Data assimilation via error subspace statistical estimation. Part I: Theory and schemes

A rational approach is used to identify efficient schemes for data assimila tion in nonlinear ocean-atmosphere models. The conditional mean, a minimum of several cost functionals. is chosen for an optimal estimate. After stati ng the present goals and describing some of the existing schemes, the const raints and issues particular to ocean-atmosphere data assimilation are emph asized. An approximation to the optimal criterion satisfying the goals and addressing the issues is obtained using heuristic characteristics of geophy sical measurements and models. This leads to the notion of an evolving erro r subspace. of variable size, that spans and tracks the scales and processe s where the dominant errors occur. The concept of error subspace statistica l estimation (ESSE) is defined. In the present minimum error variance appro ach, the suboptimal criterion is based on a continued and energetically opt imal 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 fo recasts integrate the error subspace in time. The smoothing is based on a s tatistical 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 t he validation of specific assumptions for tailored applications. The subspa ce approach is useful for a wide range of purposes. including nonlinear fie ld and error forecasting, predictability and stability studies, objective a nalyses, data-driven simulations, model improvements, adaptive sampling, an d parameter estimation.