Ionospheric bending correction for GNSS radio occultation signals

Transcription

1 RADIO SCIENCE, VOL. 46,, doi:10.109/010rs004583, 011 Ionospheric bending correction for GNSS radio occultation signals M. M. Hoque 1 and N. Jakowski 1 Received 30 November 010; revised 1 April 011; accepted 8 April 011; published 6 July 011. [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 higher order nonlinear refraction effects on GNSS signals which cannot be fully removed through a linear combination of dual frequency observables. In this paper, higher order 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 GPS CHAMP 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. Citation: Hoque, M. M., and N. Jakowski (011), Ionospheric bending correction for GNSS radio occultation signals, Radio Sci., 46,, doi:10.109/010rs Introduction [] 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 first order effect can be eliminated through a linear combination of dualfrequency observables. However, inhomogeneous plasma distribution and anisotropy cause higher order nonlinear effects which are not removed in this approach. Mainly the 1 Institute of Communications and Navigation, German Aerospace Center, Neustrelitz, Germany. Copyright 011 by the American Geophysical Union /11/010RS second and third order 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 higher order ionospheric effects and developing correction for them. Since then higher order 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, 00; Kedar et al., 003; Fritsche et al., 005; Hawarey et al., 005; Hoque and Jakowski, 006, 007, 008; Hernández Pajares et al., 007; Datta Barua et al., 008; Morton et al., 009]. They found that higher order ionospheric terms are less than 1% of the first order term at GPS frequencies; however, they represent millimeter/centimeter level errors in geodetic measurements. Hernández Pajares et al. [007] found submillimeter level shifting in receiver positions along southward direction for low latitude receivers and northward direction for high latitude receivers applying the second order term correction. Fritsche et al. [005] found centimeter level correction in GPS positions considering higher order ionospheric terms. Recently Petrie et al. [010] investigated the potential effects of the bending terms described by Hoque and Jakowski [008] on global GPS network. They found the bending correction for the dual frequency linear L1 L combination to exceed the 3 mm level in the equatorial region. All these studies were conducted to compute higher order ionospheric effects on GNSS signals for ground based reception. [5] Recently Hoque and Jakowski [010] investigated ionospheric impact on GPS occultation signals received onboard Low Earth Orbiting (LEO) CHAMP (Challenging 1of9

2 Minisatellite Payload) satellite. LEO satellites have the opportunity to receive signals from occulting GNSS satellites using onboard limb pointing 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), SAC C (Satelite de Aplicaciones Cientificas C), CHAMP, GRACE (Gravity Recovery And Climate Experiment) and COSMIC satellite network (Constellation Observing System for Meteorology, Ionosphere and Climate, also known as FORMOSAT 3) 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 dual frequency 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 [010] 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, 010] for selected GPS CHAMP 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.7 m, and the second and third order ionospheric terms to be about 13 cm and.1 cm, respectively, for the GPS L signal for an electron density profile with vertical TEC of about 167 TEC units (1 TEC unit = electrons/m ). We found the separation between the GPS L1 and L raypaths to exceed the kilometer level and errors in the GPS dual frequency 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.. Higher Order Refraction Terms [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 n ¼ 1 f p f f p f g cos Q f 3 f " # p fp 4f 4 þ f g 1 þ cos Q where f p = n e e /(4p " 0 m) and f g = eb/(pm) 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, " 0 is the permittivity of the free space, Q 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 p ¼ 40:3 Z path Z t ¼ 437 Z d I ¼ ð1 nþds ¼ p f þ q f 3 þ t 3f 4 ð1þ ðþ n e ds ¼ 40:3TEC ¼ 40:3ðTEC LoS þ DTEC bend Þ ð3þ q ¼ : Z n e ds þ 4:74 10 Z n e B cos Qds ð4þ n e B 1 þ cos Q ds ð5þ where ds is the raypath element, the quantities p/f, q/(f 3 ) and t/(3f 4 ) are the first, second and third order ionospheric phase delays, respectively. The integral R n e ds path 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 DTEC 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 Z d len ¼ ds where r is the geometric distance between the transmitter and the receiver. 3. Estimate of Higher Order Refraction Terms [10] We have used a two dimensional ray tracing program [Hoque and Jakowski, 008] 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, 000] has been used. ð6þ of9

