### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. An Ionospheric Forward Model for GAIM
- 3. Optimization Methods
- 4. Observation System Simulation Experiments
- 5. Conclusions and Discussions
- Acknowledgments
- References

[1] A global assimilative ionospheric model (GAIM) has been developed to improve the modeling of ionospheric weather. GAIM adopts a fixed three-dimensional Eulerian grid following a geomagnetic dipole configuration. A four-dimensional variational approach (4DVAR) with the adjoint technique is presented, which attempts to minimize the difference between modeled and measured line-of-sight total electron content (TEC) using nonlinear least squares minimization. The minimization is achieved by solving for corrections to the initial (climatological) model drivers so that the density state becomes consistent with the observations. The 4DVAR approach is exercised with GAIM in an observation system simulation experiment (OSSE) conducted for estimating the weather behavior of **E** × **B** drift at low latitudes. The OSSE takes the constellation of global positioning system (GPS) satellites and an existing global GPS receiver network as the observation system. The effectiveness of the 4DVAR technique with such an observation system is assessed in the experiment, which indicates that one can solve for the low-latitude **E** × **B** drift and improve the density modeling using ground-based, integrated line-of-sight (TEC) measurements from a relatively small number of stations.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. An Ionospheric Forward Model for GAIM
- 3. Optimization Methods
- 4. Observation System Simulation Experiments
- 5. Conclusions and Discussions
- Acknowledgments
- References

[2] Numerous first-principles ionospheric models, including those uncoupled and coupled with the thermosphere, have been developed in the past three decades [e.g., *Anderson*, 1973; *Schunk and Walker*, 1973; *Fuller-Rowell and Rees*, 1980; *Sojka and Schunk*, 1985; *Roble et al.*, 1988; *Torr et al.*, 1990; *Richmond et al.*, 1992; *Bailey et al.*, 1993; *Huba et al.*, 2000]. The success of such modeling depends on accurate knowledge of parameters that include solar EUV radiation, thermospheric densities, composition and temperature, electric fields, neutral winds, and auroral precipitation. These drivers enter into the collisional plasma hydrodynamic equations as inputs and control the ionization, dynamical and chemical processes. This dependence creates a major challenge in modeling ionospheric weather due to the difficulty in acquiring the required knowledge of the model inputs continuously, globally, and in real-time. There have been efforts in the past to use measured ionospheric total electron content (TEC, which corresponds to integrated electron density along line-of-sight) at distributed middle latitudes to constrain such a theoretical model by estimating perturbed **E** × **B** drift and meridional wind [e.g., *Pi et al.*, 1993]. However, those investigations were mainly intended to understand the physics processes, i.e., the electric field penetration and thermospheric circulation processes under geomagnetic storm and substorm conditions, while the model used was not designed for assimilating data.

[3] Data-driven techniques provide an alternative to the first-principles modeling approach. These techniques include computerized ionospheric tomography (CIT) [e.g., *Austin et al.*, 1988] and TEC mapping [e.g., *Mannucci et al.*, 1998; *Iijima et al.*, 1999]. The data-driven techniques apply mathematical means to estimate the ionospheric state variables (i.e., electron density or vertical TEC) from inversion of line-of-sight TEC measurements. The fundamental limitation of the data-driven techniques is the lack of prediction capability because they do not deal with the physics processes.

[4] Data assimilation is a powerful tool to bridge models and data. It has been practiced over many decades in the meteorology research community for the Numerical Weather Prediction (NWP) program. In meteorologic or atmospheric data assimilation practice, measurements are assimilated into theoretical models to bring the model in line with the data, so that the model predictions can be more accurate. This approach, however, only became possible and effective when numerous data were available continuously. In the case of the ionosphere, this has not been possible until recently when global and regional networks of instruments were established and the management of data collection and archive was centralized. One example is the constellation of global positioning system (GPS) satellites and the global GPS receiver network. The latter is overseen by the International GPS Service (IGS) for Geodynamics and currently includes more than 300 globally distributed permanent ground stations, and the number of stations is still rapidly increasing.

[5] There are several components to a data assimilation system: (1) A forward model which is the dynamical ionospheric model that propagates the state in time based on the physics; (2) a data processor which edits and prepares the data to be assimilated; (3) an observation operator which maps the model state to the observations; and (4) an optimization module which adjusts the model state and/or drivers so that the difference between the model predicted measurements and the true measurements is minimized. Various optimization schemes exist and they include a four-dimensional variational approach (4DVAR), which is the subject of this paper, and recursive filtering techniques such as the Kalman filter described by *Hajj et al.* [2000].

