Cycling the Representer Method With Nonlinear Models
NAVAL RESEARCH LAB STENNIS SPACE CENTER MS OCEANOGRAPHY DIV
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Realistic dynamic systems are often strongly nonlinear, particularly those for the ocean and atmosphere. Applying variational data assimilation to these systems requires the linearization of the nonlinear dynamics about a background state for the cost function minimization, except when the gradient of the cost function can be analytically or explicitly computed. Although there is no unique choice of linearization, the tangent linearization is to be preferred if it can be proven to be numerically stable and accurate. For time intervals extending beyond the scales of nonlinear event development, the tangent linearization cannot be expected to be sufficiently accurate. The variational assimilation would, therefore, not be able to yield a reliable and accurate solution. In this paper, the representer method is used to test this hypothesis with four different nonlinear models. The method can be implemented for successive cycles in order to solve the entire nonlinear problem. By cycling the representer method, it is possible to reduce the assimilation problem into intervals in which the linear theory is able to perform accurately. This study demonstrates that by cycling the representer method, the tangent linearization is sufficiently accurate once adequate assimilation accuracy is achieved in the early cycles. The outer loops that are usually required to contend with the linear assimilation of a nonlinear problem are not required beyond the early cycles because the tangent linear model is sufficiently accurate at this point. The combination of cycling the representer method and limiting the outer loops to one significantly lowers the cost of the overall assimilation problem. In addition, this study shows that weak constraint assimilation is capable of extending the assimilation period beyond the time range of the accuracy of the tangent linear model.
- Operations Research