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Electron density retrieval from occulting GNSS signals using a gradient-aided inversion technique
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Advances in Space Research 47 (2011) 289–295 www.elsevier.com/locate/asr
Electron density retrieval from occulting GNSS signals using a gradient-aided inversion technique Igor Kulikov a,*, Anthony J. Mannucci a, Xiaoqing Pi a, Carol Raymond a, George A. Hajj b a
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA b RBS Sempra Commodities, Stamford, CT, USA Received 29 April 2010; received in revised form 2 July 2010; accepted 4 July 2010
Abstract In the coming years, opportunities for remote sensing of electron density in the Earth’s ionosphere will expand with the advent of Galileo, which will become part of the global navigation satellite system (GNSS). Methods for accurate electron density retrieval from radio occultation data continue to improve. We describe a new method of electron density retrieval using total electron content measurements obtained in low Earth orbit. This method can be applied to data from dual-frequency receivers tracking the GPS or Galileo transmitters. This simulation study demonstrates that the method significantly improves retrieval accuracy compared to the standard Abel inversion approach that assumes a spherically symmetric ionosphere. Our method incorporates horizontal gradient information available from global maps of Total Electron Content (TEC), which are available from the International GNSS Service (IGS) on a routine basis. The combination of ground and space measurements allows us to improve the accuracy of electron density profiles near the occultation tangent point in the E and F regions of the ionosphere. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Radio occultation; Electron density data retrieval
1. Introduction Radio occultation (RO) technique is a powerful method for the study of the Earth’s ionosphere (e.g., Yunck et al., 2001). At present time GPS and Russia’s GLONASS constellations together with a large number of commercial satellites in high orbits provide intense radio illumination on the Earth. Europe’s Galileo and China’s Compass constellations will also join the global navigation satellite system (GNSS) in near future. The use of low Earth orbiting satellites (LEOs) receiving these radio signals, CITRIS instruments, and DORIS system allows us to investigate unique peculiarities of the Earth’s ionosphere (Bernhardt et al., 2006; Auriol and Tourain, 2010). RO is the only spaceborne technique to provide electron density information spanning the full range of altitudes, above and below the *
Corresponding author. Tel.: +1 818 393 5369. E-mail address: [email protected] (I. Kulikov).
density peak. RO is currently performed in receiver constellations, providing the advantages of global coverage. Several studies using the FORMOSAT-3/COSMIC constellation observing system have provided new discoveries about the electron density distribution in the ionosphere (Wang et al., 2004; Arras et al., 2008; Burns et al., 2008; Liang et al., 2008; Luan and Solomon, 2008; Zeng et al., 2008; Aragon-Angel et al., 2009; Arras et al., 2009; Lin et al., 2009; Liu et al., 2009; Mayer and Jakowski, 2009; Pedatella et al., 2009; Ram et al., 2009; Thampi et al., 2009; Tsai et al., 2009). The FORMOSAT-3/COSMIC (referred as COSMIC hereafter) is a joined Taiwan and the United States mission that includes six LEO satellites carrying ionospheric and atmospheric remote sensing instruments and flying in circular orbits at about 800 km altitude. The satellites were launched in the spring of 2006. The science payloads on each satellite include GPS RO receiver, tiny ionospheric photometer, and tri-band beacon transmitter. The detailed information about the
0273-1177/$36.00 Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.07.002
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mission is referred to http://www.ucar.edu/news/releases/ 2006/cosmicfacts.shtml. Even earlier, the CHAMP satellite provided the first long-term RO measurements from space (e.g., Wu et al., 2005), whereas the GPS/MET satellite was used to prove the concept (Hajj and Romans, 1998). Despite the power of this method, the limitations of radio occultation have been noted, particularly when retrieving electron density structure below the F-region peak density (Hysell, 2007; Lei et al., 2007; Wu et al., 2009a; Yue et al., 2010). The standard Abel inversion method to retrieve electron density from total electron content is analytically correct in the case of a spherically symmetric ionosphere (electron density depends only on radial distance). Fully exploiting remote sensing opportunities with the expanding GNSS and other possible RO signals requires that methods be developed to accommodate the horizontal electron density variation. Several recent studies describe techniques that improve upon the standard Abel inversion by combining radio occultation data with other electron density or total electron content information (Garcia-Fernandez et al., 2004, 2005; Aragon-Angel et al., 2009; Tsai et al., 2009; Wu et al., 2009b). Interest in these methods continues to grow since each method has its own strengths and weaknesses, and applicability to particular situations will vary. We note that direct inversion of RO TEC data (or bending angle, see Schreiner et al., 1999) contrasts with assimilating radio occultation directly into data assimilation models (Scherliess et al., 2004; Wang et al., 2004; Scherliess et al., 2006; Pi et al., 2009). Data