Abstract
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[1] Ionospheric propagation effects on Global Navigation Satellite Systems (GNSS) signals are the most pronounced during radio occultation due to long ionospheric travel paths of the received signal on low Earth orbiting satellites. Inhomogeneous plasma distribution and anisotropy cause higherorder nonlinear refraction effects on GNSS signals which cannot be fully removed through a linear combination of dualfrequency observables. In this paper, higherorder ionospheric effects due to straight line of sight (LOS) propagation assumption such as the excess path length of the signal in addition to the LOS path and the total electron content difference between the curved path and the LOS path have been investigated for selected GPSCHAMP occultation events. Based on simulation studies we have derived correction formulas for computing raypath bending effects as functions of signal frequency, tangential height of the raypath, ionospheric parameters such as the maximum ionization and total electron content. If these parameters are known, the proposed correction method is able to correct on an average about 65–80% bending errors of GNSS occultation signals.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[2] The ionospheric propagation medium is characterized by a refractive index which is different from that of the free space. The phase refractive index is less than unity resulting in a phase velocity that is greater than the speed of light. However, the group refractive index is greater than unity resulting in a group velocity that is less than the speed of light. Therefore, when the Global Positioning System (GPS) signals propagate through the ionosphere, the carrier experiences a phase advance and the code experiences a group delay. The carrier phase pseudoranges are measured too short and the code pseudoranges are measured too long compared to the geometric range between the satellite and the receiver. Accurate range estimations between a receiver and four or more satellites enable accurate position determination of GNSS users in space and time.
[3] The ionospheric range error is at first order directly proportional to the total number of free electrons along the path of the signal from the satellite to the receiver. It can vary from a few meters to tens of meters at the zenith [Klobuchar, 1996]. Since the ionosphere is a dispersive medium, the magnitude of the ionospheric delay depends on the signal frequency and therefore, the firstorder effect can be eliminated through a linear combination of dualfrequency observables. However, inhomogeneous plasma distribution and anisotropy cause higherorder nonlinear effects which are not removed in this approach. Mainly the second and thirdorder ionospheric terms (in the expansion of the refractive index) and errors due to bending of the signal remain uncorrected. They can be several tens of centimeters of range error at low elevation angles and during high solar activity conditions [Klobuchar, 1996].
[4] Early work was done by Brunner and Gu [1991] in computing higherorder ionospheric effects and developing correction for them. Since then higherorder ionospheric effects have been studied by different authors during last two decades [e.g., Bassiri and Hajj, 1993; Jakowski et al., 1994; Strangeways and Ioannides, 2002; Kedar et al., 2003; Fritsche et al., 2005; Hawarey et al., 2005; Hoque and Jakowski, 2006, 2007, 2008; HernándezPajares et al., 2007; DattaBarua et al., 2008; Morton et al., 2009]. They found that higherorder ionospheric terms are less than 1% of the firstorder term at GPS frequencies; however, they represent millimeter/centimeter level errors in geodetic measurements. HernándezPajares et al. [2007] found submillimeter level shifting in receiver positions along southward direction for low latitude receivers and northward direction for high latitude receivers applying the secondorder term correction. Fritsche et al. [2005] found centimeter level correction in GPS positions considering higherorder ionospheric terms. Recently Petrie et al. [2010] investigated the potential effects of the bending terms described by Hoque and Jakowski [2008] on global GPS network. They found the bending correction for the dualfrequency linear L1L2 combination to exceed the 3 mm level in the equatorial region. All these studies were conducted to compute higherorder ionospheric effects on GNSS signals for groundbased reception.
[5] Recently Hoque and Jakowski [2010] investigated ionospheric impact on GPS occultation signals received onboard Low Earth Orbiting (LEO) CHAMP (Challenging Minisatellite Payload) satellite. LEO satellites have the opportunity to receive signals from occulting GNSS satellites using onboard limbpointing antennas. The observations have the dual purpose of studying electron densities in the ionosphere as well as temperature and moisture in the neutral atmosphere. A number of satellite and minisatellite missions such as GPS/MET (GPS Meteorology Instrument), SACC (Satelite de Aplicaciones CientificasC), CHAMP, GRACE (Gravity Recovery And Climate Experiment) and COSMIC satellite network (Constellation Observing System for Meteorology, Ionosphere and Climate, also known as FORMOSAT3) carry onboard GPS receivers for the GPS radio sounding of the Earth. Although occultation measurements are not usually used for positioning or navigation, it is worthy to know the ionospheric impact on accurate range estimation using these measurements. This paper investigates raypath bending effects in dualfrequency range and total electron content (TEC) estimation and proposes correction for mitigating such effects.
