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Keywords:

  • ionosphere;
  • GPS/GNSS;
  • TEC estimation;
  • observability;
  • Kalman filtering

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[1] A method is presented for the removal of ionospheric effects from single-frequency radio navigation data. It uses data from a separate single collocated dual-frequency Global Positioning System (GPS) receiver to estimate the states of a local ionospheric total electron content (TEC) model. These states can be used to estimate the TEC along lines of sight to other types of satellite-based radio navigation transmitters. Two local ionosphere models are considered. The first model is a modified version of the classical thin-shell model, where the altitude and altitude variations of the shell with respect to latitude and longitude are added as new estimable quantities. The second model is a new thick-shell model, where the thin shell has been expanded with a Chapman electron density profile. The states of the Chapman profile are allowed to vary with respect to latitude and longitude and must be estimated. States are estimated using a Kalman filter and GPS observables. Truth-model simulations are used to test the observability of the states in each of the local ionosphere models. The subset of observable local models are then tested using real GPS data. The TEC estimates produced by the local models are compared to the results produced by an existing multireceiver network and tomographic model. The local thin-shell model is shown to compare well to the tomographic model, and significant improvement is shown over the classical single-receiver fixed-altitude thin-shell model, but the thick-shell model is shown to be unobservable.

Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[2] A method for the removal of ionospheric effects from single-frequency non-GPS radio navigation data is desired. The strategy adopted uses data from a single-collocated dual-frequency Global Navigation Satellite System (GNSS) receiver, in this case a GPS receiver. The strategy uses a single receiver because the proposed system is intended to be used by elements of a global network of remote orbit determination reference receiver stations for a particular constellation of Low Earth Orbit (LEO) satellites. Some of the stations will be on small islands in large oceans and will not have access to data from receiver arrays. In situations where additional receivers are available, it might be preferable to employ currently existing multireceiver array-based ionospheric total electron content (TEC) models. Although there exist data-based global TEC models, their accuracy can be limited due to a lack of data assimilated near an orbit determination reference station, hence the need for the present developments when operating in remote locations.

[3] The GPS receiver data are used in conjunction with a local ionosphere model to estimate the TEC along different lines of sight in the ionosphere. The concept is illustrated in Figure 1, where inline image is the receiver location and inline image and inline image are the TEC values calculated along the lines of sight to the GNSS satellites. The TEC can be measured along the lines of sight to the GNSS satellites because they operate on two frequencies, e.g., the GPS L1 and L2 frequencies. The TEC cannot be measured along the line of sight to the non-GNSS satellite because it only operates on a single frequency.

image

Figure 1. Schematic showing dual-frequency measurements of the ionosphere from GNSS satellites and one receiver, used to estimate TEC along other lines of sight to the non-GNSS satellite.

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[4] There are several survey papers on the capabilities of GPS as an ionospheric and space weather sensor [Coker et al., 1996; Coster and Komjathy, 2008], and only an overview is presented in this paper. Ionospheric TEC estimation using GPS has been verified with the Faraday rotation technique [Lanyi and Roth, 1988], incoherent scatter radar [Jakowski et al., 1996], and TOPEX data [Orus et al., 2002, 2003]. Although mostly understood, determining the receiver interfrequency biases is still a topic of current research [Rideout and Coster, 2006; Rao, 2007; Ciraolo et al., 2007], and multipath effects still present a challenge [Bishop et al., 1985; Ciraolo et al., 2007].

[5] Many different TEC estimation methods exist in the literature, but perhaps, the most widely known ionosphere model used in the area of GPS is the simple Klobuchar model [Klobuchar, 1975]. Another early TEC estimation method for single receivers is the “circus tent” local ionosphere model [Coster et al., 1991, 1992]. This model can predict TEC along nonmeasured lines of sight by computing TEC parameters for different azimuthal regions about the receiver. The circus tent model can be significantly different from the real ionosphere, but it has the advantage of simplicity. Bishop's Self-Calibration Of (pseudo)-Range Errors or SCORE method can be used to calculate TEC along GPS satellite lines of sight [Bishop et al., 1995a, 1995b, 1996, 1997], but it does not use an ionosphere model and does not prescribe a method to predict TEC along non-GPS satellite lines of sight.

[6] Many authors have made the assumption of a thin-shell ionosphere and corresponding mapping function, of which a moderate selection are listed [Bishop et al., 1987; Lanyi and Roth, 1988; Clynch et al., 1989; Coco et al., 1991; Ciraolo and Spalla, 1997; Arikan et al., 2003; Skone, 2006; Morton et al., 2007; Nayir et al., 2007; Anghel et al., 2009]. Some of the authors fit the data with polynomial-type TEC variation parameters [Lanyi and Roth, 1988; Coco et al., 1991; Ciraolo and Spalla, 1997; Skone, 2006; Morton et al., 2007; Anghel et al.,2009], which enables predictions of TEC along nonmeasured lines of sight, while others did not [Bishop et al., 1987; Clynch et al., 1989; Arikan et al., 2003; Nayir et al., 2007]. The result of different thin-shell heights was investigated in Komjathy and Langley [1996]. The fundamental errors inherent in the assumption that all the TEC is compressed at one height were investigated in Smith et al. [2008]. The fixed-altitude thin-shell models are included and generally expanded in the present work. Therefore, this work's comparisons between its different models implicitly compares its new methods with the many existing methods that are subsumed within its forms.

[7] Some authors have also investigated the use of multiple shells at different altitudes for ionospheric TEC estimation [Komjathy et al., 2002; Anghel et al., 2009]. Multiple shells allow the ionospheric TEC to be assigned in discrete quantities to different altitudes, creating a higher fidelity approximation of the true ionosphere. One of the local ionosphere models presented in this paper captures similar height distribution effects in the electron density profile but differs from the multiple shell models in that it estimates a continuous Chapman profile's [Chapman, 1931] mean altitude and scale height instead of splitting the TEC between discrete shells. When multiple receiver stations are used in an array, as in [Komjathy et al., 2002], multiple layers can improve TEC estimation results. When only a single receiver is used, as in [Anghel et al., 2009], multiple layers do not typically improve TEC estimation results. The failure of a single receiver to improve TEC estimation results using multiple layers can be inferred from the analysis in section 6; the multiple shell height states are not strongly observable with only a single receiver.

[8] There is also an extensive body of work using large arrays of receivers, which can yield both higher accuracy and better spatial resolution due to the increased number of measurements and an increase in the spatial distribution of their lines of sight. The authors of Mannucci et al. [1995] used a grid of receivers to estimate TEC in a shell-type parameterization of the ionosphere and checked their results first with a truth-model simulation, and then again with TOPEX data [Mannucci et al., 1998]. Both Otsuka et al. [2002] and Ma and Maruyama [2003] used hundreds of receivers in Japan to estimate the regional ionospheric TEC. The 3-D global Chapman electron density profiles [Feltens, 1998] and neural networks [Yilmaz et al., 2009] have also been used to model ionospheric TEC.

