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Keywords:

  • assimilative ionospheric modeling;
  • ionospheric data assimilation;
  • remote sensing of the ionosphere using GPS;
  • observation system simulation experiments

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Ionospheric Forward Model for GAIM
  5. 3. Optimization Methods
  6. 4. Observation System Simulation Experiments
  7. 5. Conclusions and Discussions
  8. Acknowledgments
  9. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Ionospheric Forward Model for GAIM
  5. 3. Optimization Methods
  6. 4. Observation System Simulation Experiments
  7. 5. Conclusions and Discussions
  8. Acknowledgments
  9. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Ionospheric Forward Model for GAIM
  5. 3. Optimization Methods
  6. 4. Observation System Simulation Experiments
  7. 5. Conclusions and Discussions
  8. Acknowledgments
  9. 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,

  • equation image
  • equation image

where ni and vi are the volume density and vector velocity of the ith ion species; Pi and Li are the ion production and loss rates (the former includes both photoionization and auroral production to be discussed later); mi is the ion mass; the subscript j associated with velocity denotes different species including neutral wind; Ti 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., equation image. For a single ion model, which is the version of GAIM being discussed here, ne= ni.

[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

  • equation image

where Φ(λ) denotes the intensity of EUV radiation as a function of wavelength λ, σi,k the ionization and absorption cross-sections of the ith and kth neutral gas, respectively, Nk 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]

  • equation image

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

  • equation image

and

  • equation image

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

  • equation image

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

  • equation image

where Nj and σj are the neutral density and ionization cross-section for the jth species, respectively, and the summation in the denominator takes major species (N2, O2, 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,

  • equation image

where dσj/dW and σj are the differential and total cross sections respectively.

image

Figure 1. Altitude profiles of ionization rates for N2+ (black), O2+ (red), and O+ (blue) with a flux of 108 electrons cm−2s−1 at several values of initial energy.

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[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+, N2, and O2. The loss rate (Li) is then determined by

  • equation image

where li is the loss rate coefficient. Detailed chemical reactions and associated coefficients to compute li 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):

  • equation image

where Ωi 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:

  • equation image

where ne and Te are the electron density and temperature, respectively, νij the collision frequencies between different species including ion-neutral collision, vj∣∣ includes the neutral wind component, and τi 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

  • equation image

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 r0 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 ni 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 (ni) 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

  • equation image

and

  • equation image

where x is an independent variable, Γ is a constant, and qmax 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

  • equation image

using equation (15) to achieve desired spatial resolutions at the equator and middle latitudes, where ΔqEQ and ΔqMIDLAT are the step lengths of q at the magnetic equator and the middle latitude of interest (which determines qMIDLAT). With appropriately specified Γ, qmax, 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.

image

Figure 2. The GAIM 3-D fixed grid is defined by constant geomagnetic field lines, constant geomagnetic potential lines and longitudinal planes, where the B field is defined by a tilted dipole.

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[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 35 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 Bz < 0 and By > 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.

image

Figure 3. An empirical pattern of incident electron and ion energy flux for level 5 of the hemispheric power input [Fuller-Rowell and Evans, 1987].

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image

Figure 4. An empirical pattern of a characteristic electron energy for level 5 of the hemispheric power input [Fuller-Rowell and Evans, 1987].

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image

Figure 5. A pattern of electric potential computed from an empirical model [Heppner and Maynard, 1987].

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image

Figure 6. An example of global vertical TEC obtained from a GAIM forward model run.

