The geomagnetic field is an important parameter in space physics. The International Geomagnetic Reference Field (IGRF) is a model of the Earth's main field. Current sources in the ionosphere, the interplanetary magnetic field, annual and diurnal variation, and other sporadic and unmodeled effects may alter the actual B field distribution. We investigate whether the model is nevertheless sufficiently accurate for computing ionospheric effects on radio frequency signals. Detailed analysis of scalar intensity is presented based on direct measurements of the magnetic field taken from UARS, SAC-C, Ørsted, and CHAMP, all satellites with magnetometers orbiting between 200 and 1000 km altitude. Our results indicate that the IGRF model is within 1% accuracy of the measured ionosphere B field, 92.80% of the time. Quality control issues associated with the scalar data are also discussed.
 The geomagnetic field B is a fundamental parameter in the magnetoionic theory. The dependency of the ionosphere refractive index on the B field vector impacts performance of satellite-based communication, navigation, and surveillance systems by causing signal group delays, carrier phase advances, signal distortions and bending, Doppler frequency shifts, and Faraday rotation [Ratcliffe, 1959]. In particular, this paper was motivated by the need to characterize the Global Positioning System (GPS) range measurement error due to higher order ionosphere group delay and carrier advances. Satellite navigation signals traversing the ionosphere have range measurement errors that can be represented as follows:
where Iρ and Iϕ represent the range error associated with a GPS receiver's code and carrier observables respectively. RX denotes the receiver, and SV the satellite. nρ is the signal group refractive index and is related to the carrier refractive index n by the relationship
e is electron charge, me is electron mass, ɛ0 is the permittivity of free space, θB is the angle between the signal and the magnetic field, and TEC is total electron content.
 The terms involving q, s, and r are functions of , , and in Equations (4) and (5) and are commonly referred to as the first, second, and third order ionosphere error terms. The first order error does not depend on the geomagnetic field and can be eliminated from the range measurements by linearly combining the observables obtained at two different frequencies, available to any dual-frequency GPS receiver [Klobuchar, 1996]. The second and third order terms, however, are functions of the geomagnetic field and electron density distributions along the signal propagation path and are collectively referred to as the higher order ionosphere error in GPS measurement error budget.
Lowes  and Lowes and Olsen  give in-depth analyses of potential errors in IGRF-type models, and the magnitude of possible variance in the coefficients from a variety of sources. For an engineering application, however, we may be more interested in the worst case difference in real-world measured magnetic field strength versus prediction, due to whatever unmodeled, sporadic, or even unknown effects are operating. These include external and induced fields, ionospheric currents, magnetic storms, short-wavelength fields, and other events not accounted for by the model. Indeed, the creators of geomagnetic main field models go to great lengths to exclude the effects of ionospheric currents and interplanetary fields, which could be important in signal propagation.
 The objective of this work is to validate the IGRF model and its secular variation using space-based magnetometer measurements. These measurements are taken by magnetometers on board Low Earth Orbit (200–1000 km altitude) (LEO) satellites from 1991 through 2010. An excess of 70 GB of raw measurement data have been processed to provide magnetic field measurements in the ionosphere under a wide range of conditions, including all Kp levels, times of day, and times of year. The IGRF model is used to generate the B field values corresponding to the measurements' location and time. Extensive analysis of the differences between the model and measurement is presented in the paper. Our results show that the IGRF provides remarkable accuracy in predicting the magnetic field values in the ionosphere measured in the time since its release. Accuracy does decline near the end of a generation's 5 year purview.
Section 2 provides a brief summary of the IGRF model and characteristics of the global B field distribution as represented by the IGRF model. Section 3 describes the satellite missions, instruments, and data for the magnetic field measurements, and section 4 describes our methods of filtering and processing the data. Section 5 presents the results and analysis of the comparison between the IGRF model and the measurements, and section 6 discusses the results in more detail. Section 7 introduces potential further work, and section 8 concludes the findings of this study.