[6] During occultation both transmitter and receiver are out of the Earth's atmosphere. When the transmitted signal approaches the Earth, the closest point of approach to the Earth's surface is known as the tangential point and the altitude corresponding to this point is known as the tangential height. The perpendicular distances of the signal path from the straight line of sight (LOS) propagation are defined as the raypath deviations. Interested readers are referred to Figure 1 of Hoque and Jakowski [2010] for an exaggerated view of GPS frequencies geometric paths during radio occultation. Radio wave traverses a long ionospheric limb path and therefore, the refraction effects are the most pronounced during radio occultation. The refraction effects depend on the actual condition of the ionospheric ionization and as well as on the raypath geometry.
[7] Our previous investigation [Hoque and Jakowski, 2010] for selected GPSCHAMP occultation events shows that the straight line propagation assumption errors such as the excess path length of the signal compared to the LOS propagation, raypath deviations and TEC differences along the curved and LOS paths significantly vary with the ionospheric profile shape and raypath geometry. We found the maximum estimates of the excess path length to be about 2.7 m, and the second and thirdorder ionospheric terms to be about 13 cm and 2.1 cm, respectively, for the GPS L2 signal for an electron density profile with vertical TEC of about 167 TEC units (1 TEC unit = 10^{16} electrons/m^{2}). We found the separation between the GPS L1 and L2 raypaths to exceed the kilometer level and errors in the GPS dualfrequency range estimation and TEC estimation to exceed the meter and 10 TEC units level, respectively.
[8] In this paper, simulation studies have been done to determine the straight line propagation assumption errors as functions of the signal frequency, different ionospheric parameters such as the maximum ionization and TEC, and geometrical parameters such as the tangential height of the raypath. Based on simulation studies we have proposed correction formulas for computing the excess path length and TEC difference between the signal and LOS paths.
2. HigherOrder Refraction Terms
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[9] Ionospheric impact on the radio wave propagation is well described in the literature [e.g., Budden, 1985; Davies, 1990; Rawer, 1993; Leitinger and Putz, 1988]. For a right hand circularly polarized signal (e.g., GPS signals), the ionospheric phase refractive index can be written in terms of the inverse power of signal frequency (f) as
where f_{p}^{2} = n_{e}e^{2}/(4π^{2}_{0}m) and f_{g} = eB/(2πm) are the plasma frequency and gyro frequency, respectively. The quantity n_{e} is the electron concentration, e and m are the electron charge and mass, respectively, ɛ_{0} is the permittivity of the free space, Θ is the angle between the Earth's magnetic field vector B and the propagation direction. The ionospheric phase delay for the GPS signal can be written as
in which
where ds is the raypath element, the quantities p/f^{2}, q/(2f^{3}) and t/(3f^{4}) are the first, second and thirdorder ionospheric phase delays, respectively. The integraln_{e}ds along a signal path is defined as the total electron content TEC. The raypath bending causes different TEC estimation along curved and LOS paths. Therefore, TEC in equation (3) is separated into TEC_{LOS} along the straight LOS and ΔTEC_{bend} which is the difference between TEC along the curved and LOS paths. The bending effect additionally causes a longer curved path length compared to the LOS path length. The excess path of the signal in addition to the LOS path length d^{len} can be written as
where ρ is the geometric distance between the transmitter and the receiver.
3. Estimate of HigherOrder Refraction Terms
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[10] We have used a twodimensional ray tracing program [Hoque and Jakowski, 2008] to simulate refraction effects on GPS signals for typical occultation geometries. The ray tracing program assumes that the ionosphere is composed of numerous thin spherical layers in each of which the ionization is homogeneous, i.e., the horizontal gradient of the ionosphere is ignored. To take into account the effect of the Earth's magnetic field on the radio wave propagation the International Geomagnetic Reference Field (IGRF) model [Mandea and Macmillan, 2000] has been used.
