- Email: [email protected]
Global ionospheric maps from GPS observations using modip latitude
Advances in Space Research 38 (2006) 2324–2331 www.elsevier.com/locate/asr
Global ionospheric maps from GPS observations using modip latitude F. Azpilicueta a
a,*
, C. Brunini a, S.M. Radicella
b
Facultad de Ciencias Astrono´micas y Geofı´sicas, Universidad Nacional de La Plata, Paseo del bosque s/n, 1900 La Plata, Argentina b Aeronomy and Radiopropagation Laboratory, Abdus Salam ICTP, Strada Costiera 11, I-34014 Trieste, Italy Received 1 November 2004; received in revised form 6 July 2005; accepted 16 July 2005
Abstract In recent years, the new possibility of estimating the global distribution of free electrons in the ionosphere by means of GPS has opened a very active and promising field of research. Nowadays, there is a variety of approaches for processing dual-frequency GPS observations to produce global ionospheric maps of vertical total electron content (vTEC) with temporal resolution of 2 h or less. At the University of La Plata, we have developed the so-called La Plata Ionospheric Model (LPIM), based on a spherical harmonic expansion. For a better modelling of the variability of the vTEC, a solar-fixed coordinate system and the geomagnetic latitude are usually adopted. Early investigations by Rawer, K. [Encyclopedia of Physics, Geophysics III, Part VII. Springer-Verlag, pp. 389–391, 1984.] demonstrated the benefit of using the modified dip latitude (modip) to describe free electron distribution on the F2 layer at mid and low latitude. We have updated the LPIM to use modip latitude instead of the geomagnetic one. Exhaustive comparisons against Topex vTEC measurements provided by Aviso – performed for 90 days of 1999 – have demonstrated that the use of modip significantly improved the agreement between GPS and Topex. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Ionosphere; GPS data; vTEC global distribution; Total electron content
1. Introduction In recent years, the new possibility of estimating the global distribution of free electrons within the ionosphere by means of GPS has opened a very active and promising field of research. Currently, a variety of approaches exists for processing dual-frequency GPS observations to produce near real-time global ionospheric maps of the so-called vertical total electron content (vTEC) with a temporal resolution of 2 h or less (Schaer, 1999; Brunini et al., 2004). Most of these researches fully develop the potentialities of the worldwide *
Corresponding author. Tel.: +54 221 423 6593; fax: +54 221 423 6591. E-mail address: [email protected] (F. Azpilicueta).
tracking network from more that 400 GPS stations overseen by the International GPS Service (IGS) (Beutler et al., 1999). Present data coverage is far from homogeneous, as tracking stations are clustered in regions such as Europe and the United States of America, fewer stations are located in the southern hemisphere and large oceanic areas have no coverage. All these conditions lead to limitations both in temporal and spatial resolution achievable with global ionospheric maps based on GPS measurements. Data gaps are filled using appropriate temporal–spatial interpolation techniques, whose efficiency strongly depends on the use of a temporal–spatial coordinate system in which the ionospheric variability is as smooth as possible. The usual solution to this problem has been the adoption of the Local Time (LT) and the
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.07.069
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
geomagnetic latitude as the temporal–spatial coordinate system for global ionospheric mapping (Manucci et al., 1999). Early investigations (reported, e.g., in Rawer, 1984) demonstrated the benefit of using the modified dip latitude (modip) (introduced in Rawer, 1963) to describe the variability of the densest part of the ionosphere, particularly at mid and low latitudes. We have adopted the modip –LT coordinate system for the present work and found that results have improved considerably, allowing a better representation of the main structures in the vTEC distribution. For the purpose of validating the results, vTEC measurements made by Topex dual frequency radar altimeter on board of Topex/Poseidon satellite have been used as control values (see Aviso, 2004). The nominal precision of the Topex vTEC is in the order of 2 Total Electron Content units (TECu, 1 TECu = 1 · 1016 m2) thus making Topex vTEC measurements a unique reference for our analysis. In spite of this, many reports have pointed out that Topex vTEC is overestimated by about 3–4 TECu, see for example (Ablain et al., 2004) and (Menard and Haines, 2001).
