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[1] We present new capabilities of our system for monitoring the ionosphere over a fixed geographical area with dimensions of the order of several thousand kilometers. The system employs a nonlinear representation for electron density that ensures a nonnegative solution. The multidimensional nonlinear inverse problem is efficiently solved using a combination of the Newton-Kontorovich method and Tikhonov's regularization technique for ill-posed problems. The system is able to utilize a variety of types of ionospheric data, which are as follows: networks of ground- and space-based (satellite mounted) dual-frequency GPS receivers provide time series of oblique absolute total electron content (TEC) and/or relative TEC data (directly calculated from the raw dual-frequency group delays and phase delays, respectively), TEC data from ground- or space-based receivers operating with dual-frequency beacons mounted on low-Earth orbit (LEO) satellites, vertical TEC data from orbiting radio altimeters (such as Jason satellite), in situ electron density data from plasma probes on LEO satellites (such as Challenging Minisatellite Payload for Geophysical Research and Application), and electron density profiles from sounders. The resulting solution for the distribution of electron density is guaranteed to be smooth in space and time and to agree with all input data within errors of measurement. Real time performance is attained on a single personal computer with 5 min data refreshment period. Operation of the system is tested on real data with various data types simultaneously present. A new form of the stabilizing functional is developed to ensure reasonable assimilation of the in situ electron density data.

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[2] We present new capabilities of our system for monitoring the ionosphere over a fixed geographical area. The system was originally presented by Fridman et al. [2006]. The inversion technique utilized in this system originated from the technique previously developed by our group for the Coordinate Registration Enhancement by Dynamic Optimization (CREDO) project [Fridman, 1998, Fridman and Nickisch, 2001; Nickisch et al., 1998] (a software package for inverting the vertical sounding, backscatter sounding, and satellite total electron content (TEC) data for over the horizon radar). This system used conventional sounding data (both vertical and oblique backscatter) in addition to relative TEC data from a number of TRANSIT receivers, and demonstrated robust performance and flexibility in assimilating data from diverse sensors. The core of this technique is Tikhonov's methodology for solving ill-posed problems. We extended the method to multidimensional nonlinear inverse problems and developed techniques for fast numerical solution. The resulting solution for the ionospheric distribution of electron density is guaranteed to be smooth in space and time and to agree with all input data within errors of measurement. The system was designed to be able to work with raw data of dual-frequency group delays and phase delays provided by a network of GPS receivers. In typical operation the bulk of ionospheric data for inversion comes from GPS receivers, so the program is named GPS Ionospheric Inversion (GPSII).

[3] The updated version of GPSII presented here is able to utilize absolute and relative TEC data from ground- and space-based GPS receivers including occultation data, TEC data obtained using beacons mounted on low-Earth orbit (LEO) satellites, vertical TEC data measured by LEO altimeters (such as Jason satellite), electron density profiles from vertical sounders, and in situ measurements of plasma density. In this paper we describe the updated algorithm and evaluate its performance with simulated data as well as with real measurements of various types.

2. Inversion Procedure

[4] We adopted the following representation for the three-dimensional, time varying distribution of electron density in the ionosphere

where n_{0}(r,t) is a background model of the ionosphere, and u(r,t) is an arbitrary function which will be determined as a result of the inversion procedure. The background model may be a model based on physical principles as well as an empirical one. Results presented here are obtained with n_{0}(r,t) represented by the Parameterized Ionosphere-Plasmasphere Model (PIM) [Daniell et al., 1995]. The representation given by (1) guaranties that the electron density will remain positive. The numerical solution will be performed over a four-dimensional spatial-temporal grid. The vector of values of u(r, t) in all nodes of this grid will be denoted as U.

[5] The unknown vector U is related to the vector of available measured quantities Y

Here M is a nonlinear operator that relates the ionospheric model to each of the measured quantities, and vector η represents the noise of measurements. We assume that the noise covariance matrix S = 〈ηη^{T}〉 is known.