assimilation might be viewed as the preferred method for generating electron density retrievals from RO data. Assimilation models are designed to combine RO and ground data where available, thereby accounting for horizontal gradients. However, methods that are not bound to specific data assimilation schemes are also valuable, for these reasons: direct retrieval is computationally much simpler and independent of particular physics-based or climatological assumptions in the models. In this work, we present a new method of electron density profile retrieval that does not assume a spherically symmetric ionosphere. According to our simulations, this method should produce improved accuracy of electron density profile retrieval in the E and F regions compared to the Abel inversion method. We demonstrate the value of our method using a simulation approach based on the International Reference Ionosphere (IRI) climatological model of electron density (Bilitza and Reinisch, 2008), characteristic of solar maximum and equinox conditions. The date of the run corresponds to March 21, 2000; the daily solar 10.7 cm radio flux index is 228.9. Our method augments space-borne occultation TEC measurements with regularly-spaced vertical TEC measurements obtained from a ground-based vantage point. The ground-based TEC links can be obtained from vertical TEC maps, for example available from the International GNSS Service (IGS) (Mannucci et al., 1998; Iijima et al., 1999; Orus
et al., 2002; Dow et al., 2009; Hernandez-Pajares et al., 2009). The space-to-space and space-to-ground radio links are restricted a two-dimensional plane. In Section 2, we present the results for a discretized version of the Abel inversion method that assumes a spherically symmetric ionosphere. The input data to this method are restricted to TEC links from space. The output of the Abel inversion method is a vertical electron density profile, which we compare to the corresponding IRI profile used to generate the simulated data. We examine retrieval accuracy in the E and F regions of the ionosphere. In Section 3, we develop and apply a new method for electron density profile retrieval adding an additional set of data – TEC along space-to-ground raypaths. For the second simulation, we use TEC for 120 ground rays with 0.5° latitude separation between the rays. In Section 4, we compare results between the Abel inversion method and the method using horizontal gradient information. 2. Electron density retrieval with the Abel inversion method TEC along the straight ray path l is the linear integral of electron density ne as follows (Hajj et al., 1994; Schreiner et al., 1999): Z TECray ¼ ne ðlÞ dl ð1Þ ray
We assume straight-line propagation of raypaths to retrieve electron density profiles from total electron content measurements. TEC is computed from the differential time delay of the GPS signal at the L1 and L2 frequencies, as described in Mannucci et al. (1999). We seek to solve the integral equation (1) for unknown electron density ne along the ray path. We first discretize the integral equation to form a system of linear simultaneous equations that relate the TEC measurements to the electron density. In this section, we assume spherically symmetric ionospheric structure dependent only on the radial coordinate. The set of simultaneous equations relating measurements TECi to electron density in each layer nj is: TECi ¼
M X
Aij nj ;
i ¼ 1; . . . ; N
ð2Þ
j¼1
assuming there are N measurements from space and M ionospheric layers. Electron density ne is found by solving for the nj from this matrix equation. We assume that the rays are parallel, straight lines for simplicity as shown in Fig. 1. The solution of the system (2) can be obtained with the Least Squares Method (LSM) (Twomey, 1996; Rodgers, 2000). The solution for electron density content is written in matrix form as 1 d ^n ¼ AT A AT TEC ð3Þ Electron density obtained with Eq. (3) depends only on radial coordinate. For computation of TEC and the elements of matrix A, we consider 389 parallel rays. The ray separa-
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Fig. 1. Space ray geometry used for electron density content determination.
tion in height is taken as 0.5 km between rays at heights from 50 to 100 km, 1 km ray separation at heights from 100 to 200 km, 2 km separation at heights from 200 to 400 km and 5 km ray separation from 400 to 885 km. TEC for each ray is computed from a simulated twodimensional electron density field using a Chapman profile for the vertical electron density distribution (Hajj and Romans, 1998). Synthesized TEC for 389 parallel rays versus ray number for the IRI electron density distribution is shown in Fig. 2. The element Aij of the matrix A in Eq. (3) is defined as the length of the ith ray in jth layer. The spherical symmetry assumption restricts nj to be uniform within the layer j. To compute matrix A, we divide the ionosphere into M = 49 layers of 16.3 km thickness each from 90 to 890 km altitude. Knowing the position of each ray (straight-line propagation), we compute elements of the matrix A. Inserting A and the measured TEC values into Eq. (3), we obtain the electron density ne in each layer. Fig. 3 compares the retrieved electron density ^n (curve with open circles) to the “true” electron density from the IRI distribution (solid curve) at the location of the occultation tangent point (x = 0 in Fig. 1). The electron density profile obtained with the spherically symmetric inversion method deviates from the IRI profile at the E and lower
Fig. 2. TEC as a function of ray number from the space rays.