2. La Plata ionospheric model Ionospheric observable obtained from the combination of the GPS observations has been widely discussed in the literature (Lanyi and Roth, 1988; Manucci et al., 1998). Basically, when simultaneous carrier phase observations in both frequencies (U1 and U2) are subtracted, the satellite-receiver geometrical range and all frequency independent biases are removed and the so-called geometry free linear combination (U4) is obtained U4 ¼ U1 U2 ¼ k sTEC þ C R þ C S þ C SR þ e;
ð1Þ
where k is a constant; sTEC is the slant total electron content (the electron density integrated along the slant path of the signal from the receiver to the satellite); CR and CS are the so-called differential code biases (DCBs) due to electronic delays in the receiver and satellite hardware respectively; C SR is a constant related with the carrier phase ambiguities in both frequencies; and e is the resultant measurement error. To deal with the ambiguities, we pre-process the raw observations to detect jumps in the carrier phase; then, we estimate a C SR value for every continuous arc, averaging the differences between carrier phase and P-code observations; and finally, we subtract the average value from the corresponding carrier phase observations. In this way, every continuous arc of carrier phase observations is leveled (on average) to the P-code observations and the ambiguities are removed from the problem. DCBs for both receivers and satellites remain as unknowns and we estimate daily constant values as well as the ionospheric information sought. There are evi-
2325
dences that satellite DCBs are quite stable for periods of several months (Schaer, 1999). The stability of the receiver DCBs is more complicate to assess. Calibration for most receivers of the IGS network is performed once per day (Beutler et al., 1999) and we followed the same criterion in this work. We assume that the overall electron content is concentrated in an infinitesimal thin shell located close to the height where the ionosondes detect the higher electron density, namely the F2 region. We adopted the widely used ‘‘thin shell’’ obliquity factor, M, (Lanyi and Roth, 1988; Coco et al., 1991; Brunini et al., 2004) to relate slant and vertical TEC 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 2 sTEC 4 R ffi 1 cosðEÞ 5 ; M¼ ð2Þ vTEC RþH where E is the elevation angle of the satellite, R is the mean radius of the Earth and H is the height of the shell that we fixed at 450 km. The central approximation of the ‘‘shell-ionosphere’’ model is to assume that there are no horizontal variations of the electron distribution along the ray-path of the signal from satellite to receiver. The errors involved in this approximation may worsen for low-elevation observations, particularly during dusk local times or in the equatorial regions. In order to map the vTEC over the shell, a sun-fixed coordinate system is adopted. This system is geocentric and rotates westward is such a way that the x-axis always points to the direction of the sun. Using this coordinate system, the ionospheres variability seen from an earth fixed coordinate system is almost eliminated (Schaer, 1999). The distribution of the vTEC over the shell is represented with a spherical harmonic expansion up to a maximum degree and order values equal to 15, as shown in Eq. (3). vTECðh; /Þ ¼
15 X 1 X l¼0 m¼0
mh mh þ blm sin 2p alm cos 2p 24 24
P lm ðsin /Þ;
ð3Þ
where h is Local Time (LT) and / the latitude coordinate, (geomagnetic latitude or modip); P lm ðsin /Þ are the Legendre associated functions; and alm and blm are the expansion coefficients which we fit using GPS observations. Introducing Eqs. (2) and (3) into Eq. (1) leads to Eq. (4), which links the observations and the unknowns, and represents the observation equation for the problem: U4 C SR e ¼ k M vTECðalm ; blm Þ þ C R þ C S . ð4Þ Eq. (4) leads to a rank deficient design matrix because it is impossible to solve all the satellite and receiver DCBs independently. To avoid this problem a condition is introduced that the sum of satellite DCBs must be zero.
2326
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Every day is split into 12 periods of 2 h and a full set of expansion coefficients is adjusted for every period. As mentioned in Section 1, GPS data coverage is far from being homogeneous and leaves large areas with no data at all. This lack of data could lead to unrealistic fluctuations between coefficients adjusted for consecutive time periods. To prevent this situation, the differences between coefficients of consecutive periods are constrained to vary within a range. The constraining conditions can be expressed in the following equations: 2
E½almiþ1 almi ¼ 0;
E½ðalmiþ1 almi Þ ¼ r2 ;
E½blmiþ1 blmi ¼ 0;
E½ðblmiþ1 blmi Þ ¼ r2 ;
2
ð5Þ
where E[ Æ ] is the expectation operator, indices l and m refer to the degree and order of the spherical harmonic and i refers to the time period. Using observations from 110 IGS stations (downloaded by anonymous ftp from the Scripps Orbit and Permanent Array Center at Scripps Institution of Oceanography, University of California, San Diego, lox.ucsd. edu.) we estimate the unknowns by means of a least squares algorithm. The criterion for selecting the stations was to achieve a worldwide coverage of observations as homogeneous as possible. In order to avoid using redundant information a reduced set of stations was selected in those regions where dense tracking networks exist (USA,
Fig. 1a. Representation of the vTEC (in TECu) distribution obtained using LPIM-geomagnetic for day 070, 1999.
