Radio Science

Development of the Global Assimilative Ionospheric Model

Authors


Abstract

[1] This paper provides an overview of the development of the Global Assimilative Ionospheric Model (GAIM) by a team of investigators from the University of Southern California (USC) and the Jet Propulsion Laboratory (JPL). The USC/JPL GAIM utilizes data assimilation techniques, which are widely used in meteorological applications, for the purpose of monitoring and forecasting Earth's ionosphere. We discuss the general structure of GAIM, which includes a first-principles model of the ionosphere, a series of auxiliary models for the driving forces, a data processing subsystem, and an optimization subsystem. Two techniques for the estimation of electron density and driving forces in the ionosphere are presented: The four-dimensional variational method and the Kalman filter. Some validation methods and results are also presented. These results demonstrate the potential of GAIM in providing accurate specification of the ionosphere.

1. Introduction

[2] The impact of Earth's ionospheric conditions on social, economical, and national defense activities has become increasingly significant as satellite navigation and wireless communication systems become part of our daily lives. For example, significant degradation of positioning accuracy in GPS-based navigation systems can be caused by disturbed ionospheric conditions. Similarly, the propagation of a high-frequency (HF) radio wave can be strongly affected by ionospheric irregularities. The need for a reliable and accurate ionospheric specification and forecast system is widely recognized by the space weather community. Indeed, the National Space Weather Program (NSWP) implementation plan drafted in 1995 has set the understanding of the day-to-day variability of the large-scale ionospheric features and small-scale plasma density irregularities as one of major goals of the NSWP. In 1999 the Department of Defense identified global ionospheric data assimilation as one of 12 topics for the Multidisciplinary University Research Initiative (MURI) program.

[3] Our research efforts on ionospheric data assimilation began in 1996 as a collaborative research project involving the Jet Propulsion Laboratory (JPL), University of Southern California (USC), Cornell University, and University of Sheffield. As an exploratory research project, our goal was to explore the feasibility of combining ground and space-based GPS total electron current (TEC) measurements with the theoretical Sheffield Plasmaspheric and Ionospheric Model (SUPIM) [Bailey and Sellek, 1990; MacPherson et al., 2000] to produce ionospheric specifications and forecasts. In 1999 a MURI project supported the development of the Global Assimilative Ionospheric Model (GAIM) by two consortia, one led by USC/JPL and the other led by Utah State University. GAIM's purpose is to assimilate multiple types of ionospheric measurements including ground and space-based GPS, ionosonde profiles, UV airglow radiances, in situ electron and neutral densities, plasma drifts, neutral winds, neutral densities, or other types of measurements creating the equivalent of numerical weather prediction (NWP) models. The assimilation of ionospheric measurements into mature first-principles ionospheric models will produce physically consistent accurate ionospheric analysis, as well as the important ionospheric drivers, therefore enabling the generation of more accurate ionospheric weather forecasts.

[4] One of our research goals is to develop a reliable and accurate global ionospheric weather monitoring and forecasting system that can serve as a prototype for an operational system. To achieve that, we leverage the body of knowledge gained from data assimilation research in the meteorological community. However, the difference in the physics and data types between the ionosphere and the lower atmosphere necessitates adaptation of data assimilation techniques for ionospheric research. In this paper, we provide a summary of the mathematical methods used in the development of the USC/JPL GAIM.

2. General Structure of a GAIM

[5] There are two ways to describe a data assimilation system. The first is to describe the sequence of processing steps that form the data assimilation cycle [Daley, 1991]. We can call this description a temporal description or operational description. The second consists of the description of the functional components in the data assimilation system. We call this description the functional description. In this section we shall first use the operational description to present the operational environment for GAIM. Then we shall describe the different functional components of GAIM.

2.1. Operational Description of GAIM

[6] The current operational objective of GAIM is to provide real time or near-real time ionosphere specification (i.e., the ionospheric condition at the time when measurements were taken) as well as short-term (1 hour to 1 day) ionospheric forecast. The main tasks to be performed in a data assimilation cycle are: (1) data processing and quality control; (2) data assimilation cycle, which includes determination of the ion densities and other ionospheric variables including driving forces such as neutral wind, E × B drift, etc.; (3) statistical data analysis; and (4) computation of forecast for the subsequent data assimilation cycles.