Fig. 3. Electron density profile obtained by spherically symmetric inversion method (open circles). The solid curve corresponds to the input IRI electron density profile.
F regions as well as near the peak density. Fractional electron density errors are particularly large, exceeding 700% in the middle of the E-region. 3. Electron density retrieval method using horizontal gradients The previous section has shown that the spherical symmetry assumption inherent in the Abel inversion causes large electron density errors in the E and lower F regions, as well as near the peak density. To improve these results, we modify the Abel method by allowing horizontal gradients into the retrieval scheme. Such horizontal gradients for atmospheric delay and for ionospheric TEC mapping, respectively, can be obtained from measurements made using individual and a network of ground-based GPS receivers (e.g., Bar-Sever et al., 1998; Mannucci et al., 1998; Iijima et al., 1999; Orus et al., 2002; Hernandez-Pajares et al., 2009). Let us assume that such TEC data are available uniformly in the horizontal direction, as shown in Fig. 4. We assume, as in the previous case, that tangent points for each of the space rays have the latitude #0 and that the space rays are straight and parallel to each other. The ground rays are equally distributed in 0–60° latitude interval. Evenly distributed direct measurements of the ion-
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Fig. 4. Space and ground ray geometry used for electron density content determination.
osphere TEC are not generally available in practice. However, such measurements can be synthesized from global ionospheric maps (GIM) (Mannucci et al., 1998). In our extension of the Abel approach, we define an analytical model of electron density that includes radial and latitudinal variation. The latitudinal variation is modeled as a quadratic polynomial in latitude #. The electron density n(r, #) is expressed as the sum of two terms: nðr; #Þ ¼ n0 ðrÞ þ ðcð#Þ þ cÞ þ að#Þ# þ bð#Þ#2 f ðrÞ ð4Þ The first term is the electron density profile at the tangent point latitude #0 (i.e., at x = 0 in Fig. 4). The second term includes deviations from the tangent point profile due to electron density horizontal gradients that vary up to second order in latitude (#2). We develop an inversion method to estimate coefficients c(#), a(#), b(#) from vertical TEC data. The vertical TEC data from 120 regularly-spaced locations at different latitudes obtained by path integration with the use of IRI electron density profiles is shown in Fig. 5. Note that we are not requiring data from 120 regularly spaced ground-based receivers. Rather, we are assuming that the TEC data can be estimated, for example, from a GIM. The results here do not account for possible errors in the vertical TEC data source. In a future work, we will assess the impact of errors that arise in vertical TEC maps
Fig. 5. TEC curve of ground rays as a function of latitude.
by applying the JPL GIM mapping technique to simulated measurements from IRI, assuming a realistic distribution of ground-based GPS receivers. Error estimates of these maps appear in the literatures (Iijima et al., 1999; Ho et al., 1997), but the impact of these errors on our retrieval technique requires additional simulation work. The coefficients c(#), a(#), b(#) are computed by discretizing the 0–60° latitudinal interval (Fig. 5) into 20 strips of 3° width each. Values c(#), a(#), b(#) in each strip are found through second-order polynomial fitting to TEC values in each strip. In other words, the TEC horizontal map defines a latitudinal variation of TEC within each of the 20 strips. A second-order polynomial, different for each strip, is fitted to this latitudinal variation within each strip. Thus the coefficients have unique values within each strip but may vary from strip to strip. Angular dependences of the coefficients c(#k), a(#k) and b(#k), and where k = 1, . . ., 20, are generated by fitting the vertical TEC data to a second-order polynomial. The dependence of the coefficients c, a and b on the latitude is shown in Fig. 6 by the curves which are marked by triangles, circles and crosses correspondingly. The analytic equation for electron density (Eq. (4)) contains a function that depends on altitude, f(r) (so-called “altitude shaping” function). This function multiplies the assumed quadratic dependence of electron density on lati-
Fig. 6. Coefficients c(#k), a(#k), b(#k) as functions of latitude.