[6] An effort is being made under the sponsorship of the U.S. Department of Defense through a MultiDisciplinary University Research Initiative to develop assimilative ionospheric models capable of assimilating a variety of ground-based and satellite-based observations. Two consortia, one led by the Utah State University (USU) and one by the University of Southern California (USC), have been formed and are currently investigating various ionospheric data assimilation techniques. In this paper we describe the USC/Jet Propulsion Laboratory effort to build a Global Assimilative Ionospheric Model (GAIM) (to be distinguished from the Global Assimilation of Ionospheric Measurements—GAIM—effort by the USU consortium). In particular, we describe the development of a new ionospheric forward model to circumvent some of the limitations of traditional ionospheric models when applied to data assimilation. The details of this model and the motivation behind it are explained in section 2. We also discuss the 4DVAR technique (section 3) and apply it to an observation system simulation experiment (OSSE) for assimilating GPS observables (section 4). This OSSE is an initial attempt to assess the effectiveness of the 4DVAR data assimilation technique in improving low-latitude ionospheric modeling by estimating the weather behavior of plasma **E** × **B** drift. Conclusions are given in section 5.

### 2. An Ionospheric Forward Model for GAIM

- Top of page
- Abstract
- 1. Introduction
- 2. An Ionospheric Forward Model for GAIM
- 3. Optimization Methods
- 4. Observation System Simulation Experiments
- 5. Conclusions and Discussions
- Acknowledgments
- References

[7] Most ionospheric models developed in the past were constructed to solve for ion and electron densities in one dimension (1-D) along single geomagnetic field lines in Lagrangian frames (i.e., frames moving with plasma drift perpendicular to the magnetic field lines; see the references cited at the beginning of the introduction). Ion and electron densities in a fixed two-dimensional or three-dimensional frame can be obtained by interpolation of a number of 1-D solutions computed in a moving frame. The major advantage of this approach is that the model is simplified to a 1-D problem. However, ionospheric data assimilation in general requires dealing with 3-D geometry to map modeled quantities to observations at their precise locations. A typical example is assimilation of line-of-sight TEC that can be readily measured using a dual-frequency GPS receiver on the ground or from a low-Earth orbiting satellite. Such GPS observation links pass through large regions in various directions. One-dimensional modeling in a moving Lagrangian frame along single field lines introduces complications in implementing optimization techniques. In particular, the density state must be interpolated onto an Earth-fixed grid before one can compute the observation operator for the GPS links. For this reason, we have constructed a new fully three-dimensional model in a fixed Eulerian frame.

[8] Our starting point was to reconstruct the model developed at the University of Sheffield [*Bailey et al.*, 1993] for middle and low latitudes on a fixed grid. The model has later been extended to high latitudes [*Pi et al.*, 2001] to become a global model including the physics processes of plasma convection and auroral particle precipitation driven by the coupling of the solar wind and magnetosphere. The model is based upon the physics principles of collisional plasma hydrodynamics. It numerically solves for ion and electron densities through the continuity and momentum equations for individual ions,

where *n*_{i} and **v**_{i} are the volume density and vector velocity of the *i*th ion species; *P*_{i} and *L*_{i} are the ion production and loss rates (the former includes both photoionization and auroral production to be discussed later); *m*_{i} is the ion mass; the subscript *j* associated with velocity denotes different species including neutral wind; *T*_{i} is the ion temperature; *k* is the Boltzmann's constant; and *e* is the elementary charge. The continuity and momentum equations for electrons can be written similarly by replacing subscript *i* with *e* in equations (1) and (2) and changing the sign of the Lorentz force term. For electrons, the density is the sum of densities of all individual ions, i.e., . For a single ion model, which is the version of GAIM being discussed here, *n*_{e}= *n*_{i}.