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3. Optimization Methods

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. An Ionospheric Forward Model for GAIM
  5. 3. Optimization Methods
  6. 4. Observation System Simulation Experiments
  7. 5. Conclusions and Discussions
  8. Acknowledgments
  9. 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 n0 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 yk at time tk, the assimilation problem can be cast as an optimization problem where we search for n0 and α which minimize the following cost functional subject to the constraints of the dynamical model equations

  • equation image

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 (Hk, known as the observation matrix, maps the model state nk to the measurements yk both at the time tk), (2) the difference between the state n0 and an apriori guess of n0, and (3) the difference between α and an apriori guess of α. γn and γF are regularization coefficients in which the subscripts n and F denote the state and driving forces, respectively. The coefficients γn and γF act as regularization factors, which prevent the solutions of n0 and α from deviating too far from their apriori values. In the present study we set γn to zero and choose not to minimize J with respect to n0. 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 γn 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

  • equation image

where equation image is the Hessian matrix associated with the cost functional, and ωi 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 t0 to tN. 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

  • equation image

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

  • equation image

where ηk are defined by the adjoint equation

  • equation image

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, vi(LT), by a small number of parameters (nine as a test for this study), αj, in the following sum

  • equation image

where φj(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.

image

Figure 7. The 9 cubic spline basis functions used in the parameterization of equatorial vertical plasma drift are plotted as a function of local time.

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[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.

4. Observation System Simulation Experiments

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

[21] We have conducted observation system simulation experiments (OSSEs) which are studies using simulated data, generated for specific observation systems, to test the impact of data and assimilation methods. Figure 8 is a schematic of an OSSE, which can be described as follows:

image

Figure 8. A schematic diagram describing the Observation System Simulation Experiment (OSSE).

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Producing the Simulated Data

  1. Run the forward model with perturbed drivers (electric fields, winds, neutral densities, EUV, particle precipitation, etc.). This run is used as the “truth” or “weather” run.
  2. Generate data (using results of the model “truth” run) specific to the instrument and observation sampling and geometry under consideration.
  3. Add noise to the data to account for both measurement and representativeness errors.

Assimilation

  1. Generate an initial condition for the density state based on a forward model integration using the unperturbed drivers (i.e., empirical drivers derived from climatology).
  2. Assimilate the synthetic data in the manner described in the optimization section.
  3. Compare the assimilation run to the “truth” run to assess the observability of the system, meaning the ability of the observation system to capture the “true” features of the physical system, with certain levels of resolution and accuracy.

[22] Such OSSEs test the success or failure of the observation system to recover the “ionospheric weather”, and reveal the degree of sensitivity of the overall system to the ionospheric state and drivers. OSSEs therefore can be performed at different levels of complexity, and provide a controlled experimental environment where the strength of the data is precisely understood.

[23] Our observation system consists of 24 to 30 (counting spare ones) DoD GPS satellites, transmitting two L-band signals at 1575.42 MHz (19 cm) and 1227.6 MHz (24.4 cm), and an existing global network of ground-based receivers (the network of the International GPS Service for geodynamics). The satellites are distributed in 6 circular orbits at ∼20,200 km altitude with a period of ∼12 hours so that a ground receiver at any place and any time can receive GPS signals from at least 4 satellites in different directions simultaneously. The present IGS network consists of more than 300 permanent receiver sites. Dual-frequency GPS observables, including satellite-to-receiver pseudorange and carrier phase, are sampled every 30 seconds by each receiver, and such data are collected on a daily and near real-time (hourly) basis at a few data archive centers. Figure 9 shows the coverage of the IGS GPS network as of the year 2001. From the GPS observables one can derive absolute line-of-sight total electron content (TEC) with an accuracy of ∼1 – 2 TEC units and precision of <0.1 TECU (1 TECU = 1016 electrons/m2) using well-developed techniques.

image

Figure 9. The IGS GPS receiver network. Circles represent the field-of-view coverage at a subionospheric height of 450 km and an elevation mask of 10 degrees.

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4.1. Simulation Experiment Setting

[24] A low to midlatitude portion (between ±40° geographic latitudes and covering all longitudes) of the IGS GPS network is used in our experiment. Besides testing the assimilation algorithms, our experiments are also designed to examine the following effects: (1) the observability of the system with a relatively small number of observation links; (2) the assimilation cycle length and number of cycles required to reach optimal solutions under specified conditions. It is worth mentioning that the success of assimilative modeling with a small number of observation sites will have a significant impact on practical operations, including both real-time data acquisition and computational issues related to their assimilation. Although more than 50 sites are available in the region specified above, 27 are selected to test the observability of the system. Figure 10 is a map of the distribution of the 27 stations with a mask of measurement coverage. The actual station coordinates and the ephemeris of GPS satellites are used to compute 3-D observation links at 5-minute intervals for the entire experiment period. Random noise with a standard deviation of 5 TECU, with zero mean and normal distribution, is added to the simulated data.

image

Figure 10. A subset of 27 stations selected from the IGS GPS network used in the OSSE.