Fig. 1b. Representation of the vTEC (in TECu) distribution obtained using LPIM-modip for day 070, 1999.
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Europe). Although the coverage provided by IGS tracking stations is global, there exist many regions (mainly over ocean regions) with no data coverage. We process the observations on a day to day basis and estimate independent DCBs for every day as well as a set of spherical harmonic expansion coefficients every 2-h UT. For further details about the La Plata Ionospheric Model (LPIM), see (Brunini et al., 2002; Meza et al., 2002).
2327
(see, for example, Rawer, 1984). As mentioned before the efficiency of approaches such as LPIM strongly depends on using a coordinate system in which the ionospheric variability is as smooth as possible in space and time. The usual approach is to adopt a geomagnetic sun-fixed coordinate system, while this work proposes a modip sun-fixed coordinate system. There follows a summary description of the main features of each system. 3.1. Geomagnetic latitude
3. Coordinate systems In the literature of ionospheric research and modeling, several coordinate systems are proposed for describing the electron density distribution of the different layers
Geomagnetic dipole is constructed as a best fit to the magnetic field (Fraser-Smith, 1987). North and South Geomagnetic Poles are the points where the geomagnetic dipole intersects the Earths surface. Their coordinates are N79.53/W71.65 and S79.53/E108.35. The
Fig. 2a. Mean bias between LPIM_geomagitc and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
Fig. 2b. Mean bias between LPIM_modip and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
2328
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
geomagnetic equator is the locus of points formed by passing a plane through the Earths center perpendicular to the geomagnetic dipole. Geomagnetic latitude is determined by the rotation that occurs when aligning the geomagnetic axis, i.e., the axis defined by the geomagnetic dipole and the geographic axis, as expressed in Eq. (6). sin /gm ¼ sin / sin /0 þ cos / cos /0 cosðk k0 Þ ð/; kÞ are the geographic coordinates to be converted; ð/0 ; k0 Þ are the geodetic coordinates of the
Rawer (1984) proposed a new coordinate for modeling the F2-layer and the top-side ionosphere, adapted to the real magnetic field, e.g., to the magnetic inclination (dip). This coordinate is now called modified dip or modip l and is defined by Eq. (7). I tan l ¼ pffiffiffiffiffiffiffi
cos /
l is the modip latitude; I is the true magnetic dip; usually at a height of 350 km;
North Geomagentic Pole; ð/gm ; kgm Þ are the geomagnetic coordinates:
3.2. Modip
ð6Þ
/ is the geographic latitude ð7Þ
Fig. 3a. Standard deviation of the bias between LPIM_geomagnetic and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
Fig. 3b. Standard deviation of the bias between LPIM_modip and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Modip equator is the locus of points where the magnetic dip (or inclination) is 0. In the equatorial zone, the lines of constant modip are practically identical to those of the magnetic inclination but as latitude increases they deviate and come nearer to those of constant geographical latitude. The poles are identical to the geographic ones.
4. Description of the work GPS data from a set of 110 IGS stations with world wide coverage were processed with LPIM algorithm using both latitude systems thus obtaining a LPIM_geomagnetic and a LPIM_modip solution for every analyzed day. The processes were performed for 90 days in 1999 (March, April and May), which is the time period required for Topex to provide a set of vTEC measurements with complete local time and space coverage. As examples of the two solutions, Fig. 1a shows the global vTEC distribution adjusted with LPIM_geomagnetic solution and Fig. 1b, the vTEC distribution corresponding to LPIM_modip solution for day 070 1999 and UT range from 16:00 to 18:00. From these figures, it is clear that LPIM-modip represents better the double peak structure of the Equatorial Anomaly. For every day analyzed, a set of 12 pairs of maps like this has been computed and compared with TOPEX measurements.
