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 Precise navigation and positioning using GPS/GLONASS/Galileo require the ionospheric propagation errors to be accurately determined and corrected for. Current dual-frequency method of ionospheric correction ignores higher order ionospheric errors such as the second and third order ionospheric terms in the refractive index formula and errors due to bending of the signal. The total electron content (TEC) is assumed to be same at two GPS frequencies. All these assumptions lead to erroneous estimations and corrections of the ionospheric errors. In this paper a rigorous treatment of these problems is presented. Different approximation formulas have been proposed to correct errors due to excess path length in addition to the free space path length, TEC difference at two GNSS frequencies, and third-order ionospheric term. The GPS dual-frequency residual range errors can be corrected within millimeter level accuracy using the proposed correction formulas.
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 The ionosphere is a significant error source of the GNSS error budget. Ionosphere-induced range errors vary from a few meters to tens of meters at the zenith [Klobuchar, 1996]. Since the ionosphere is a dispersive medium, the most part of the ionospheric delay can be eliminated through linear combinations of dual-frequency observables; therefore, only the higher order ionospheric errors will remain in the range estimation. However these errors can reach several tens of centimeters at low elevation angles and during extreme space weather conditions (i.e., ionospheric storms), which represent large errors in geodetic measurements [Klobuchar, 1996]. Satellite positions can also be improved to the centimeter level when higher order ionospheric corrections are applied [Fritsche et al., 2005].
 Early work is done by Brunner and Gu  in developing correction models for higher order ionospheric errors, especially to correct the second-order term and raypath bending error. Their approach requires the knowledge of the actual ionosphere and the average value of the longitudinal component of the Earth's magnetic field along raypaths. In practical cases, these parameters are not easy to estimate accurately.
 Later Bassiri and Hajj  have done similar work assuming an Earth-centered tilted dipole approximation for the geomagnetic field. They consider the ionosphere as a single (thin) layer at a certain altitude (300 km) and compute the magnetic field vector at the ionospheric pierce point (IPP). However investigation by Hawarey et al.  for Very Long Baseline Interferometry (VLBI) shows that further improvements for the second-order ionospheric term can be achieved using a more realistic magnetic field model such as the International Geomagnetic Reference Field (IGRF) instead of a dipole model.
Strangeways and Ioannides  presented a method for the range correction to less than 1 cm accuracy by employing three factors: one representing the ratio of the slant TECs experienced by the two frequencies and the other two representing the ratio of their geometric path lengths with respect to the true range. However, these parameters are not easy to estimate accurately in practical cases.
 In our previous work [Hoque and Jakowski, 2006, 2007], we proposed models for the second-order ionospheric error correction as a function of geographic latitude and longitude and geometrical parameters such as elevation and azimuth angles for GNSS users in Germany and Europe (latitude range 30–65°N, longitude range 15°W–45°E).
 In the present work, we have derived approximation formulas to estimate and correct dual-frequency range errors due to excess path length of the signal in addition to the free space path length, TEC difference at two GNSS frequencies, and third-order ionospheric term.
where the index i = 1, 2,…, refers to the GNSS carrier frequencies (in case of GPS, f1 = 1575.42 MHz and f2 = 1227.6 MHz), ρ is the geometric distance between the transmitter and the receiver also known as the true range, ne is the electron concentration, ds is the raypath element, Θ is the angle between the ray direction and the magnetic field vector B, and B is the magnitude of B. The integral ∫neds along the raypath is the total electron content (TEC) often measured in TEC units (1 TECU = 1016 electrons/m2). The additional term dib(length) at the right-hand side of equations (1) and (2) corresponds to the excess path length of the signal in addition to the free space path length.
 Due to the dispersive nature of the ionosphere, GNSS signals travel along different raypaths through the ionosphere and TEC along f1 path will be different from that along f2 path and from that along the straight line of sight (LoS). Considering this, TEC in equation (3) is separated into TECLoS, which is along the straight LoS, and ΔTECi, which is the difference between TEC along the LoS and the raypath.