[6] The problem of resolving equation (2) with respect to U typically turns out to be an ill-posed problem and some kind of regularization technique needs to be applied. We are solving (2) using the Tikhonov regularization technique with the residual principle [Tikhonov and Arsenin, 1977]. In application to our problem this method may be formulated as the following optimization problem [Fridman et al., 2006]:

The left hand side of equation (3) is the weighed normalized mean square residual error between the measured quantities and their simulated values calculated in accordance with the ionospheric model (1). The left hand side of (4) is called the stabilizing functional. The pseudocovariance matrix P is required to be positive definite and it should be selected in such a way so that the stabilizing functional would tend to take on larger values for unreasonably behaving solutions. Thus, the stabilizing functional may be interpreted as a measure of reasonableness (smaller values correspond to more reasonable solutions). The nonlinear optimization problem (7) and (4) is solved iteratively. In our experience it typically requires 2–4 iterations for equation (7) to be satisfied within 10%. All observed cases of slow convergence (more than 10 iterations) seem to be associated either with errors in RINEX data files or with poorly performing GPS receivers.

[7] The algorithm is able to handle simultaneously data from various sources characterized by different noise levels. The presence of the noise covariance matrix in expression (7) ensures that less noisy data samples have greater effect on the solution than more noisy samples.

3. Tests With Realistic Simulated Data

3.1. Generation of Synthetic Data

[8] GPSII validation and performance testing requires the generation of realistic synthetic data so that the “truth” is absolutely known. Our approach is to generate synthetic data corresponding to the target ionosphere which is formed by a known PIM ionosphere with an added known perturbation, and then compare the GPSII solution to that “truth.” GPSII is initialized using the unperturbed PIM model.

[9]Figures 1 and 2 show samples of our synthetic data. The data is generated as follows. First we analyze actual observations from a number of GPS receivers with the goal to estimate the standard deviation of data noise σ_{ij}(t) as a function of time for each pair of receiver i and transmitter j. The sliding average technique with 7.5-min sliding window is employed at this stage. Then the noise-free synthetic TEC time series C_{ij}(t) is calculated by line-of-sight integration of electron density in the target ionosphere model using real GPS orbit information and real GPS receiver positions. Finally, the noisy “measured” TEC is formed as

where ξ_{ij}(t) is a sequences of pseudorandom normally distributed numbers with unit variance, η_{i}^{Rbias} and η_{j}^{Rbias} are random constants simulating receiver and transmitter biases (these two terms are applicable only to absolute TEC data). Thus, we are using time-varying noise variances estimated from real measured data to corrupt our synthetic TEC data. Discontinuities in the synthetic data mimic discontinuities in the observed data flow. Note that although Figure 2 shows the phase data as absolute TEC, in GPSII only TEC increments between sampling times are ever used. This is consistent with the approach of Andreeva et al. [1992] for resolving the well-known ambiguity of phase TEC data.

[10] Note that GPSII requires no leveling of GPS phase TEC because leveling is, in effect, performed within GPSII solution as it utilizes both absolute and relative TEC data. The data are synthesized with 30 s time sampling to be consistent with typical real GPS data. Synthetic Vertical Ionogram (VI) data is also generated using vertical electron density profiles in the target (perturbed) PIM ionosphere model, and we assume realistic error tolerances for these vertical profiles in the GPSII solution.

3.2. Validation and Performance Testing

[11] Validation and performance testing are done by comparing GPSII output from a fitted ionosphere to the same quantity derived from a known “target” ionosphere model. An important feature of GPSII processing is that it requires no a priori information about GPS transmitter and receiver biases, though it can use these if available. GPSII makes its own estimates of transmitter and receiver biases and improves these over time. Figure 3 shows examples of GPSII convergence on receiver and transmitter biases when the program was running with real GPS observations. Note that receiver biases converge very quickly. For the transmitter, which is on board the satellite, GPSII must wait until that satellite is observable to make a bias estimate, so the process takes longer. GPSII will have obtained good estimates of the biases for all transmitters within a day. For operational use, it can be assumed that GPSII will have good initial bias estimates and will simply continue to correct the bias estimates in response to drifts.

