Real-time reconstruction of the three-dimensional ionosphere using data from a network of GPS receivers



[1] We present a system that processes phase and group delay time series from a network of dual-frequency GPS receivers and produces a dynamic ionospheric model that is consistent with all the input data. The system is intended for monitoring the ionosphere over a fixed geographical area with dimensions of the order of several thousand kilometers. The inversion technique utilized in this system stems from the inversion technique previously developed by our group within the Coordinate Registration Enhancement by Dynamic Optimization (CREDO) project (a software package for inverting the vertical sounding, backscatter sounding, and satellite total electron content (TEC) data for over-the-horizon radar). 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 input data consist of time series of absolute TEC and relative TEC (directly calculated from the raw dual-frequency group delays and phase delays, respectively). The system automatically estimates the measurement noise and receiver-transmitter biases. We test the system using archived data from dual-frequency GPS receivers in the southern California Scripps Orbit and Permanent Array Center (SOPAC) network and data from a vertical sounder.

1. Introduction

[2] Accurate determination of HF and low-VHF propagation characteristics for purposes like direction finding or analyzing signal penetration likelihoods requires an accurate determination of the ionospheric electron density distribution in three dimensions. Vertical and oblique HF sounding is certainly useful in this regard, but sounders are often not available in the region of interest or they may be too geographically sparse to get a good representation of the ionosphere over a broad region. GPS receivers are both relatively inexpensive and fairly ubiquitous, and since the total electron content (TEC) of the ionosphere between a GPS satellite transmitter and a ground receiver can be determined from the GPS two-frequency signals, GPS presents a potentially powerful tool for determining the state of the ionosphere when insufficient sounding data are available.

[3] We are developing a capability for inverting the data recorded by an array of GPS receivers, vertical sounders, and electron density sensors to recover the three-dimensional electron density distribution of the ionosphere in the region pierced by the GPS lines of sight. Since each GPS receiver typically views from four to six GPS satellites and since it is possible to have many GPS receivers in a region of interest, many intersecting lines of sight can be formed. On each line of sight, TEC can be determined to some accuracy. Obtaining the three-dimensional electron density distribution that best reproduces these many (intersecting) TEC data requires inverse processing. We concentrate on building a robust algorithm for regional real-time monitoring of the three-dimensional ionosphere using only the computational power of a personal computer. The system should be capable of processing raw data from an arbitrary number of GPS receivers, vertical sounders, and in situ electron density sensors available in the area of interest. The area of interest may be of the order of several thousand kilometers in diameter, while horizontal spatial resolution may be of the order or less than 50 km.

[4] The task of reconstructing the three-dimensional electron density from multisensor data has been addressed by a number of researchers, notably by Mitchell and Spencer [2003] and Bust et al. [2004]. Our approach stems from the inversion technique previously developed by our group within the CREDO project [Fridman, 1998; Fridman and Nickisch, 2001]. This system utilized conventional sounding data (both vertical and oblique) 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 effort reported here is to extend our methodology to solve the task of inverting TEC data from GPS receivers combined with vertical sounding data.

[5] We apply the Tikhonov regularization technique and the residual principle combined with a Newton algorithm to solve the full three-dimensional inverse problem. This approach may be interpreted as finding the most reasonable distribution of electron density that agrees with all available measurements within errors of measurements. The algorithm is described in section 2. Performance of the algorithm is discussed in section 3.

2. Inversion Procedure

2.1. Model Representation of the Ionosphere

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

equation image

where n0(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. The representation given by (1) guarantees 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.

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

equation image

Here M is a nonlinear operator which we will call the measurement operator. This operator relates the ionospheric model to each of the measured quantities, and vector η represents the noise of measurements. We will assume that the noise covariance matrix S = 〈ηηT〉 is known.

2.2. Regularization Technique and the Residual Principle

[8] 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 described by the following relations:

equation image
equation image

The left-hand side of equation (3) is essentially the normalized mean square residual error between the measured quantities and their predicted values calculated in accordance with the ionospheric model (1). Expression (4) is called the stabilizing functional. The matrix P is required to be positive definite, and it should be selected so that the stabilizing functional will tend to take on larger values for unreasonably behaving solutions. We will call matrix P the pseudocovariance matrix. Thus the stabilizing functional may be interpreted as a measure of reasonableness (smaller values correspond to more reasonable solutions). We will discuss the pseudocovariance matrix in section 2.3.

[9] Solving the problem (3), (4) leads to the intermediate task of minimizing the following smoothing functional:

equation image

where α is a positive regularization parameter. The minimization task (5) is solved iteratively using the Newton method:

equation image

where the matrix Ln is the linearization of the operator M in the vicinity of Un.