 The first-order (1/f2) ionospheric error can be eliminated by constructing a linear combination of dual-frequency observables. In case of GPS, this approach is known as ionosphere-free linear L3 combination. However the second (1/f3)- and third-order (1/f4) ionospheric terms, the error due to TEC difference along f1 and f2 paths, and the excess path length dib(length) will not be canceled out in this approach. Therefore, dual-frequency (f1 and f2) code and phase combinations can be written as (using equations (1) and (2))
where TEC1 and TEC2 are the total electron content along f1 and f2 paths, and these TEC values are larger than TECLoS by amounts of ΔTEC1 and ΔTEC2, respectively, the terms Δs2 and Δs3 are referred to as the dual-frequency second and third-order residual error, respectively, Δsb(length) and Δsb(TEC) are referred to as the dual-frequency residual errors due to excess path length and TEC difference (along f1 and f2 paths), respectively. Adding all these terms, (RRE)gr and RRE are obtained, which correspond to the dual-frequency total residual range error for code and phase combinations, respectively (see equations (6) and (7)). The only way to investigate all these errors is to trace rays through the ionosphere.
3. Ray Tracing Through the Ionosphere
 A two-dimensional ray tracing program is developed to trace rays through a user specified model of the ionosphere. We assume models for the electron density ne and the Earth's magnetic field B. The electron density at an ionospheric height h can be well described by a Chapman layer and written as [Rishbeth and Garriott, 1969]
where z = (h − h0)/H, H is the atmospheric scale height, χ is the sun's zenith angle, and N0 is the peak electron density at the altitude h0 at zenith (χ = 0). It can be seen that ne(h) has a maximum value Nm = N0 at the altitude hm = h0 + H ln(sec χ) [Budden, 1985]. In our previous paper [Hoque and Jakowski, 2007], ne profiles are modeled as the sum of three Chapman layers describing electron densities of the ionospheric F2, F1, and E layers and a superposed exponential decay function for the plasmasphere. The layer parameters are chosen from wide ranges (for F2 layer hmF2 range 250–400 km, HF2 range 60–80 km; for F1 layer hmF1 range 180–220 km, HF1 range 40–60 km, and for E layer hmE range 100–120 km, HE range 5–15 km). The same ionosphere modeling is used here. For the Earth's magnetic field, the IGRF model [Mandea and Macmillan, 2000] is used to calculate B vectors along raypaths.
 However the limitations of modeling the ionosphere with Chapman layers may limit the accuracies obtained in the higher order ionospheric term estimation. Therefore, a large number (about 20,000) of ne profiles reconstructed from GPS radio occultation measurements onboard the CHAMP satellite (available at http://w3swaci.dlr.de/) are used to validate the results obtained by Chapman profiles. The radio occultation measurements onboard low Earth orbiting (LEO) satellites can provide vertical electron density profiles of the ionosphere from satellite orbit heights down to the bottomside. The retrieval algorithm uses an adaptive model for estimating the topside ionosphere and plasmasphere above the CHAMP orbit height [Jakowski et al., 2002; Jakowski, 2005].
 Now the refraction of the ray can be calculated in realistic ways assuming the ionosphere to be composed of a large number of thin spherical shells, in each of which the medium is homogeneous (see Figure 1). The refraction angle at an arbitrary jth layer can be determined by Bouguer's law (Snell's law in spherical surface) of refraction as [Croft and Hoogasian, 1968]
where δ0 is the zenith angle (π/2− elevation) at the Earth's surface where the refractive index n0 = 1, δj is the refraction angle at the jth layer (where nj ≠ 1), and r0 and rj are the radial distances (from the Earth's center) at the surface and at the jth layer, respectively. In general, for a radio wave of frequency f, the refractive index n is given by the Appleton–Hartree formula [Budden, 1985] as
in which X = nee2/(4π2ɛ0mf2), Y = eB/(2πmf) where e and m are the electron charge and mass, respectively, and ɛ0 is the permittivity of the free space.
 Regardless of the vertical distribution of the refractive index, the angle of refraction is given by equation (13). However, when the medium is anisotropic, Bouguer's law cannot readily be used. In this case, the unknown angle δj could be found from equation (13) if nj were known, but nj depends on the angle Θ between the wave normal and the geomagnetic vector B (see equation (14)), and this angle in turn depends on δj.