2.1.1. Data Processing

[7] The design of GAIM foresees an operational environment where diverse types of ionospheric data are available. A detailed list of ionospheric measurements considered for use by GAIM is given in Table 1. All of these measurements should first be preprocessed and edited to reduce the outliers and noise. For example, the delays in signals received from GPS satellites by the ground stations are used to derive the TEC between the satellites and the ground receivers. In this phase of processing, data with gross abnormalities are singled out and in some cases removed. Calibrations between different instruments are also performed to remove known systematic biases. In order to limit the scope of discussion in this paper, we will assume that data providers have carried out most of the preliminary data processing and quality control and most importantly assigned an uncertainty to each data point.

Table 1. Potential Data Sources for GAIM
Sensor TypeMeasurementsData CoverageHeritage/Current Instruments
Ionosondebottom side profile, NmF2, Hmf2localDISS, SHIN
Ground GPS receiverline-of-sight TEC from ground receiver to GPSlocal/regional/globalTENET, CORS
Spaceborne GPS receiverline-of-sight TEC From LEO to GPS satelliteregional/globalGPS/MET, Champs, SAC/C, IOX, COSMIC
Limb EUV sensorO+ and neutral particle density profileson satellite orbital planeLORAAS, SSULI
Cross-track EUV scannerintegrated and profiles of neutral particle and O+ densityglobalGUVI, SSUSI
Various sensors for satellite in situ measurementsin situ electron density, electrical field, and aurora particle precipitationglobal along satellite orbitSSIES, SSJ/4
Radio beaconline-of-sight TEC from satellite to ground stationregionalC/NOFS

[8] For each data type to be assimilated into GAIM the mapping between the model variables (e.g., ion densities) and the measurement is computed. This mapping is usually referred to as the observation operator in system control theory. For example, in the case of TEC data collected by GPS ground stations, the computation of the observation operator requires accurate determination of the positions of the GPS satellites and identification of the elements in the modeling domain along the lines of site. At the end of data processing, each of the TEC measurements has the form of a triplet: (tk, yk, Hk) where tk is the time tag for the measurement, yk is the value of the measurement, and Hk is the observation operator.

2.1.2. Data Assimilation Cycle and Forecast

[9] Although the collection and processing of data are carried out continuously, the actual analysis or assimilation of the data can be done periodically in time. Intermittent data assimilation [Daley, 1991] with a cycle time of several hours is particularly suited for the estimation of driving forces in Earth's ionosphere. Since most of the driving forces directly influence the rate of change of the ion densities, it is often necessary to observe the changes in the ionosphere over a sufficiently long period of time in order to be able to make adjustments in the driving forces. As shown in Figure 1, the conditions of the ionosphere at the beginning of each of the data assimilation cycles are determined in the middle of the previous cycle. Data collected during the second half of the previous cycle are reused, and newly collected data are introduced. The reuse of a certain portion of the data introduces correlation between the errors in the initial conditions and the errors in the subsequent observations. Modifications to standard data assimilation techniques which assume independence between the two error sources are necessary. In our current implementation, intermittent data assimilation is primarily used with the Four-Dimensional Variational (4DVAR) technique, which does not require the independence between the two error sources mentioned above. Data collected over a time interval (e.g., 4 hours) are used to adjust the driving forces as well as the ionospheric initial condition at the beginning of the data assimilation cycle. These adjustments are made so that the model predicted values for the observations match the measured values. The overlap of the data assimilation cycles has the effect of stabilizing the estimates for the driving forces as well as for the ionospheric conditions. The underlying first-principle model is used to extend the ionospheric beyond the time when the last measurement is made. Thus a forecast of 2 hours is produced. At the present time, only short-term forecast is considered by GAIM.

Figure 1.

Example schedule of intermittent data assimilation cycles.

[10] Intermittent data assimilation when used with 4DVAR techniques allows us to consider the full nonlinear aspect of the model parameter estimation problem. However, recursive estimation techniques such as the Kalman filter or extended Kalman filter take advantage of linearization of the model equation to produce statistically optimal estimates of the ionospheric parameters as data are collected. The recursive nature of the techniques does not require the storage of a large quantity of observations. Since data assimilation approaches based upon recursive estimation can analyze the data continuously as they are collected, we refer to these methods as continuous data assimilation methods.

