The ionosphere is a dominant source of Global Positioning System receiver range measurement error. Although dual-frequency receivers can eliminate the first-order ionospheric error, most second- and third-order errors remain in the range measurements. Higher-order ionospheric error is a function of both electron density distribution and the magnetic field vector along the GPS signal propagation path. This paper expands previous efforts by combining incoherent scatter radar (ISR) electron density measurements, the International Reference Ionosphere model, exponential decay extensions of electron densities, the International Geomagnetic Reference Field, and total electron content maps to compute higher-order error at ISRs in Arecibo, Puerto Rico; Jicamarca, Peru; and Millstone Hill, Massachusetts. Diurnal patterns, dependency on signal direction, seasonal variation, and geomagnetic activity dependency are analyzed. Higher-order error is largest at Arecibo with code phase maxima circa 7 cm for low-elevation southern signals. The maximum variation of the error over all angles of arrival is circa 8 cm.
 It is well known that ionization encountered by Global Navigation Satellite System (GNSS) signals introduces delays into the code and advances into the carrier phases, and that standard models and broadcast parameters for determining these propagation errors are generally inadequate for precise position correction [Klobuchar, 1996]. Dual-frequency receivers, by taking the difference between code and/or carrier phase measurements on the two frequencies, can estimate and eliminate the first-order error. The majority of the higher-order errors, however, remain in both the code and carrier observables [Morton et al., 2009a]. Typical quiet time second- and third-order range errors are on the order of a few cm and a fraction of a mm, respectively [Bassiri and Hajj, 1993; Brunner and Gu, 1991; Datta-Barua et al., 2008; Kedar et al., 2003]. Under severe geomagnetically active conditions, the second-order error may reach tens of cm [Hoque and Jakowski, 2008; Wang et al., 2005] and the third-order error may become comparable to the second-order error [Morton et al., 2009a].
 Many scientific, military, and civilian applications require continuous high-accuracy positioning at the mm to cm levels on a global scale. Earlier studies of the higher-order ionospheric error were limited to the use of idealized model electron density profiles and average ionospheric magnetic field values to infer average behavior or error bounds for the higher-order errors [Brunner and Gu, 1991; Datta-Barua et al., 2008; Hoque and Jakowski, 2008]. Some studies used a simple dipole field model for the magnetic field [Bassiri and Hajj, 1993; Kedar et al., 2003]. Recently, Morton et al. [2009a, 2008, 2009b] carried out comprehensive assessments of the higher-order errors and the resulting position solution errors based on electron density measurements obtained by the Arecibo incoherent scatter radar (ISR) and the 10th generation International Geomagnetic Reference Field (IGRF) model. These recent studies, however, are only concerned with the higher-order error at one particular site at Arecibo, Puerto Rico (latitude: 18° 20′ N; longitude: 66° 45′ W). Because the ionosphere is a nonhomogeneous medium characterized by wide variations in electron density and the magnetic field on the global scale, the higher-order error characteristics will vary at diverse geographical locations. The objective of this paper is to extend our previous studies at Arecibo to two other sites: Jicamarca, Peru and Millstone Hill, Massachusetts. These three sites were selected for two reasons. First, all three sites are equipped with powerful ISRs which can measure electron density profiles covering the most significant portion of the ionosphere. Second, the unique geographical locations of these sites are characteristic of the North American sector ranging from magnetic equatorial (Jicamarca), to low-latitude to midlatitude (Arecibo), and midlatitude to high-latitude regions (Millstone Hill).
 In this paper, we improve the methodology developed in our earlier publications [Morton et al., 2009a, 2009b, 2010] and apply it to the analysis of higher-order error at all three sites. Major contributions of the current work include: (1) Extrapolating the ISR measured electron density profiles to 10,000 km height so that the error computation is more accurate. In our previous studies, we only used the ISR electron density measurements up to 500 km altitude [Morton et al., 2009a, 2008], and extended the profiles to 2000 km using the International Reference Ionosphere (IRI) model [Morton et al., 2009b, 2010]. (2) Spatial distributions and comparison of the higher-order error for all three sites at given date and time. (3) Analysis of diurnal, seasonal, and solar activity dependency of the higher-order error for all three sites using multiyear ISR data.
Section 2 presents the mathematics and detailed methodology developed to compute higher-order errors. Section 2.3 describes the experimental facilities and multiyear data acquisition involved in the study. Section 3 provides the results. Section 4 summarizes our findings.