 Since the anisotropic plasma is doubly refracting (indicated by the ± sign in refractive index equation (14)) there are actually two waves. The wave with the upper (+) sign is usually called the ordinary wave, whereas the lower (−) sign is related to the extraordinary wave. The ordinary mode is left-hand circularly polarized, while the extra-ordinary mode is right-hand circularly polarized [Hartmann and Leitinger, 1984]. However, since GPS signals are transmitted in the right-hand polarization [Parkinson and Gilbert, 1983], only the results of extraordinary mode are considered here.
 To solve the refractive index equation in an anisotropic plasma, a two-dimensional Cartesian coordinate system (ζ, υ) with ζ horizontal and υ vertically upward (i.e., along the radius) axes is considered here. The introduction of such a coordinate system to solve this problem is already described by Budden  for the plane ionosphere. To take into account the effect of the Earth's curvature on ray propagation, this concept is used here for the spherically layered ionosphere.
 In the jth stratum, the wave normal makes an angle δj with the υ axis (see Figure 1). The refractive index is considered as a vector nj with magnitude nj and in the direction of the wave normal, and B field components are computed along ζ and υ axes. Taking dot products of nj and B, the angle Θ between them can be determined. In this approach, two unknowns (nj and Θ) in equation (14) can be replaced by one unknown, which results in a quartic equation known as Booker quartic [Booker, 1949]. For details of this approach, we refer to Budden, 1985.
 Ray tracing determines the raypath between a known transmitter and a receiver for a given frequency in a user specified ionosphere model. In general, ray tracing starts from specified angles of elevation and azimuth, so the ray may miss the target. This problem is known as homing-in problem. To solve this problem Nelder–Mead [Nelder and Mead, 1965] simplex algorithm is implemented in the ray tracing program. It has been found that the simplex method using Nelder–Mead algorithm works very reliably when solving the raypath homing-in problem [Strangeways, 2000; Strangeways and Ioannides, 2002].
 The ray tracing program is checked by comparing its results with the results given by Brunner and Gu . Brunner and Gu used a three-dimensional ray tracing technique based on differential equations [Haselgrove, 1963]. It has been found that excess path lengths estimated by the two methods do not differ more than one millimeter even at low elevation (e.g., 7.5°) and high vertical TEC values (455 TECU) at GPS frequencies. The ray tracing program is used to develop empirical formulas and also to verify analytical solutions of different higher order ionospheric corrections proposed in this paper.
4. Second Order Ionospheric Term
 The effect of the Earth's magnetic field on radio wave propagation, which is known as the magneto-ionic effect or second-order ionospheric effect, is investigated in our previous work [Hoque and Jakowski, 2006, 2007]. Empirical formulas are developed to estimate and correct the second-order ionospheric effect on GNSS users in Germany and Europe.
 Since tracing of the magneto-ionic interaction (between the ionosphere and the geomagnetic field) along raypaths is very cumbersome, a common practice is to assume the ionosphere as a single (thin) layer at a certain altitude and compute the magnetic field vector B at the IPP. Thus, the second-order error coefficient q (see equation (4)) can be written as
where B and Θ will be estimated at the IPP. Bassiri and Hajj  were the first to propose such a single layer thin ionosphere assumption for the second-order ionospheric correction; they choose a representative global average peak height as 300 km. Later Kedar et al.  assumes the ionosphere as a single layer at a 400-km altitude and estimates the effect of the second-order GPS ionospheric correction on receiver positions. Recently, Hernandez-Pajares et al.  considers the ionosphere as a single layer at a 450-km altitude to estimate impacts of second-order ionospheric term on geodetic estimates.
 However such assumption for the B cos Θ computation introduces errors in the second-order ionospheric term estimations. In the following, these errors will be investigated by estimating the GPS dual-frequency (L1–L2) residual error Δs2 by two ways: (1) computing B cos Θ along raypaths and (2) computing B cos Θ only at the IPP (400 km). Then, the differences between the two estimations (Δs2 by first method −Δs2 by second method) are derived at elevation 90° and 5° for azimuth 0° and 180° over the globe. These values are plotted in Figure 2. The ionosphere is modeled by a Chapman layer with a maximum ionization of about 4.96 × 1012 electrons/m3 at an altitude of 400 km and TEC in vertical direction (TECV) is estimated to be about 143 TECU.