[9] Array-based ionospheric tomography is another common TEC estimation technique, and it has been shown to be more accurate than the thin-shell models [Meggs et al., 2004]. A short selection of early tomographic techniques include the two-slab methods of Juan et al. [1997] and Hernandez-Pajares et al. [1999] and the three slab method of Mannucci et al. [1999]. Ionospheric tomography has been validated using synthesized measurements from the International Reference Ionosphere [Hernandez-Pajares et al., 1998; Bilitza and Reinisch, 2008], with LEO GPS data [Hernandez-Pajares et al., 1998], and with TOPEX data [Hernandez-Pajares et al., 2002]. Real-time tomographic ionospheric TEC estimates have also been shown to have applications in integer ambiguity resolution in carrier-phase differential GPS [Hernandez-Pajares et al., 2000, 2002]. The International GNSS Service (IGS) has also created TEC maps using a variety of methods [Feltens, 2003; Hernandez-Pajares et al., 2009]. One of the more recent models was implemented by the National Ocean and Atmospheric Administration (NOAA) [Spencer et al., 2004; Araujo-Pradere et al., 2007; Minter et al., 2007], and it uses tomographic techniques, assumed electron density profiles, and approximately 90 receivers to create a map of TEC over the continental United States, denoted US-TEC. A full survey of tomographic techniques is presented in Bust and Mitchell [2008]. Unfortunately, these powerful techniques cannot be used in the application envisioned for the current developments.

[10] This paper's three main contributions are the following: a pair of new local ionosphere models, an evaluation of these model's Kalman filter-type observability, and tests of the corresponding Kalman filters using real data. The first model is a generalization of the classical thin-shell model. It incorporates vertical TEC (VTEC) variations with respect to latitude and longitude. Additional estimated quantities include the height of the shell and its height variations with respect to latitude and longitude. The second model generalizes these ideas to allow ionospheric thickness via an expansion of the TEC layer into an electron density distribution. Additional estimated states of this model are its nominal scale height (i.e., thickness scale) and its scale height variations with respect to latitude and longitude.

[11] The models are first evaluated with truth-model simulations to test their observability, or the ability of each model state to be uniquely reconstructed from the measurements produced by that state. A model that is shown to be observable in simulation is not guaranteed to be observable when using measurements from the real ionosphere, because the true ionosphere will differ from the local model. Therefore, the models are also evaluated using real data to estimate the states of the local ionosphere. Then, the local model predictions of TEC are compared to those predicted by the NOAA ionosphere model.

[12] This paper is organized as follows. The relevant GPS observables equations are derived in section 2. The thin-shell and thick-shell local ionosphere models are developed in section 3. Section 4 presents an extended Kalman filter (EKF) method of estimating each model's states. Section 5 discusses practical implementation issues for those EKFs. The theoretical observabilities of the local ionosphere models are evaluated using truth-model simulations in section 6. Section 7 estimates the states of the local ionospheric models using real GPS data and compares those results to NOAA's results over the same days. Conclusions are drawn in section 8.

Ionospheric Effects on GPS Observables

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[13] To most GNSS users, the ionospheric effects on electromagnetic (EM) waves [Klobuchar, 1983] are typically regarded as a nuisance parameter that degrades the fidelity of the navigation solution. The ways in which the ionosphere affects the GPS observables, through group delay and phase advance, have been described by many authors and will be presented here only briefly [Komjathy and Langley, 1996; Arikan et al., 2008; Misra and Enge, 2006]. Similar effects would occur on any L1 band non-GPS radio navigation data.

[14] The ionosphere acts on the pseudorange or carrier-phase of a signal by delaying or advancing by an amount that is inversely proportional to the square of its frequency, f in Hertz, as shown in the first-order approximation [Liao, 2000]:

  • display math(1)

where inline image is the TEC to satellite j from the receiver in units of electrons per meter squared (e/m2), and inline image is given in meters. TEC is defined as the integrated electron density between the receiver and satellite:

  • display math(2)

The quantity ρelec is the electron density at the desired location defined by α and LoS, where α is the physical distance along the given line of sight (LoS).

[15] The delay in (1) appears in the model's pseudorange and carrier-phase observables:

  • display math(3)
  • display math(4)

where inline image and inline image are the pseudorange and carrier-phase accumulated delta range from satellite j to the receiver as measured at the GPS L1 frequency. The term rj is the actual physical range, inline image is the receiver clock error, δtj is the satellite clock error, inline image is the tropospheric delay, and c is the speed of light in meters per second (m/s). Many terms are frequency dependent, as designated by inclusion of the L1 subscript: inline image, a satellite-dependent bias inline image, and a receiver-dependent bias inline image. λL1 is the wavelength, inline image is the phase bias for satellite j, inline image is the pseudorange noise, and inline image is the range-equivalent carrier-phase noise term, all at the L1 frequency. The equations are the same for the L2 observables, but with L2 subscripts.

[16] The paper develops its TEC Kalman filter by using measurement models for the single differences of the pseudorange and carrier-phase equations, with differences taken between the L1 and L2 frequencies. The differencing removes the unneeded navigation term rj, and nuisance parameters inline image, δtj, and inline image. The resulting single differences are the following:

  • display math(5)
  • display math(6)

where the term inline image is satellite j's interfrequency bias, and inline image is the receiver interfrequency bias. In addition, ΔΦjis the difference in the range-equivalent carrier-phase biases at the L1 and L2 frequencies, and the terms ΔνP and ΔνL are the differenced noise terms.

[17] The differenced pseudorange equation provides noisy measurement of TEC that includes the effects of hardware bias on the satellite and receiver. The phase measurement is much more precise, with about 2 orders of magnitude less noise, but it includes the additional delta phase bias ΔΦj. The receiver interfrequency bias inline image and the satellite interfrequency bias inline imageboth drift very slowly and will be assumed to stay constant over a full day. The static assumption was verified by Coco et al. [1991], which showed that the day-to-day variations of the satellite's differential interfrequency code biases over a 5 week span were less than 0.3 ns, and the receiver bias variations were on the order of 0.5 ns. Expressed in TEC units (TECU), where 1 TECU is equal to 1016 e/m2, the satellite and receiver differential bias variations were, respectively, 0.8 and 1.4. The satellite interfrequency bias, inline image, is provided by the Center for Orbit Determination in Europe (CODE) for each satellite, with approximately 1 month latency. The satellite biases will be assumed to have been removed in the rest of this paper, although the use of old satellite biases could introduce errors into the final filter results. This assumption is made because a single dual-frequency GNSS receiver cannot estimate the satellite biases very accurately in a single pass. The reliance on CODE's satellite biases is for ease of comparison only.

Local Ionosphere Models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[18] The GPS signals provide a biased measurement of TEC along GPS satellite lines of sight but give no information about TEC along other lines of sight. Local ionosphere models can provide the means to predict TEC along other lines of sight. This section presents two local ionosphere models that serve this purpose, a thin-shell model and a thick-shell model. The thin-shell model will be presented first, and then an electron density profile will be added to create the thick-shell model.

Local Thin-Shell Model With Altitude Variations

[19] In the classical local thin-shell ionosphere, the thickness of the ionosphere is assumed to be compressed into an infinitesimal layer at a given fixed altitude of maximum electron density, hmax. Some thin-shell models parametrize a spatial second-order Taylor series expansion on VTEC [e.g., Skone, 2006]. In this paper, a second Taylor series expansion is added for the height of maximum electron density, so that hmax is allowed to vary with respect to latitude and longitude. Each Taylor series is a two-dimensional function of latitude and longitude perturbation from the receiver location on the WGS-84 Ellipsoid:

  • display math(7)
  • display math(8)

where hmax is the height of the thin shell, VTEC is the vertical TEC, Δθpp and Δφpp are, respectively, the latitude and longitude differences between the line of sight pierce point at the ionosphere and the receiver location. Figure 2 depicts the relevant geometry. This includes the variable hmax surface, the line of sight vector, whose normalized unit direction vector is inline image, the receiver location inline image, the pierce point location rpp, and the angle between the line of sight and the normal to the hmax surface, z. In fact, VTEC is not truly the vertical TEC in this model. Rather, it is the TEC along a line of sight that pierces the thin-shell with angle z=0.

image

Figure 2. Geometry of the local ionosphere thin-shell layer and of TEC evaluation at the line-of-sight pierce point.