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tude. The altitude shaping function f(r) is introduced in Eq. (4) to modulate the horizontal electron density gradient as a function of altitude. For example, electron density variations versus latitude will be smaller at high altitudes compared to altitudes near the F2 peak density. This altitudinal variation is a simple consequence of the fact that electron density is largest near the F2-peak, with decreased density in the top and bottom side ionosphere. The altitude shaping function is normalized as Z hmax f ðhÞ dh ¼ 1 ð5Þ hmin
The function f(r) used in our study was computed by modification of electron density vertical profile for spherically symmetric case with the use of Eq. (5). The function is shown in Fig. 7. The constant c in Eq. (4) is chosen as c ¼ TEC#0 , where TEC#0 is TEC of the ground ray at central latitude #0. This constant is required to obtain the TEC curve (Fig. 5) when we integrate Eq. (4) along the vertical rays. The electron density profile n0(r) at #0 latitude is found by rewriting Eq. (4) in the form ð6Þ n0 ðrÞ ¼ nðr; #Þ ðcð#Þ þ cÞ þ að#Þ# þ bð#Þ#2 f ðrÞ
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In this equation, Tb is a N-element array constructed from the right hand side of the system (7). The simulation result for this new retrieval method is shown in Fig. 8. In Fig. 8, the solid curve corresponds to the IRI electron density profile. The starred curve corresponds to the electron density profile computed using knowledge of horizontal gradients. 4. Comparison of two methods The IRI electron density profile, the electron density profile obtained with the Abel inversion method and the electron density profile computed with the use of horizontal gradients are shown in Fig. 9. As follows from this plot, incorporating horizontal gradients yields significantly more accurate retrievals, particularly in the E and F1 regions as well as around the density peak. To assess quantitatively the two algorithms described above, we compute the ratios:
Since the coefficients c(#k), a(#k), b(#k) have been determined from the ground rays, Eq. (6) can be integrated along the space rays (horizontally oriented) and rewritten as Z M X Aij n0j ¼ TECi ðc þ cÞ þ a# þ b#2 f ðrÞ dl; j¼1
space rayðiÞ
i ¼ 1; . . . ; N
ð7Þ
where TECi for ray i is given by Eq. (1). Since the right hand side of Eq. (7) can be computed along each space ray, we solve for the unknown array n^0 with the least squares method as 1 T ^ n0 ¼ AT A ð8Þ A Tb
Fig. 7. Altitude dependence of a normalized f(r) function.
Fig. 8. Electron density profiles: the solid curve corresponds to IRI electron density profile; the starred curve describes the electron density obtained with the use of horizontal ionospheric gradients.
Fig. 9. Electron density profiles: the solid curve corresponds to IRI electron density profile; the circled curve corresponds to Abel inverted electron density profile and the starred curve describes the electron density obtained with the use of horizontal gradients.
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Administration. Sponsorship of NASA Earth Surface and Interior Focus Area program (Grant No. 101248 622.74.10.24) is gratefully acknowledged. References
Fig. 10. The altitude dependences of the ratios R1 (the solid curve) and R2 (the dashed curve).