2329
5. Results A total set of more than 160,000 Topex observations were used for the present work. For each Topex measurement, the corresponding LPIM_geomagnetic and LPIM_modip values were computed, and then the biases between both LPIM solutions and Topex values were calculated. In order to understand the effect on the vTEC spatial distribution caused by the change in the coordinate system, the Earths surface was tessellated into spherical triangles of 9 sides and for each triangle the mean value and the standard deviation of the biases were computed. The global distribution of the mean biases between LPIM solutions and Topex are shown in Fig. 2a for LPIM_geomagnetic and in Fig. 2b for LPIM_modip. Similarly, the global distribution of the standard deviations of the biases is shown in Fig. 3a for LPIM_geomagnetic solution and in Fig. 3b for LPIM_modip solution. The analysis of Figs. 3a and b shows a clear improvement in favor of the LPIM_modip solution. The decrease on the mean and standard deviation observed in the equatorial region is significantly important. Histograms of the biases are shown in Fig. 4a for LPIM_geomagnetic and in Fig. 4b for LPIM_modip. They show a 10% increase in the percentage of differences in the range [10,10] TECu for the LPIM_modip. A bias of about 4 TECu is observed on both histograms, but as
Fig. 4a. Histogram of the bias between LPIM_geomagnetic and Topex.
2330
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Fig. 4b. Histogram of the bias between LPIM_modip and Topex.
was previously mentioned, this could be associated to a calibration problem of the Topex radar altimeter.
This work has been partially supported by the National University of La Plata and ICTP under the TRIL program.
6. Conclusions The use of modip has probed to be a more suitable coordinate than the geomagnetic latitude for the LPIM model. Modip provides a coordinate system with smooth variability in space and time thus validating the condition imposed on the variability of the coefficients. Taking Topex as reference, the use of modip improves the accuracy of the global vTEC representation in more than 10% and reduces the mean and the standard deviation of the differences, particularly in the region of the equatorial anomaly, which is the most important and difficult region to model.
Acknowledgments The authors thank Pierdavide Coı¨sson and Bruno Nava from the staff of the Aeronomy and Radiopropagation Laboratory – Abdus Salam ICTP staff for their help in understanding the features of modip and for providing the necessary softwares. The authors are grateful to Dr. Wolfgang Bosch from the Deutsches Geoda¨tisches Forschungsinstitut, Munich, for providing the Topex vTEC measurements.
References Ablain, M., Mertz F. Dorandeu, J., TOPEX/Poseidon validation activities 11 years of T/P data (GDR-Ms), SALP-RP-P2-EA15161-CLS, pp. 37–38, March 2004. Aviso, Available from:, 2004. Beutler, G., Rothacher, M., Schaer, S., Springer, T.A., Kouba, J., Neilan, R.E. The international GPS service (IGS): an interdisciplinary service in support of earth sciences. Advances in Space Research 23 (4), 631–653, 1999. Brunini, C., VanZele, M.A., Meza y, A., Gende, M. Quiet and perturbed ionospheric representation according the electron content from GPS signals. Journal of Geophysical Research 108 (A2), doi:10.1029/2002JA009346, 2002. Brunini, C., Meza, A., Azpilicueta, F., Van Zele, A., Gende, M., Diaz, A. A new ionosphere monitoring technology based on GPS. Astrophysics and Space Science 290, 415–429, 2004. Coco, D.S., Coker, C., Dahlke, S.R., Clynch, J.R. Variability of GPS satellite differential group delays biases. IEEE Transactions on Aerospace and Electronic Systems 27, 931–938, 1991. Fraser-Smith, A.C. Centered and eccentric geomagnetic dipoles and their poles 1600–1985. Reviews of Geophysics 25 (1), 1–16, 1987. Lanyi, G.E., Roth, T. A comparison of mapped and measured total ionospheric electron content using Global Positioning System and beacon satellite observations. Radio Science 23 (4), 483–492, 1988.
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331 Manucci, A.J., Wilson, B.D., Yuan, D.N., Ho, C.M., Lindqwister, U.J., Runge, T.F. A global mapping technique for GPS-derived ionospheric total electron-content measurements. Radio Science 33 (3), 565–582, 1998. Manucci, A.J., Iijima, B.A., Lindqwister, U.J., Pi, X., Sparks, L., Wilson, B.D. GPS and ionosphere, URSI Reviews of Radio Science. Jet Propulsion Laboratory, Pasadena, California, March 1999. Menard, Y., Haines, B. Jason-1 CALVAL Plan, Ref: TP2-J0-PL974-CN, report to CNES and JPL, NASA, pp. 13–14, October 2001.