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[20] Depending on the number of terms used in the Taylor series expansion of the height states in (7), a number of different types of thin-shell models can be identified. Figure 3 illustrates these differences by showing the different models' possibilities for their thin height contours. Model 1 is a fixed known altitude model (typically around 350 or 400 km) with full second-order variations with respect to latitude and longitude of VTEC. Model 1 is very similar to those used by Coco et al. [1991] and Skone [2006]. If the latitude and longitude expansion of VTEC were taken out to third-order terms then the model would be similar to those used by Lanyi and Roth [1988] and Anghel et al. [2009]. If only the first-order derivatives of VTEC were used, then the model would be similar to that used by Morton et al. [2007], and if only the latitude term were used, then the model would be closer to that used by Ciraolo and Spalla [1997]. Model 2 is the same as Model 1, but with an uncertain altitude that must be estimated. Models 3 and 4 are the same as Model 2, but with uncertain nonzero first- and second-order derivatives of height that also must be estimated. Model 3 only considers first derivatives, and Model 4 considers both first and second derivatives.

image

Figure 3. Representative height contours for four types of thin-shell ionosphere models.

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[21] The TEC of a given line of sight can be evaluated in this model without using an integration scheme to approximate (2). This is because the electron density is all concentrated at one altitude, hmax(θ,φ). The TEC along any given line of sight is equal to the VTEC at the pierce point, rpp, multiplied by a mapping function. The VTEC can be determined using the Taylor series expansion in (8), after calculating

  • display math(9)
  • display math(10)

and then the mapping function inline imageyields the following:

  • display math(11)

[22] There are other forms of the mapping function [Komjathy, 1997], and other authors have looked at using tomography to approximate mapping functions [Hernandez-Pajares et al., 2005], but they are not a focus of this paper.

[23] The latitude and longitude of the pierce point, θpp and φpp, and the angle z are determined based on the known receiver location inline image, the known unit direction vector inline image, and the height model in (7). This determination starts by recognizing that the pierce point can be written in the form inline image, where α is the distance along the line of sight from the receiver to the thin shell. A nonlinear equation is developed that solves for the exact αof the intersection of the line of sight and the thin-shell ionosphere. The nonlinear equation involves calculation of the latitude, longitude, and altitude relative to the WGS-84 ellipsoid for any given α. The resulting height relative to the ellipsoid is differenced with the thin-shell ionosphere's height at that same latitude and longitude. The correct αdrives this distance to zero. A Newton-Raphson procedure is used to determine this α, and with it θpp, φpp, and rpp.

[24] The quantity cos(z) equals the dot product between inline image and the normal to the thin-shell ionosphere, with this normal evaluated at rpp. The normal to the ionosphere can be calculated by taking the gradient of an altitude error function with respect to Earth-centered Earth-fixed (ECEF) position r. This error function is like the one used to determine α, except that it equals the difference between the WGS-84 altitude of an arbitrary Cartesian position vector r and the altitude of the thin-shell ionosphere at the WGS-84 latitude and longitude of that same arbitrary point r.

Local Thick-Shell Model With Altitude and Thickness Variations

[25] The thin-shell model can be modified to create a thick-shell model that more closely approximates the actual ionosphere. It expands the electron density about hmax in a Chapman profile [Chapman, 1931]. An example Chapman profile, shown in Figure 4, is created using the vertical Chapman profile equation:

  • display math(12)

where

  • display math(13)

is a non-dimensional altitude measured relative to the altitude of peak density in units of scale heights. The term h is the height at which the profile is being evaluated, hmax is the altitude of maximum electron density, hscale is the width parameter of the profile, and ρmax is the maximum electron density. This profile can also be formulated using VTEC instead of ρmax:

  • display math(14)
image

Figure 4. An example Chapman profile, with the parameters hmax=350 km, hscale=75 km, and ρmax=1.5·1012 e/m3.

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[26] The thick-shell model is obtained by modifying the electron density profile of (14) with a Taylor series to allow variations in hmax, VTEC, and hscalewith respect to latitude and longitude, as in (7), (8), and

  • display math(15)

[27] The model is represented graphically in Figure 5, where the curve above the receiver, inline image, is the surface of maximum electron density, and the dots along inline image are points at which the electron density must be evaluated in order to compute the TEC integral of (2). As in the thin-shell model, α is the physical distance along the line of sight, from the receiver to the satellite, at which one wishes to evaluate the electron density. It is important to realize that in this local model, the Chapman profile is defined not along the local vertical, but along the local normal to the hmax surface, as shown in Figure 5.

image

Figure 5. Geometry of the thick-shell local ionosphere model and TEC calculation using Chapman profile evaluation and integration of electron density over the whole line of sight.

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[28] The TEC of a given line of sight can be calculated in the thick-shell model using numerical integration to approximate (2). The electron density at a point, inline image, is evaluated as follows. First, one finds the nearest point on the hmax surface inline image. Next one evaluates the Taylor series in (8) and (15) to determine VTEC and hscale. Finally, one uses these in the Chapman profile associated with that point. The value zChap is calculated using the distance inline image in place of hhmax in (13) with a positive sign when inline image is above the maximum density surface and a negative sign when it is below. Integration requires evaluation of the local model at numerous points along any given line of sight.

[29] The determination of inline image as a function of inline image is complicated. It involves nonlinear equation solving of the first-order necessary conditions of a constrained optimization. The constrained optimization finds the inline image that minimizes inline image subject to the constraint that inline imagelie on the surface of maximum electron density as defined using (7) and the WGS-84 ellipsoid. The resulting first-order necessary conditions are complicated and have been omitted for the sake of brevity.

Extended Square Root Information Filtering Components

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[30] This project used an Extended Square Root Information Filter (ESRIF), which is a more numerically stable form of the extended Kalman filter, to estimate the state of the local ionosphere models based on dual-frequency GPS observables. The reader is assumed to have some familiarity with this type of filter, and its formulation is not presented here. Further information on extended Kalman filters can be found in Bar-Shalom et al. [2001], and square root information formulations in Bierman[1977].

[31] There are three components to any type of EKF: the state vector, the dynamic model, and the measurement model. The state information is propagated through time using the dynamic model and updated with new information using the measurement model.

Filter State

[32] Depending on the ionospheric model used, the ionospheric state is defined explicitly as

  • display math(16)

where the first state definition is used for the thin-shell ionosphere model, and the second definition is used for the thick-shell model. The vector inline image consists of the height model Taylor series coefficients in (7), xVTEC contains the integrated electron density coefficients from (8), and inline imagecontains the scale height Taylor series coefficients of (15). These coefficients are listed in Table 1.

Table 1. Ionospheric States, xIono
inline imagexVTECinline image
hmax 0VTEC0hscale 0
inline imageinline imageinline image
inline imageinline imageinline image
inline imageinline imageinline image
inline imageinline imageinline image
inline imageinline imageinline image

[33] The state dimension can be reduced by further truncating the Taylor series to be only first-order or even zeroth-order, or by fixing some of the coefficients listed in the table at a priori known values.

[34] The full system state x includes the ionospheric state vector xIono and the nuisance parameters of receiver inter-frequency bias, inline image, and vector of delta phase ambiguities, ΔΦ:

  • display math(17)

Recall that these bias parameters are defined in connection with (5) and (6).

