## 1. Introduction

[2] It is now well known that the ionosphere displays both a background state (climatology) and a disturbed state (weather) [*Rishbeth and Mendillo*, 2001]. The ionospheric climatology has been successfully represented by both empirical and theoretical models developed in the past [e.g., *Anderson*, 1973a, 1973b; *Bilitza*, 2001; *Bailey et al.*, 1997; *Fuller-Rowell et al.*, 2002; *Huba et al.*, 2000; *Richards and Torr*, 1985; *Roble et al.*, 1988; *Schunk et al.*, 1998; *Sojka and Schunk*, 1985]. However, it has been much more difficult to model the ionospheric weather with both types of models because empirical models are statistically constructed based on observations, whereas the accuracy of theoretical models often depends on the accuracy of external drivers. The ionospheric weather can have detrimental effects on several human activities and operational systems, including high-frequency communications, over-the-horizon radars, and survey and navigation systems using Global Positioning System (GPS) satellites. To avoid the destructive effects of ionospheric weather on military and civilian systems, there is a growing need to more accurately represent and forecast the ionosphere. Recently, many groups attempt to incorporate observations into ionospheric models by using optimization schemes, which are known as data assimilation methods, to give specific representation of ionosphere [*Hajj et al.*, 2000, 2004; *Pi et al.*, 2003, 2004a, 2004b; *Schunk et al.*, 2003, 2004, 2005; *Wang et al.*, 2004; *Scherliess et al.*, 2004, 2006a, 2006b; *Mandrake et al.*, 2005; *Thompson et al.*, 2006]. Data assimilation methods incorporate continuous observations into models, providing a global/regional description of a system that is optimally consistent with both model and data. These methods have been highly successful in weather and climate modeling in meteorology and oceanography, and form the basis of Numerical Weather Prediction (NWP) models routinely used for weather forecastings. The standard data assimilation cycle consists of quality control (to check the validity of the observations), objective analysis (to generate a complete analysis field with the model results and observations), and forecast (to propagate the state forward in time) [*Bust et al.*, 2004; *Daley*, 1991]. Several necessary components to build a data assimilation model include a forward model (which can propagate the state forward in time), observations (which will be mapped to the model state by an observation operator), and an optimization method (which can give an optimal analysis by ingesting observations into the model) [see *Daley*, 1991; *Pi et al.*, 2003]. Various optimization methods, such as optimal interpolation, variational methods (include three-dimensional and four-dimensional variation), and Kalman filter (KF), have been widely used by meteorologists and oceanographers for several decades and encouraging achievements have been obtained [*Bouttier and Courtier*, 1999]. A more detailed description of the terminology and methods can be found in the works of *Bouttier and Courtier* [1999], *Daley* [1991], *Kalnay* [2003], *Maybeck* [1979], and *Talagrand* [1997]. With the significant increase in ionospheric observations and the speed of the computers, data assimilation also became a powerful technique, providing a better specification and forecasting of the global ionosphere [*Schunk et al.*, 2004; *Wang et al.*, 2004].

[3] There is a growing interest in assimilating empirical models [*Angling and Cannon*, 2004; *Bust et al.*, 2004; *Howe et al.*, 1998; *Schlüter et al.*, 2003; *Spencer et al.*, 2004; *Stolle et al.*, 2006]. The major limitation of data assimilation based on empirical model is lack of prediction capability because they do not deal with the physical processes and so can not propagate the state forward in time. There are many techniques that can be used to propagate the state forward including using Gauss-Markov Kalman filter and using a physics based model. The Gauss-Markov process assumes that the errors of the state grow at an exponential rate with time, allowing the state to relax back to climatology according to a specific timescale if observations have become unavailable [*Fuller-Rowell et al.*, 2004; *Scherliess et al.*, 2006a, 2006b]. A physical based model can advance the state from one time to the next on the basis of several dynamical equations and external drivers. Assimilative models that use first-principles physical models attempt to merge the benefits of observations and physical models and provide an improved forecasting. These ionospheric data assimilation models include the University of Southern California/the Jet Propulsion Laboratory Global Assimilative Ionospheric Model (USC/JPL GAIM) [*Hajj et al.*, 2000, 2004; *Mandrake et al.*, 2005; *Pi et al.*, 2003, 2004a, 2004b; *Wang et al.*, 2004], the Utah State University Global Assimilation of Ionospheric Measurements (USU GAIM) [*Scherliess et al.*, 2004, 2006a, 2006b; *Schunk et al.*, 2003, 2004, 2005; *Thompson et al.*, 2006], and the IonoNumerics model [*Khattatov et al.*, 2004, 2005]. At the same time, the observations including electron densities, plasma drifts, magnetometer measurements, and so on are also assimilated into models to accomplish some scientific researches [*Eccles*, 2004; *Lei et al.*, 2004b, 2004c; *Retterer et al.*, 2005; *Richmond and Kamide*, 1988; *Richmond et al.*, 1988; *Sojka et al.*, 2001, 2003; *Zhang et al.*, 1999, 2001, 2002, 2003]. In the ionospheric data assimilation applications, a majority of the employed data assimilation methods are 3DVAR [*Bust et al.*, 2004], 4DVAR [*Pi et al.*, 2003, 2004a, 2004b], and Kalman filter or its approximations [*Fuller-Rowell et al.*, 2004; *Hajj et al.*, 2004; *Scherliess et al.*, 2004]. In comparison with 3DVAR, Kalman filter can propagate the background error covariance forward dynamically and this is its primary reason why many researchers prefer it to other methods [*Kalnay*, 2003]. Kalman filter is proved to be equivalence to 4DVAR, except that Kalman filter externally forwards the error covariance while 4DVAR internally; Kalman filter does not need an adjoint, and 4DVAR relies on the hypothesis that the model is perfect [*Bouttier and Courtier*, 1999]. Therefore Kalman filter has been more widely used in ionospheric data assimilation investigations than other methods. However, the Kalman filter takes an assumption that the model should be linear which most ionospheric models do not satisfy [*Scherliess et al.*, 2004]. Though the extended Kalman filter has been developed to overcome this issue, it can only make a suboptimal estimation and may result in filtering divergence problem when the system is highly nonlinear [*Welch and Bishop*, 2004]. If we want to use Kalman filter in a nonlinear system, the model should be linearized. This is not practical for a highly nonlinear system. Furthermore, full implementation of the Kalman filter is not practical for large dimensional problems, and therefore many approximations such as band-limited Kalman filter or reduced-state approximation are usually used [*Hajj et al.*, 2004; *Scherliess et al.*, 2004].