2331
Meza, A., Brunini, C.A., Klueusberg, A. Global behavior of the ionosphere electron density using GPS observations. Advance in Space Research (ASR) 30 (2), 401–407, 2002. Rawer, K. In: Landmark, B. (Ed.), Meteorological and Astronomical Influences on Radio Wave Propagation. Pergamon Press, Oxford, pp. 221–250, 1963. Rawer, K. (Ed.). Encyclopedia of Physics, Geophysics III, Part VII. Springer-Verlag, pp. 389–391, 1984. Schaer, S., Mapping and Predicting the Earths Ionosphere Using the Global Positioning System, PhD thesis, 164 pp, Bern University, Bern, Switzerland, 1999.
Global ionospheric maps from GPS observations using modip latitude F. Azpilicueta a
a,*
, C. Brunini a, S.M. Radicella
b
Facultad de Ciencias Astrono´micas y Geofı´sicas, Universidad Nacional de La Plata, Paseo del bosque s/n, 1900 La Plata, Argentina b Aeronomy and Radiopropagation Laboratory, Abdus Salam ICTP, Strada Costiera 11, I-34014 Trieste, Italy Received 1 November 2004; received in revised form 6 July 2005; accepted 16 July 2005
Abstract In recent years, the new possibility of estimating the global distribution of free electrons in the ionosphere by means of GPS has opened a very active and promising field of research. Nowadays, there is a variety of approaches for processing dual-frequency GPS observations to produce global ionospheric maps of vertical total electron content (vTEC) with temporal resolution of 2 h or less. At the University of La Plata, we have developed the so-called La Plata Ionospheric Model (LPIM), based on a spherical harmonic expansion. For a better modelling of the variability of the vTEC, a solar-fixed coordinate system and the geomagnetic latitude are usually adopted. Early investigations by Rawer, K. [Encyclopedia of Physics, Geophysics III, Part VII. Springer-Verlag, pp. 389–391, 1984.] demonstrated the benefit of using the modified dip latitude (modip) to describe free electron distribution on the F2 layer at mid and low latitude. We have updated the LPIM to use modip latitude instead of the geomagnetic one. Exhaustive comparisons against Topex vTEC measurements provided by Aviso – performed for 90 days of 1999 – have demonstrated that the use of modip significantly improved the agreement between GPS and Topex. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Ionosphere; GPS data; vTEC global distribution; Total electron content
1. Introduction In recent years, the new possibility of estimating the global distribution of free electrons within the ionosphere by means of GPS has opened a very active and promising field of research. Currently, a variety of approaches exists for processing dual-frequency GPS observations to produce near real-time global ionospheric maps of the so-called vertical total electron content (vTEC) with a temporal resolution of 2 h or less (Schaer, 1999; Brunini et al., 2004). Most of these researches fully develop the potentialities of the worldwide *
Corresponding author. Tel.: +54 221 423 6593; fax: +54 221 423 6591. E-mail address: [email protected] (F. Azpilicueta).
tracking network from more that 400 GPS stations overseen by the International GPS Service (IGS) (Beutler et al., 1999). Present data coverage is far from homogeneous, as tracking stations are clustered in regions such as Europe and the United States of America, fewer stations are located in the southern hemisphere and large oceanic areas have no coverage. All these conditions lead to limitations both in temporal and spatial resolution achievable with global ionospheric maps based on GPS measurements. Data gaps are filled using appropriate temporal–spatial interpolation techniques, whose efficiency strongly depends on the use of a temporal–spatial coordinate system in which the ionospheric variability is as smooth as possible. The usual solution to this problem has been the adoption of the Local Time (LT) and the
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.07.069
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
geomagnetic latitude as the temporal–spatial coordinate system for global ionospheric mapping (Manucci et al., 1999). Early investigations (reported, e.g., in Rawer, 1984) demonstrated the benefit of using the modified dip latitude (modip) (introduced in Rawer, 1963) to describe the variability of the densest part of the ionosphere, particularly at mid and low latitudes. We have adopted the modip –LT coordinate system for the present work and found that results have improved considerably, allowing a better representation of the main structures in the vTEC distribution. For the purpose of validating the results, vTEC measurements made by Topex dual frequency radar altimeter on board of Topex/Poseidon satellite have been used as control values (see Aviso, 2004). The nominal precision of the Topex vTEC is in the order of 2 Total Electron Content units (TECu, 1 TECu = 1 · 1016 m2) thus making Topex vTEC measurements a unique reference for our analysis. In spite of this, many reports have pointed out that Topex vTEC is overestimated by about 3–4 TECu, see for example (Ablain et al., 2004) and (Menard and Haines, 2001).