[9] The ion production rate is determined by the photoionization due to absorption of solar EUV radiation and the energy deposition of auroral particle precipitation in the thermosphere at high latitudes. The former process is specified by

where Φ(λ) denotes the intensity of EUV radiation as a function of wavelength λ, σ_{i,k} the ionization and absorption cross-sections of the *i*th and *k*th neutral gas, respectively, *N*_{k} the concentration of the corresponding neutral gas, *H* the scale height, *Ch* the Chapman function, and χ the solar zenith angle. GAIM adopts the cross section values given by [*Torr and Torr*, 1982] at 37 solar EUV spectrum intervals for calculations of ion production due to photoionization. The ion production due to the precipitation of charged particles in the auroral regions is based on the ionization rate given by *Rees* [1989]

where *F* is the incident electron flux (in cgs units) at initial energy *E* (in eV); ε is the energy loss per ionization and its value is given to be 35 eV; *z* is the atmospheric depth at the altitude of interest; *R* is the effective penetration range of precipitating electrons; Λ is the normalized energy dissipation distribution function for a pitch angle distribution; and ρ(*z*) is the atmospheric density at the altitude *h* where *z* is computed. The atmospheric depth *z* and penetration range *R* are determined by

and

[10] GAIM adopts a general form of energy distribution for the incident electrons,

where the parameter ψ can be chosen to represent various forms of distributions (such as a Maxwellian distribution). The ionization rate *q* integrated over an energy range is then computed using equations (4) and (7). The ionization rate for individual ions can be computed by

where *N*_{j} and σ_{j} are the neutral density and ionization cross-section for the *j*th species, respectively, and the summation in the denominator takes major species (N_{2}, O_{2}, and O) into account. Figure 1 shows altitude profiles of the ionization rate for three ion species at several values of initial energy. The auroral production rate of individual ions can be computed from the individual ionization rates with ionization cross sections for the involved species,

where *d*σ_{j}*/dW* and σ_{j} are the differential and total cross sections respectively.

[11] For the GAIM runs considered here the electron density within the *F* region is taken to be that of the O^{+} density. This simplifies our optimization problem since the only density equation that needs to be solved is that arising from the O^{+} ion. Thus the ion loss processes are primarily attributed to the chemical reactions of charge exchange between ions and neutrals, i.e., between O^{+}, N_{2}, and O_{2}. The loss rate (*L*_{i}) is then determined by

where *l*_{i} is the loss rate coefficient. Detailed chemical reactions and associated coefficients to compute *l*_{i} are described in [*Torr and Torr*, 1979].

[12] Dynamical processes in the momentum equation include diffusion and convection that are controlled by the pressure gradient, gravitation (**g**), the Lorentz force, and collisions between species, particularly between ions and neutral particles. The plasma drift velocity can be divided into components perpendicular and parallel to the geomagnetic field (**B**) lines. From equation (2) the perpendicular component can be expressed as (neglecting relatively small terms):

where Ω_{i} is the ion gyro frequency, and **b** is the unit vector of the magnetic field. In the *F* region, the collision frequencies are much smaller than the gyro frequency and the **E** × **B** term on the right hand side dominates. The parallel component can be expressed as:

where *n*_{e} and *T*_{e} are the electron density and temperature, respectively, ν_{ij} the collision frequencies between different species including ion-neutral collision, **v**_{j∣∣} includes the neutral wind component, and τ_{i} is the ion stress tensor due to the possible temperature difference between parallel and perpendicular directions [*Schunk*, 1975]. Equation (12) is derived from a combination of the momentum equation (2) for both ions and electrons by neglecting the inertial terms on the left hand side and other relatively small terms.

[13] Our present model adopts a *p-q-l* coordinate system for the entire globe, which is defined as

where θ and ϕ denote magnetic latitude and longitude, respectively, *r* is the radius of the point of interest from the origin of the coordinate system which can have an offset from the center of the Earth, and *r*_{0} is a reference radius (chosen to be the equatorial radius of the Earth). The coordinates are orthogonal and follow a dipole magnetic field configuration, in which gradients ∇*p* and ∇*l* are perpendicular to **B** in meridional planes and in zonal directions, respectively, while ∇*q* is parallel to **B** along the magnetic flux tube. Using the *p-q-l* coordinate system, the partial differential equations are discretized using a finite volume scheme. The discretized equations are reorganized to form algebraic equations relating the state variables at time steps and at adjacent volume elements or voxels. With specified initial and boundary conditions as well as the forcing terms, the ion state is solved forward in time using a hybrid explicit (to update *n*_{i} with convection)-implicit time integration scheme. The model can be run in a dipole, tilted dipole, or eccentric tilted dipole magnetic frame, and the parameters provided in geographic coordinates are converted to the corresponding magnetic frame.