2.1. Mathematical Background
 The observables recorded by a GPS receiver can be written as
where ρ is the code phase, i indexes each frequency received, ψ is the true range, I is the ionospheric error, M is the multipath error, T is the troposphere propagation error, c is the speed of light in free space, δtu is the receiver clock error, δts is the satellite clock error, δψs is the satellite orbit error, ε is noise, ϕ is the carrier phase measurement, λ is the wavelength, and N is the carrier cycle integer ambiguity. We shall represent the nondispersive errors as
The frequency-dependent ionospheric error may be expanded into an inverse frequency power series:
Bassiri and Hajj  derived formulas for these coefficients in terms of the electron density Ne and magnetic field vector B along the signal path, utilizing the Appleton-Hartree equation for the refractive index of an electromagnetic field in a cold collision-free plasma, an approximation to the ionosphere [Ratcliffe, 1959].
Here TEC is the total electron content ∫SVRXNedl. Its units are electrons per square meter.
where θB is the angle between the magnetic field vector and the signal propagation path,
e is electron charge (−1.602 × 10−19C), me is electron mass (9.109 × 10−31kg), and ε0 is the permittivity of free space (8.854 × 10−12 F/m).
 An L1/L2 dual-frequency receiver corrects the ionospheric error by approximating q with (ρ1 − ρ2)f12f22 / (f22 − f12), where f1 is the L1 frequency 1.57542 GHz and f2 is the L2 frequency 1.22760 GHz. This code phase difference may be smoothed utilizing the similarly defined carrier phase differences.
 However, recalling equation (4), the code phase difference results in:
Equation (12) indicates that the first-order error estimation is actually contaminated with scaled higher-order ionospheric error and amplified multipath and receiver noise error. A numerical estimate of the multipath amplification is a factor of 2.55.
 Triple frequency receivers may estimate both the first- and second-order error by similar means. However, since
these estimates contain third- and higher-order ionospheric residual errors and further amplified multipath and receiver errors.
2.2. Error Computation
 The integrals (7) and (8) are evaluated numerically by taking the sum of the values evaluated at altitudes from 56 km to 10,000 km multiplied by the altitude interval between successive data. This interval is dependent on the incoherent scatter radar measurement altitude resolution. It is about 600 m for Arecibo and about 15 km for Jicamarca and Millstone.
 The values for Ne are taken from the ISR measurements where available. Among the data we obtained for the computation, Jicamarca’s results extend to 1635 km, but the data from Arecibo and Millstone Hill only extend up to circa 500 km. In our previous studies, the IRI model was used to extend the Arecibo ISR measurements to 2000 km altitude [Morton et al., 2009a, 2010]. In this study, we applied the IRI model to the Arecibo and Millstone measurements and extended the measurement profiles to 1635 km, matching that of Jicamarca. It is known that beyond the 1500–2000 km altitude range, there is an additional 10–20% TEC [Makela et al., 2000]. To take this into consideration, we determined an exponential decay function with constant decay rate k by a nonlinear least squares fit to the topside:
where x represents altitude, and x0 is the altitude of the uppermost available electron density value y0. This extension reflects the fact that the ionosphere scale height 1/k is nearly constant above the so-called transition height of 1000–2000 km [Reinisch et al., 2007]. The decay factor k is chosen to minimize the deviation from the last five measurements or model values.
Figure 1 shows an example of extended electron density profiles compared with the originals. This is a typical profile chosen for exhibition. The thick black line shows the measurements, the medium line shows the extension by the IRI, and the thin line shows the exponential decay extension, which continues to the 10,000 km altitude range. This has been plotted on a log scale so that values at high altitudes remain distinguishable.
 The values of the magnetic field vector B were determined by the tenth generation IGRF model. The IGRF models the core field, neglecting crustal fields, fields due to ionospheric currents, tidal forces, solar wind, diurnal and seasonal variation, and other modelled effects, and includes a predictive “secular variation” component of unknown accuracy [Macmillan and Maus, 2005; Lowes, 2000; Lowes and Olsen, 2004]. Whether it is close enough to actual ionospheric magnetic field values for use in accurate error computation might therefore be questionable. To address this, we analyzed magnetometer field measurements from low Earth orbiting satellites. We found that the IGRF was sufficiently accurate for our purposes. More detailed analysis of the IGRF performance in modeling the magnetic field in the ionosphere is the subject of a separate publication (N. A. Matteo and Y. T. Morton, Ionosphere geomagnetic field: Comparison of IGRF model prediction and satellite measurements, manuscript in preparation, 2010.)