2. La Plata ionospheric model Ionospheric observable obtained from the combination of the GPS observations has been widely discussed in the literature (Lanyi and Roth, 1988; Manucci et al., 1998). Basically, when simultaneous carrier phase observations in both frequencies (U1 and U2) are subtracted, the satellite-receiver geometrical range and all frequency independent biases are removed and the so-called geometry free linear combination (U4) is obtained U4 ¼ U1 U2 ¼ k sTEC þ C R þ C S þ C SR þ e;
ð1Þ
where k is a constant; sTEC is the slant total electron content (the electron density integrated along the slant path of the signal from the receiver to the satellite); CR and CS are the so-called differential code biases (DCBs) due to electronic delays in the receiver and satellite hardware respectively; C SR is a constant related with the carrier phase ambiguities in both frequencies; and e is the resultant measurement error. To deal with the ambiguities, we pre-process the raw observations to detect jumps in the carrier phase; then, we estimate a C SR value for every continuous arc, averaging the differences between carrier phase and P-code observations; and finally, we subtract the average value from the corresponding carrier phase observations. In this way, every continuous arc of carrier phase observations is leveled (on average) to the P-code observations and the ambiguities are removed from the problem. DCBs for both receivers and satellites remain as unknowns and we estimate daily constant values as well as the ionospheric information sought. There are evi-
2325
dences that satellite DCBs are quite stable for periods of several months (Schaer, 1999). The stability of the receiver DCBs is more complicate to assess. Calibration for most receivers of the IGS network is performed once per day (Beutler et al., 1999) and we followed the same criterion in this work. We assume that the overall electron content is concentrated in an infinitesimal thin shell located close to the height where the ionosondes detect the higher electron density, namely the F2 region. We adopted the widely used ‘‘thin shell’’ obliquity factor, M, (Lanyi and Roth, 1988; Coco et al., 1991; Brunini et al., 2004) to relate slant and vertical TEC 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi31 2 sTEC 4 R ffi 1 cosðEÞ 5 ; M¼ ð2Þ vTEC RþH where E is the elevation angle of the satellite, R is the mean radius of the Earth and H is the height of the shell that we fixed at 450 km. The central approximation of the ‘‘shell-ionosphere’’ model is to assume that there are no horizontal variations of the electron distribution along the ray-path of the signal from satellite to receiver. The errors involved in this approximation may worsen for low-elevation observations, particularly during dusk local times or in the equatorial regions. In order to map the vTEC over the shell, a sun-fixed coordinate system is adopted. This system is geocentric and rotates westward is such a way that the x-axis always points to the direction of the sun. Using this coordinate system, the ionospheres variability seen from an earth fixed coordinate system is almost eliminated (Schaer, 1999). The distribution of the vTEC over the shell is represented with a spherical harmonic expansion up to a maximum degree and order values equal to 15, as shown in Eq. (3). vTECðh; /Þ ¼
15 X 1 X l¼0 m¼0
mh mh þ blm sin 2p alm cos 2p 24 24
P lm ðsin /Þ;
ð3Þ
where h is Local Time (LT) and / the latitude coordinate, (geomagnetic latitude or modip); P lm ðsin /Þ are the Legendre associated functions; and alm and blm are the expansion coefficients which we fit using GPS observations. Introducing Eqs. (2) and (3) into Eq. (1) leads to Eq. (4), which links the observations and the unknowns, and represents the observation equation for the problem: U4 C SR e ¼ k M vTECðalm ; blm Þ þ C R þ C S . ð4Þ Eq. (4) leads to a rank deficient design matrix because it is impossible to solve all the satellite and receiver DCBs independently. To avoid this problem a condition is introduced that the sum of satellite DCBs must be zero.