. IRI n R1 ¼ nIRI ngradients e e e inv . IRI sym R2 ¼ ne nsph nIRI e e inv
ð9Þ
The first ratio R1 includes electron density data ngradients at e inv the central latitude h0 obtained with the algorithm that includes horizontal gradients; the second ratio R2 includes sym electron density data nsph obtained with a spherical syme inv metry assumption for electron density distribution. The altitude dependences of the ratios R1 and R2 are given by the solid and dashed curves in Fig. 10. As shown in Figs. 9 and 10, the inclusion of horizontal gradients in the equation for the electron density distribution (Eq. (6)) allows us to improve the electron density profile retrieval compared to the spherically symmetric model. 5. Conclusion In this work, we have developed a new method to retrieve electron density profiles from GPS occultation measurements as acquired by GPS receivers onboard low Earth orbiters. The new method takes into account horizontal gradients in the ionosphere and reduces retrieval error compared to the Abel inversion, which assumes horizontal uniformity of the ionosphere. Numerical simulations have been conducted to examine the effectiveness of the method. In the simulations, vertical TEC data from the ground are used to estimate the horizontal gradients, which, together with the space TEC measurements, allow us to improve the accuracy of electron density prediction in both E and F regions. Horizontal gradients are computed from two-dimensional vertical TEC maps using a polynomial fit up to second order in a given horizontal interval. Acknowledgements This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space
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Electron density retrieval from occulting GNSS signals using a gradient-aided inversion technique Igor Kulikov a,*, Anthony J. Mannucci a, Xiaoqing Pi a, Carol Raymond a, George A. Hajj b a
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA b RBS Sempra Commodities, Stamford, CT, USA Received 29 April 2010; received in revised form 2 July 2010; accepted 4 July 2010
Abstract In the coming years, opportunities for remote sensing of electron density in the Earth’s ionosphere will expand with the advent of Galileo, which will become part of the global navigation satellite system (GNSS). Methods for accurate electron density retrieval from radio occultation data continue to improve. We describe a new method of electron density retrieval using total electron content measurements obtained in low Earth orbit. This method can be applied to data from dual-frequency receivers tracking the GPS or Galileo transmitters. This simulation study demonstrates that the method significantly improves retrieval accuracy compared to the standard Abel inversion approach that assumes a spherically symmetric ionosphere. Our method incorporates horizontal gradient information available from global maps of Total Electron Content (TEC), which are available from the International GNSS Service (IGS) on a routine basis. The combination of ground and space measurements allows us to improve the accuracy of electron density profiles near the occultation tangent point in the E and F regions of the ionosphere. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Radio occultation; Electron density data retrieval
1. Introduction Radio occultation (RO) technique is a powerful method for the study of the Earth’s ionosphere (e.g., Yunck et al., 2001). At present time GPS and Russia’s GLONASS constellations together with a large number of commercial satellites in high orbits provide intense radio illumination on the Earth. Europe’s Galileo and China’s Compass constellations will also join the global navigation satellite system (GNSS) in near future. The use of low Earth orbiting satellites (LEOs) receiving these radio signals, CITRIS instruments, and DORIS system allows us to investigate unique peculiarities of the Earth’s ionosphere (Bernhardt et al., 2006; Auriol and Tourain, 2010). RO is the only spaceborne technique to provide electron density information spanning the full range of altitudes, above and below the *
Corresponding author. Tel.: +1 818 393 5369. E-mail address: [email protected] (I. Kulikov).
density peak. RO is currently performed in receiver constellations, providing the advantages of global coverage. Several studies using the FORMOSAT-3/COSMIC constellation observing system have provided new discoveries about the electron density distribution in the ionosphere (Wang et al., 2004; Arras et al., 2008; Burns et al., 2008; Liang et al., 2008; Luan and Solomon, 2008; Zeng et al., 2008; Aragon-Angel et al., 2009; Arras et al., 2009; Lin et al., 2009; Liu et al., 2009; Mayer and Jakowski, 2009; Pedatella et al., 2009; Ram et al., 2009; Thampi et al., 2009; Tsai et al., 2009). The FORMOSAT-3/COSMIC (referred as COSMIC hereafter) is a joined Taiwan and the United States mission that includes six LEO satellites carrying ionospheric and atmospheric remote sensing instruments and flying in circular orbits at about 800 km altitude. The satellites were launched in the spring of 2006. The science payloads on each satellite include GPS RO receiver, tiny ionospheric photometer, and tri-band beacon transmitter. The detailed information about the
0273-1177/$36.00 Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.07.002
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mission is referred to http://www.ucar.edu/news/releases/ 2006/cosmicfacts.shtml. Even earlier, the CHAMP satellite provided the first long-term RO measurements from space (e.g., Wu et al., 2005), whereas the GPS/MET satellite was used to prove the concept (Hajj and Romans, 1998). Despite the power of this method, the limitations of radio occultation have been noted, particularly when retrieving electron density structure below the F-region peak density (Hysell, 2007; Lei et al., 2007; Wu et al., 2009a; Yue et al., 2010). The standard Abel inversion method to retrieve electron density from total electron content is analytically correct in the case of a spherically symmetric ionosphere (electron density depends only on radial distance). Fully exploiting remote sensing opportunities with the expanding GNSS and other possible RO signals requires that methods be developed to accommodate the horizontal electron density variation. Several