[12] For this analysis, we took the following approach to generating the starting and target ionospheres. The target ionosphere is PIM with an added ionospheric distortion that consists of a Gaussian shaped enhancement in electron density offset from a reference site by 400 km to the West and with a Gaussian 1/e diameter of 2200 km. The relative strength of the enhancement was taken to be 50% above the background ionosphere at the center of the Gaussian region. The starting model was taken to be the undistorted PIM model. Runs were started at 0000 UT and run for 24 h. Figure 4 displays plots of critical frequency at 2100 UT (around 1300 LT in this region) and show the starting ionosphere (Figure 4a), the GPSII solution for the distorted ionosphere using 10 GPS receivers (Figure 4b), the GPSII solution using 5 receivers plus three vertical ionosondes (Vis; Figure 4c), and the truth ionosphere (Figure 4d). The GPS receivers are shown as diamonds, the VI locations as circles, and the reference site is indicated by the circle evident in Figures 4a and 4c. Plus symbols indicate ionospheric pierce points at 400 km altitude for the GPS lines-of-sight (symbols are shown for this instant; GPSII uses signals continuously over time to develop the solution for any particular instant, so the plus symbols do not indicate all the data that actually goes into the solution for this time, though the symbols roughly define the data region). The disagreement between the solution critical frequency and the truth is within 2.5%, whereas the disagreement between PIM and the truth is 22.5% at the center of the perturbation. For critical frequency, the inclusion of the vertical sounders provides a somewhat more accurate result, though not substantially so. However, we have not found that the absence of vertical sounder data significantly affects the prediction of vertical TEC. This is shown in Figure 5, which displays a similar collection of plots for vertical TEC calculated to 1000 km altitude. For vertical TEC the disagreement between the solution and the truth is within 2.5%, whereas the disagreement between PIM and the truth is 50% at the center of the perturbation.

4. Testing of GPSII With Various Combinations of Data Sources

4.1. Diverse Input Data Sources

[13] We are conducting tests of GPSII with its new options and data sources. Apart from testing the code, this activity is important for preparing GPSII to handle anomalies and errors which commonly occur in data acquired from various sources. Figure 6 presents a sequence of images from a GPSII solution utilizing GPS TEC, vertical TEC from Jason, in situ electron density from Challenging Minisatellite Payload for Geophysical Research and Application (CHAMP; see description of the mission by Reigber et al. [2000]), and initial biases of all satellites and some of the receivers (the initial bias estimates were provided by the Astronomy Institute at the University of Bern). This sequence shows the passage of CHAMP over the solution region. This demonstrates to our satisfaction that the introduction of additional data from CHAMP does not cause any noticeable abrupt effects in the dynamics of the solution.

[14] We have continued testing GPSII with real data supplied simultaneously by multiple instruments. GPSII showed reasonable and robust performance most of the time. Figure 7 presents an example of GPSII solution with GPS TEC and in situ data compared to the unperturbed PIM model.

[15]Figure 8 shows two consecutive frames of the GPSII solution for vertical TEC integrated to 1000 km altitude obtained using data from 36 stationary GPS receivers and 4 GPS receivers mounted on Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) satellites. The geographical area of interest is periodically visited by COSMIC satellites. For example, COSMIC-5 flew over the area of the solution from 1930 to 1945 UT. A string of triangles extending from 16N 109W to 48N 96W in the right plate indicate sequential positions of the satellite separated by the 40 s data sampling interval. One can see that introduction of the new data does not violate spatial and temporal smoothness of the solution.

4.2. Adjusting the Algorithm to Address Problems Revealed at Testing

[16]Figure 9 illustrates a problem that occasionally occurred in the presence of in situ electron density data from a satellite, which we have fixed. In this case the CHAMP satellite enters the geographical area covered by the GPSII solution at 2100 UT (Figure 9a). The satellite reports values of electron density that are considerably lower than electron densities predicted by the background model (PIM in this case) at the altitude of the satellite (approximately 370 km). The resulting distribution of the critical frequency as predicted by GPSII is shown in Figure 9a. The vertical distribution of plasma in a close vicinity of the orbital plane of CHAMP is shown in Figure 9b. One can see that GPSII has adjusted the model to accommodate the in situ data by introducing a smooth depletion of electron density around the CHAMP orbit, causing an unrealistic split of the F layer. This split is most evident in Figure 9c which shows the GPSII plasma frequency profile over the geographical location of CHAMP at 2100 UT.