[14] To better support the needs of data assimilation, our model has been constructed with several features that are different from the traditional approach [e.g., *Anderson*, 1973; *Schunk and Walker*, 1973; *Bailey et al.*, 1993; *Huba et al.*, 2000]. Two major features are: (1) the equations are defined and solved in a Eulerian grid, i.e., the coordinate system is fixed in space unlike Lagrangian frames traditionally adopted that move with plasma; (2) the grid is fully three-dimensional. The typical Lagrangian coordinate frame is dictated by the plasma drift perpendicular to geomagnetic field lines. Our Eulerian approach makes it suitable to apply standard data assimilation approaches that properly account for the state transition and the state covariance, and to apply the adjoint method described later. The fixed 3-D grid makes it convenient to construct the geometry-related observation matrix that maps model state (*n*_{i}) to measurements and affects modeling optimization.

[15] Using a fixed grid requires that boundary conditions be set appropriately. At the lower altitude boundary, chemical equilibrium is assumed. An open upper boundary is adopted where the state and ion flux are handled by extrapolation. This can also be augmented with known inward flux. To maintain computational efficiency while still achieving good resolution in the latitude and altitude dimensions, the *p*-*q* grid is chosen to be nonuniform. At low-to-middle latitudes, the *p* interval increases at higher altitudes while *q* is given by

and

where *x* is an independent variable, Γ is a constant, and *q*_{max} is determined by equation (13) to define the modeled region up to a specified middle latitude (with a specified low altitude boundary). Γ can be determined by

using equation (15) to achieve desired spatial resolutions at the equator and middle latitudes, where Δ*q*_{EQ} and Δ*q*_{MIDLAT} are the step lengths of *q* at the magnetic equator and the middle latitude of interest (which determines *q*_{MIDLAT}). With appropriately specified Γ, *q*_{max}, *dx*, and the range of *x*, as well as *p* values, the *p*-*q* grid so determined satisfies the required latitude and altitude resolutions in the magnetic dipole frame for the modeled low and middle latitudes. This varying *p*-*q* scheme applies up to the specified middle latitude. At higher latitudes *p* and *q* are specified such that the latitude and altitude spacing are either evenly distributed or varying to satisfy special resolution requirements in particular regions. The model grid is also made flexible either to extend to the plasmasphere with complete magnetic flux tubes or to include only partial flux tubes. Figure 2 shows an example of the grid with mixed full and partial flux tubes.

[16] GAIM uses the following well-developed empirical models for various input parameters: thermospheric densities and winds (MSIS [*Hedin*, 1991], HWM [*Hedin et al.*, 1996]), solar EUV [*Tobiska*, 1991], electric fields [e.g., *Fejer et al.*, 1991; *Heppner and Maynard*, 1987; *Scherliess and Fejer*, 1999], and electron energy precipitation flux [*Fuller-Rowell and Evans*, 1987]. As examples, Figures 3–5 show empirical patterns of the incident electron energy flux and characteristic electron energy for level 5 of the hemispheric power input, as well as the convection electric potential. Figure 6 shows an example of a global model run for conditions of solar maximum, the March equinox, interplanetary magnetic field (IMF) components *B*_{z} < 0 and *B*_{y} > 0, and energy level 5 for auroral precipitation. The GAIM forward model has been run under various geophysical conditions and the results have been compared with independent techniques, such as TEC measurements from the TOPEX altimeter, global ionospheric maps derived from GPS measurements, electron density profiles retrieved from GPS-Low Earth Obiter occultations or satellite airglow limb scans, other models, etc. These results will be described in separate papers.

### 3. Optimization Methods

- Top of page
- Abstract
- 1. Introduction
- 2. An Ionospheric Forward Model for GAIM
- 3. Optimization Methods
- 4. Observation System Simulation Experiments
- 5. Conclusions and Discussions
- Acknowledgments
- References

[17] A common method for performing data assimilation is the variational approach (four-dimensional variational approach with time domain as the 4th dimension, briefly 4DVAR). In 4DVAR, the data assimilation is formulated as a problem of minimizing a nonlinear functional under a system of constraints. In ionospheric data assimilation, the unknowns for the optimization problem are the ion densities **n**_{0} at time 0, corresponding to the initial state of a data assimilation cycle, and the parameters α which specify driving forces such as the electric field or plasma drift, solar EUV flux, thermospheric wind, and/or neutral densities (bold letters are used here to represent vectors). Given a set of measurements **y**_{k} at time *t*_{k}, the assimilation problem can be cast as an optimization problem where we search for **n**_{0} and α which minimize the following cost functional subject to the constraints of the dynamical model equations

