Data assimilation is the process of merging measurement data with a model to estimate the states of a system that are not directly measured. By means of data assimilation, we can expand the effectiveness of limited measurements by using the model and, at the same time, increase the accuracy of model estimates using the measurements.

In this dissertation, we survey and develop data assimilation algorithms that are applicable to large-scale nonlinear systems. Very high order dynamics, nonlinearity, and input uncertainties are addressed since they characterize the problems associated with large-scale data assimilation. Specifically, we focus on developing the data assimilation algorithms for the ionosphere-thermosphere using the Global Ionosphere-Thermosphere Model (GITM).

For developing computationally tractable algorithms, we obtain finite-horizon optimal reduced-order estimators for time-varying linear systems, and, subsequently, develop linear suboptimal reduced-complexity estimators. The suboptimal estimators are based on localization and the reduced-rank square root of the error covariance.

To deal with nonlinearity, we use the unscented Kalman filter and ensemble Kalman filter. We apply suboptimal reduced-complexity algorithms developed for linear systems based on the unscented Kalman filter. Also, we develop the ensemble-on-demand Kalman filter, which can be used for the special case of a single global disturbance, and which avoids propagating the ensemble members for all of the time steps. Furthermore, we show that the ensemble size of the ensemble Kalman filter does not have to be unnecessarily large if the statistics of the disturbance sources are identified.

Finally, we apply the ensemble-on-demand Kalman filter and ensemble Kalman filter to data assimilation based on GITM for uncertain solar EUV flux and geomagnetic storm conditions, respectively. We present data assimilation results, through extensive numerical investigations using simulated measurements. While performing simulations, we observe that poor correlations between states should be set to zero to avoid filter instability. In addition, ionosphere and thermosphere measurements can be used together with an appropriate region of data injection to guarantee overall good estimation performance. With those constraints, we show that good estimation results can be obtained using a small ensemble size for each ensemble filter.