3 Figure 1. Ionospheric refraction effects on GPS L signal during occultation. [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. [00] and Jakowski [005]. 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. The case 1 profile has a maximum ionization NmF of m 3 at an altitude hmf of 417 km (see Figure 1a, plotted until 1000 km altitude) at the geographic latitude 5.7 S, longitude 16 W and 15.1 h local time (LT). The case profile has a maximum ionization NmF of m 3 at hmf = 313 km (see Figure 1b, plotted until 1000 km altitude) at 7.6 N,. W and 16.4 LT. 3of9

4 The corresponding vertical TEC up to GPS height is estimated as about 16 and 167 TEC units, respectively. The actual GPS and CHAMP positions over phase connected arcs of the specific case 1 and case events have been used to trace signals for GPS CHAMP 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. [1] Figures 1a and 1b show electron density profiles used for case 1 and case, 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 DTEC 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 third order 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 higher order terms with the TEC and TEC rate dtec/dh T, we found that bending related effects such as the d len, DTEC bend, and raypath deviations are more sensitive to TEC rate changes than to the absolute TEC level. However, the second and third order 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, 010]. The variation of different higher order terms had been plotted as a function of the measurement time and described in that paper. [14] The maximum estimates of different higher order terms found for case 1 and case have been summarized in Table 1 for the GPS L signal. Their dependency on the signal frequency is also given in the Table 1. [15] We found that bending effects such as DTEC 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 [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. n e ðhþ ¼ NmF expð0:5ð1 z expðzþþþ ð7þ where NmF is the maximum ionization and z =(h hmf)/h in which hmf 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 hmf = 50, 350 and 450 km. The signal frequency f = 17.6 MHz, parameters H =70km and NmF = m 3 are kept constant in each case. The total electron content in the vertical direction will be the same since VTEC 4.13HNmF 143 TEC units [Hoque and Jakowski, 008]. The obtained excess path length d len, TEC and dtec/dh T have been plotted as a function of tangential height in Figures a c, respectively. [18] Figure a shows that the peak of the d len plot moves to the higher tangential height and its value decreases significantly with the increase of hmf from 50 km to 450 km. Therefore, the magnitude of the d len depends on the magnitude of hmf 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 NmF and hmf are kept constant as 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 hmf. [0] 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, NmF and hmf. 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. d len ¼ TEC 0ðdTEC=dhT ÞB f 4 A 1 cos 0km < h T < h Tml 400 For fof 8 MHz For 8 < fof 5 MHz B ¼ c 1 þ c fof þ c 3 fof ð8þ ð9þ A ¼ c 4 c 5 expðfof=1:1881þ ð10þ A ¼ c 6 c 7 expðfof=:56þ ð11þ dtec dh T ¼0:031 þ 0:007foF 0:016foF ð1þ max pffiffiffiffiffiffiffiffiffiffiffiffi where fof 8.98 NmF is the critical plasma frequency and measured in Hz when maximum ionization NmF is 4of9