[11] As already mentioned, radio occultation observations are used to derive electron densities in the ionosphere. The reconstruction technique has been described by Jakowski et al. [2002] and Jakowski [2005]. The retrieval algorithm assumes a spherically layered ionosphere and uses an adaptive Chapman layer [Rishbeth and Garriott, 1969] superposed by an exponential decay function for estimating the topside ionosphere and plasmasphere above the CHAMP orbit. The reconstructed profiles contain electron densities from E layer (about 90 km from the Earth's surface) up to the GPS orbit. In the present work, electron densities below the E layer have been calculated using an adaptive Chapman layer. Two electron density profiles retrieved from CHAMP measurements have been used as the input of the ray tracing program. The simulation results are referred to as case 1 and case 2. The case 1 profile has a maximum ionization NmF2 of 5.33 × 10^{12} m^{−3} at an altitude hmF2 of 417 km (see Figure 1a, plotted until 1000 km altitude) at the geographic latitude 25.7°S, longitude 126°W and 15.1 h local time (LT). The case 2 profile has a maximum ionization NmF2 of 4.93 × 10^{12} m^{−3} at hmF2 = 313 km (see Figure 1b, plotted until 1000 km altitude) at 27.6°N, 2.2°W and 16.4 LT. The corresponding vertical TEC up to GPS height is estimated as about 216 and 167 TEC units, respectively. The actual GPS and CHAMP positions over phase connected arcs of the specific case 1 and case 2 events have been used to trace signals for GPSCHAMP paths. Obtained refraction effects have been plotted in Figure 1 as a function of the tangential height. We have found that refraction effects experienced by the received signal vary with the tangential height of the received signal.
[12] Figures 1a and 1b show electron density profiles used for case 1 and case 2, respectively, and Figures 1 (left) and 1 (right) below Figures 1a and 1b represent corresponding simulation results. Figures 1c and 1d show the variation of the excess path length d^{len} and ΔTEC_{bend} with the tangential height (see left and right scales, respectively). Figures 1e and 1f show the variation of the maximum raypath deviation and deviation at the tangential height, Figures 1g and 1h give the variation of the second and thirdorder phase delays, Figures 1i and 1j show the variation of the TEC and dTEC/dh^{T} (see left and right scales, respectively). The quantity dTEC/dh^{T} is defined as the TEC rate and determined by dividing the TEC difference between two measurement epochs by the corresponding tangential height difference.
[13] Comparing the variation of different higherorder terms with the TEC and TEC rate dTEC/dh^{T}, we found that bending related effects such as the d^{len}, ΔTEC_{bend}, and raypath deviations are more sensitive to TEC rate changes than to the absolute TEC level. However, the second and thirdorder ionospheric phase delays are found to be more closely related to TEC than to TEC rate. For detailed description of different curves plotted in Figure 1 we refer to our previous paper [Hoque and Jakowski, 2010]. The variation of different higherorder terms had been plotted as a function of the measurement time and described in that paper.
[14] The maximum estimates of different higherorder terms found for case 1 and case 2 have been summarized in Table 1 for the GPS L2 signal. Their dependency on the signal frequency is also given in the Table 1.
Table 1. Maximum Estimate of HigherOrder Propagation Effects for Case 1 and Case 2Maximum Propagation Effects  Case 1, L2  Case 2, L2  Frequency Dependency 

Excess path length (m)  1.35  2.71  1/f^{4} 
Secondorder delay (m)  −0.15  −0.13  1/f^{3} 
Thirdorder delay (m)  0.02  0.02  1/f^{4} 
ΔTEC_{bend} (TECU)  9.5  17.9  1/f^{2} 
Maximum raypath deviation (km)  2.05  2.86  1/f^{2} 
Deviation at tangential height (km)  1.82  2.69  1/f^{2} 
[15] We found that bending effects such as ΔTEC_{bend} and raypath deviation are inversely proportional to the square of the signal frequency whereas the excess path length is inversely proportional to the quartic power of the frequency (see Table 1).