 We see that at 180° azimuth (see Figure 2(a)), Δs2 is positive for the northern hemisphere and also for a large part of the southern hemisphere (at low and mid latitudes) and negative for the rest of the globe. The maximum Δs2 is estimated to be about 28.8 mm in the northern hemisphere at about 45°N latitude and 100°E longitude.
 At 0° azimuth (see Figure 2(c)), Δs2 is negative for the southern hemisphere and also for a large part of the northern hemisphere (at low latitudes) and positive for the rest of the globe. The maximum (absolute value) Δs2 is estimated to be about −28.5 mm in the southern hemisphere at about (35°S, 120°E).
 The Δs2 difference plots in Figure 2(b, d) show that large differences are occurred along the geomagnetic equator. In case of 180° azimuth, the maximum (absolute value) difference is found to be about −1.9 mm at about (10°N, 100°E) whereas in case of 0° azimuth, the maximum difference is found to be about 1.9 mm at about (5°N, 95°E).
Figure 2(e) shows that at zenith propagation (90° elevation), Δs2 values are directly proportional to the vertically downward component of the geomagnetic field, that is, these values are large at poles and vanish at equator [also see Hoque and Jakowski, 2006]. The difference plot (see Figure 2(f)) shows that Δs2 values are overestimated (i.e., negative) in the northern hemisphere and underestimated in the southern hemisphere by the second method. The large differences are observed in the poles and not in the equatorial regions. The percentage error (Δs2 difference/Δs2 by first method) × 100% is found to be about 10% at elevation 5° and about 5% at zenith on an average. Therefore, computing B cos Θ only at the IPP does not give the most accurate Δs2 estimation. Considering this, B cos Θ is computed along raypaths for Δs2 estimation in later sections.
5. Third-Order Ionospheric Term
 To correct the third-order ionospheric term, a correction formula based on analytical integration of Chapman function will be derived here. The third-order ionospheric phase error term can be separated from other higher order terms as (see equation (2))
 The first and second terms in the right-hand side of equation (16) are estimated separately by the ray tracing program for GPS L2 frequency at a receiver location with geographic latitude 48°N and longitude 15°E. Their values are plotted in Figure 3. In these polar plots, the third-order phase error components are plotted as radial distances from the center of the polar plot while (receiver-to-satellite) azimuth angle is varied from 0° to 360° for three elevations β = 1°, 45°, and 90°.
 Since the horizontal gradient of ne is ignored in simulation process, the first term of dI(3) is independent of azimuth (see Figure 3(a)), but the second term depends on azimuth (due to dependency on B) and has the maximum magnitude in the North-South direction. The second term of dI(3) is found to be less than the first term by about 1–2 orders of magnitude. Therefore, the second term will not be considered in determining correction formula for the third-order term. Thus, equation (16) can be simplified as
 Now using the expression ∫ne2ds ≈ 0.6577NmTEC, the dual-frequency third-order residual error (equation (10)) can be simplified as
 To check the accuracy of the correction formula (equation (21)), the residual error Δs3 is estimated by the ray tracing program and by the correction formula. These values are plotted with elevation angle in Figure 4.
Figure 4 shows that the differences between Δs3 estimated by the two methods are in the submillimeter level and can be ignored. If the ionosphere is modeled by adding a number of Chapman layers, for each layer the third-order phase error can be determined by the approximation formula and added together for the resultant error.
 As already mentioned, the limitation of modeling the ionosphere with Chapman layers limits the accuracies obtained in the higher order term estimation. However, ray tracing simulations done for CHAMP reconstructed profiles show that for such profiles the correction formula (equation (21)) can correct about 80–85% Δs3 error on an average.