[17] In order to avoid such layer split problems we decided to modify the definition of the smoothness functional. The original definition of the smoothing functional favors smooth and localized modifications of electron density. Our goal is to design a smoothing functional which would equally favor nonlocalized modifications that look like vertical displacements of the whole F layer. Properties of the smoothing functional are characterized by virtue of the pseudocovariance matrix which is defined as follows [see Fridman et al., 2006, section 2.3]

Here

Here x_{1} and x_{2} are generalized geographical coordinates, λ_{1} and λ_{2} are the user-defined correlation scales of the solution along each of the horizontal dimensions, and x_{3} is the scaled altitude, x_{3}(h) = [dh′/H(h′)], H(h) is a user-defined distribution of the ionospheric-scale height [see Fridman et al., 2006, section 2.3]. We need to modify only the term Φ_{3} in (5). Let us estimate the shape of the covariance matrix associated with height variations of an ionospheric layer. Given the representative electron density profile of a single layer as N_{0} (h) = f(h), the electron density for the same profile shifted up by p will be N_{p} (h) = f(h − p), then the variation of electron density associated with a small displacement p is

The above expression has been linearized in the assumption that the representative profile is smooth and that the displacement p is small compared to the F-layer-scale height characteristics (i.e., the half-width of the F layer).

[18] Then the covariance matrix associated with layer height variation is

With this result in mind we modify the altitudinal factor Φ_{3} of the pseudocovariance matrix as

Here β is a user-defined nonnegative parameter, and C_{LH} is calculated using a representative functional form for the F layer. Note that The first term in the right-hand side of (7) represents localized perturbations of the vertical profile. If only this term is present, then, in view of (6), perturbations which are well separated in height are uncorrelated with each other. The second term in (7) introduces the long-range correlation between perturbations, the shape of this correlation is approximately consistent with modifications amounting to vertical displacement of the whole F layer. We obtained satisfactory results using a Gaussian function for the representative profile and assuming layer height of 350 km, and layer width of 100 km.

[19]Figure 10 illustrates the GPSII solution obtained using the same data that were used to obtain the solution displayed in Figure 10, but employing expression (7) for the vertical component of the pseudocovariance matrix. One can see that the new algorithm does not suffer from the split F layer problem. The new solution achieves agreement with the low-electron density values reported by CHAMP by moving down the original F layer provided by PIM and by reducing its maximum. In the present version of GPSII the height and width parameters of the Gaussian representative profile are chosen automatically to match those of the background model of the ionosphere.

5. Conclusions

[20] We have extended the capability of the GPSII algorithm to utilize data from diverse instruments. Performance of the algorithm was tested with simulated and real data; the algorithm was adjusted and tuned to achieve robust performance with real data sources.

[21] The program is now able to use external estimates for initiating GPS receiver and transmitter biases. It can utilize in situ electron density measurements from LEO satellites as well as vertical TEC data from spacecraft-based altimeters. These capabilities were tested with data from the CHAMP and Jason satellites, respectively. Automated noise estimation for these data types is performed within GPSII.

[22] Another new capability is the utilization of GPS TEC data from on-board satellite GPS receivers (occultation-type data). Performance with this data type is being tested with observations from GPS receivers mounted on CHAMP and COSMIC satellites.

[23] Tests with in situ electron density data from CHAMP revealed that this data occasionally caused unrealistic electron density profiles in GPSII solutions. We successfully resolved this issue by designing a new form of GPSII's pseudocovariance matrix.

Acknowledgments

[24] This work has been supported by the Air Force Research Laboratory Space Weather Center of Excellence (AFRL/RVBXP) under contracts FA9718-06-C-0034 and FA8718-07-C-0015.