Since objective analysis of meteorological data was first attempted about 50 years ago, data assimilation methods have been successfully applied to various Earth's systems. Data assimilation has benefited by the recent increase of (real-time) global observing systems and computing resources, and promises to increase our understanding and eventual prediction of those systems.

The field of space weather research is also part of this trend and requires the adaptation of modern data assimilation techniques developed in meteorology and oceanography to the near-Earth space regimes. Space weather defines the instantaneous state of the near-Earth space environment. It is a consequence of the eruptive releases of electromagnetic energy and plasma material from the sun and the subsequent dynamic interactions with the Earth's magnetosphere and upper atmosphere. As modern society increasingly relies on technological systems in space, the contribution of space weather research to society is becoming more important. The proposed research concerns algorithmic development of data assimilation methods for the Earth's polar ionosphere, where the most dynamic electromagnetic energy and momentum exchange processes between the upper atmosphere and the magnetosphere occur. The proposed research will help to specify the major forcing of the system; therefore it can potentially bring significant advancement into the field of space weather research when completed.

However, it is not necessarily straightforward to adapt atmospheric and oceanic data assimilation methods to the ionosphere, whose time scale that concerns ``weather'' of the system (a half day) is significantly shorter than for meteorological (a week) and oceanic (1 month) systems. Furthermore, characteristics of spatial coherence for the ionospheric electrodynamic quantities are rather unique because of the nature of electromagnetic interactions. Challenged additionally by relatively sparse observational data, the problem demands rigorous examinations of statistical properties of the system. The validity of these statistical assumptions are crucial to enable the data assimilation analysis to optimally extract information from observational data.

Assimilative Mapping of Ionospheric Electrodynamics (AMIE) procedure, developed by Richmond and Kamide [1988], carries out an objective multivariate functional analysis of high-latitude ionospheric electrodynamic variables: electric fields, electric potential, ionospheric currents, and magnetic field perturbations. The scheme is essentially based on Optimal Interpolation (OI) theory [e.g., Lorenc, 1986]. In the current AMIE procedure, each variable is presumed to be expanded in a series of I basis functions, which are chosen as modified spherical harmonic functions. For the problems to which the AMIE procedure is usually applied the size of state vector (I=244) is smaller than the size of observation vector (300 to 1000). In this paper some technical improvements upon the traditional implementation of the OI method are demonstrated for the storm period of January 9-11, 1997. One of the improvements is to use the set of 11 Empirical Orthogonal Functions (EOFs) of Matsuo et al. [2002] as basis functions (I=11). This extension incorporates realistic spatial coherence of the electric field on large scales into the analysis and also reduces the background error covariance to a diagonal matrix. Furthermore, both observational and background error covariances are parameterized with respect to 4 free parameters, and these parameters are estimated at each analysis time-step using the maximum-likelihood method [Dee, 1995]. As has been demonstrated by Dee [1995], the error covariance parameter estimation can be performed in the innovation covariance space by matching a covariance model to the actual innovation vector. As the size of the problem is relatively small, on-line covariance parameter estimation is feasible. In this way the state dependence of the background error covariance is incorporated systematically into the procedure without resorting to a dynamic data assimilation methodology such as Kalman-filter.

Finally, as an attempt to overcome the difficulties mentioned above, I propose several approaches to augment a current data assimilation procedure for the Earth's polar ionosphere. First, development of a wavelet-based covariance model using observational and simulated data: wavelets are versatile multi-resolution bases to characterize the stochastic features of a spatial field. Since high-latitude electrodynamic quantities exhibit highly non-stationary, inhomogeneous, and anisotropic characteristics, wavelet-based covariance modeling and use of wavelets as basis functions in the procedure are especially promising. This approach opens up the possibility of adaptive localized analysis in future. A second approach is the statistical modeling of relationships between electrodynamic quantities using model-free nonparametric methods. Nonparametric methods are powerful in detecting possible nonlinearities and in exploring functional relationships. For example, once the relationship between the electric field and the auroral conductance is established, the analysis of the electric field will be constrained from the auroral conductance whose spatial distribution is relatively well monitored by imaging from satellites.