6. Excess Path Length
 The difference in length between the curved path and the true range ρ is defined as the excess path length dIb(length), and it can be written as
 It has been found that the excess path length depends not only on the TEC value but also on the ionosphere profile shape [Jakowski et al., 1994]. In Figure 5, two ionosphere profiles (PROF 1 and PROF 2) are plotted, which have the same TEC value in vertical direction, but their shapes are different. At L2 frequency, the excess paths and TECs estimated by the ray tracing program using these two profiles are given in Table 1.
Table 1. Excess Path Length Error at L2 Frequency
Table 1 shows that, at L2 frequency and elevation 5°, the excess path lengths are about 27.5 mm for PROF 1 and about 7.6 mm for PROF 2. The corresponding TEC along raypaths are estimated to be about 322.1 and 257.1 TECU, respectively. The excess paths per square TEC (dIb(length)/TEC2) are estimated to be about 2.65 × 10−4 and 1.15 × 10−4 mm/TECU2, respectively, for the two profiles at 5° elevation. The difference in dIb(length)/TEC2 estimation indicates that the large variations in (slant) TEC measurement are not alone responsible for the large variations in the excess path length for the two profiles. The amount of curvature of the two profiles also influences the excess path length estimations. It has been found that the excess path is higher for a thin profile with large maximum ionization than for a thick profile with small maximum ionization.
 The ray tracing calculations have been carried out to compute dIb(length) for a large number of geometrical and ionospheric conditions. Functional dependencies have been studied separately for each parameter to develop the empirical formula. Thus, the following formula has been obtained.
The excess path dIb(length) will be measured in meters when TEC is in TEC units, frequency f in GHz, F2 layer scale height HF2 and peak ionization height hmF2 in kilometers, and elevation β is in radians. The dual-frequency residual error Δsb(length) can be estimated by equations (11) and (23).
 To check the accuracy of the correction formula (equation (23)), Δsb(length) is estimated for GPS L1–L2 signals in two ways: (1) by the ray tracing program using a large number of profiles (about 16200) and (2) by the correction formula (using equations (11) and (23)) knowing the TEC and the ionosphere parameters of F2 layer of those profiles. The profiles are generated in such a way that TECs in vertical direction are same (in this case 100 TECU) for all profiles but their shapes will be different. The detail of this procedure is described in Hoque and Jakowski . The differences between excess paths estimated by the two methods are then calculated. These values are termed as remaining errors (Δsb(length) by first method −Δsb(length) by second method) and their statistical estimates such as root mean squared (RMS) error and maximum (MAX) error are given in Table 2 together with ray tracing results.
 We see that at 1° elevation, the RMS remaining error is less than 2 mm for the combined L1–L2 signal. The maximum error in the correction formula is found to be about 5.7 mm, whereas the maximum excess path can be as big as about 22.8 mm without correction (see Table 2). It has been found that on an average about 85% error will be corrected using the proposed correction formula (equations (23) and (11)). At high elevations (>60°), the excess paths are found to be small (<1 mm) at GPS frequencies and can be ignored.
 For CHAMP profiles, it has been found that the proposed correction formula (equations (23) and (11)) can correct about 70% Δsb(length) error on an average. The F2 layer scale height HF2 in the correction formula (equation (23)) is calculated by equation (19) knowing elevation, TEC, hmF2 and NmF2 values. The parameters hmF2 and NmF2 are known from the radio occultation data.
7. Range Error Due to TEC Difference at Two Frequencies
 As already mentioned, due to the dispersive nature of the ionosphere, two radio signals do not follow the same curved path. The traversed curved path through the ionosphere by a signal using Bouger's law of refraction can be written as [Chen et al., 1990]
where r0 is the radial distance (from the Earth's center) of the receiver, β0 is the elevation angle at the receiver, r and n are the radial distance and refractive index, respectively, at any point on the curved path, and dr is the raypath element. Equation (24) can be extended binomially as (detail is given in Appendix C)
where K = 40.3 and a = r0 cos β0. If the curved path is known, TEC can be estimated along that path multiplying it by the electron concentration ne as
We see that the first term on the right-hand side of equation (26) does not have frequency dependency. Therefore, for two different operating frequencies (e.g., f1, f2) this term will be equal and canceled out during difference method (see equation (8)). The third term is proportional to the inverse quartic power of frequency and its contribution to the TEC difference estimation is found to be negligible at L-band frequencies. Thus, the TEC difference can be estimated with sufficient accuracy by the second term of equation (26) at GNSS frequencies.