2326
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Every day is split into 12 periods of 2 h and a full set of expansion coefficients is adjusted for every period. As mentioned in Section 1, GPS data coverage is far from being homogeneous and leaves large areas with no data at all. This lack of data could lead to unrealistic fluctuations between coefficients adjusted for consecutive time periods. To prevent this situation, the differences between coefficients of consecutive periods are constrained to vary within a range. The constraining conditions can be expressed in the following equations: 2
E½almiþ1 almi ¼ 0;
E½ðalmiþ1 almi Þ ¼ r2 ;
E½blmiþ1 blmi ¼ 0;
E½ðblmiþ1 blmi Þ ¼ r2 ;
2
ð5Þ
where E[ Æ ] is the expectation operator, indices l and m refer to the degree and order of the spherical harmonic and i refers to the time period. Using observations from 110 IGS stations (downloaded by anonymous ftp from the Scripps Orbit and Permanent Array Center at Scripps Institution of Oceanography, University of California, San Diego, lox.ucsd. edu.) we estimate the unknowns by means of a least squares algorithm. The criterion for selecting the stations was to achieve a worldwide coverage of observations as homogeneous as possible. In order to avoid using redundant information a reduced set of stations was selected in those regions where dense tracking networks exist (USA,
Fig. 1a. Representation of the vTEC (in TECu) distribution obtained using LPIM-geomagnetic for day 070, 1999.
Fig. 1b. Representation of the vTEC (in TECu) distribution obtained using LPIM-modip for day 070, 1999.
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Europe). Although the coverage provided by IGS tracking stations is global, there exist many regions (mainly over ocean regions) with no data coverage. We process the observations on a day to day basis and estimate independent DCBs for every day as well as a set of spherical harmonic expansion coefficients every 2-h UT. For further details about the La Plata Ionospheric Model (LPIM), see (Brunini et al., 2002; Meza et al., 2002).
2327
(see, for example, Rawer, 1984). As mentioned before the efficiency of approaches such as LPIM strongly depends on using a coordinate system in which the ionospheric variability is as smooth as possible in space and time. The usual approach is to adopt a geomagnetic sun-fixed coordinate system, while this work proposes a modip sun-fixed coordinate system. There follows a summary description of the main features of each system. 3.1. Geomagnetic latitude
3. Coordinate systems In the literature of ionospheric research and modeling, several coordinate systems are proposed for describing the electron density distribution of the different layers
Geomagnetic dipole is constructed as a best fit to the magnetic field (Fraser-Smith, 1987). North and South Geomagnetic Poles are the points where the geomagnetic dipole intersects the Earths surface. Their coordinates are N79.53/W71.65 and S79.53/E108.35. The
Fig. 2a. Mean bias between LPIM_geomagitc and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
Fig. 2b. Mean bias between LPIM_modip and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
2328
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
geomagnetic equator is the locus of points formed by passing a plane through the Earths center perpendicular to the geomagnetic dipole. Geomagnetic latitude is determined by the rotation that occurs when aligning the geomagnetic axis, i.e., the axis defined by the geomagnetic dipole and the geographic axis, as expressed in Eq. (6). sin /gm ¼ sin / sin /0 þ cos / cos /0 cosðk k0 Þ ð/; kÞ are the geographic coordinates to be converted; ð/0 ; k0 Þ are the geodetic coordinates of the
Rawer (1984) proposed a new coordinate for modeling the F2-layer and the top-side ionosphere, adapted to the real magnetic field, e.g., to the magnetic inclination (dip). This coordinate is now called modified dip or modip l and is defined by Eq. (7). I tan l ¼ pffiffiffiffiffiffiffi
cos /
l is the modip latitude; I is the true magnetic dip; usually at a height of 350 km;
North Geomagentic Pole; ð/gm ; kgm Þ are the geomagnetic coordinates:
3.2. Modip
ð6Þ
/ is the geographic latitude ð7Þ
Fig. 3a. Standard deviation of the bias between LPIM_geomagnetic and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
Fig. 3b. Standard deviation of the bias between LPIM_modip and Topex in TECu (solid line indicates modip equator and dashed line geomagnetic equator).
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Modip equator is the locus of points where the magnetic dip (or inclination) is 0. In the equatorial zone, the lines of constant modip are practically identical to those of the magnetic inclination but as latitude increases they deviate and come nearer to those of constant geographical latitude. The poles are identical to the geographic ones.
4. Description of the work GPS data from a set of 110 IGS stations with world wide coverage were processed with LPIM algorithm using both latitude systems thus obtaining a LPIM_geomagnetic and a LPIM_modip solution for every analyzed day. The processes were performed for 90 days in 1999 (March, April and May), which is the time period required for Topex to provide a set of vTEC measurements with complete local time and space coverage. As examples of the two solutions, Fig. 1a shows the global vTEC distribution adjusted with LPIM_geomagnetic solution and Fig. 1b, the vTEC distribution corresponding to LPIM_modip solution for day 070 1999 and UT range from 16:00 to 18:00. From these figures, it is clear that LPIM-modip represents better the double peak structure of the Equatorial Anomaly. For every day analyzed, a set of 12 pairs of maps like this has been computed and compared with TOPEX measurements.