 The IGRF model is a function of time t, geocentric colatitude θ, longitude λ, and radius r:
where R is the reference radius of the Earth 6371.2 km, nmax is 13 since 2000 (and 10 before), Pnm is a Schmidt seminormalized associated Legendre function, and the coefficients gnm (t) and hnm (t) vary linearly between values given for each 5 years from 1900 through 2005 by the IGRF tenth generation [Macmillan and Maus, 2005].
 The value ∣B∣ cos θB is equal to B · v where v is the unit vector pointing from the satellite to the receiver. Given azimuth α and elevation ɛ at receiver (geocentric) latitude ϕ and longitude λ,
This is a coordinate system transformation. The vector is the satellite position in a local system, and the matrix rotates it into Earth-centered Earth-fixed Cartesian coordinates.
 To compute the range error in directions other than zenith, we took the approach described by Morton et al. [2009b], in which we assume that the “shape” of the profile is homogeneous throughout the local region, but scale the profile based on the TEC values. TECs were determined by use of the TEC map IONEX files distributed by the International GNSS Service (IGS), or those distributed by the Jet Propulsion Laboratory if the IGS maps were not available. These provide TEC values at points on a grid at intervals of 2.5 degrees latitude, 5 degrees longitude, and 2 h time. We interpolated between these using rotated time interpolation and 4-point grid interpolation, as recommended by Schaer et al. .
 Thus, for a time t, in hours, between grid time points T1 and T2, latitude ϕ between grid latitudes ϕ1 and ϕ2, and longitude λ between grid longitudes λ1 and λ2,
where TEC0 is a provided grid data point.
 IGS TEC maps are only available later than May 1998. For data earlier than this, such as the majority of our Arecibo data, we selected a date with a TEC map available in a similar point of the 11 year solar cycle, a similar time of year, and with a similar Kp value at the given time of day.
 Kp is a planetary index of geomagnetic activity level. Its value ranges from 0 to 9, with values above 5 considered to indicate “high” geomagnetic activity.
 Specifically, we selected the time t which minimizes
where Y(t) is the year, Kp(t) is the Kp index normalized onto a 0–27 integral scale, and D(t) is day of year. The coefficients 8 and 5 are weights for these three criteria; they were selected so that finding a closer match in the solar cycle is worth moving a week away in time of year, while a significantly better match in Kp index (by two steps or more) outweighs either.
 With this method, in addition to computing the higher-order error along the radar signal path (which was near zenith for all our data), we computed the error for 10° elevation in the north, east, south, and west directions for all records. In addition, error was computed for all signal directions over an azimuth and elevation grid of about 3° resolution for a few records of interest. A record during a typical quiet day, a record during geomagnetic activity (high Kp level), and the records with maximal first-, second-, or third-order error were examined in this way.
2.3. Experimental Facilities and Data Acquisition
 Arecibo Observatory is in Puerto Rico. Over 2660 h of electron density data were acquired over a 14 year period from 1986 to 2000. Data were available in every month of the year, hour of the day, and for Kp values from 0 to 9. The processing done to convert raw radar data to electron density and eliminate outliers is given by Zhou et al. . Figure 2 shows an example set of electron densities from Arecibo by local time and altitude taken on 13 April 2000.
 Jicamarca Radio Observatory is in Peru. A total of 700 h of electron density data, between 2002 and 2009, were acquired. Data were available in every hour of the day and in every month except January or November. Kp values for the data ranged from 0 to 5+. Figure 3 shows an example set of electron densities from Jicamarca by local time and altitude taken on 15 September 2004.
 The Millstone Hill Observatory is in Westford, Massachusetts. About 3030 h of electron density data between 2002 and 2009 were acquired. Data were available in every month of the year and hour of the day. Kp values for the data ranged from 0 to 9−. Figure 4 shows an example set of electron densities from Millstone by local time and altitude taken on 2 May 2007. The strip-like appearance is due to the interpolation methods used at Millstone in order to produce the validated data which we used (J. M. Holt, Explanation of fitted gridded data, http://madrigal.haystack.mit.edu/madrigal/experiments/2008/mlh/17jan08g/explain.html, 2009). Several E layer “spikes” are visible in Figure 4. These spikes have characteristics of sporadic E, which is an intensive and narrow layer of electrons, typically seen in the nighttime E region [Mathews, 1998]. Since this reflects actual electron content, we did not remove the data. Spikes also occur in the Jicamarca data, but much larger in electron count (up to 1013 electrons per cm3) These are called “150 km echoes” and are an artifact of the electrojet current which runs in the equatorial ionosphere [Balsley, 1964; Chau and Kudeki, 2006]. The nature of these echoes is not fully understood, but it seems likely that they do not reflect actual electron content. Therefore, we have discarded any spikes from the Jicamarca electron density profiles.