2.1.3. Quality Control

[11] The main tool for automatic data quality control is statistical analysis. Recursive methods such as the Kalman filter rely explicitly on probabilistic assumptions for the observation errors and the system-model errors. Using these assumptions, recursive methods provide rigorous predictions for the probability distribution of the adjustments to the model variables. For example, if the model variables represent ion densities at different locations in space, and if we assume that we know the statistical reliability of the ionospheric model and the observation system, we can compute the expected variance of the corrections to the ion densities. Therefore monitoring of the statistics of the corrections allows one to examine the accuracy of the statistical assumptions. Some data assimilation techniques do not explicitly make statistical assumptions for the physical model and for the observation system. This is the case for most variational methods. For these methods, one needs to build a statistical database and derive the statistical properties of the model adjustment empirically. Since the significance of the statistics relies on the accumulation of a large quantity of data, the statistical analysis must be done in an intermittent fashion. As a result, even in the case of continuous data assimilation, data assimilation cycles can still exist. In this case the primary role of the data assimilation cycle is to statistically validate the data stream and the corrections to the model variables.

2.2. Functional Description of GAIM

[12] The temporal description of the data assimilation system given above shows the steps or phases in data assimilation. However, many functional components of the system are used in several phases of the data assimilation. For example, the underlying ionospheric model is used to determine what size of adjustments to the ion densities and driving forces are necessary to match the observations. The same model is also used to provide the ionospheric forecast. From a functional point of view, an ionospheric data assimilation system typically has the following components: (1) forward ionospheric model, (2) auxiliary parameterized models for driving forces, (3) data processing subsystem, (4) optimization algorithms, and (5) statistical analysis tools. The diagram in Figure 2 illustrates the connections between these components.

Figure 2.

Main components of a data assimilation model.

[13] Central to a first principles-based data assimilation model is the forward model. In the case of the USC/JPL GAIM the forward ionospheric model is a global three-dimensional time-dependent model in which the main model variables are the ion densities. To simplify implementation, the current version of GAIM uses a single ion (O+) model of the ionosphere. The equations of conservation of mass and momentum are solved on an Earth-fixed Eularian grid. The volume elements have surfaces parallel to either the magnetic field and potential lines or the geomagnetic meridional planes. An example of the grid used in GAIM is shown in Figure 3. A finite volume technique is used for the discretization of the model equations in spatial variables. A hybrid implicit and up-winding explicit method is used for the time discretization. The resulting forward model is unconditionally stable and preserves the positivity of electron density. The forward model uses a series of empirical models to define the driving forces. Because these models are mostly climatological, it is essential to be able to capture deviations of the driving forces from the background climatology using efficient parameterization. The main purpose of such a parameterization is to reduce the dimension of the optimization vector when the values of the driving forces are to be estimated and their correlation to be captured. For example, the equatorial plasma vertical drift (E × B drift) in the ionosphere, a crucial parameter for the ionospheric dynamics, can be parameterized using a scalar function of the magnetic longitude. It is important to note that all a priori values for the driving forces, as well as the model initial state, are derived from climatological values. Thus the forward model is primarily a climatological model. Thus far, forward model validation has been performed by comparing a model-predicted vertical TEC map to a data-driven Global Ionospheric Map (GIM) of TEC [Mannucci et al., 1998]. On a day of moderate magnetic activity the forward model predicted vertical TEC agrees with those of GIM to within 20% of peak TEC values. The spatial TEC distribution is also similar. The data processing subsystem is responsible for the preliminary preprocessing of data. This consists of removal of gross outliers and calibration of data. Many auxiliary items are also generated by this subsystem, including the determination of the data collection time, computation of the transmitter and receiver positions, and computation of the observation operator for each data point collected. This last and most important function of the data processing system requires a careful modeling of the sensors that are used for collecting data. The role of the sensor model is to establish the connection between the ionospheric variables, such as the electron and ion densities, and the output of the sensor instrument. In the case of GPS measurements the delays in the signals at the L1 and L2 frequencies are caused by the presence of the ionosphere and atmosphere. The sensor model establishes the connection between the signal delays at the two frequencies and the TEC along the satellite-to-receiver line-of-sight. The effect of the atmosphere on the GPS signal delay is removed by forming a linear combination which isolates the ionospheric delay. However, both ground- and space-based GPS measurements have be used in meteorological applications to retrieval atmospheric profiles or total water content. A good sensor model must also identify possible biases and perturbations contained in the measurements and processing algorithms. Calibration steps are then performed to remove the known biases from the measurements. Finally, the sensor model must assign an uncertainty to each measurement that accounts for random noise in the instrument, geolocation errors, and uncalibrated systematic errors. The level of uncertainty in the measurement values estimated by the sensor model will be used in the optimization system to balance the size of adjustments to different model variables.