2329
5. Results A total set of more than 160,000 Topex observations were used for the present work. For each Topex measurement, the corresponding LPIM_geomagnetic and LPIM_modip values were computed, and then the biases between both LPIM solutions and Topex values were calculated. In order to understand the effect on the vTEC spatial distribution caused by the change in the coordinate system, the Earths surface was tessellated into spherical triangles of 9 sides and for each triangle the mean value and the standard deviation of the biases were computed. The global distribution of the mean biases between LPIM solutions and Topex are shown in Fig. 2a for LPIM_geomagnetic and in Fig. 2b for LPIM_modip. Similarly, the global distribution of the standard deviations of the biases is shown in Fig. 3a for LPIM_geomagnetic solution and in Fig. 3b for LPIM_modip solution. The analysis of Figs. 3a and b shows a clear improvement in favor of the LPIM_modip solution. The decrease on the mean and standard deviation observed in the equatorial region is significantly important. Histograms of the biases are shown in Fig. 4a for LPIM_geomagnetic and in Fig. 4b for LPIM_modip. They show a 10% increase in the percentage of differences in the range [10,10] TECu for the LPIM_modip. A bias of about 4 TECu is observed on both histograms, but as
Fig. 4a. Histogram of the bias between LPIM_geomagnetic and Topex.
2330
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331
Fig. 4b. Histogram of the bias between LPIM_modip and Topex.
was previously mentioned, this could be associated to a calibration problem of the Topex radar altimeter.
This work has been partially supported by the National University of La Plata and ICTP under the TRIL program.
6. Conclusions The use of modip has probed to be a more suitable coordinate than the geomagnetic latitude for the LPIM model. Modip provides a coordinate system with smooth variability in space and time thus validating the condition imposed on the variability of the coefficients. Taking Topex as reference, the use of modip improves the accuracy of the global vTEC representation in more than 10% and reduces the mean and the standard deviation of the differences, particularly in the region of the equatorial anomaly, which is the most important and difficult region to model.
Acknowledgments The authors thank Pierdavide Coı¨sson and Bruno Nava from the staff of the Aeronomy and Radiopropagation Laboratory – Abdus Salam ICTP staff for their help in understanding the features of modip and for providing the necessary softwares. The authors are grateful to Dr. Wolfgang Bosch from the Deutsches Geoda¨tisches Forschungsinstitut, Munich, for providing the Topex vTEC measurements.
References Ablain, M., Mertz F. Dorandeu, J., TOPEX/Poseidon validation activities 11 years of T/P data (GDR-Ms), SALP-RP-P2-EA15161-CLS, pp. 37–38, March 2004. Aviso, Available from:
F. Azpilicueta et al. / Advances in Space Research 38 (2006) 2324–2331 Manucci, A.J., Wilson, B.D., Yuan, D.N., Ho, C.M., Lindqwister, U.J., Runge, T.F. A global mapping technique for GPS-derived ionospheric total electron-content measurements. Radio Science 33 (3), 565–582, 1998. Manucci, A.J., Iijima, B.A., Lindqwister, U.J., Pi, X., Sparks, L., Wilson, B.D. GPS and ionosphere, URSI Reviews of Radio Science. Jet Propulsion Laboratory, Pasadena, California, March 1999. Menard, Y., Haines, B. Jason-1 CALVAL Plan, Ref: TP2-J0-PL974-CN, report to CNES and JPL, NASA, pp. 13–14, October 2001.
2331
Meza, A., Brunini, C.A., Klueusberg, A. Global behavior of the ionosphere electron density using GPS observations. Advance in Space Research (ASR) 30 (2), 401–407, 2002. Rawer, K. In: Landmark, B. (Ed.), Meteorological and Astronomical Influences on Radio Wave Propagation. Pergamon Press, Oxford, pp. 221–250, 1963. Rawer, K. (Ed.). Encyclopedia of Physics, Geophysics III, Part VII. Springer-Verlag, pp. 389–391, 1984. Schaer, S., Mapping and Predicting the Earths Ionosphere Using the Global Positioning System, PhD thesis, 164 pp, Bern University, Bern, Switzerland, 1999.