Filter Dynamic Model

[35] The dynamic model for this system is not based on ionospheric physics. The states do not contain enough information to allow direct application of the physical principles that drive ionospheric electron density dynamic variations. As a reasonable alternative, this model attempts to capture the temporal correlation between the local ionospheric states through the use of a first-order Gauss-Markov process to model the perturbations of the states from nominal values. The nuisance parameters, inline image and ΔΦ, are considered to be constants. The Gauss-Markov model for the ionospheric states takes the following form:

  • display math(18)

where τi is the correlation time constant of the ith element of xIono, and σSS,i is the steady state standard deviation of that element from its nominal value in the ith element of inline image. The vector w is the white noise, zero-mean unit-power-spectral-density process noise input vector. The elements of the reference vector, inline image, are chosen to give a reasonable nominal local ionosphere. Specific values for the model quantities, τi, σSS,i, and inline image are discussed in section 6. The values τi and σSS,i can be considered dynamic model tuning parameters, which characterize the behavior of the system.

Filter Measurement Model

[36] The measurement model for this filter is effectively a nesting of one of the local ionosphere measurement models inside of a GPS single-differenced observables measurement model. It takes the form:

  • display math(19)

where the measurement function hP(x) is defined by the single-differenced pseudorange equation (5); and hL(x) is defined by the single-differenced carrier phase equation (6).

[37] Each row of the hP(x) function equals the sum of all of the terms on the extreme right-hand side of equation (5) except for the very last term, ΔνP. Likewise, each row of the hL(x) function is defined to equal the extreme right-hand side of equation (6), after removing ΔνL. The term TECinline image in each of these expressions depends on the xIonopart of xas per the thin-shell or thick-shell calculations of section 3. The remaining terms in (5) and (6) can be written directly in terms of known satellite calibration parameters or the receiver bias elements of x, inline image and ΔΦj.

[38] EKFs require the measurement sensitivity matrix, composed of the Jacobians inline image and inline image, in order to determine the optimal state estimate. The measurement sensitivity matrix takes the form:

  • display math(20)

where I is the identity matrix, 0 is a matrix of zeros, the vector eis defined as

  • display math(21)

and the Jacobian matrix inline image is

  • display math(22)

where inline image is the local ionosphere model's measurement sensitivity row vector for line of sight j.

Filter Practicalities

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[39] There are three important practical considerations of the filter that deserve to be mentioned briefly.

[40] The first consideration is that the filter could be provided erroneous measurements that could cause it to incorrectly estimate the height of the ionosphere to be below the surface of the Earth or above a feasible ionospheric height. Although this behavior has never been observed, it is still prudent to enforce a constraint on the altitude of the ionosphere. The model naturally enforces this constraint via the dynamic model, but one could add an altitude pseudomeasurement if necessary.

[41] The second consideration is a modification to the ionospheric states to account for numerical issues that arise near the poles of the Earth. The modification changes states of the form inline image to the form inline image, where ∗ denotes either hmax, VTEC, or hscale, and corresponding changes are made to the Taylor series in equations (7), (8), and (15). This modification was made solely to improve handling of possible numerical issues, and the distinction will be dropped in the remainder of the paper.

[42] A third consideration is that of computation time and numerical integration accuracy of the thick-shell model. The thin-shell model and its Jacobians can be computed in approximately the same time required for a single integration point along the thick-shell model's line of sight and without computation of its Jacobians. The thick-shell implementation in this paper used 125 quadrature integration points, resulting in a minimum increase in computation time of 125 times that of the thin-shell model. The quadrature points have been chosen judiciously, concentrated near the peak electron density, in a way that yields a numerical integration error of no more than 1 part in 108.

The Question of Local Ionosphere Model Observability

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[43] The practical observability of both the thin-shell and thick-shell ionosphere models has been investigated in order to determine the preferred model for the current application. If the system is not observable, its predictions of TEC along nonmeasured lines of sight can have large errors.

[44] Observability is a mathematical concept that addresses the question of whether a particular dynamics and measurement model can be used to uniquely reconstruct the system's state from a time series of measurements. Observability is a property of the model under analysis and can be tested independently of real data. Observability answers the question of whether model inversion is possible. A forward model starts with a known initial internal state and propagates it in time using a dynamics model. The state time history is then propagated through a measurement model to predict the measurements. Model inversion tries to start from actual measurements and reconstruct/observe the unknown initial internal state. For an observable system, this inversion produces a unique initial internal state estimate. For an unobservable system, the state estimates are not unique; therefore, any derived estimate cannot be trusted. Mathematical tests for observability are typically related to rank tests on matrices that have at least as many rows as columns, as in linear least squares problems. In fact, the simplest observability analyses apply directly to such systems of equations. Some observability tests are binary/yes-no tests. In the present context, practical tests will be used, ones that also consider practical convergence time and accuracy of the model inversion. A thorough exploration of the concept of observability is beyond the scope of this paper and only an overview is presented here. Further information can be found in Kailath[1980].

[45] A truth-model simulation, using the same states, the same Markov dynamics model, and the same measurement model as that used by the Kalman filter, can test a necessary but not sufficient condition for practical observability. Markov processes have the convenient property that their observability can be tested by a simple comparison. The comparison is made between the corresponding Kalman filter's theoretical standard deviation for the given state, σest,i, and the open-loop standard deviation of the Markov model, σSS,i. If σest,i is significantly smaller than σSS,i, then the state is practically observable. The standard deviation σSS,i characterizes the steady state uncertainty in the absence of measurements. Therefore, a filter is useful only to the extent that its measurement updates can reduce the uncertainty well below this level.

[46] The above test is only a necessary condition for practical observability due to differences between the Kalman filter's states, dynamics, and measurement model and that of the real ionosphere. The results presented in this section do not consider the additional issues that arise due to model mismatch between the true ionosphere and the ionosphere as understood by the Kalman filter. The performance of the proposed filter in the presence of model mismatch could be tested with a more sophisticated simulation, but no simulation will be a better test than the true ionosphere. Therefore, the performance of the filter is tested using real data in the next section. It is possible that new states could be added to the current models to include information about ionospheric physics, which may change the observability results. Despite the lack of physics-based information in the current model, it approximates the TEC behavior of the local ionosphere to a level that is sufficient for the current application.

[47] The practical observability of each of the Kalman filter's local ionosphere models was tested using numerous truth-model simulations, with the initial states set to the reference states listed in Table 2, inline image, and the initial uncertainties set to the steady state values in the same table, σSS,i. The vertical TEC is given in TECU.

Table 2. Truth-Model Simulation Parameters for the Local Ionosphere Models
States of inline imageinline imageσSS,iUnits
∗=hmax3503km
inline image,inline image01.5km/rad
inline image,inline image,inline image00.4km/rad2
∗=VTEC101TECU
inline image,inline image00.25TECU/rad
inline image,inline image,inline image00.125TECU/rad2
∗=hscale500.55km
inline image,inline image00.26km/rad
inline image,inline image,inline image00.065km/rad2

[48] The reference states, the entries of inline image, were set to what are believed to be reasonable levels for a quiet ionospheric day. The results of a similar analysis on geomagnetically disturbed days or days with solar storms are beyond the scope of this paper. The steady state uncertainties, σSS,i, for hmax were originally selected in the same manner, but were later tuned to improve results on training data. The training data were real data from physical receivers and were not included in the results section of this paper. The chosen levels of σSS,i for the hmax states are much lower than indicated by observations of the real ionosphere, effectively forcing the filter to believe that the ionospheric height is at 350 km unless the data tells the filter with a high level of certainty that the height should be a different value. The σSS,i values for VTEC were originally taken from Skone [2006], but were halved based on results using the training data mentioned previously, again yielding values lower than observations of the real ionosphere would indicate. The σSS,i values for hscale were selected to force the ionosphere's scale height to vary slowly, in order to test a best case scenario for observability of the thick-shell model. If the hscale states cannot be estimated when there is only minor time variation, then it is less likely that they can be estimated with a greater magnitude of process noise.