Figure 3.

Example of grid used by the forward model of GAIM.

2.2.1. Optimization Subsystem

[14] Using the latest update of the ionospheric state and the driving forces, the forward model propagates the ionospheric state to a future time. The observation operator is then used to produce a prediction of the future measurement data, and the predicted measurement values are compared to the actual measurements when they become available. The difference between the two sets of values is referred to as the innovation vector. The optimization subsystem is responsible for adjusting the values of the model variables as well as the driving force vector so that the predicted measurements based on the adjusted values are closer the actual measurement. For example, when the Kalman filter is used, the optimization strategy estimates the electron density on the basis of the innovation vector to produce a statistically minimum variance estimate of the electron density. However, in our 4DVAR implementation, only the driving force parameters are adjusted to produce a least square estimate of the driving forces as well as electron density. This subsystem relies upon the forward model to determine the size of the adjustment as well as the ultimate tradeoff between the confidence level and the sensitivity of the adjustments. The main guiding factor for this tradeoff is the statistical information on the various uncertainties, including model and measurement uncertainties. The updates of the model variables and the driving force vector are strongly influenced by the statistical information. In fact, erroneous covariances can lead to erroneous solutions. The statistical analysis package is used to continuously monitor the validity of the statistical information.

2.2.2. Statistical Analysis

[15] Since actual values of the ionospheric states such as electron and ion densities are generally not available, the statistical package primarily monitors the consistency between the statistical assumptions and the observed statistics of the innovation vector. For most recursive filtering techniques, if appropriate linearization is made and if model and observation random errors are assumed unbiased, the implied covariance matrix of the innovation vector can be explicitly derived. The comparison of the theoretical covariance matrix with the empirical covariance matrix of the innovation vector will lead to necessary modifications of the error statistical information. The key to the statistical inference using the discrepancies between the implied statistics and the empirical statistics is the proper modeling of the errors.

[16] The combination of the forward model, the driving force models, and the data processing subsystem forms a complete ionospheric observation simulation system. As we have discovered, the availability of a simulation system offers extremely valuable tools for the validation of data assimilation methods. In fact, this simulation system has been used in a series of Observation System Simulation Experiments (OSSE) to verify our implementation of the data assimilation algorithms. This system can also be used to evaluate the possible impact of new data types from space weather remote sensing missions on the accuracy of the global monitoring and forecasting of ionospheric weather.

3. Four-Dimensional Variational Approach

[17] The variational approach formulates data assimilation as a problem of minimization over a set of functions. Simply stated, the objective of ionospheric data assimilation is to identity the best ion density distribution and driving forces over time and space. The criterion for the selection is usually formulated in two parts: The objective function and the constraints. The objective function (also known as the cost function) represents the measure of the closeness between the observations and the model predictions. The formulation of the objective function also requires the selection of the set of independent variables, referred to as the optimization variables (or control parameter). In our current implementation of GAIM the optimization variables consist of coefficients in the parameterization of the ionospheric driving forces. These driving forces include the E × B equatorial drift, ion production rates, and neutral wind. The variational data assimilation algorithm uses an objective function of the following form:

equation image

where

ni,0

ion density distribution at time t0;

q

driving force vector;

Nobs

total number of observation vectors to be assimilated into the model;

Rk

error covariance matrix for the k observation vector which is measured at time tk;

yk

observation vector at time tk;

Hk

the mapping between the ion density distribution at time tk and the observation vector;

ni,k

ion density distribution at time tk;

Q

covariance matrix for the initial ion density distribution;

equation image

a priori value of the initial ion density distribution;

W

covariance matrix for the driving force vector;

equation image

a priori value of the driving force vector.

[18] The variational algorithm is formulated in a deterministic sense; the covariance matrices can be viewed as weighting factors between the terms in the objective function. The first term is easily understood: It is simply the weighted difference between observed values and the model predicted values for the measurements. The last two terms are referred to as regularization terms. Appropriate selection of the regularization terms is needed to stabilize the estimated values for the optimization variables. Since the a priori values represent the climatological average of the corresponding quantity, the regularization terms help to limit the deviation of the estimate from the climatological values.