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[25] Solar maximum (F10.7 cm = 229) and equinox (March 21, 2000) as well as geomagnetically quiet conditions (magnetic index Ap = 4) are chosen for the modeling and assimilation experiments. These conditions specify the ionospheric model inputs of solar EUV radiation flux, thermospheric densities and temperature, and winds computed from the corresponding empirical models. Equatorial plasma vertical drift in the F region (driven by the zonal electric field) is selected as the only dynamical forcing term to be adjusted because of its dominant effect at low latitudes. This is based on the following considerations: (1) the drift plays a dominant role in the fountain effect that produces the well-known equatorial anomaly, the most prominent feature in the low-latitude ionosphere; (2) the drift is a critical parameter which affects the electron density gradients (∇ne) at low latitudes and fosters the collisional plasma Rayleigh-Taylor instability and ionospheric irregularities during evening hours; (3) the drift measurements are not readily available in most longitude sectors. Thus the present OSSE is formulated as a first step toward optimizing the modeling of ionospheric weather associated with the E × B drift at low latitudes. A solar minimum pattern of the drift is intentionally chosen to simulate the “climatology” condition, and the pattern is obtained from an average of vertical drifts measured using the Jicamarca incoherent scatter radar for the specified conditions [Fejer et al., 1991]. This drift pattern is then perturbed with a typical pre-reversal enhancement pattern (during the evening hours) to simulate the weather behavior. Figure 11 shows the simulated “climatology” and “weather” for the equatorial vertical plasma drift.

image

Figure 11. A climatology pattern and simulated weather perturbations of the equatorial vertical plasma drift.

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4.2. Experiment Results

[26] The 3-D forward model was run separately for the climatology and weather conditions, yielding ne at low and middle latitudes in the p-q-l frame. Line-of-sight (LOS) TEC along real GPS links was then computed by line integration of ne. Computed LOS TEC from the weather run was used to simulate GPS measurements as the assimilation data source. Figure 12 shows snapshots of vertical TEC obtained from the simulated climatology and weather runs. Maps of absolute and relative TEC difference between the two conditions are also included in the figure (right column). The major weather feature is the enhanced TEC in the conjugate equatorial anomaly regions and the TEC depletion at lower latitudes. These effects are due to the strong pre-reversal enhancement in the upward plasma drift.

image

Figure 12. Modeled vertical TEC with empirical driver inputs (upper left) and simulated weather conditions (lower left). The absolute (upper right) and relative (lower right) TEC differences are shown for the modeled region (within ±30° latitudes). White dots mark the locations of ground GPS receiver sites.

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[27] In our assimilation experiment, the forward model is first run with the empirical inputs to reach the climatology initial state. Then simulated data for the first assimilation cycle (2 hours following the initial state) are assimilated, the cost functional J is computed, and the drift (vi) is adjusted to minimize J. The newly adjusted drift is used to compute the initial state of the next assimilation cycle. This process is repeated in subsequent assimilation cycles.

[28] In principle both the initial state and driving forces in each cycle can be adjusted by use of 4DVAR. However, because the state equation constraints are strictly enforced, adjustment of driving forces alone will automatically lead to a self-consistent initial state in subsequent assimilation cycles. For this reason we intentionally set the state regularization coefficient γn to zero and limit the adjustment of parameters to the driving forces alone. This allows us to investigate how many assimilation cycles are required for the state and drift to converge to the “truth”. The value of coefficient γF for the present OSSE is chosen to be a constant within each assimilation cycle, and it is decreased linearly with subsequent cycles. γF starts at 0.5, decreases to a minimum value of 0.1 at the 8th cycle, and is kept constant afterward.