[49] The truth-model simulations used 6 h of simulated data, but they used actual line-of-sight geometry to GPS satellites from a single receiver station located at Key West in Florida. The receiver location was selected because of the availability of real data from the same location, which is analyzed in the next section. The lines of sight were for the day of 22 April 2011, starting at 6:00 A.M. and ending at 12:00 P.M. local time (noon). All of the nuisance parameters, inline image and ΔΦ, were included in the simulation. The simulation also included zero-mean, white Gaussian process and measurement noise. The process noise was identity covariance and the measurement noise used standard deviations of 7 m on pseudorange and 1 cm on phase.

[50] Based on the results of the truth-model simulations, each local ionospheric state was classified as not observable, weakly observable, moderately observable, or strongly observable. All the states in the two local ionosphere models are sorted into four observability categories in Table 3. Representative plots of σest,itime histories for each observability type are shown in Figures 6 and 7. This type of figure has been used to make the classifications listed in Table 3. Rapid convergence of σest,i to a value much lower than σSS,i indicates strong observability, as in Figure 6 (top). Failure of the σest,i values to move below σSS,i by any noticeable amount indicates that the state is not observable, as in Figure 7 (bottom). Moderate and weak observability are indicated by σest,itime histories in between these two extremes, as indicated in Figure 6 (bottom) and Figure 7 (top). A lower value of σest,i indicates a lower estimated state uncertainty and therefore a higher level of practical observability.

Table 3. Classifications of Different Local Ionospheric States Into Observability Categories
NotWeaklyModeratelyStrongly
ObservableObservableObservableObservable
inline imageinline imagehmax 0VTEC0
inline imageinline image inline image
inline imageinline image inline image
hscale 0inline image  
inline imageinline image  
inline image   
inline image   
inline image   
inline image   
image

Figure 6. Time histories of σest,i and σSS,i for a strongly observable state, VTEC0, and a moderately observable state, hmax0.

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image

Figure 7. Time histories of σest,i and σSS,i for a weakly observable state, inline image, and a state that is not observable, inline image.

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[51] All of the hscale states have been classified as not observable. The cause of the unobservability is that a ground-based observer only receives data in the form of electron density integrated along the full line of sight, which contains only minor amounts of information about the electron density distribution within that line of sight. If the TEC measurement did not pass through the full Chapman profile, and instead started at a significant nonzero altitude, then the measurement might provide useful information about profile scale height. Unfortunately, this is not the case in the current study. The lack of practical observability, and the drastic increase in the thick-shell model's computation time, implies that one should forgo the use of the thick-shell model and instead use the thin-shell model. The only states in the thin-shell model that are not practically observable are the second derivatives of hmax, or inline image, inline image, and inline image. These results indicate that Models 1–3 from section 3.1, which are illustrated in Figure 3, are observable. Model 4, however, is unobservable. Therefore, only Models 1–3 are evaluated using real data. Additional testing with significantly different values for inline image and larger values of σSS,idid not dramatically change the conclusions about practical observability from the truth-model simulations. It should again be noted that these results are for a quiet ionospheric day and that future work would be required to analyze the practical observability of the local models on a more interesting ionospheric day.

Results From Real Dual-Frequency GPS Data

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[52] The accuracy of each of the thin-shell models is investigated by comparing the local ionosphere model's predicted slant TEC (STEC) values for various lines of sight to those predicted by NOAA's model, US-TEC. US-TEC is based on tomography, and it estimates the amplitude of electron density profiles derived from a singular value decomposition of the electron densities in the IRI95 model. The model assumes a spatial correlation of the profiles between latitude and longitude based on a Gaussian function. The US-TEC model is composed of a data-driven component and an assumed climatological component, the IRI95 model. The resulting state estimate is formed from a weighting of these two components, with 90% of the weight given to the data. The US-TEC product has been validated in several papers, Araujo-Pradere et al. [2007] and Minter et al. [2007], and has an average root-mean-square error of 2.4 TECU over a 6 month period. The comparison between the NOAA model and the local models is a reasonable first comparison, but further validation work should be carried out in the future because the NOAA model itself is imperfect and because this paper's new model may degrade for disturbed ionospheric conditions.

[53] The following results are from two representative days in 2011 with quiet ionospheric conditions, separated by 1 month, 23 April, denoted Day 1, and 23 May, denoted Day 2. The 24 h average Kp index for Day 1 was approximately 1.6, and for Day 2, the value was approximately 0.8. The results are computed using data from two ground stations, one located at Key West in Florida, denoted Rx1, and the other at Philadelphia in Pennsylvania, denoted Rx2.

[54] The comparisons attempt to characterize the difference between the local and NOAA models' TEC predictions for the full field of view around each ground-based GPS receiver. The comparisons start by using approximately 100 lines of sight spaced evenly in azimuth and elevation around the receiver, with a 15° elevation mask. The lines of sight about Rx1 are shown in Figure 8.

image

Figure 8. Unit vectors/lines-of-sight about Rx1.

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[55] All three thin-shell local ionosphere models, Models 1–3 of Figure 3, are considered. Each model uses a full second-order Taylor series expansion of their VTEC variations with respect to latitude and longitude. The principal differences between the models concern which coefficients are estimated in the Taylor series expansion of the ionospheric height state.

[56] The dynamic model parameters are listed in Table 2. The ionosphere contains greater TEC at lower latitudes, but the model parameters used in this study were the same at both receivers despite their differing latitudes. The results presented in this section could be improved with custom tuning of the dynamic model parameters to account for the different average magnitude TEC and TEC fluctuation experienced at each station.

[57] The results are presented in three parts. The first part investigates the daily behavior of the difference in STEC predicted by the local models and NOAA's model. The second part presents the average of the STEC difference over a full day. The third part considers the practical observability of the thin-shell altitude of the local models using real data.

Results Part 1: Local Model Daily Behavior

[58] The local ionosphere model results are first investigated by plotting the average magnitude of the differenced STEC predictions between the local and NOAA models for Day 2. Figure 9 plots the mean absolute values of the STEC differences averaged over the 100 lines of sight in 15 min intervals. Similarly, Figure 10 plots the mean of the absolute percent difference along the 100 lines of sight. The STEC difference values have been converted to a percent of the NOAA TEC along the lines of sight.

image

Figure 9. The mean absolute differences in the STEC prediction between the local thin-shell models and NOAA's model, averaged over 100 lines of sight on Day 2.

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image

Figure 10. The mean absolute differences in the STEC prediction between the local thin-shell models and NOAA's model, expressed as a percent of the NOAA value and averaged over 100 lines of sight on Day 2.

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[59] In both figures, the blue line with diamonds indicates results from comparing Model 1 to NOAA's model, the red line with squares is for Model 2, and the black line with triangles is for Model 3. Although the NOAA model estimates 3-D electron density profiles, the expected model estimation error values are provided for VTEC only. The NOAA VTEC uncertainty is converted to STEC uncertainty using the mapping function and an assumed ionosphere altitude. The altitude is set to the height at which 50% of the electron content has been accumulated along a vertical line of sight for a typical electron density profile taken from the NOAA model on Day 1, approximately 340 km. The green line with circles plots this expected NOAA STEC estimation error standard deviation averaged over the same 100 lines of sight as used in the comparisons.