 The attributes of the various radars, and the data obtained from each, are summarized in Table 1. We would like to point out that these three radars operate with different power levels, sensitivity, and frequencies. Furthermore, the data obtained from these sites have undergone extensive processing, including calibration, signal-to-noise ratio reduction, interference mitigation, and spurious data point removal. We have not accounted for these differences in data quality in this study.
Table 1. ISR Site and Data Information
Millstone Hill Observatory
Jicamarca Radio Observatory
100 km to 548 km
56 km to −500 km
59 km to −1635 km
Dates of data used
16 Apr 2002 to 14 May 2009
23 Sep 1986 to 15 Apr 2000
13 Aug 2002 to 26 Apr 2009
Number of days Kp < 5
Number of days Kp ≥ 5
 It is possible that the difference in altitude resolution for the different sites could impact the comparability of the results. To address this concern, we computed the error for a selected date from Arecibo once using all the available data, as the other computations were done, and again using only one data point per 15 km of distance, to match the altitude resolution of the other ISRs. The selected date was 17 December 1990, a typical day of moderate geomagnetic activity. The results are summarized in Table 2. The columns labeled “relative” are the results of the previous column, as a proportion of the mean magnitude of the computed error. The “mean difference” is the mean absolute difference.
Table 2. Sensitivity of Error Computation to Altitude Resolution at Arecibo
Root Mean Square
 These deviations are several orders of magnitude smaller than the absolute corresponding higher-order errors, so the sensitivity of analysis to the altitude resolution of the data should not impact the conclusions which we draw.
Figure 5 compares the second- and third-order ionospheric code phase pseudorange error, terms s/fi3 and r/fi4, respectively, of equation (4), along with their sum for signals from the south at 10° elevation at Jicamarca. Light dots represent the range error in cm for each experiment with Kp < 5, plotted against local time of day. Dark dots (there are very few for Jicamarca) represent experiments for which Kp was 5 or higher. The black circles represent the average error for each half-hour of the day, while the bars show standard deviation. As expected, the second-order error is considerably larger than the third-order error. In this case, the two agree in direction, resulting in a larger sum, but this is not universally true.
Figure 6 compares the second-order error at each site for signals from the zenith, signals from the north at 10° elevation, and signals from the south at 10° elevation. Once again, light dots represent ISR experiments when the Kp index was below 5, and dark dots represent ISR experiments when the Kp was 5 or greater. Black lines and bars represent the average error and its standard deviation in each half hour of local time. The tendency of north error to be negative and of south error to be positive can be clearly seen. This tendency is due to differences in the ordinary or extraordinary mode of GPS signal propagation, as discussed extensively by Morton et al. .
 Jicamarca’s second-order error is nearly zero at the zenith, since it is very near the magnetic equator and so zenith GPS signals are nearly perpendicular to the magnetic field there. Thus, for zenith signals, the third-order error (not shown) can be comparable to second-order error at Jicamarca. In fact, average third-order error at the zenith is above 0.01 cm at some times, rivaling the average second-order zenith error of 0.02 cm. By comparison, in Figure 5 average third-order error at 10° south reaches only 0.03 cm while average second-order error exceeds 0.8 cm.
 Also visible in Figure 6 is the near equality of the error from low elevations to the north and south at Jicamarca, due to the symmetry of the magnetic field distribution at that location.
 At Millstone Hill, where the magnetic field dip angle is relatively high, the zenith error is larger than the error from low elevation in the north. At both Arecibo and Millstone Hill, the error from 10° south is larger than that from zenith or 10° north. At Arecibo, the error at low elevation in the south could reach 6 cm. We can speculate that at higher latitude such as Poker Flat in Alaska where the magnetic zenith is very close to the geographical zenith, the zenith error will be considerably larger relative to those from low elevations. At such high latitude, GPS satellites do not pass overhead, so the zenith error is academic. However, even at the 65° north latitude of Poker Flat, GPS satellites are seen at 78° elevation. Note that the Arecibo results show larger magnitude of error compared with those presented by Morton et al. [2009a]. This is due to the larger altitude range included in the present study.