5 Table 1. Maximum Estimate of Higher Order Propagation Effects for Case 1 and Case Maximum Propagation Effects Case 1, L Case, L Frequency Dependency Excess path length (m) /f 4 Second order delay (m) /f 3 Third order delay (m) /f 4 DTEC bend (TECU) /f Maximum raypath /f deviation (km) Deviation at tangential height (km) /f 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 fof and given by equations (9) (11). Since tropospheric refraction becomes a dominating source of raypath bending in lower tangential height (<0 km tangential height), we limit the usage of the correction formula to 0 km < h T < h Tml (see equation (8)). The quantity h Tml is the maximum tangential height limit at Figure 3. Variations of excess path length with tangential height for different scale heights H = 60, 70, and 80 km. which dtec/dh T becomes equal to the maximum TEC rate (dtec/dh T ) max defined by equation (1). The maximum TEC rate depends on fof and is measured in TECU/km units when fof is in MHz. The excess path length d len is measured in meters by equations (8) (1) when signal frequency f is measured in GHz, fof 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. The residual error in the fitting procedure is found about 5% of ray tracing results on average. [1] 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 fof and dtec/dh T have been described by equations (8) (1). If the above ionospheric parameters and coefficients in Table are known, the excess path length for occultation signals can be determined using the proposed correction formula. [] 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 ) 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 010. Table. Correction Coefficients Are Dimensionless Numbers Number Figure. Variations of (a) excess path length, (b) TEC, and (c) dtec/dh T with tangential height. Dotted vertical lines show the occurrences of the highest excess paths and corresponding TEC and dtec/dh T e e e e e e e e e e e e+001 d c 1 c c 3 c 4 c 5 c 6 c 7 d 1 d d 3 d 4 d 5 5of9

6 GRACE was launched initially into an orbit of 500 km and decreased to 460 km by the end of 010. The six FORMOSAT 3/COSMIC spacecrafts are orbiting at different heights ranging from 746 to 845 km (information based on February 011). 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. [3] 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 L signal has been traced for single layered Chapman profiles with H = 70 km and different hmf values of 50, 350 and 450 km. [4] In order to utilize the correction formula we need to know the parameters NmF, slant TEC and corresponding tangential height to calculate dtec/dh T. The NmF 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 dual frequency measurements, and NmF can be obtained from available ionospheric electron density models such as International Reference Ionosphere [Bilitza, 001] and NeQuick [Nava et al., 008]. In fact occultation measurements contain NmF information which can be obtained by a standard reconstruction technique [e.g., Jakowski et al., 00]. [5] 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. [6] The maximum estimate of ray tracing results is found about.6,.0 and 1. m, for Chapman profiles with hmf values of 50, 350 and 450 km, respectively. The maximum correction errors are computed about 0.6, 0.4 and 0. m and the root mean squared (RMS) percentage errors are about 6%, 19% and 8% for hmf values of 50, 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) (1)). Our investigation shows that if fof is wrong by 0%, the RMS percentage error is increased to about 45%. [7] In the previous section we have seen that the bending effects such as d len, maximum raypath deviation, deviation at the tangential height, DTEC bend are all correlated. The d len versus DTEC 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 DTEC bend can be expressed in terms of d len and signal frequency as DTEC bend ¼ d 1 þ d f þ d 3 f 1 þ d4 d len þ d 5 d len þd6 d len 3 ð13þ where DTEC 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. As already given in the Table 1, DTEC 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 DTEC bend in addition to the d len dependency. [8] To assess the performance of the d len and DTEC bend correction formulas for non Chapman profiles, we have computed correction errors (see Table 3) for GPS L1 and L signals using case 1 and case profiles. For CHAMP profiles NmF 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 dual frequency GNSS measurements and NmF can be obtained either from standard ionospheric models or by processing occultation measurements which contain NmF information. The ray tracing and correction results for the L signal have been plotted in Figure 5. [9] In case of d len, the maximum correction error for the L signal is found about 0.5 and 0.6 m for case 1 and case, respectively, whereas without correction the maximum estimate of d len (i.e., ray tracing results) is about 1.4 and.7 m, respectively. The RMS percentage error is found about 34% and 19%, respectively. In case of DTEC bend computation, the maximum correction error is found about 4.1 and 4.9 TECU for case 1 and case, 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 0%, respectively. We have found that on an average about 65 80% errors can be corrected by the proposed d len and DTEC bend correction (equations (8) (13)). [30] We see that the correction performance for case 1 is not as good as case. The case 1 profile gives a high vertical TEC of 16 TECU and we find the scale height H =98km for Chapman layer approximation VTEC 4.13HN m F [Hoque and Jakowski, 008]. Studies by different authors [Budden, 1985; Kelley, 1989; Davies, 1990; Norman, 003; Stankov and Jakowski, 006] show that Chapman layers with km scale height can well describe typical ionospheric F 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. Higher Order Correction of TEC Estimation [31] The dual frequency TEC LOS expression can be written in terms of carrier phase difference and higher order ionospheric terms as [Hoque and Jakowski, 010] TEC LoS ¼ f 1 f ð F 1 F Þ K f1 f DTEC bend1 DTEC nd þ DTEC len ð14þ 6of9