[60] The local models were not allowed to settle before the comparison plots were made, but the NOAA model is a continuous real-time model. Therefore, the local models' results contain transient convergence errors over the first few hours which would not exist if the local ionosphere filter were allowed to run for a period of time before the comparisons were made.

[61] The comparisons between the NOAA and local ionosphere models presented in this paper allow for a number of observations to be made. The first observation is that the results of the single-receiver local thin-shell ionosphere models agree fairly well with the NOAA model. The theoretically optimal agreement from this type of comparison would be if the local models had no error and all of the STEC differences were solely due to the error in the NOAA model. In other words, if the average errors of Models 1–3 were to equal the NOAA σest levels. If that were the case, then the curves for Models 1–3 in Figures 9 and 10 would lie on top of the green-circles curve, and often that is the case, particularly for the higher latitude receiver located in Philadelphia, Rx2. This implies reasonable agreement between the local models and NOAA model estimates of STEC. This is encouraging given the large difference in the number of receiver stations used in the present thin-shell model, 1, and the number used in the NOAA model, about 90, and given the difference in the type of ionosphere model, thin-shell versus tomographic.

[62] The second observation is that thin-shell models that estimate height parameters have improved results throughout the day, especially during the night-time period. The night-time improvement is most easily noticed in Figure 10 (bottom) during hours 10–17. The improvement in the STEC prediction abilities of Models 2 and 3 is attributed to the varying altitude of maximum electron density in the true ionosphere during these times. Models 2 and 3 allow the GPS measurements to adjust the heights of their thin shells, allowing them to more closely approximate the behavior of the true ionosphere.

[63] Although not plotted, the peak difference of STEC, maximized over the 100 lines of sight, tends to vary about a constant value throughout the day, with a slight increase at night. Average peak difference values are discussed in the following subsection. Generally, Model 1 tends to have slightly higher peak differences than Models 2 and 3, and Models 2 and 3 tend to have approximately the same peak differences. The peak differences consistently come from lines of sight with the lowest elevations. The reason for this is that the low elevation lines of sight pass through a longer path of the ionosphere, creating larger TEC measurements than the higher elevation lines of sight, and corresponding larger TEC differences. The peak percent difference is uninformative, as the night-time values dominate the results. The reason for this behavior is that the peak TECU difference is roughly constant throughout the day, but the nominal TEC drops significantly at night.

Results Part 2: Local Model Average Behavior

[64] The second set of results present the comparisons of the local and NOAA STEC estimates averaged over a full day. The compiled results are listed in Table 4 for the two filter days considered. The numbers in the first pair of columns are the averaged STEC differences in TECU over 100 lines of sight and one full day, and the numbers in the second pair of columns are the same averages converted to percentages of the corresponding NOAA values. The conversion to percent error provides a more equal weighting of the results over the full day, as the day-time TEC values tend to be much larger than the night-time values. The averages were computed after ignoring the first 2 h of results during which the filter transients decayed.

Table 4. Average and Peak Magnitude Differences Between NOAA and the Local Ionosphere Models
 Average Difference Peak Difference
 STECRelative Peak, 1 Day Average  Peak, 1 Day Maximum
 (TECU)(%) (TECU)(TECU)
    Day 1    
 Rx1Rx2Rx1Rx2 Rx1Rx2Rx1Rx2
Model 14.22.813.914.2 13.310.923.522.9
Model 23.52.711.812.2 12.810.626.820.9
Model 33.92.613.511.5 13.310.327.821.3
NOAA σest4.013.7013.016.6 
    Day 2    
 Rx1Rx2Rx1Rx2 Rx1Rx2Rx1Rx2
Model 16.83.625.016.5 20.613.935.433.6
Model 24.82.516.511.6 19.59.638.022.8
Model 35.12.517.711.6 19.49.437.022.1
NOAA σest3.963.6713.616.9 

[65] The differences in the ionospheric TEC estimates are on average significantly reduced between Models 1 and 2, but Models 2 and 3 are comparable. The results of Table 4imply that the accuracy of two of the new thin-shell versions, Models 2 and 3, are better than the widely used classical thin-shell model, Model 1.

[66] Table 4also indicates that the performance of Models 2 and 3 is better at the Philadelphia receiver location, Rx2, than at the Key West receiver location, Rx1. This is true for both the average differences and peak differences. Of course, the peak are larger than the average differences, but not by as much in Philadelphia. Part of the improved performance at Philadelphia may stem from the lower nominal VTEC at this higher latitude. This nominal difference, however, does not explain the reduced average percent error at Philadelphia versus Key West in three of the four cases that encompass Models 2 and 3 and the two filtering days. Perhaps the percentage improvement indicates that the tuning parameters from Table 2are better suited to Philadelphia.

Results Part 3: Local Model Ionospheric Shell Altitude

[67] The last set of results considers the ionospheric shell altitude, specifically its observability. The estimated altitudes of the thin-shell ionosphere models are shown in Figure 11 for Day 1, where the plotted curves correspond to the same models as before. The figure allows for a number of observations to be made.

image

Figure 11. Ionospheric shell heights for Models 1–3 on Day 1.

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[68] The first observation concerns the height of maximum electron density. If the filter was not able to add information about the ionospheric altitude, then the altitude state would simply converge onto the dynamics model's reference state of 350 km. This is not the case in the figure, where Model 1 stays at its fixed altitude of 350 km, but Models 2 and 3 vary over a range of more than 300 km and are roughly centered at 500 km. This implies that the measurements the filter assimilates contain some nontrivial amount of information about the height of maximum electron density and that the ionospheric shell altitude is practically observable. This result correlates with both investigation of local ionosphere model observability in section 6 and with the previous subsection's results, where Models 2 and 3 outperform Model 1. Additionally, the transition behavior of the height of maximum electron density is visible in the rises of ionospheric altitude after dusk (around hours 7 or 10) and in the dips at dawn (around hour 17), despite a noisy altitude estimate.

[69] The second observation concerns the first derivatives of the height of maximum electron density with respect to latitude and longitude. Although not shown, the estimated state values of the shell height's first derivatives remain small. The small values may imply weak observability, if they are observable at all, and are consistent with the truth-model simulation results of section 6. The ionospheric shell height of Model 3 follows a very similar trajectory as that of Model 2 because the estimates of the latitude and longitude derivatives on shell height are nearly zero, in effect making Models 2 and 3 similar.

[70] Of course, all of these evaluations against the US-TEC model assume that it is “truth.” This is not the case. It could be that the new model should be used to correct the US-TEC model locally, or when US-TEC and the new model agree, both might have significant errors.

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References

[71] Two new local ionosphere models have been considered for use in the removal of ionospheric effects from single-frequency non-GPS radio navigation data, a thin-shell model and a thick-shell model. The novel aspects of the thin-shell model are a variable ionospheric shell height and variable height derivatives with respect to latitude and longitude. The thick-shell model is similar except for an expansion of its thin electron sheet into a Chapman electron density profile.

[72] The local ionosphere models were evaluated using a set of truth-model simulations, which were allowed for two conclusions to be drawn about the observability of each model. The first conclusion is that most of the states of the thin-shell model are observable, with the only exceptions being the shell height's second-order derivatives with respect to latitude and longitude. The second conclusion is that all of the Chapman scale height states of the thick-shell model are unobservable, precluding its use in applications where only a single ground-based dual-frequency GNSS receiver is available.