4. Raypath Bending Correction
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[16] One approach to mitigate raypath bending effects is the introduction of empirical formulas, which compute refraction effects by using ionospheric parameters such as TEC, scale height, maximum ionization, its height, and geometrical parameters such as elevation angle or tangential height of the raypath. For modeling purposes a singlelayered Chapman profile [Rishbeth and Garriott, 1969] has been considered for electron density distribution n_{e} as a function of height h in the ionosphere.
where NmF2 is the maximum ionization and z = (h − hmF2)/H in which hmF2 is the height of maximum ionization and H is the atmospheric scale height.
[17] In the previous section, we have found that refraction effects experienced by occultation signals depend on the tangential height of the raypath. Therefore, tangential height is a parameter that characterizes occultation raypaths. The excess path length between a GPS and a LEO satellite at 450 km altitude has been computed for typical occultation geometries by the ray tracing program considering Chapman profiles with different hmF2 = 250, 350 and 450 km. The signal frequency f = 1227.6 MHz, parameters H = 70 km and NmF2 = 4.96 × 10^{12} m^{−3} are kept constant in each case. The total electron content in the vertical direction will be the same since VTEC ≈ 4.13HNmF2 ≈ 143 TEC units [Hoque and Jakowski, 2008]. The obtained excess path length d^{len}, TEC and dTEC/dh^{T} have been plotted as a function of tangential height in Figures 2a–2c, respectively.
[18] Figure 2a shows that the peak of the d^{len} plot moves to the higher tangential height and its value decreases significantly with the increase of hmF2 from 250 km to 450 km. Therefore, the magnitude of the d^{len} depends on the magnitude of hmF2 as well as on the tangential height of the raypath.
[19] For the same occultation geometries, the excess path lengths have been computed considering Chapman profiles with different scale heights H = 60, 70 and 80 km. In each case the NmF2 and hmF2 are kept constant as 4.96 × 10^{12} m^{−3} and 450 km, respectively. The obtained d^{len} has been plotted as a function of the tangential height in Figure 3. We see that the peak of the d^{len} plot moves to the lower tangential height and its value decreases with the increase of H from 60 km to 80 km. Therefore, the d^{len} is inversely proportional to H; however the H dependency of the d^{len} is not as prominent as its dependency on hmF2.
[20] Considering the d^{len} dependency on the signal frequency, ionospheric profile shape and raypath geometry, ray tracing calculation has been carried out to compute d^{len} for different geometrical and ionospheric conditions. By Chapman layers a broader variety of ionospheric conditions are created varying layer parameters H, NmF2 and hmF2. Functional dependencies have been studied separately for different parameters to develop correction formulas. Thus, the following formula has been obtained for the excess path length correction.
For foF_{2} ≤ 8 MHz
For 8 < foF_{2} ≤ 25 MHz
where foF2 ≈ 8.98 is the critical plasma frequency and measured in Hz when maximum ionization NmF2 is measured in m^{−3}. The d^{len} dependency on the TEC rate dTEC/dh^{T} has been modeled by a cosine function whose phase and amplitude are functions of foF2 and given by equations (9)–(11). Since tropospheric refraction becomes a dominating source of raypath bending in lower tangential height (<20 km tangential height), we limit the usage of the correction formula to 20 km < h^{T} < h^{Tml} (see equation (8)). The quantity h^{Tml} is the maximum tangential height limit at which dTEC/dh^{T} becomes equal to the maximum TEC rate (dTEC/dh^{T})_{max} defined by equation (12). The maximum TEC rate depends on foF2 and is measured in TECU/km units when foF2 is in MHz. The excess path length d^{len} is measured in meters by equations (8)–(12) when signal frequency f is measured in GHz, foF2 is in MHz, tangential height h^{T} is in kilometers and slant TEC is in TECU. The polynomial coefficients c_{1}–c_{7} are derived based on a nonlinear fit with ray tracing results in least square senses and given in Table 2. The residual error in the fitting procedure is found about 25% of ray tracing results on average.