[10] The smoothing functional (5) is minimized for a range of values of the regularization parameter α, and each minimizing solution Uα is tested against the condition (3). The optimum regularization parameter is the largest α such that Uα still satisfies (3).

2.3. Pseudocovariance Matrix

[11] There is considerable freedom in the choice of the stabilizing functional and its pseudocovariance matrix. In our previous applications of the Tikhonov method to ionospheric reconstruction problems [Fridman, 1998; Fridman and Nickisch, 2001], we defined the stabilizing functional as a finite difference approximation to the following integral expression:

equation image

This definition is satisfactory in that it correctly reflects the physical nature of ionospheric variations: It ensures that results of inversions are always smooth and bounded, and at the same time it allows sufficient freedom for matching diverse ionospheric data. The main disadvantage of this definition is its prohibitively high demand for computer resources when dimensionality of the problems is higher than two.

[12] We found that a factorized (separable) representation of the pseudocovariance matrix allows a substantial increase of the computation speed while considerably decreasing the computer memory usage. In the case of inversion in three-dimensional space (x1, x2, x3), we use the following form of the pseudocovariance matrix:

equation image

where function Φ is selected in a way that would ensure positive definiteness of the matrix P as well as smoothness of the resulting solution. In the results presented in this paper we use

equation image

It can be shown that this function produces a positive definite pseudocovariance matrix and that the resulting solution will be continuous along with its first derivatives. Parameters λ1, λ2, λ3 in (6) define the correlation scales of the solution along each of the dimensions. Physical considerations should be used in choosing suitable values of these parameters.

2.4. Recurrent Processing

[13] Our goal is to develop a tool for real-time monitoring of electron density in the ionosphere. We are developing here a system that would allow continuous updates of the solution as new ionospheric measurements become available. In order to accomplish this, we need to specify an evolution equation that tentatively relates solution vectors on consecutive temporal layers.

[14] In this paper we assume that the evolution of the solution obeys the following rule:

equation image

where γ is an attenuation coefficient and v(r, tk) is the correction vector at the time step k.

[15] In order to ensure proper treatment of the TEC data on the basis of phase measurements, we adopt a solution vector containing two time layers U(k) = {u(r, tk), u(r, tk−1)}, and we solve for the vector of corrections to both layers V(k) = {v+(r, tk), v(r, tk)}.

[16] In the beginning of a time step k, an initial approximation to the state vector is formed as Uinit(k) = {γu(r, tk−1), u(r, tk−1)}, and the starting correction vector Vinit(k) = 0. This initialization goes into the iterative process described in section 2.2 which produces the correction vector V(k). The updated state vector is calculated as U(k) = Uinit(k) + V(k). The component u(r, tk) of this vector is utilized to produce the ionospheric model for t = tk.

[17] Equations (6) and (7) specify the spatial part of the pseudocovariance matrix. The temporal part in this case is a 2 × 2 matrix which we adopted as diag(1, equation image), where 0 equation image ɛ equation image 1 is a parameter of the inversion process.

2.5. Measurement Operator and Its Linearization

[18] The components of vector Y may be sorted into three distinct groups that correspond to three types of measurements utilized by our system. They are the absolute TEC (or group delay–based TEC) data, the relative TEC (or phase delay–based TEC) data, and the electron density values at specified locations (vertical profile data). There are three components of the measurement operator that correspond to the above data types. We will outline below how these three components of the operator are constructed.

[19] Our definition of the measurement operator for the absolute TEC data type as well as the linearization of this operator are described by Fridman and Nickisch [2001]. Essentially, for a given receiver-transmitter path, the TEC value is approximated as a linear combination of electron density at the nodes of the spatial grid with subsequent substitution of nodal values of electron density in accordance with (1). Thus the value of TEC between receiver i and transmitter j at t = tk may be expressed as

equation image

[20] The phase delay–based TEC measurements are utilized in the form of the TEC rate equal to the TEC increment ΔCij(tk) = Cij(tk) − Cij(tk−1) between sequential time steps of our solution. It is convenient to use this combination because the differencing eliminates the unknown additive constant inherent to phase measurements of TEC. Substituting (9) into this definition, we obtain the expression that defines Mequation image, the measurement operator for relative TEC:

equation image

where U(tk) = {u(·, tk), u(·, tk−1)} is the state vector of the system. A linearized version of Mequation image is derived from the linearized Mequation image in an obvious manner.

[21] The electron density data type assumes that electron density is measured for a number of known locations r and time t. The measurement operator MN(r,t) for one datum of this data type is constructed by expressing the electron density at the given location as a linear combination of electron densities at the nodes of the spatial-temporal grid by employing linear interpolation along each dimension with subsequent substitution of the nodal values of electron density in accordance with (1). Derivation of the linearized version of MN(r,t) is straightforward.