[73] A comparison of the predicted TEC from the local ionosphere models to that predicted by the NOAA ionosphere model allowed for a number of conclusions to be drawn about the local model's effectiveness in a real-world scenario. The first conclusion was that using local ionosphere models to predict TEC about a single receiver can be an effective alternative to array-based tomographic models, if a multireceiver array is not available. The second conclusion was that modifying the classical thin-shell ionosphere model to allow the height of the ionosphere to be an unknown estimated quantity can significantly improve TEC estimation because the ionospheric height state is practically observable. The third conclusion was that allowing the thin-shell model to have first-order variations of shell height with respect to latitude and longitude does not significantly improve TEC prediction results.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Ionospheric Effects on GPS Observables
  5. Local Ionosphere Models
  6. Extended Square Root Information Filtering Components
  7. Filter Practicalities
  8. The Question of Local Ionosphere Model Observability
  9. Results From Real Dual-Frequency GPS Data
  10. Conclusions
  11. Acknowledgments
  12. References
  • Anghel, A., C. Carrano, A. Komjathy, A. Astilean, and T. Letia (2009), Kalman filter-based algorithms for monitoring the ionosphere and plasmasphere with GPS in near-real time, J. Atmos. Sol. Terr. Phys., 71, 158174, doi:10.1016/j.jastp.2008.10.006.
  • Araujo-Pradere, E. A., T. J. Fuller-Rowell, P. S. J. Spencer, and C. F. Minter (2007), Differential validation of the US-TEC model, Radio Sci., 42, RS3016, doi:10.1029/2006RS003459.
  • Arikan, F., C. B. Erol, and O. Arikan (2003), Regularized estimation of vertical total electron content from Global Positioning System data, J. Geophys. Res., 108(A12), 1469, doi:10.1029/2002JA009605.
  • Arikan, F., H. Nayir, U. Sezen, and O. Arikan (2008), Estimation of single station interfrequency receiver bias using GPS-TEC, Radio Sci., 43, RS4004, doi:10.1029/2007RS003785.
  • Bar-Shalom, Y., X. Rong Li, and T. Kirubarajan (2001), Estimation With Applications to Tracking and Navigation, pp. 371395, John Wiley and Sons, New York.
  • Bierman, J. (1977), Factorization Methods for Discrete Sequential Estimation, pp. 68112, Dover Publications, New York.
  • Bilitza, D., and B. W. Reinisch (2008), International Reference Ionosphere 2007: Improvements and new parameters, Adv. Space Res., 42, 599609, doi:10.1016/j.asr.2007.07.048.
  • Bishop, G. J., J. A. Klobuchar, and P. H. Doherty (1985), Multipath effects on the determination of absolute ionospheric time delay from GPS signals, Radio Sci., 20(3), 388396, doi:10.1029/RS020i003p00388.
  • Bishop, G. J., D. J. Jacavanco, D. S. Coco, C. E. Coker, J. A. Klobuchar, E. J. Weber, and P. H. Doherty (1987), An advanced system for measurement of transionospheric radio propagation effects using GPS signals, Environmental Research Papers, No. 989, Air Force Geophys. Lab., Hanscom Air Force Base, Mass.
  • Bishop, G., A. Mazzella, and E. Holland (1995a), Self-calibration of pseudorange errors by GPS two-frequency receivers, Proceedings of the 1995 National Technical Meeting of the Institute of Navigation, pp. 251–259, Anaheim, CA,.
  • Bishop, G., A. Mazzella, and E. Holland (1995b), Application of SCORE techniques to improve ionospheric observations, Proceedings of the 8th International Technical Meeting of the Satellite Division of the Institute of Navigation, pp. 1209–1218, Palm Springs, CA,.
  • Bishop, G., A. Mazzella, E. Holland, and S. Rao (1996), Algorithms that use the ionosphere to control GPS errors, Position Location and Navigation Symposium, 1996 IEEE, pp. 145–152, Atlanta, GA.
  • Bishop, G., A. Mazzella, S. Rao, A. Batchelor, P. Fleming, N. Lunt, and L. Kersley (1997), Validations of the SCORE process. Proceedings of the 1997 National Technical Meeting of the Institute of Navigation, pp. 289–296, Santa Monica, CA.
  • Bust, G. S., and C. N. Mitchell (2008), History, current state, and future directions of ionospheric imaging, Rev. Geophys., 46, RG1003, doi:10.1029/2006RG000212.
  • Chapman, S. (1931), The absorption and dissociative or ionizing effect of monochromatic radiation of an atmosphere on a rotating Earth, Proc. Phys. Soc., 43(1), 2645.
  • Ciraolo, L., and P. Spalla (1997), Comparison of ionospheric total electron content from the Navy Navigation Satellite System and the GPS, Radio Sci., 32(3), 10711080, doi:10.1029/97RS00425.
  • Ciraolo, L., F. Azpilicueta, C. Brunini, A. Meza, and S. M. Radicella (2007), Calibration errors on experimental slant total electron content (TEC) determined with GPS, J. Geod., 81(2), 111120, doi:10.1007/s00190-006-0093-1.
  • Clynch, J., D. Coco, C. Coker, and G. Bishop (1989), A versatile GPS ionospheric monitor: High latitude measurements of TEC and scintillation, Proceedings of ION GPS-89; the Second International Technical Meeting of the Satellite Division of the Institute of Navigation, pp. 445–450, Washington, D. C.
  • Coco, D. S., C. Coker, S. R. Dahlke, and J. R. Clynch (1991), Variability of GPS satellite differential group delay, IEEE Trans. Aerosp. Electron. Syst., 27(6), 931938, doi:10.1109/7.104264.
  • Coker, C., D. Coco, and T. Gaussiran II (1996), Emerging capabilities for GPS as an ionospheric sensor, Proceedings of the 1996 Ionospheric Effects Symposium, pp. 391–397, Alexandria, Va..
  • Coster, A. J., E. M. Gaposchkin, and L. E. Thornton (1991), Real-time GPS ionospheric monitoring system at Millstone-initial results, Proceedings of the 1991 Space Surveillance Workshop, MIT Lincoln Laboratory Project Report STK-175, 39–45.
  • Coster, A., E. Gaposchkin, and L. Thornton (1992), Real-time ionospheric monitoring system using the GPS, Navigation, 39(2), 191204.
  • Coster, A., and A. Komjathy (2008), Space weather and the Global Positioning System, Space Weather, 6, S06D04, doi:10.1029/2008SW000400.
  • Feltens, J. (1998), Chapman profile approach for 3-D Global TEC representation, IGS presentation, in Proceedings of the 1998 IGS Analysis Centers Workshop, pp. 285297, Eur. Space Oper. Centre, Darmstadt, Germany, 9–11 Feb.
  • Feltens, J. (2003), The International GPS Service (IGS) Ionosphere Working Group, Adv. Space Res., 31(3), 635644, doi:10.1016/S0273-1177(03)00029-2.
  • Hernandez-Pajares, M., J. M. Juan, J. Sanz, and J. G. Sole (1998), Global observation of the ionospheric electronic response to solar events using ground and LEO GPS data, J. Geophys. Res., 103(A9), 20,78920,796.
  • Hernandez-Pajares, M., J. M. Juan, and J. Sanz (1999), New approaches in global ionospheric determination using ground GPS data, J. Atmos. Sol. Terr. Phys., 61(16), 12371247.