 Now to estimate the TEC difference by equation (27), we have to assume an ionosphere model for ne. It has been found that L1 and L2 paths are largely separated (e.g., about 60 m at 7.5° elevation and TECV = 138 TECU [Brunner and Gu, 1991]) at and near peak electron density heights. Their separations on other parts through the ionosphere are not significant. As a result, the differences in TEC measurement along L1 and L2 paths mostly come from the ionosphere region at and near peak electron density heights.
 Therefore, it will be worthy looking for a function which can represent ne at and near peak density heights very well and by which the integral equation (27) can be solved analytically. It has been found that a single quasi parabolic (QP) layer can be used for such a purpose.
 It should be mentioned here that the QP layer assumption is done only for deriving the TEC difference between two signal paths. The individual TEC measurement along L1 or L2 paths cannot be determined correctly in this approach. The vertical distribution of electron density given by a QP layer is [Chen et al., 1990]
in which A = −(Nm · rm2 · rb2) / ym2, B = 2Nm · rm · rb2/ym2, and C = Nm − Nm · rb2/ym2. The QP layer is only valid for rb < r < rt where rb = rm − ym, rt = rmrb / (rb − ym), r is the radial distance from the Earth's centre, rm is the radial distance from the Earth's centre to the height of maximum ionization, Nm is the value of maximum ionization, and ym is the layer semithickness.
Therefore, if the ionosphere parameters rm, Nm, and ym are known, the TEC difference can be estimated by equations (29) and (30). Thus, knowing the TEC difference, the residual error Δsb(TEC) can be determined by equation (8).
 From studies of ionospheric electron distribution [e.g., Rishbeth and Garriott, 1969; Davies, 1990], it is known that the QP layer can describe ne well at and near peak electron density height; but it cannot describe ne well at other heights. Although a Chapman layer can describe ne along height very well, the equation (27) cannot be solved analytically using a Chapman layer. For this, a single QP layer representation of ne is sought here.
 Now the parameters describing the QP layer will have to be determined carefully. It has been seen that, at the peak electron density height, the curvature of a Chapman layer with scale height H is about the same as that at the apex of a parabola of semithickness ym = 2H [Budden, 1985]. However, it has been found that, for the assumption ym = 2H + 15 (measured in km), the TEC differences (between L1 and L2 paths) estimated by the analytical integration method using a single layer QP function (equations (29) and (30)) have better convergence with those estimated by the ray tracing program using a Chapman layer. The maximum ionization and its height are considered to be same for both layers. Figure 6(a) shows a Chapman layer and the corresponding QP layer assumption. The residual error Δsb(TEC) is estimated by the analytical integration method using the QP layer and by the ray tracing program using the Chapman layer. These values are plotted in Figure 6(b).
 It has been found that at low elevation angles, Δsb(TEC) can be as big as about 4 cm during typical high ionosphere conditions (TECV = 143 TECU). The differences between Δsb(TEC) estimated by two methods are found to be less than 1 mm.
 However, the analytical solution for determining TEC difference is based on a single Chapman layer. But the actual ionosphere cannot always be described by a single Chapman layer; rather a number of layers are needed for this. Considering this, based on extended simulation studies using a large variety of ionosphere profiles, another correction formula has been developed to determine the excess TEC in addition to the LoS TEC at a given frequency.
where ΔTECi will be measured in electrons/m2 when F2 layer scale height HF2 and peak ionization height hmF2 are in kilometers, frequency fi is in Hertz, (slant) TEC is in electrons/m2, and elevation β is in radians. Now, the residual error Δsb(TEC) can be estimated in meter by equation (8) in conjunction with equation (31).
 The residual error Δsb(TEC) is estimated by the ray tracing program and by the correction formula (equations (8) and (31)) for a large number of ionosphere profiles (about 3600) having the same TECV = 100 TECU. Then their differences (Δsb(TEC) by ray tracing −Δsb(TEC) by correction formula) are estimated and defined as remaining errors. The RMS and maximum (MAX) value of remaining errors are determined and given in Table 3 together with ray tracing results for selected elevation angles.