[4] To overcome the difficulties that Kalman filter has been confronted with, many meteorologists and oceanographers attempted to introduce Monte Carlo based methods to approximate Kalman filter (EnKF) [*Anderson and Anderson*, 1998; *Evensen*, 1994, 2003; *Evensen and van Leeuwen*, 1996; *Houtekamer and Mitchell*, 1998, 2001; *Mitchell and Houtekamer*, 2000; *Mitchell et al.*, 2002]. In the EnKF, the error statistics are calculated from an ensemble of the forward model forecasts which run in parallel. The EnKF was first introduced by *Evensen* [1994] and has become popular recently because of its simple conceptual formulation and relative ease of implementation. It requires neither derivation of a tangent linear operator or adjoint equations nor integrations backward in time [*Evensen*, 2003]. The EnKF has been widely examined and applied in meteorology and oceanography in the past decade. In this paper, we will explore the feasibility of assimilating incoherent scatter radar (ISR) observations into a theoretical ionospheric model by this method. To validate this method, we will also compare the assimilated results between 3DVAR and EnKF.

[5] Although there are many outstanding ionospheric data assimilation models, it is of great importance for us to continue the investigation about ionospheric data assimilation for methodical, practical, and scientific reasons as follows. (1) Every data assimilation method, such as 3DVAR, 4DVAR, KF, or EnKF, has its own advantage and shortcoming. To give a better performance of ionospheric data assimilation model, we should investigate the adaptability of different methods. EnKF, as a recent data assimilation method, has only been investigated in ionospheric or thermospheric data assimilation by few researchers [*Codrescu et al.*, 2004; *Scherliess et al.*, 2006a, 2006b]. So it is significant for us to implement EnKF in ionospheric assimilation for a methodological reason. In this investigation, questions associated with the influence of ensemble size on the assimilation results, and the representation of model errors, will be studied and discussed in detail. (2) The ionospheric observations are usually distributed asymmetrically in space. To provide a specific nowcasting of the ionosphere, it is important for the assimilation model to extend the influences of observations from data-rich regions to data-sparse regions. This task is usually carried out by background error covariance. Usually, 3DVAR method background error covariance and initial forecast error covariance of Kalman filter are obtained by an empirical correlation model [*Bust et al.*, 2004]. However, the correlation coefficients of ionosphere may have temporal and spatial variations. In this investigation, we will give the altitude and local time variations of the correlation coefficients of electron densities derived from the sample covariance. The ISR observations are used to confirm its reliability. (3) As we know, the initial conditions in the theoretical model are not as so important as the model drivers because of the short time constants for electron density changes in E and F regions [*Scherliess et al.*, 2004]. To achieve a better prediction performance, the external driving forces should also be adjusted simultaneously under the real weather conditions [*Pi et al.*, 2003, 2004b; *Schunk et al.*, 2004]. In this investigation, we will improve the prediction ability of the assimilation model by using the equivalent wind method [*Liu et al.*, 2003, 2004].

[6] The theoretical model and the observations used are presented in section 2. In section 3 we give a brief description about the assimilation methods including 3DVAR, KF, and EnKF and the representations of model and observation error. The comparisons between 3DVAR and EnKF are given in section 4. In section 5, we present the assimilated results by EnKF. Finally, discussions and conclusions are given in sections 6 and 7.