The three terms on the RHS correspond respectively to (1) the difference between the actual measurements and those predicted by the model at all observation times during the assimilation cycle (*H*_{k}, known as the observation matrix, maps the model state **n**_{k} to the measurements **y**_{k} both at the time *t*_{k}), (2) the difference between the state **n**_{0} and an apriori guess of **n**_{0}, and (3) the difference between α and an apriori guess of α. γ_{n} and γ_{F} are regularization coefficients in which the subscripts *n* and *F* denote the state and driving forces, respectively. The coefficients γ_{n} and γ_{F} act as regularization factors, which prevent the solutions of **n**_{0} and α from deviating too far from their apriori values. In the present study we set γ_{n} to zero and choose not to minimize *J* with respect to **n**_{0}. The reason is that as the assimilation cycle proceeds, adjustment of driving forces alone will automatically lead to a self-consistent initial state in subsequent assimilation cycles because the state equation constraints are strictly enforced. In addition, setting γ_{n} to zero also keeps the number of variational parameters to a manageable size.

[18] In principle, *J* can be minimized using a gradient-based iterative search technique such as the quasi-Newton method whose *i* + 1th iteration is given by the expression

where is the Hessian matrix associated with the cost functional, and ω_{i} are nonnegative weights. There are however major technical challenges in practice. First, *J* is highly nonlinear and each evaluation of the functional requires model integration from time *t*_{0} to *t*_{N}. Note that each iteration step in equation (17) requires the computation of the gradient of the cost functional at the current estimate of the unknown parameters. In the absence of an adjoint model, this is typically achieved via finite difference approximation. If α is of dimension *m*, then each optimization iteration requires *m* + 1 evaluations of the cost functional *J*, and consequently *m* + 1 forward integrations of the model equations. If the dimension of the parameter vector is large this can be computationally prohibitive. In our data assimilation practice, this problem is solved by the use of an adjoint method that provides an elegant and efficient means of computing the gradient of *J*. We describe the adjoint method with the following set of equations (more detailed mathematical description is given by *Rosen et al.* [2001]). If the model equations are written in a discretized form as

then the partial derivatives of the cost functional can be derived by differentiating the cost functional and the state equation as

where η_{k} are defined by the adjoint equation

The adjoint method for computing the gradient of the least squares cost functional then consists of the following steps: (1) integrate the model equation forward in time (equation (19)); (2) integrate the adjoint equation backward in time (equation (21)); compute the gradient of *J* (equation (20)). Using the adjoint method, the computational burden (i.e. the number of required integrations of the model equations) remains essentially fixed and independent of the number of parameters under investigation. This is in contrast to a method that employs a finite difference based approach for computing the gradient. In this case, the number of required integrations of the model, in each iteration of the optimization, increases linearly with the number of unknown parameters to be identified.

[19] Another major concern is that when the number of parameters to be estimated becomes too large, the nonlinear minimization becomes ill-posed. The solution to the second problem is achieved by parameterizing the driving forces with the smallest number of parameters needed to faithfully reproduce the range of the drivers. One example of such a parameterization is applied under this study to the **E** × **B** drift component at the magnetic equator. By neglecting altitude dependence, we describe the vertical ion drift velocity as a function of local time, v_{i⟂}(*LT*), by a small number of parameters (nine as a test for this study), α_{j}, in the following sum

where φ_{j}(*LT*) are 24-hour periodic cubic polynomial spline basis functions. These spline basis functions (shown in Figure 7) are chosen so that a linear combination can represent drift patterns for all longitudes or local time sectors for a given UT interval under various geophysical conditions. The drift at the equator is then mapped to middle latitudes along magnetic flux tubes based on the assumptions of electric equipotential and magnetic flux conservation. More generalized parameterizations can be done to include height dependence and to model other drivers with the intent of keeping the number of parameters to a manageable size.

[20] We would like to point out that there are no explicit probabilistic models used in the formulation of the presented 4DVAR approach. As a consequence, ad hoc constructions of covariance for estimated parameters lack the theoretical rigors. A further development of theoretical framework to address this issue will be reported in following works, though beyond the scope of this manuscript.