Table 2. Correction Coefficients Are Dimensionless Numbers  Number 

c_{1}  −1.114e001 
c_{2}  6.440e002 
c_{3}  1.124e001 
c_{4}  9.548e004 
c_{5}  −6.296e001 
c_{6}  2.9985e005 
c_{7}  −3.414e002 
d_{1}  1.095e003 
d_{2}  −7.293e004 
d_{3}  3.619e003 
d_{4}  1.384e+003 
d_{5}  −5.7e+001 
d_{6}  3.541 
[21] Equation (8) shows that the d^{len} is proportional to the square power of the slant TEC and inversely proportional to the quartic power of the signal frequency. The dependencies on foF2 and dTEC/dh^{T} have been described by equations (8)–(12). If the above ionospheric parameters and coefficients in Table 2 are known, the excess path length for occultation signals can be determined using the proposed correction formula.
[22] The LEO orbit is kept constant at 450 km for d^{len} computation by the ray tracing. Therefore, the estimated coefficients c_{1}–c_{7} (see Table 2) are only valid for occultation raypaths with LEO orbit 450 km. However, LEO orbits are not the same for different missions and even the orbit for the same LEO reduces with time due to atmospheric drag (if not uplifted by external means). The CHAMP orbit was 454 km when it was launched and decreased to below 300 km before the CHAMP mission was ended on 19 September 2010. GRACE was launched initially into an orbit of 500 km and decreased to 460 km by the end of 2010. The six FORMOSAT3/COSMIC spacecrafts are orbiting at different heights ranging from 746 to 845 km (information based on 22 February 2011). For a higher or lower LEO orbit instead of a 450 km orbit, the d^{len} computation by the ray tracing will be different and therefore the estimated coefficients c_{1}–c_{7} will be different. However, our limited investigation shows that d^{len} relationships with TEC and f will not be changed for higher or lower LEO orbits. The same technique can be applied to calculate a new set of coefficients for another LEO orbit height.
[23] To assess the performance of the correction formula the d^{len} has been computed for occultation raypaths by the ray tracing program and also by the correction formula and plotted in Figures 4a–4c. For this the GPS L2 signal has been traced for singlelayered Chapman profiles with H = 70 km and different hmF2 values of 250, 350 and 450 km.
[24] In order to utilize the correction formula we need to know the parameters NmF2, slant TEC and corresponding tangential height to calculate dTEC/dh^{T}. The NmF2 is known for Chapman profiles. The unknown slant TEC and corresponding h^{T} are considered as known and therefore, taken from the ray tracing results. However, in practical cases TEC and TEC rate can be estimated using dualfrequency measurements, and NmF2 can be obtained from available ionospheric electron density models such as International Reference Ionosphere [Bilitza, 2001] and NeQuick [Nava et al., 2008]. In fact occultation measurements contain NmF2 information which can be obtained by a standard reconstruction technique [e.g., Jakowski et al., 2002].
[25] The differences between the ray tracing results and the correction results have been computed and defined as the correction error (ray tracing result  correction result). Then the maximum absolute correction error and the percentage error (correction error/ray tracing result) × 100% have been determined.
[26] The maximum estimate of ray tracing results is found about 2.6, 2.0 and 1.2 m, for Chapman profiles with hmF2 values of 250, 350 and 450 km, respectively. The maximum correction errors are computed about 0.6, 0.4 and 0.2 m and the root mean squared (RMS) percentage errors are about 26%, 19% and 28% for hmF2 values of 250, 350 and 450 km, respectively. We have found that for Chapman based ionospheric profiles on an average about 70–80% excess path error can be corrected by the proposed correction (equations (8)–(12)). Our investigation shows that if foF2 is wrong by 20%, the RMS percentage error is increased to about 45%.
[27] In the previous section we have seen that the bending effects such as d^{len}, maximum raypath deviation, deviation at the tangential height, ΔTEC_{bend} are all correlated. The d^{len} versus ΔTEC_{bend} plots drawn in Figures 4d–4f also prove this. Therefore, there is a possibility to derive approximation formulas for other effects from d^{len}. We have found that the ΔTEC_{bend} can be expressed in terms of d^{len} and signal frequency as
where ΔTEC_{bend} is measured in TECU, f is measured in GHz and d^{len} is in meters. The polynomial coefficients d_{1}–d_{6} are derived based on a nonlinear fit with ray tracing results in least square senses and given in Table 2. As already given in the Table 1, ΔTEC_{bend} is proportional to the inverse square power of the frequency whereas d^{len} is proportional to the inverse quartic power of the frequency. Due to this reason, equation (13) shows frequency dependency of ΔTEC_{bend} in addition to the d^{len} dependency.