2.6. Data Noise Model

[22] In order to specify the noise covariance matrix S, we need to formulate the statistical model of measured data. Our system is intended for utilization of raw data from any receiver that may become available. We assume that no leveling of the data has been performed and that estimates for receiver and transmitter biases may not be readily available.

[23] The following is the model of the group delay–based TEC measured between receiver i and transmitter j [Coster et al., 1999]:

equation image

Here Cij(t) is the actual value of the TEC, ηiRbias is the unknown receiver bias, ηjTbias is the transmitter bias, and ηijNoise is the measurement noise. We assume that the biases are slowly varying quantities and that the noise may be treated as white noise. Then the covariance between receiver-transmitter ij and the receiver-transmitter kl takes the form

equation image

where ViRbias and VlTbias are variances of the receiver and transmitter biases and RijTEC(t) is the variance of the measurement noise.

[24] We utilize the phase delay–based TEC measurements in the form of TEC increments Δequation imageij(t) = equation imageij(t) − equation imageij(t − Δt) between sequential time steps of our solution. An appropriate model for this signal is

equation image

where ηij(t) is the white noise, so that the covariance matrix for this type of measurements is

equation image

[25] Our system performs automated sliding window estimates of the variances RijTEC(t) and RijΔTEC(t) of the white noise components for (11) and (12). Results presented below use a sliding window of length 450 s applied to raw data series sampled with 30 s intervals. The noise power is estimated as the mean square deviation from the linear trend. The latter is estimated using the least squares method applied to data samples belonging to the window.

[26] The matrix of the measurement noise S is constructed with the assumption that the noise processes of different types of measurements are statistically independent from each other. Suppose that for a given time instant we have NTEC of absolute TEC measurements, NΔTEC of relative TEC measurements, and Nequation image of data points of electron density. Then matrix S will have size (NTEC + NΔTEC + Nequation image)2. Matrices (11) and (12) will occupy two square blocks along the main diagonal of S. These blocks will cover NTEC +NΔTEC elements of the main diagonal. Remaining Nequation image diagonal elements of S are determined by the noise of electron density measurements that should be specified by the user. All other off-diagonal elements of S should be set to zero (under the assumption that noise components of electron density measurements are mutually independent).

[27] The receiver and transmitter biases ηiRbias and ηjTbias are estimated after each temporal step using the maximum likelihood approach. These are then utilized for correcting input data on the subsequent step.

3. Results of the Analysis

3.1. Background Ionosphere Model

[28] We have adopted a fairly coarse model of the background ionosphere n0(r, t). The background electron density below 1500 km is generated using the International Reference Ionosphere (IRI) 2000 model. The background electron density above 2000 km and up to 20,000 km is calculated analytically assuming that the plasma consists of electrons, protons, and helium ions and that the particles are in diffuse equilibrium in the Earth's gravitational field. The interval from 1500 to 2000 km is a transition region where electron density is calculated as weighted average between IRI and the analytical formula. The relative weight of the IRI output falls linearly in this region from 1 at 1500 km to 0 at 2000 km. Although the IRI model is known to overestimate electron density values at altitudes above 1000 km, we will still proceed with employing this model for our initial test, keeping in mind that our inversion algorithm is able to correct the ionosphere on the basis of available data.

3.2. Results and Conclusions

[29] We will demonstrate performance of our inversion system using data recorded by 16 receivers of the Scripps Orbit and Permanent Array Center (SOPAC) network ( All selected receivers lie within 1000 km radius from 36°N, 120°W. The group and phase delay data were read directly from the files (receiver-independent exchange format) recorded by the receivers and transformed into absolute and relative TEC estimates. In addition, we used bottomside vertical profile data from the Point Arguello vertical sounder.

[30] The temporal step of the solution was set to 15 min (the same as the interval between vertical soundings). The solution covers a 20° × 20° region in geographical coordinates. The horizontal grid is uniform and rectangular and contains 50 × 50 nodes. The vertical grid extends from 90 to 20,065 km, and it has 135 nonuniformly spaced nodes with step varying from 5 to 200 km. The output of the solution is a three-dimensional table of electron density produced for every time step.

[31] The pseudocovariance needs to be calculated on the three-dimensional grid in accordance with (6) and (7). Latitudinal and longitudinal grid values are substituted for x1 and x2 in (6). A scaled altitude variable

equation image

is substituted for x3. Here H(h) is a variable vertical scaling distance. Our adopted H(h) increases from 20 to 1000 km in the interval of heights from 90 to 2000 km, and the scale remains constant at h ≥ 2000 km. In this solution we used λ1 = λ2 = 6° and λ3 = 1. The initial variances of receiver and transmitter biases ViRbias and VlTbias in (11) were set to correspond to 30 and 15 total electron content units (TECU), respectively.