recent studies describe techniques that improve upon the standard Abel inversion by combining radio occultation data with other electron density or total electron content information (Garcia-Fernandez et al., 2004, 2005; Aragon-Angel et al., 2009; Tsai et al., 2009; Wu et al., 2009b). Interest in these methods continues to grow since each method has its own strengths and weaknesses, and applicability to particular situations will vary. We note that direct inversion of RO TEC data (or bending angle, see Schreiner et al., 1999) contrasts with assimilating radio occultation directly into data assimilation models (Scherliess et al., 2004; Wang et al., 2004; Scherliess et al., 2006; Pi et al., 2009). Data assimilation might be viewed as the preferred method for generating electron density retrievals from RO data. Assimilation models are designed to combine RO and ground data where available, thereby accounting for horizontal gradients. However, methods that are not bound to specific data assimilation schemes are also valuable, for these reasons: direct retrieval is computationally much simpler and independent of particular physics-based or climatological assumptions in the models. In this work, we present a new method of electron density profile retrieval that does not assume a spherically symmetric ionosphere. According to our simulations, this method should produce improved accuracy of electron density profile retrieval in the E and F regions compared to the Abel inversion method. We demonstrate the value of our method using a simulation approach based on the International Reference Ionosphere (IRI) climatological model of electron density (Bilitza and Reinisch, 2008), characteristic of solar maximum and equinox conditions. The date of the run corresponds to March 21, 2000; the daily solar 10.7 cm radio flux index is 228.9. Our method augments space-borne occultation TEC measurements with regularly-spaced vertical TEC measurements obtained from a ground-based vantage point. The ground-based TEC links can be obtained from vertical TEC maps, for example available from the International GNSS Service (IGS) (Mannucci et al., 1998; Iijima et al., 1999; Orus
et al., 2002; Dow et al., 2009; Hernandez-Pajares et al., 2009). The space-to-space and space-to-ground radio links are restricted a two-dimensional plane. In Section 2, we present the results for a discretized version of the Abel inversion method that assumes a spherically symmetric ionosphere. The input data to this method are restricted to TEC links from space. The output of the Abel inversion method is a vertical electron density profile, which we compare to the corresponding IRI profile used to generate the simulated data. We examine retrieval accuracy in the E and F regions of the ionosphere. In Section 3, we develop and apply a new method for electron density profile retrieval adding an additional set of data – TEC along space-to-ground raypaths. For the second simulation, we use TEC for 120 ground rays with 0.5° latitude separation between the rays. In Section 4, we compare results between the Abel inversion method and the method using horizontal gradient information. 2. Electron density retrieval with the Abel inversion method TEC along the straight ray path l is the linear integral of electron density ne as follows (Hajj et al., 1994; Schreiner et al., 1999): Z TECray ¼ ne ðlÞ dl ð1Þ ray
We assume straight-line propagation of raypaths to retrieve electron density profiles from total electron content measurements. TEC is computed from the differential time delay of the GPS signal at the L1 and L2 frequencies, as described in Mannucci et al. (1999). We seek to solve the integral equation (1) for unknown electron density ne along the ray path. We first discretize the integral equation to form a system of linear simultaneous equations that relate the TEC measurements to the electron density. In this section, we assume spherically symmetric ionospheric structure dependent only on the radial coordinate. The set of simultaneous equations relating measurements TECi to electron density in each layer nj is: TECi ¼
M X
Aij nj ;
i ¼ 1; . . . ; N
ð2Þ
j¼1
assuming there are N measurements from space and M ionospheric layers. Electron density ne is found by solving for the nj from this matrix equation. We assume that the rays are parallel, straight lines for simplicity as shown in Fig. 1. The solution of the system (2) can be obtained with the Least Squares Method (LSM) (Twomey, 1996; Rodgers, 2000). The solution for electron density content is written in matrix form as 1 d ^n ¼ AT A AT TEC ð3Þ Electron density obtained with Eq. (3) depends only on radial coordinate. For computation of TEC and the elements of matrix A, we consider 389 parallel rays. The ray separa-
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Fig. 1. Space ray geometry used for electron density content determination.
tion in height is taken as 0.5 km between rays at heights from 50 to 100 km, 1 km ray separation at heights from 100 to 200 km, 2 km separation at heights from 200 to 400 km and 5 km ray separation from 400 to 885 km. TEC for each ray is computed from a simulated twodimensional electron density field using a Chapman profile for the vertical electron density distribution (Hajj and Romans, 1998). Synthesized TEC for 389 parallel rays versus ray number for the IRI electron density distribution is shown in Fig. 2. The element Aij of the matrix A in Eq. (3) is defined as the length of the ith ray in jth layer. The spherical symmetry assumption restricts nj to be uniform within the layer j. To compute matrix A, we divide the ionosphere into M = 49 layers of 16.3 km thickness each from 90 to 890 km altitude. Knowing the position of each ray (straight-line propagation), we compute elements of the matrix A. Inserting A and the measured TEC values into Eq. (3), we obtain the electron density ne in each layer. Fig. 3 compares the retrieved electron density ^n (curve with open circles) to the “true” electron density from the IRI distribution (solid curve) at the location of the occultation tangent point (x = 0 in Fig. 1). The electron density profile obtained with the spherically symmetric inversion method deviates from the IRI profile at the E and lower
Fig. 2. TEC as a function of ray number from the space rays.
