## Data Assimilation and Kalman Filters

As a practical matter, there is a natural tendency to trust data over theory if there is some assurance that the data truly represents the quantity being observed. In practice, data is obtained from measurements by imperfect instruments against a background of noise and other factors that may camouflage the true values be sought. Even if we are convinced that the data is of high quality, the application of multi-station data in development of an ionospheric "map" can be problematic for a variety of reasons. Some include: a paucity of observations over oceanic areas, and an irregular spacing of data derived from observing platforms in both space and time.

Many attempts have been made to assimilate various data types (i.e., foF2 from ionosondes) to produce instantaneous maps over regions where observing platforms are sufficiently dense. The European COST 238 and COST 251 Actions come to mind, and these will be discussed in Section 3.12.3. American sector, European region and polar maps of GPS-derived TEC have also been produced with some success. Worldwide maps can be produced by piecing together local "maps". But how can we make the individual maps consistent with each other in the intervening regions where there is little data upon which to develop a data-driven model? One approach is to develop a transitional region that gracefully merges with climatology after a certain distance. This appears to be a method used in the COST Actions pertaining to the ionosphere and radio communications.

The Kalman filter was developed decades ago [Kalman, 1960], and has found many engineering and military applications [Anderson and Moore, 1979]. The IEEE has reprinted a number of classic papers [Sorenson, 1985]. Of special interest is the so-called extended Kalman filter [Costa and Moore, 1991], The main purpose of the Kalman filter in the present context is to determine the state of a system (e.g., the ionosphere) from measurements that contain random errors. There are a number of error sources in ionospheric measurement systems, and the error variances will depend upon the type of measurement. In simple terms, the question addressed by a Kalman filter is as follows: Given our knowledge of the general physical behavior of the ionosphere, and given all of our diverse measurements at our disposal, what is the best estimate of specified ionospheric variables. We know how the ionosphere should behave on the basis of a physical model, and we have a number of measurements of specified parameters, so how can one evaluate the complete state of the system? It seems obvious that we should be able to do better than simply take the measurements as the only basis for system state estimation, most especially if there is an abundance of measurement noise. The Kalman filter is an algorithm that minimizes the estimation error. The Kalman filter equations may be formulated in a number of ways, and it is beyond the scope of this manuscript to present them. The Kalman filter algorithm involves a considerable amount of matrix algebra in its application, but the reader can review this field of mathematics in books by Golub and Van Loan [1989] and Horn and Johnson [1985]. In the GAIM programs, described in Section 3.12.2, a Kalman filter approach is used.

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