[19] The constraints for the optimization require that the ion densities satisfy the first-principles model of the ionosphere. In other words, the constraints implicitly define the dependence of the ion density at a time tk as a function of the initial ion density distribution and the driving force vector. The constraints can be written in the following form:

equation image

where Φk,0 represents the mapping between the initial ionospheric state and the state at time tk. The operator Φk,0 is also referred to as the state propagation operator. When the ionospheric model equations are treated as the constraints for the optimization, the resulting data assimilation algorithm strictly satisfies the underlying physics model. Alternatively, one can treat the ion density distributions at every tk as independent optimization variables and include a term in the objective function to penalize the state equation error of the form:

equation image

[20] The advantage of this formulation is that it allows us to take into account the errors in the ionospheric model. The main difficulty of the penalty function approach is the significant increase in the dimension of the optimization variables. In the present version of GAIM the model equations are strictly satisfied. Since the main goal of the 4DVAR approach for GAIM is to estimate the driving forces vector q, constraint (2) forces the resulting electron density distribution to be smooth in both space and time. This is in part owing to the fact that our forward model is based on a climatological large-scale model. In the case of densely spaced measurements, additional refinement using a recursive estimation technique may improve the accuracy of the model analysis.

[21] The minimization method used in our 4DVAR implementation is a quasi-Newtonian method, the Broyden Fletcher Goldfarb Shanno nonlinlinear optimization method (BFGS) [Liu and Nocedal, 1989]. The iterative unconstrained minimization method is given by

equation image

where H(J) represents an approximation of the Hessian matrix of the functional J, ∇qJ and represents the gradient vector of J. The optimization routine computes the matrix and the optimal step size λk. Although the BFGS algorithm also has the option to approximate the gradient vector ∇qJ using a finite difference approximation, the computation requires a large number of evaluations of the functional J. Each evaluation of the functional J corresponds to a forward integration of the ionospheric physics model. A well-known technique for efficient computation of the gradient vector is the adjoint method. In our implementation of GAIM we use the adjoint method to compute the gradient vector. The adjoint method ensures that the computational effort required for one iteration of BFGS is equivalent to only two integrations of the forward model.

[22] We have tested our implementation of the 4DVAR approach through OSSEs. In particular, we have focused on the estimation of equatorial E × B drift velocity. The perturbation to the climatological E × B drift is parameterized by polynomial spline functions of local time. The perturbed values of the drift velocity are used in the simulation to generate the synthetic measurements. The initial estimates of the electron density and the drift velocity are obtained from the climatological models. Our preliminary results indicate that the 4DVAR method can successfully estimate the appropriate perturbation and at the same time provide improved determination of electron density. In Figure 4, a typical case of the estimation of E × B drift velocity is shown. The climatology curve represents the initial guess and a priori of the drift velocity as a function of local time given by climatological model. The “weather” curve corresponds to the perturbed drift velocity that is used in the simulation to generate the synthetic TEC measurements. Note that the perturbed drift has a much larger prereversal enhancement than the climatology. The “estimation” curve represents the GAIM-estimated drift velocity using the 4DVAR approach, and it matches the weather curve quite well. It is a significant result that the E × B vertical drift, which along with the neutral wind drives the amplitude and height of the density peaks in the equatorial anomaly, can be estimated using only integrated TEC measurements from ground GPS. For a more detailed description of this 4DVAR OSSE, see Pi et al. [2003]. The details of the implementation of the adjoint method can be found in the work of Rosen et al. [2001].

Figure 4.

Estimation of equatorial E × B drift velocity.

[23] There are many interesting issues that can be investigated using OSSEs. One important issue is the identifiably of the driving force vector for a given set of ionospheric measurements. For example, when only ground-based GPS TEC measurements are available, can equatorial E × B drift be accurately estimated? Moreover, can the effects of perturbed E × B drift velocity be separated from that of perturbation of neutral wind? Our preliminary OSSE results demonstrate that on the basis of ground GPS TEC measurements, the E × B drift can be estimated. Mathematically, the question can be answered by examining the rank of the observability matrix for the linearized model of the ionosphere and the observation operator [Kuo, 1980].