  • Hernandez-Pajares, M., J. M. Juan, J. Sanz, and O. L. Colombo (2000), Application of ionospheric tomography to real-time GPS carrier-phase ambiguities resolution, at scales of 400–1000 km and with high geomagnetic activity, Geophys. Res. Lett., 27(13), 20092012.
  • Hernandez-Pajares, M., J. M. Juan, J. Sanz, and O. L. Colombo (2002), Improving the real-time ionospheric determination from GPS sites at very long distances over the equator, J. Geophys. Res., 107(A10), 1296, doi:10.1029/2001JA009203.
  • Hernandez-Pajares, M., J. M. Juan, J. Sanz, and M. Garcia-Fernandez (2005), Towards a more realistic ionospheric mapping function, XXVIII URSI General Assembly.
  • Hernandez-Pajares, M., J. M. Juan, J. Sanz, R. Orus, A. Garcia-Rigo, J. Feltens, A. Komjathy, S. C. Schaer, and A. Krankowski (2009), The IGS VTEC maps: A reliable source of ionospheric information since 1998, J. Geod., 83(3–4), 263275, doi:10.1007/s00190-008-0266-1.
  • Jakowski, N., E. Sardon, E. Engler, A. Jungstand, and D. Klahn (1996), Relationships between GPS-signal propagation errors and EISCAT observations, Ann. Geophys., 14, 14291436, doi:10.1007/s00585-996-1429-0.
  • Juan, J. M., A. Rius, M. Hernandez-Pajares, and J. Sanz (1997), A two-layer model of the ionosphere using Global Positioning System data, Geophys. Res. Lett., 24(4), 393396, doi:10.1029/97GL00092.
  • Kailath, T. (1980), Linear Systems, pp. 80111, Prentice-Hall, New Jersey.
  • Klobuchar, J. A. (1975), A First-Order, Worldwide, Ionospheric, Time-Delay Algorithm (No. AFCRL-TR-75-0502), Air Force Cambridge Res. Lab., Hanscom Air Force Base, Mass.
  • Klobuchar, J. A. (1983), Ionospheric Effects on Earth-Space Propagation (No. AFGL-TR-84-0004), Air Force Geophys. Lab., Hanscom Air Force Base, Mass.
  • Komjathy, A. (1997), Global ionospheric total electron content mapping using the Global Positioning System, PhD thesis, Dep. of Geod. and Geomatics Eng., Univ. of New Brunswick, Fredericton, New Brunswick, Canada.
  • Komjathy, A., and R. Langley (1996), An assesment of predicted and measured ionospheric total electron content using a regional GPS network, in Proceedings of the 1996 National Technical Meeting of the Institute of Navigation, pp. 615624, Santa Monica, Calif., 22–24 Jan.
  • Komjathy, A., B. D. Wilson, T. F. Runge, B. M. Boulat, A. J. Mannucci, L. Sparks, and M. J. Reyes (2002), A new ionospheric model for wide area differential GPS: The multiple shell approach, in Proceedings of the 2002 National Technical Meeting of The Institute of Navigation, pp. 460466, San Diego, Calif., 28–30 Jan.
  • Lanyi, G. E., and T. Roth (1988), A comparison of mapped and measured total ionospheric electron content using global positioning system and beacon satellite observations, Radio Sci., 23(4), 483492, doi:10.1029/RS023i004p00483.
  • Liao, X. (2000), Carrier phase based ionosphere recovery over a regional area GPS network, MS thesis, Dep. of Geomatics Eng., Univ. of Calgary, Calgary, Alberta, Canada.
  • Ma, G., and T. Maruyama (2003), Derivation of TEC and estimation of instrumental biases from GEONET in Japan, Ann. Geophys., 21, 20832093, doi:10.5194/angeo-21-2083-2003.
  • Mannucci, A. J., B. D. Wilson, and D. N. Yuan (1995), An improved ionospheric correction method for wide-area augmentation systems, in Proceedings of the 8th International Technical Meeting of the Satellite Division of The Institute of Navigation, pp. 11991208, Palm Springs, Calif., 12–15 Sept.
  • Mannucci, A. J., B. D. Wilson, D. N. Yuan, C. H. Ho, U. J. Lindqwister, and T. F. Runge (1998), A global mapping technique for GPS-derived ionospheric total electron content measurements, Radio Sci., 33(3), 565582, doi:10.1029/97RS02707.
  • Mannucci, A. J., B. Iijima, L. Sparks, X. Pi, B. Wilson, and U. Lindqwister (1999), Assessment of global TEC mapping using a three-dimensional electron density model, J. Atmos. Sol. Terr. Phys., 61, 12271236, doi:10.1016/S1364-6826(99)00053-X.
  • Meggs, R. W., C. N. Mitchell, and P. S. J. Spencer (2004), A comparison of techniques for mapping total electron content over Europe using GPS signals, Radio Sci., 39, RS1S10, doi:10.1029/2002RS002846.
  • Minter, C. F., D. S. Robertson, P. S. J. Spencer, A. R. Jacobson, T. J. Fuller-Rowell, E. A. Araujo-Pradere, and R. W. Moses (2007), A comparison of Magic and FORTE ionosphere measurements, Radio Sci., 42, RS3026, doi:10.1029/2006RS003460.
  • Misra, P., and P. Enge (2006), Global Positioning System, Signals, Measurements, and Performance, pp. 164169, Ganga-Jamuna Press, Lincoln, Mass.
  • Morton, J., Q. Zhou, and M. Cosgrove (2007), A floating vertical TEC ionosphere delay correction algorithm for single frequency GPS receivers, in Proceedings of the 63rd Annual Meeting of The Institute of Navigation, pp. 479484, Cambridge, Mass., 23–25 April.
  • Nayir, H., F. Arikan, O. Arikan, and C. B. Erol (2007), Total electron content estimation with Reg-Est, J. Geophys. Res., 112, A11313, doi:10.1029/2007JA012459.
  • Orus, R., M. Hernandez-Pajares, J. Juan, J. Sanz, and M. Garcia-Fernandez (2002), Performance of different TEC models to provide GPS ionospheric corrections, J. Atmos. Sol. Terr. Phys., 64, 20552062, doi:10.1016/S1364-6826(02)00224-9.
  • Orus, R., M. Hernandez-Pajares, J. Juan, J. Sanz, and M. Garcia-Fernandez (2003), Validation of the GPS TEC maps with TOPEX data, Adv. Space Res., 31(3), 621627, doi:10.1016/S0273-1177(03)00026-7.
  • Otsuka, Y., T. Ogawa, A. Saito, T. Tsugawa, S. Fukao, and S. Miyazaki (2002), A new technique for mapping of total electron content using GPS network in Japan, Earth Planet Space, 54, 6370.
  • Rao, G. S. (2007), GPS satellite and receiver instrumental biases estimation using least squares method for accurate ionosphere modelling, J. Earth Syst. Sci., 116(5), 407411.
  • Rideout, W., and A. Coster (2006), Automated GPS processing for global total electron content data, GPS Solutions, 10, 219228, doi:10.1007/s10291-006-0029-5.
  • Skone, S. (2006), TECMODEL operating manual, version 1.0, Univ. of Calgary, Calgary, Alberta, Canada.
  • Smith, D. A., E. A. Araujo-Pradere, C. Minter, and T. Fuller-Rowell (2008), A comprehensive evaluation of the errors inherent in the use of a two-dimensional shell for modeling the ionosphere, Radio Sci., 43, RS6008, doi:10.1029/2007RS003769.
  • Spencer, P. S. J., D. S. Robertson, and G. L. Mader (2004), Ionospheric data assimilation methods for geodetic applications, in Proceedings of IEEE PLANS 2004, pp. 510517, Montery, Calif., 26–29 April.
  • Yilmaz, A., K. E. Akdogan, and M. Gurun (2009), Regional TEC mapping using neural networks, Radio Sci., 44, RS3007, doi:10.1029/2008RS004049.