[29] Ideally, a longer assimilation cycle is desirable for better data coverage and improved sensitivity; however, this must be considered against computational efficiency and computer memory requirements. With the present low-latitude coverage of the IGS GPS network, we have chosen 2 hours as the assimilation cycle interval. The gradient of J is computed using the adjoint method and the minimization of J was conducted using a gradient-based iterative search technique [Liu and Nocedal, 1989].

[30] Figure 13 plots the drift as a function of local time for the climatology, weather, and assimilation results obtained in cycle 3. Note the ability of the assimilation system to capture the strong pre-reversal enhancement in spite of the fact that this feature is nonexistent in the assumed climatology. Once the adjusted drift is obtained, the corresponding electron density is computed and vertical TEC maps are derived. In Figure 14 we compare vertical TEC maps obtained at the end of assimilation cycle 3 to the weather at the same UT. In contrast to Figure 12 (obtained based on climatology), Figure 14 shows dramatic improvements in capturing ionospheric weather and the equatorial anomaly.

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Figure 13. Comparisons of UT snapshots of the climatology, weather, and assimilation results for the equatorial vertical drift.

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Figure 14. Assimilation results for vertical TEC (lower left) compared with the simulated weather conditions (upper left) at the end of assimilation cycle 3. The absolute (upper right) and relative (lower right) TEC differences are also shown.

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[31] Figure 15 captures the improvements in determining the drift and electron density as a function of assimilation cycle. Each diamond (circle) in the figure corresponds to the root-mean square (RMS) difference between the simulated weather and estimated vi averaged over all local times for the corresponding UT interval (estimated ne averaged over the entire modeled region at the end of each cycle). Note that the initial large error in the force estimation in cycle 1 is due to the fact that the initial guess of the state (climatology) is so different from the weather that extra forcing is needed to satisfy the minimization requirement. When the initial guess of the state becomes close to the weather in later cycles, the drift estimation is much improved. In practice, one could discard the estimation in the first few cycles and take the results from later cycles. We have performed additional experiments starting with the weather initial state to estimate the drift, and the assimilation retrieves the correct drift in only one cycle as expected.

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Figure 15. The OSSE results for the root-mean square of ne and vi⟂ differences between the weather (“truth”) and assimilation at each assimilation cycle step.

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5. Conclusions and Discussions

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

[32] A global assimilative ionospheric model (GAIM) has been developed with a fixed 3-D Eulerian p-q-l grid following the dipole geomagnetic field. The model is being exercised in this study with simulated line-of-sight TEC observations that in practice can be readily obtained from the GPS observation system. The OSSE conducted using this model and the GPS observation system demonstrates that ground-based TEC measurements can be used in improving ionospheric weather modeling. The model optimization is conducted with the 4DVAR and adjoint techniques for a low-latitude region allowing estimation of the weather behavior of the equatorial E × B drift. The OSSE performed here demonstrates that the 4DVAR technique has the potential of solving for the equatorial E × B drift and improving the density state. It is promising that one could do so using only ground-based, integrated line-of-sight (TEC) measurements, and a relatively small number of stations.

[33] As an initial effort to assess the effectiveness of 4DVAR, the presented OSSE was simplified such that only zonal electric field perturbations were considered. The electric field was chosen for this first experiment because it is the most effective dynamical force determining the major features in the low-latitude ionosphere. More sophisticated OSSEs adjusting multiple forces, including neutral wind and production rates, are currently being assessed and will be presented in the future.

Acknowledgments

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

[34] We thank Arthur Richmond and Gang Lu at the National Center for Atmospheric Research for their assistance in the effort of building GAIM. This study is supported in part by the Department of Defense through a Multiple-disciplinary University Research Initiative under grant N00014-99-1-0743 and by the National Science Foundation under grant ATM-9613947. The research conducted at the Jet Propulsion Laboratory, California Institute of Technology, is under a contract with the National Aeronautics and Space Administration.

[35] Lou-Chuang Lee and Chin S. Lin thank Bruce Howe and another referee for their assistance in evaluating this paper.

References

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