4. Recursive Estimation Approach

[24] Recursive estimation techniques produce a sequence of estimations of the model variables in time. These model variables may include electron and ion densities as well as values for the driving forces. More precisely, we can consider a time sequence {tk}. At each time instant tk, an a priori estimate for the model variables is computed using all prior measurements. The recursive data assimilation consists of two tasks. The first is to improve the estimation of the state variables using newly obtained measurements at time tk. The second is to produce the best a priori estimate for the state variables at the next time step tk+1 and an estimate of the error covariance. One refers to this approach as recursive estimation because the same process is then used for the subsequent time steps. The first task in the recursive data assimilation is very similar to tomographic inversion in the case of GPS TEC data assimilation. Since this approach does not involve the time evolution of the system, it is also similar to the three-dimensional variational method (3DVAR). The second task of recursive data assimilation requires the propagation of the model variables and the state error covariance matrices in time. The most well-known recursive data assimilation approach is the Kalman filter. The Kalman filter provides the optimal solution to a linear least squares recursive estimation problem. In the formulation of the linear least squares minimization problem the model for the transition of the state vector in time and the observation map are assumed to be linear, so we have

equation image

where xk and yk represent the model variables and the measurement vector, respectively. The matrices Φk and Hk are the state transition operator and the observation operator, respectively. The vector θk represents the modeling error in the state transition. The vector εk represents the cumulative effects of the measurement errors and representation error. The representation error corresponds to the inaccuracy of the observation operator. For example, the derivation of slant TEC ignores bending of signal path. More importantly, the discrete representation of the ion distribution also introduces differences between the observation and the model predicted value. Since we include representation error in the observation error, εk, the observation error can be significantly larger than the measurement error. We have empirically estimated the size of the representation error as follows. First, the ionospheric dynamic equations are solved with a very high spatial resolution (around 120,000 elements). The predicted measurement values from the high-resolution model are compared to those produced by the moderate resolution model (around 12,000 elements). Similar to the case of error of the forward equation, improvements of the estimates for the error statistics can be achieved most realistically through long-term monitoring of the performance of GAIM. The least squares estimation problem consists of the computation of the linear estimator of the state variable equation image = Kkyk that minimizes the functional

equation image

where E represents the expected value. The minimization of the variance of the estimation error requires knowledge of the covariance of the modeling errors θk and εk measurement errors, as well as the error covariance of the prior estimation equation image given by the past measurements. The basic probabilistic assumptions include zero bias for the error vectors and uncorrelated error vectors. The optimal solution is given by

equation image

where Pk and Rk are the error covariance matrices of the a priori estimate equation imagexk and the measurement errors, respectively. The error covariance matrix of equation imagexk is given by

equation image

[25] The a priori estimate for the model variable and the error covariance matrix at the next time step are given by

equation image

where Qk is the error covariance matrix of the state transition model. We note that equations (5) and (6) are precisely the minimum variance estimation of the state variables in a tomographic inversion. The computationally costly steps are the inversions of the operator HkPkHkT + Rk and the matrix operations in equations (6) and (7). Although the correct evaluation of the matrix Kk and the evaluation of equation (7) are necessary, it important to remember that the optimality of the Kalman filter solution is subject to the validity of the assumptions made on the various error probabilities. As in the case of tomography, selection of the proper error covariances Pk, Rk, and Qk constitute a major difficulty for developing a reliable retrieval algorithm.

[26] Another major difficulty is the computational complexity of the matrix algebra. Several variations of equations (6) and (7) have been tested to speed up the computations, including optimal interpolation and two approximations to the full Kalman filter [Hajj et al., 2000, 2004]. Our current implementation of “Kalman-like” recursive data assimilation uses a banded matrix approximation of the error covariance matrix Pk, where correlations between grid elements that are separated by a distance larger than a preset value are discarded. Preliminary testing of the implementation has demonstrated that it is highly efficient and produces significant reductions in the postfit residuals. Figure 5 shows an example of an assimilation in which simulated ground GPS TEC measurements are used by the recursive estimation method. The plot on the left shows the difference between the model predicted (using a climate run) and “measured” (using a weather run) slant TEC prior to assimilating the synthetic data. In this OSSE the weather is created by perturbing the climate neutral density and E × B drifts [Hajj et al., 2000]. The bimodal distribution in this figure can be because perturbation of the neutral density has less of an effect on the night side than on the dayside ionosphere. This is further confirmed by inspecting differences in vertical TEC maps between the climate and weather runs. The plot on the right shows the residuals after data assimilation with a reduction of nearly 80%. This demonstrates that our band-limited approximation to the recursive Kalman filter can produce electron densities that are significantly more consistent with the observations while considerably reducing the computation time.

