## Abstract

A rational approach is used to identify efficient schemes for data assimilation in nonlinear ocean–atmosphere models. The conditional mean, a minimum of several cost functionals, is chosen for an optimal estimate. After stating the present goals and describing some of the existing schemes, the constraints and issues particular to ocean–atmosphere data assimilation are emphasized. An approximation to the optimal criterion satisfying the goals and addressing the issues is obtained using heuristic characteristics of geophysical measurements and models. This leads to the notion of an evolving error subspace, of variable size, that spans and tracks the scales and processes where the dominant errors occur. The concept of error subspace statistical estimation (ESSE) is defined. In the present minimum error variance approach, the suboptimal criterion is based on a continued and energetically optimal reduction of the dimension of error covariance matrices. The evolving error subspace is characterized by error singular vectors and values, or in other words, the error principal components and coefficients.

Schemes for filtering and smoothing via ESSE are derived. The data–forecast melding minimizes variance in the error subspace. Nonlinear Monte Carlo forecasts integrate the error subspace in time. The smoothing is based on a statistical approximation approach. Comparisons with existing filtering and smoothing procedures are made. The theoretical and practical advantages of ESSE are discussed. The concepts introduced by the subspace approach are as useful as the practical benefits. The formalism forms a theoretical basis for the intercomparison of reduced dimension assimilation methods and for the validation of specific assumptions for tailored applications. The subspace approach is useful for a wide range of purposes, including nonlinear field and error forecasting, predictability and stability studies, objective analyses, data-driven simulations, model improvements, adaptive sampling, and parameter estimation.

*Corresponding author address:* Dr. Pierre F. J. Lermusiaux, Division of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138.

Email: pierrel@pacific.harvard.edu