Fig. 3. Electron density profile obtained by spherically symmetric inversion method (open circles). The solid curve corresponds to the input IRI electron density profile.
F regions as well as near the peak density. Fractional electron density errors are particularly large, exceeding 700% in the middle of the E-region. 3. Electron density retrieval method using horizontal gradients The previous section has shown that the spherical symmetry assumption inherent in the Abel inversion causes large electron density errors in the E and lower F regions, as well as near the peak density. To improve these results, we modify the Abel method by allowing horizontal gradients into the retrieval scheme. Such horizontal gradients for atmospheric delay and for ionospheric TEC mapping, respectively, can be obtained from measurements made using individual and a network of ground-based GPS receivers (e.g., Bar-Sever et al., 1998; Mannucci et al., 1998; Iijima et al., 1999; Orus et al., 2002; Hernandez-Pajares et al., 2009). Let us assume that such TEC data are available uniformly in the horizontal direction, as shown in Fig. 4. We assume, as in the previous case, that tangent points for each of the space rays have the latitude #0 and that the space rays are straight and parallel to each other. The ground rays are equally distributed in 0–60° latitude interval. Evenly distributed direct measurements of the ion-
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Fig. 4. Space and ground ray geometry used for electron density content determination.
osphere TEC are not generally available in practice. However, such measurements can be synthesized from global ionospheric maps (GIM) (Mannucci et al., 1998). In our extension of the Abel approach, we define an analytical model of electron density that includes radial and latitudinal variation. The latitudinal variation is modeled as a quadratic polynomial in latitude #. The electron density n(r, #) is expressed as the sum of two terms: nðr; #Þ ¼ n0 ðrÞ þ ðcð#Þ þ cÞ þ að#Þ# þ bð#Þ#2 f ðrÞ ð4Þ The first term is the electron density profile at the tangent point latitude #0 (i.e., at x = 0 in Fig. 4). The second term includes deviations from the tangent point profile due to electron density horizontal gradients that vary up to second order in latitude (#2). We develop an inversion method to estimate coefficients c(#), a(#), b(#) from vertical TEC data. The vertical TEC data from 120 regularly-spaced locations at different latitudes obtained by path integration with the use of IRI electron density profiles is shown in Fig. 5. Note that we are not requiring data from 120 regularly spaced ground-based receivers. Rather, we are assuming that the TEC data can be estimated, for example, from a GIM. The results here do not account for possible errors in the vertical TEC data source. In a future work, we will assess the impact of errors that arise in vertical TEC maps
Fig. 5. TEC curve of ground rays as a function of latitude.
by applying the JPL GIM mapping technique to simulated measurements from IRI, assuming a realistic distribution of ground-based GPS receivers. Error estimates of these maps appear in the literatures (Iijima et al., 1999; Ho et al., 1997), but the impact of these errors on our retrieval technique requires additional simulation work. The coefficients c(#), a(#), b(#) are computed by discretizing the 0–60° latitudinal interval (Fig. 5) into 20 strips of 3° width each. Values c(#), a(#), b(#) in each strip are found through second-order polynomial fitting to TEC values in each strip. In other words, the TEC horizontal map defines a latitudinal variation of TEC within each of the 20 strips. A second-order polynomial, different for each strip, is fitted to this latitudinal variation within each strip. Thus the coefficients have unique values within each strip but may vary from strip to strip. Angular dependences of the coefficients c(#k), a(#k) and b(#k), and where k = 1, . . ., 20, are generated by fitting the vertical TEC data to a second-order polynomial. The dependence of the coefficients c, a and b on the latitude is shown in Fig. 6 by the curves which are marked by triangles, circles and crosses correspondingly. The analytic equation for electron density (Eq. (4)) contains a function that depends on altitude, f(r) (so-called “altitude shaping” function). This function multiplies the assumed quadratic dependence of electron density on lati-
Fig. 6. Coefficients c(#k), a(#k), b(#k) as functions of latitude.