[28] To assess the performance of the d^{len} and ΔTEC_{bend} correction formulas for non Chapman profiles, we have computed correction errors (see Table 3) for GPS L1 and L2 signals using case 1 and case 2 profiles. For CHAMP profiles NmF2 is known and the unknown parameters TEC, h^{T} and thus dTEC/dh^{T} are considered as known and taken from the ray tracing results. As already mentioned, in practical cases TEC and TEC rate can be estimated using dualfrequency GNSS measurements and NmF2 can be obtained either from standard ionospheric models or by processing occultation measurements which contain NmF2 information. The ray tracing and correction results for the L2 signal have been plotted in Figure 5.
Table 3. Raypath Bending Errors With and Without Correction^{a}  Case 1  Case 2 

L1  L2  L1  L2 


d^{len}     
Maximum ray tracing result (m)  0.5  1.4  1.0  2.7 
Maximum correction error (m)  0.2  0.5  0.2  0.6 
RMS correction error (%)  33  34  19  19 
ΔTEC_{bend}     
Maximum ray tracing result (TECU)  5.8  9.5  11.1  17.9 
Maximum correction error (TECU)  2.3  4.1  2.8  4.9 
RMS correction error (%)  32  35  19  20 
[29] In case of d^{len}, the maximum correction error for the L2 signal is found about 0.5 and 0.6 m for case 1 and case 2, respectively, whereas without correction the maximum estimate of d^{len} (i.e., ray tracing results) is about 1.4 and 2.7 m, respectively. The RMS percentage error is found about 34% and 19%, respectively. In case of ΔTEC_{bend} computation, the maximum correction error is found about 4.1 and 4.9 TECU for case 1 and case 2, respectively, whereas without correction the maximum estimate is about 9.5 and 17.9 TECU, respectively. The RMS percentage error is found about 35% and 20%, respectively. We have found that on an average about 65–80% errors can be corrected by the proposed d^{len} and ΔTEC_{bend} correction (equations (8)–(13)).
[30] We see that the correction performance for case 1 is not as good as case 2. The case 1 profile gives a high vertical TEC of 216 TECU and we find the scale height H = 98 km for Chapman layer approximation VTEC ≈ 4.13HN_{m}F2 [Hoque and Jakowski, 2008]. Studies by different authors [Budden, 1985; Kelley, 1989; Davies, 1990; Norman, 2003; Stankov and Jakowski, 2006] show that Chapman layers with 60–80 km scale height can well describe typical ionospheric F2 layer conditions. The case 1 profile scale height is very high compared to the typical representative scale height. This indicates that the case 1 profile deviates much from a Chapman layer approximation. Therefore, the correction formula based on a single layer Chapman approximation has not performed well. However, still about 65% bending errors are corrected.
5. HigherOrder Correction of TEC Estimation
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[31] The dualfrequency TEC_{LOS} expression can be written in terms of carrierphase difference and higherorder ionospheric terms as [Hoque and Jakowski, 2010]
in which
where K = 80.6 m^{3}s^{−2}, Φ_{1} and Φ_{2} are the carrier phases of L1 and L2, respectively, ΔTEC_{2nd} is the secondorder term due to Earth's magnetic field, and ΔTEC_{len} and ΔTEC_{bend12} are the higherorder terms due to excess path and TEC difference along L1 and L2 paths. The quantities q is given by equation (4), ΔTEC_{bend1} and ΔTEC_{bend2} are the excess TEC due to bending in addition to the LOS TEC (see equation (3)), and d_{1}^{len} and d_{2}^{len} are the excess path for L1 and L2 signals, respectively.
[32] Equations (15) and (17) show that the proposed d^{len} and ΔTEC_{bend} correction formulas (8)–(13) can be used for computing higherorder terms in dualfrequency TEC estimation. To assess their performance, we have computed ΔTEC_{len} and ΔTEC_{bend12} by the ray tracing program and also by the correction formulas for case 1 and case 2 and the results are plotted in Figures 6a–6d.