Figure 7 (left) shows the second-order error over a “sky map” of azimuths and elevations for selected epochs at each site. The dashed circles represent curves of constant elevation angle, with the center corresponding to zenith, and the outer ring corresponding to the horizon. In Figure 7 (right) the TEC data used at the given time are shown on a map, for reference. The signal directions with zero second-order error (marked with a thick line) are determined by which signals are perpendicular to the predominant magnetic field direction, and so are dependent on latitude. Since the magnetic field in the northern hemisphere generally points north and down, signals are generally perpendicular when arriving from a northerly direction. At Jicamarca, furthest south, the zero-error curve is roughly in the middle, corresponding to latitudinal azimuths. At Arecibo, further north of the magnetic equator, the line of zero errors is for signals coming from the northwest. At Millstone, even further north, there are no points of zero error above 10° elevation. Figure 4 of Morton et al. [2009a] provides a graphical explanation of these effects.
Figure 8 shows the seasonal dependency of the diurnal second-order error from 10° elevation south at each site. As usual, light dots represent records with Kp below 5, dark dots represent records with Kp 5 or greater, and black lines are the half-hourly average. The southern error was chosen for display because it is generally large, but the patterns of error magnitude versus season are similar for the error on any angle of arrival. It should be noted that, for instance, January, February, and March correspond to winter at the Millstone and Arecibo sites, but summer at Jicamarca.
Figure 9 shows the average second-order error southward at Millstone and Arecibo when the global Kp index was at most 4, corresponding to relatively quiet geomagnetic periods, with light squares, and the average second-order error when the global Kp index was 5 or greater, corresponding to relatively active geomagnetic periods, with dark circles. The bars on each set represent standard deviation. The plots are not on the same scale, so that the differences can be appreciated in both. Active periods correspond to an increased higher-order error, as would be expected. The Jicamarca data obtained for this study included very few records corresponding to Kp of 5 or greater, so there is not enough data for a useful comparison.
 The epochs plotted in Figure 7 were chosen to match in season, local time of day, and on the Kp value of 4−. Figure 10 shows a similar map for Arecibo with the largest spatial variation in the second-order error observed at any site at a given time. The error varies by 8.1 cm, from −0.57 to 7.49 cm, over the angles of arrival examined.
 Continuous high-accuracy positioning at the mm to cm levels on a global scale are important for many frontier scientific research and military and civilian applications. Recent GPS modernization efforts brought us one step closer to achieving the high accuracy positioning solutions. The high-order ionosphere error in GPS range measurement, however, continues to be a major challenge in achieving centimeter-level accuracies.
 In this paper, we provided assessment of higher-order ionosphere error through joint use of incoherent scatter radar measurements, ionospheric models, fitted exponential decay functions, total electron content maps, and the IGRF magnetic field model. These provide a powerful means to characterize ionospheric higher-order error temporal structure. High-latitude (Millstone), midlatitude to low-latitude (Arecibo), and geomagnetic equatorial (Jicamarca) sites were examined to generate insight to higher-order error spatial distributions. Second-order errors were seen from −2.7 cm to 2.3 cm at Jicamarca; from −0.9 cm to 7.5 cm at Arecibo; and from 0 cm to 3.3 cm at Millstone. Third-order errors were seen from 0 to 0.3 cm, at Jicamarca; from 0 to 0.9 cm at Arecibo; and from 0 to 0.13 cm at Millstone. The largest spatial variation in the second-order error at a single site and a single time was 8.1 cm, and occurred after noon at Arecibo in October 1991.
 In future work, more electron density profiles will be collected at each site, including measurement extensions to higher altitudes. We will also get data from other incoherent scatter radars, including Alaska’s Poker Flat ISR (PFISR). PFISR is a steerable array-based radar and permits measurements along directions other than zenith, which will allow removal of the assumption of a homogeneous profile shape, scaled by TEC, which we have been using. Millstone also has steering capabilities.
 With these and other sources of electron density information and characterization of the higher-order errors, we hope to create a model of the higher-order error directional distribution by latitude, local time, and first-order error (TEC). If successful, this model would allow dual or triple-frequency receivers to estimate the higher-order error for each satellite in view.
 This research was funded by AFOSR grant FA-9550-07-0354 and AFRL grant FA-8650-08-D-1451. Student workshop and summer support was provided by the National Science Foundation. We would like to thank Jorge Chau, Cesar Valladares, Juan Carlos Espinoza, and David Hysell for their assistance with data and analysis at Jicamarca Radio Observatory; Anthea Coster, Philip Erickson, and John Holt for their assistance at Millstone Hill Observatory; and Mike Sulzer and Qihou Zhou for their assistance with Arecibo data.