Table 3. Statistical Estimates of Δsb(TEC)
RMS of Remaining Error (mm)
MAX of Remaining Error (mm)
RMS Δsb(TEC) by Ray Tracing (mm)
MAX Δsb(TEC) by Ray Tracing (mm)
 We have found that, at 1° elevation angle and TECV = 100 TECU, the ray tracing Δsb(TEC) varies from 12 to 46 mm; this large variation in Δsb(TEC) is due to the variation in profile shapes. At 5° elevation angle, the RMS and maximum remaining errors are estimated to be about 3 and 10 mm, respectively. The corresponding RMS and maximum value in the ray tracing results are estimated to be about 22 and 40 mm, respectively.
 It has been found that, although the ionosphere profiles are modeled by four layers with wide range of layer parameters [given in Hoque and Jakowski, 2007], the knowledge about the F2 layer scale height and peak density height alone can eliminate about 85% error on an average.
 However, in practical cases, the information about F2 layer scale height HF2 and peak density height hmF2 are not easy to estimate. It has been found that using assumptions for HF2 and hmF2, a significant amount of error can be corrected. For HF2 = 70 km and hmF2 = 350 km, about 80% error will be removed on an average. Further investigation using CHAMP profiles shows that about 65% error will be corrected on an average by the correction formula.
8. Comparisons of Higher Order Ionospheric Errors
 Different higher order ionospheric errors are plotted in Figure 7. The residual error terms Δsb(TEC), Δs2, Δs3, and Δsb(length) are estimated by the ray tracing program while RRE and (RRE)gr are estimated by both the ray tracing program and by the correction formulas given in this paper. The correction formula for Δs2 is given in Hoque and Jakowski , and Δs2 is estimated for azimuth angle 180° at a receiver position at (48°N and 15°E).
Figure 7 shows that at low elevation angles (about <15°), the error component Δsb(TEC) is the highest among other errors and it decreases very rapidly with the increase of the elevation angle. It has been seen that at low elevation angles, Δs2 is less than Δsb(TEC) but exceeds Δsb(TEC) with increasing elevation angle. The Δs2 does not change significantly with increasing elevation angle and therefore, it cannot be ignored even at zenith. The third highest error is found to be the excess path length error Δsb(length). It decreases with increasing elevation angle very rapidly and vanishes at zenith. It has been seen that the residual error Δs3 is small (∼<5 mm) but it can be bigger than Δsb(TEC) and Δsb(length) at high (about >60°) elevation angles.
 The excess path error Δsb(length) is additive to other errors in the (RRE)gr expression (see equation (6)) whereas it is subtractive in the RRE expression (see equation (7)). For this reason, (RRE)gr (absolute value) is much higher than RRE (see Figure 7). Figure 7 shows that RRE and (RRE)gr values estimated by the correction formulas converge with the ray tracing results. It has been found that for a Chapman profile the proposed correction formulas can estimate RRE and (RRE)gr within 1- to 2-mm level of accuracy.
 Although the second-order ionospheric term is considered in recent investigations, a common practice is to compute the magnetic field vector at the IPP assuming the ionosphere as a single layer at a certain altitude. Our investigation shows that computing average magnetic field component at the IPP introduces an additional error of about 0–2 mm in the GPS dual-frequency second-order error estimation.
 We have found that the dual-frequency third-order residual error can be as big as about 3 mm during high TEC values and low elevation angles such as TECV = 143 TECU and 1° elevation. The proposed third-order correction formula can correct this error within the submillimeter level of accuracy.
 The dual-frequency range error due to TEC difference at two GPS frequencies is found to be significant at low elevation angles (e.g., <15°). On the basis of analytical integration of raypath equation, an approximation formula has been derived to calculate TEC difference and consequent range error in dual-frequency measurements. It has been found that the analytical solution can correct this error within the 1-mm level of accuracy while the error without correction can exceed a 4-cm level for a Chapman layer with TECV = 143 TECU at elevation 1°. In addition, an empirical formula is developed for the same purpose.