7 Figure 4. (a c) Comparison between excess path lengths computed by the ray tracing and by the correction (equations (8) (1)) for different hmf values. (d f) The relationship between the DTEC bend and d len. in which DTEC bend1 ¼ f 1 DTEC bend f DTEC bend1 f1 ð15þ DTEC nd ¼ q f1 1f þ f K f 1 f ðf 1 þ f Þ ð16þ DTEC len ¼ K d1 len f1 f ð17þ f1 f d len where K = 80.6 m 3 s, F 1 and F are the carrier phases of L1 and L, respectively, DTEC nd is the second order term due to Earth s magnetic field, and DTEC len and DTEC bend1 are the higher order terms due to excess path and TEC difference along L1 and L paths. The quantities q is given by equation (4), DTEC bend1 and DTEC bend are the excess TEC due to bending in addition to the LOS TEC (see equation (3)), and d 1 len and d len are the excess path for L1 and L signals, respectively. [3] Equations (15) and (17) show that the proposed d len and DTEC bend correction formulas (8) (13) can be used for computing higher order terms in dual frequency TEC estimation. To assess their performance, we have computed DTEC len and DTEC bend1 by the ray tracing program and also by the correction formulas for case 1 and case and the results are plotted in Figures 6a 6d. [33] In case of DTEC len, the maximum correction error (ray tracing result correction result) is found about 3.3 and 5 TECU for case 1 and case, respectively, whereas without correction the maximum estimate of DTEC len (i.e., ray tracing results) is about 8.3 and 16.5 TECU, respectively. The RMS percentage error is found about 34% and 0%, respectively. In case of DTEC bend1 computation, the maximum correction error is found about 6.7 and 10. TECU Table 3. Raypath Bending Errors With and Without Correction a Case 1 Case L1 L L1 L d len Maximum ray tracing result (m) Maximum correction error (m) RMS correction error (%) DTEC bend Maximum ray tracing result (TECU) Maximum correction error (TECU) RMS correction error (%) a That is, ray tracing results. 7of9

8 Figure 5. (a and b) Comparison between excess path lengths computed by the ray tracing and correction formula for case 1 and case. (c and d) The corresponding DTEC bend computation. for case 1 and case, respectively, whereas without correction the maximum estimate is about 15. and 8.4 TECU, respectively. The RMS percentage error is found about 37% and 0%, 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 DTEC bend computation in Figure 5 and as well as DTEC len and DTEC bend1 computation in Figure 6. Since DTEC bend1 and DTEC len are opposite in sign in equation (14) and the magnitude of DTEC nd is very small (<1 TECU) compared to the magnitude of other terms, the total higher order contribution is less than the magnitude of DTEC bend1 alone and thus less than 1% of the first order TEC LOS which is about 107 and 177 TECU for case 1 and case, 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 NmF to run the correction calculation. However, occultation measurements contain this information and can be Figure 6. Comparison between higher order TEC terms (see equation (14)) computed by the ray tracing and correction formula for (a and c) case 1 and (b and d) case. 8of9

9 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 [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 NmF 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, NmF, 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 dual frequency GNSS measurements. Although the maximum ionization NmF is not immediately known, it can be obtained by processing occultation measurements using a reconstruction technique. [38] In our previous work [Hoque and Jakowski, 010] we found higher order 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 higher order 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 CHAMP derived 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. References Bassiri, S., and G. A. 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