Figure 5.

Comparison between (a) preassimilation residual and (b) postassimilation residual derived from a simulation experiment.

[27] Further improvement to the Kalman filter can be obtained by the use of the Kalman smoother where future data is used to estimate past electron densities and driving forces. The equivalence between the Kalman smoother technique and 4DVAR can be established for linear systems [Zhijin and Navon, 2001]. Since the driving force vector is nonlinearly coupled with electron density, linearization is required for the implementation of the Kalman filter and Kalman smoother. We have not experimented with the Kalman smoother approach in the current GAIM implementation.

5. Validation Techniques

[28] Validation of a model is the most critical and challenging step in the model development. It requires the availability of independent data sources which are dedicated to the validation. For example, if TEC data derived from GPS signals and ionosonde data are assimilated into GAIM, one may use incoherent scattering radar (ISR) data to validate the resulting electron density profiles. However, there are only a limited number of ISRs, and their availability for ionospheric measurements is also limited. Another important source of validation data is the ionosonde measurements from the DISS network. Under ideal conditions, Digisonde measurements can be used to derive the bottom-side profile of the ionosphere. We have conducted preliminary sanity checks of our electron density profile estimation against NmF2 and Hmf2 measurements derived from ionosondes, and we are encouraged by the improvement in the estimates produced by data assimilation. The primary objective of this validation was to determine whether or not the introduction of ground GPS TEC measurements into GAIM can produce improvement in the estimation of the electron density profile over the climatological model. For this evaluation, ground GPS TEC data and ionosonde measurements from 13 June 2001 were used. Figure 6 shows the distribution of the GPS receivers used in this comparison. Each receiver station reports TEC measurement every 30 s, and these data are reduced to a lower sampling rate of 1/15 min for this study. The plot on the left in Figure 7 shows the scatterplot of the F region peak density (NmF2) measured by ionosonde versus the GAIM climatological prediction. In this case, no data assimilation is performed. The plot on the right-hand side shows the comparison between measured NmF2 and the GAIM analysis result based on assimilation of the ground GPS TEC data. One can observe a visible change in the density of points near the diagonal. This indicates that assimilating the ground-based GPS TEC data improves the GAIM predictions of peak density.

Figure 6.

Location of the ground GPS receiver stations.

Figure 7.

Comparison of (a) GAIM climatological prediction and(b) GAIM analysis for NmF2 against ionosonde measurement.

[29] Another indirect verification of GAIM electron density is through the comparison between vertical TEC measured by ocean altimeters such as TOPEX and the GAIM predicted TEC by assimilating GPS data. We note that one of the challenging aspects of this validation is that most ground GPS TEC measurements are obtained over land while TOPEX TEC measurements are only available over the ocean, so we are probing regions far from the input of GPS data. However, our preliminary validation results indicate that GAIM data assimilation can produce significant improvements over the climatology from the forward model. For a more detailed discussion, see Hajj et al. [2004].

6. Conclusion

[30] The development of GAIM is still work in progress. Although the main components of the system are implemented, considerable work still needs to be carried out to fine-tune the algorithms and error statistics. In particular, the development of empirical methods to estimate the modeling error covariance matrices, and the appropriate weighting of different terms in 4DVAR cost functional are some of the challenges in transitioning the current implementation of GAIM into a reliable global ionospheric monitoring system. In addition to implementation issues, significant basic scientific investigations need to be conducted for the simultaneous estimation of multiple driving forces to better understand the combined effects of these forces. In fact, a fundamental question that needs to be answered in the development of GAIM is whether or not the currently available data sources are sufficient to estimate all of the ionospheric driving forces simultaneously. Effective methods of estimating various driving forces in different regions, where physics processes are fundamentally different, are also to be investigated. Extensive systematic validation is still to be carried out in the future. Our preliminary results are encouraging. As we continue our development of GAIM, more experience will be gained to allow us to produce reliable specifications and forecasts of Earth's ionospheric conditions.

Acknowledgments

[31] We thank Graham Bailey and Arthur Richmond for their assistance in the effort of building the GAIM. The development of GAIM is supported by the Department of Defense through the Multidisciplinary University Research Initiative. The research conducted at the Jet Propulsion Laboratory is under a contract with NASA.

Ancillary