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tude. The altitude shaping function f(r) is introduced in Eq. (4) to modulate the horizontal electron density gradient as a function of altitude. For example, electron density variations versus latitude will be smaller at high altitudes compared to altitudes near the F2 peak density. This altitudinal variation is a simple consequence of the fact that electron density is largest near the F2-peak, with decreased density in the top and bottom side ionosphere. The altitude shaping function is normalized as Z hmax f ðhÞ dh ¼ 1 ð5Þ hmin
The function f(r) used in our study was computed by modification of electron density vertical profile for spherically symmetric case with the use of Eq. (5). The function is shown in Fig. 7. The constant c in Eq. (4) is chosen as c ¼ TEC#0 , where TEC#0 is TEC of the ground ray at central latitude #0. This constant is required to obtain the TEC curve (Fig. 5) when we integrate Eq. (4) along the vertical rays. The electron density profile n0(r) at #0 latitude is found by rewriting Eq. (4) in the form ð6Þ n0 ðrÞ ¼ nðr; #Þ ðcð#Þ þ cÞ þ að#Þ# þ bð#Þ#2 f ðrÞ
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In this equation, Tb is a N-element array constructed from the right hand side of the system (7). The simulation result for this new retrieval method is shown in Fig. 8. In Fig. 8, the solid curve corresponds to the IRI electron density profile. The starred curve corresponds to the electron density profile computed using knowledge of horizontal gradients. 4. Comparison of two methods The IRI electron density profile, the electron density profile obtained with the Abel inversion method and the electron density profile computed with the use of horizontal gradients are shown in Fig. 9. As follows from this plot, incorporating horizontal gradients yields significantly more accurate retrievals, particularly in the E and F1 regions as well as around the density peak. To assess quantitatively the two algorithms described above, we compute the ratios:
Since the coefficients c(#k), a(#k), b(#k) have been determined from the ground rays, Eq. (6) can be integrated along the space rays (horizontally oriented) and rewritten as Z M X Aij n0j ¼ TECi ðc þ cÞ þ a# þ b#2 f ðrÞ dl; j¼1
space rayðiÞ
i ¼ 1; . . . ; N
ð7Þ
where TECi for ray i is given by Eq. (1). Since the right hand side of Eq. (7) can be computed along each space ray, we solve for the unknown array n^0 with the least squares method as 1 T ^ n0 ¼ AT A ð8Þ A Tb
Fig. 7. Altitude dependence of a normalized f(r) function.
Fig. 8. Electron density profiles: the solid curve corresponds to IRI electron density profile; the starred curve describes the electron density obtained with the use of horizontal ionospheric gradients.
Fig. 9. Electron density profiles: the solid curve corresponds to IRI electron density profile; the circled curve corresponds to Abel inverted electron density profile and the starred curve describes the electron density obtained with the use of horizontal gradients.
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Administration. Sponsorship of NASA Earth Surface and Interior Focus Area program (Grant No. 101248 622.74.10.24) is gratefully acknowledged. References
Fig. 10. The altitude dependences of the ratios R1 (the solid curve) and R2 (the dashed curve).
. IRI n R1 ¼ nIRI ngradients e e e inv . IRI sym R2 ¼ ne nsph nIRI e e inv
ð9Þ
The first ratio R1 includes electron density data ngradients at e inv the central latitude h0 obtained with the algorithm that includes horizontal gradients; the second ratio R2 includes sym electron density data nsph obtained with a spherical syme inv metry assumption for electron density distribution. The altitude dependences of the ratios R1 and R2 are given by the solid and dashed curves in Fig. 10. As shown in Figs. 9 and 10, the inclusion of horizontal gradients in the equation for the electron density distribution (Eq. (6)) allows us to improve the electron density profile retrieval compared to the spherically symmetric model. 5. Conclusion In this work, we have developed a new method to retrieve electron density profiles from GPS occultation measurements as acquired by GPS receivers onboard low Earth orbiters. The new method takes into account horizontal gradients in the ionosphere and reduces retrieval error compared to the Abel inversion, which assumes horizontal uniformity of the ionosphere. Numerical simulations have been conducted to examine the effectiveness of the method. In the simulations, vertical TEC data from the ground are used to estimate the horizontal gradients, which, together with the space TEC measurements, allow us to improve the accuracy of electron density prediction in both E and F regions. Horizontal gradients are computed from two-dimensional vertical TEC maps using a polynomial fit up to second order in a given horizontal interval. Acknowledgements This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space
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