[32] Figure 1 shows positions of the receivers as well as the large number of satellite-receiver paths that were used in the inversion (only inputs for the first 8 hours are shown). This example was executed on a 2.2 GHz Pentium PC. Each time step in this run required 0.5–4 min of computations, with most steps taking less than 1 min.

Figure 1.

Data sources for the first 8 hours of simultaneous inversion of TEC and vertical sounding data. Triangles indicate positions of the receivers, and dots show positions of subionospheric points (400 km altitude) for each satellite-receiver path that contributed to the first 8 hours of TEC data for the inversion.

[33] Results of the solution are illustrated by Figures 2 and 3. We emphasize that the result of our inversion is a three-dimensional electron density distribution produced for every time step. Figure 2 shows contours of the critical frequency reconstructed for 0215 UT. Figure 3 shows the distribution of plasma frequency for the altitude-latitude slice through 120.6°W longitude.

Figure 2.

Contours of ionosphere critical frequency reconstructed from TEC and vertical sounding data for 26 September 2004, 0215 UT. Position of the sounder (34.8°N, 120.5°W) is marked by a circle, triangles indicate positions of TEC receivers, and crosses show positions of subionospheric points at 0215 UT.

Figure 3.

Distribution of ionosphere plasma frequency reconstructed from TEC and vertical sounding data for 0215 UT.

[34] We have created an animated map of critical frequency reconstructed for a nearly 24 hour period (from 0015 to 2345 UT) of 14 September 2004 (see Animation 1). Notations are the same as in Figure 2. Vertical sounder data were used for the whole period of the inversion in this case. This animation demonstrates that the solution is reasonably smooth in time and in space. Observe that the central area of this geographical region is most densely pierced by satellite-receiver paths; consequently, strongest data-driven corrections occur in this area. Outside of this area, the solution tends to relaxate toward the background model.

[35] Figures 4, 5, and 6give an idea of how the measurements compare to corresponding predicted values. Figure 4 shows critical frequencies measured by the Point Arguello sounder compared to critical frequencies extracted from the results of inversion.

Figure 4.

Critical frequency (MHz) obtained by inverting GPS TEC and vertical sounding data (squares) compared to that measured by the Point Arguello sounder (solid line), 26 September 2004.

Figure 5.

Absolute TEC (gray line) measured by one of the receivers (receiver CAND, GPS satellite 23) compared to the predicted TEC (solid black line). Crosses indicate samples of the data that went into the inversion process (sampling interval was 15 min).

Figure 6.

Rate of the relative TEC (gray line) measured by one of the receivers (receiver CAND, GPS satellite 23) compared to the predicted TEC rate (solid black line). Crosses indicate samples of the data that went into the inversion process (sampling interval was 15 min).

[36] Figure 5 presents a typical example of measured absolute TEC data (the group delay-based measurements) compared to the predicted values for absolute TEC measurements. The predicted absolute TEC is estimated as Cij + ηiRbias + ηjTbias, where Cij is the TEC calculated in the reconstructed ionosphere and ηiRbias and ηjTbias are current estimates of receiver and transmitter biases. Note that all these estimates may vary with time.

[37] Figure 6 presents an example of the measured rate of relative TEC (the phase delay–based measurements; a 30 s differencing interval was used) compared to the TEC rate obtained from the solution. Note that Figure 6 refers to the same satellite-receiver pair and the same time interval as Figure 5. Figures 5 and 6 present two different data types associated with the satellite-receiver pair. Both data types are simultaneously processed by the inversion algorithm. One can see that for all data types, the predicted values agree with measurements within errors of measurements.

[38] Results presented in Figure 7 test the importance of the vertical sounding data for the ionospheric reconstruction problem. The setup of the input data is essentially the same as for the example shown in Figure 2. In this case we stopped feeding vertical profile data into the inversion process after 0700 UT. Note that the good agreement between measured critical frequencies and inversion results continues for nearly 4 hours after 0700 UT. At later times, the critical frequency is predicted with accuracy worse than 0.3 MHz, errors occasionally exceeding 1 MHz. This test stresses the importance of incorporating vertical profile data into the inversion process.

Figure 7.

Critical frequency (MHz) obtained by inverting GPS TEC and vertical sounding data (squares) compared to that measured by the Point Arguello sounder (solid line), 25 September 2004. Note that the inversion process did not use the vertical ionogram (VI) data after 0700 UT.

[39] These results suggest that the system being developed may be useful for real-time monitoring of the ionosphere over wide geographical areas using data from GPS TEC receivers and vertical sounders when they are available. Further development of the system is under way for utilizing in situ electron density data measured by orbiting satellites.