobserved physical state needs to be estimated from the observed data for a certain period. Expansion of the predictable zone caused by the effect of non-linearity on the wave group velocity is thought to be essential for this improvement. A twin experiment was conducted for non-linear deep-water waves, and the optimization efficiency of the new method-SWEAD using SVD (SWEAD-S)-was significantly greater than that of SWEAD. A new criterion for reusing the ensemble was also devised for this new method, increasing the dimension of search directions. Here, the Jacobian was first approximated by the linear dispersion relationship and successively updated to consider the non-linearity of the physical system. To concentrate ensembles on the predictable zone, we propose using singular value decomposition (SVD) of the approximated Jacobian of the squared error. However, Fourier modes are global bases and could be redundant for a narrow predictable zone confined by a dispersion relationship. SWEAD generates ensembles of search directions from Fourier modes to numerically differentiate the squared error between observed data and a physical model.

For circumstances in which observed data are too limited and sparse for oceanographic data assimilation, the surface wave reconstruction by ensemble adjoint-free data assimilation (SWEAD) method was developed in a previous study.

The ensemble-based variational method is easier to implement and parallelize than the adjoint method. Research Institute, Nippon Kaiji Kyokai, Tokyo, Japan.

Waves 12 complete 2021