[33] In case of ΔTEC_{len}, the maximum correction error (ray tracing result – correction result) is found about 3.3 and 5 TECU for case 1 and case 2, respectively, whereas without correction the maximum estimate of ΔTEC_{len} (i.e., ray tracing results) is about 8.3 and 16.5 TECU, respectively. The RMS percentage error is found about 34% and 20%, respectively. In case of ΔTEC_{bend12} computation, the maximum correction error is found about 6.7 and 10.2 TECU for case 1 and case 2, respectively, whereas without correction the maximum estimate is about 15.2 and 28.4 TECU, respectively. The RMS percentage error is found about 37% and 20%, respectively. We have found that on an average about 60–80% errors can be corrected by the proposed correction. The correction formula overestimates ray tracing results for d^{len} and ΔTEC_{bend} computation in Figure 5 and as well as ΔTEC_{len} and ΔTEC_{bend12} computation in Figure 6. Since ΔTEC_{bend12} and ΔTEC_{len} are opposite in sign in equation (14) and the magnitude of ΔTEC_{2nd} is very small (<1 TECU) compared to the magnitude of other terms, the total higherorder contribution is less than the magnitude of ΔTEC_{bend12} alone and thus less than 1% of the firstorder TEC_{LOS} which is about 1207 and 1277 TECU for case 1 and case 2, respectively. Although we have only discussed correction at GPS frequencies, the correction formulas are equally valid for the new upcoming Galileo frequencies.
[34] Our approach requires the external information of NmF2 to run the correction calculation. However, occultation measurements contain this information and can be obtained by a standard reconstruction technique. Alternatively, we can assimilate the occultation information in a global ionospheric model and compute the bending effect from the initial background model or assimilated electron densities. Although the later approach may give more consistent result, the ray tracing calculation for numerous occultation raypaths will be computationally expensive. Additionally the results will be affected by the topside electron densities of the used background model since occultation measurements give only bottom side information.
[35] It should be noted that the correction formula is derived based on a single Chapman layer assumption ignoring horizontal gradients of the ionosphere. The used profiles have no features like sporadic E which could lead to a kind of waveguide for electromagnetic signals to stay in that layer rather than continue its bending path. For such nonstandard cases the correction formula is neither derived nor tested.
6. Conclusions
 Top of page
 Abstract
 1. Introduction
 2. HigherOrder Refraction Terms
 3. Estimate of HigherOrder Refraction Terms
 4. Raypath Bending Correction
 5. HigherOrder Correction of TEC Estimation
 6. Conclusions
 References
 Supporting Information
[36] Simulation studies have been done to determine the raypath bending error of GNSS occultation signals as functions of the signal frequency f, the raypath geometry and key ionospheric parameters such as the maximum ionization NmF2 and the total electron content TEC. Based on simulation studies we have derived correction formulas for computing the excess path length and TEC difference between the signal and LOS paths as functions of f, NmF2, TEC and tangential height of the raypath. If these parameters are known, the proposed correction method is able to correct bending errors of GNSS occultation signals by about 65–80% on average.
[37] In practical cases, the actual TEC and differential change of TEC with respect to the tangential height can be estimated using dualfrequency GNSS measurements. Although the maximum ionization NmF2 is not immediately known, it can be obtained by processing occultation measurements using a reconstruction technique.
[38] In our previous work [Hoque and Jakowski, 2010] we found higherorder ionospheric errors in the GPS dualfrequency range estimation and TEC estimation using occultation signals to exceed the meter and 10 TEC units level, respectively. In the present work, we have found that higherorder TEC errors can be reduced significantly applying the proposed correction formulas. Similarly, improved range estimations can be obtained by applying the bending correction without tracing rays in the ionosphere.
[39] It should be noted that we have used single Chapman and CHAMPderived profiles to trace rays. The horizontal gradient of the ionosphere is ignored although the rays cover many degrees of latitude and longitude between the transmitter and the receiver. This may overestimate/underestimate the raypath bending estimation and also the correction.