 The excess path length of the signal in addition to the free space path length depends on ionosphere profile shapes. On the basis of simulation studies considering broader varieties of profile shapes, an approximation formula is developed to correct this error at GNSS frequencies. It has been found that, for the combined L1–L2 signal, the approximation formula corrects about 70–80% excess path length error on an average.
 In this paper, higher order ionospheric corrections are given for dual-frequency code and carrier-phase combinations. However, the noise on the code (the code noise may exceed 1 m whereas the phase noise is at the millimeter level [Langley, 1997; Feng, 2003]) may exceed the uncertainty in the higher order ionospheric errors and make the corrections less attractive for code combinations.
 The availability of the third frequency from the modernized GPS and future Galileo systems and the interoperability of GPS and Galileo systems provide opportunities to eliminate higher order ionospheric effects by double and triple differencing methods which result in triple- and quadruple-frequency combinations, respectively. Using the triple-frequency approach, the second-order ionospheric error can be removed completely, but the range error due to TEC difference at three GPS/Galileo frequencies will not be canceled out completely. Future work is planned to investigate remaining errors in triple-frequency combinations.
Appendix A:: TEC Approximation
 TEC along straight LoS can be written by the Chapman function as
where z = (h − h0)/H, and dl is the differential increment in LoS direction (see Figure A1). Since the function inside the integral is in terms of z(h), the element dl will be substituted by dz as follows. In the triangle ΔROS, we can write
where Rh is the receiver altitude. Equation (A2) can be solved for l as
where a = (Rh + RE) cos β and b = h0 + RE. Substituting x = zH + b and Hdz = dx (using differentiation), equation (A5) becomes
If a ≪ x or (Rh + RE) cos β ≪ (Rh + RE) (substituting x value at Rh), equation (A6) can be extended using binomial expansion. The assumption a ≪ x is valid at any elevation angle except β = 0. Under this assumption, equation (A6) can be expanded as
Now different terms of equation (A7) will be treated separately and added together to obtain the approximate solution for the slant TEC. The first term on the right-hand side of equation (A7) can be written after substituting = y and dx = −2Hdy (using differentiation) as
The second term on the right-hand side of equation (A7) can be written after substituting = y, x = b − 2H ln y and dz = 2Hdy as
If ∣2H ln y∣≪∣b∣ or Sh − h0 ≪ h0 + RE (substituting 2Hlny value at satellite height Sh), equation (A9) can be extended using binomial expansion. This assumption is valid only when Sh ≪ (6371 + 2h0) km. Under such assumption, equation (A9) becomes
Similarly, the third term on the right-hand side of equation (A7) can be written as
 Similarly, the second terms on the right-hand side of equations (A10) and (A11) can be added together to get I22, but for simplicity we will ignore these terms here. Therefore, the TEC expression can be simplified combining equations (A12) and (A13) and substituting the values of a, b as
Appendix B:: ∫ne2dl Approximation
 Integrating ne2 along any direction (using Chapman function)
Using the same procedure described in Appendix A for TEC approximation, equation (B1) can be manipulated and extended binomially as
The first term on the right-hand side of equation (B2) can be written after substituting y = and −Hdy = dx as
The second term on the right-hand side of equation (B2) can be written after substituting = y, x = b − H ln y and dx = −Hdy as
Similarly, the second terms on the right-hand side of equations (B5), (B6),…, can be added together to get I22, but for simplicity we will ignore these terms here. Therefore ∫ne2dl expression can be simplified as
where y = . At χ = 0 and for realistic H (range 60–80 km) and hm (range 250–450 km) values, equation (B8) can be simplified as
Appendix C:: Curved Path Length Approximation
 If r0 cos β0 ≪ rn, equation (24) can be extended using binomial expansion. This assumption is valid except at very low elevation angle (i.e., β0 = 0°). Under this assumption equation (24) can be written as
The index of refraction n (equation (14)) can be expanded in inverse powers of frequency [Bassiri and Hajj, 1993] as n = 1 − 40.3 · ne/f2 (up to the second inverse powers of frequency). Using this assumption, the second term of equation (C1) can be extended using binomial expansion as
where K = 40.3. Similarly, the third term of equation (C1) can be extended as