2. The IGRF
 The IGRF is a mathematical model designed to predict the Earth's main internal magnetic field vector at given coordinates and time, valid since 1900. Updates have been made to this model, typically at 5 year intervals, since its first release in 1968 to account for the continual variation in the Earth's main field. The latest version, the 11th generation, is a truncated spherical harmonic model of degree 13, using a total of 195 coefficients with precision of 0.1 nanoTesla (nT) [Finlay et al., 2010]. Being a global harmonic model, it cannot precisely replicate all of the irregularities, anomalies, and space weather events which affect the magnetic field. The model includes a degree 8 secular variation component to describe the shift in the field over time. This variation is linear in time and recommended for use only for five years, a decade at the outside, since the field often changes in unpredictable ways as a result of internal core field processes not well understood. The 11th generation was released in January 2010.
 The 10th generation IGRF had the same degree and precision as the 11th, and was finalized by the International Association of Geomagnetism and Aeronomy (IAGA), an Association of the International Union of Geodesy and Geophysics (IUGG), in December 2004 [Maus et al., 2005].
 The magnetic field is modeled as the negative gradient of a scalar potential V, which is in turn modeled by a truncated series expansion of a spherical harmonic:
where (θ, ϕ, r) are the geocentric coordinates of colatitude, longitude, and distance from the center of the Earth; R is the Earth reference radius, 6371.2 km; gnm and hnm are the expansion's Gauss coefficients at time t, Pnm is the Schmidt seminormalized associated Legendre function of degree n and order m, and nmax is 13 for dates since 2000, and 10 earlier [Macmillan and Maus, 2005]. Secular variation coefficients predict the linear variation in the first eight degrees of coefficients for times after the most recent values. Typical magnetic field strengths in the ionosphere range between 20,000 nT and 50,000 nT.
3. Space-Based Geomagnetic Field Measurements
 IAGA declared the decade from 1999 through 2009 to be the International Decade of Geopotential Field Research. As a result, several nations launched LEO satellites carrying instruments for magnetic field measurements. The Danish satellite Ørsted, German satellite CHAMP, and Argentinian SAC-C are all part of this effort to monitor the Earth's magnetic field in space. The US satellite UARS is an earlier LEO satellite with a magnetometer.
 The Upper Atmosphere Research Satellite (UARS), by NASA, was launched in 1991. It orbited at an altitude circa 578 km and inclination of 57 degrees, and continued returning data until August 2005. The Particle Environment Monitor (PEM) experiment on board includes a three-axis fluxgate vector magnetometer instrument, VMAG. For details on the satellite and its instruments, see Reber  and Winningham et al. . Since the magnetometer was not a focus of this satellite's design, as with the others used, more local interference and calibration issues may be present in UARS data. However, since it is the only satellite here which did not provide any data for the creation of the IGRF models, it is the most independent validator of the model. It is also the only vector magnetometer which is not an Compact Spherical Coil Fluxgate Magnetometer (FGM) from the Technical University of Denmark.
 Ørsted, by the Danish Meteorological Institute, has an elliptical orbit from 630 to 860 km altitude at 96.5 degree inclination. It has been orbiting from March 1999 to the present, and carries both a FGM for vector measurements and an Overhauser Magnetometer (OVM) for scalar magnitude readings. The FGM was a newly developed sensor for the Ørsted mission. Assisted by the Danish Meteorological Institute, we collected data at 1 Hz time resolution from 15 March 1999 through 12 January 2010. For details on the satellite's operation and instruments, see Duret et al. , Hoffmeyer , and Olsen et al. . The position information for Ørsted is obtained from GPS receivers and interpolated by an orbit model. We have only used data for which the GPS position is available.
 The German Challenging Minisatellite Payload (CHAMP), by GFZ Potsdam, was in orbit from 350 to 450 km altitude at 87 degrees inclination since July 2000 until September 2010. We obtained data at 1 Hz time resolution from May 15, 2001 through May 27, 2010. It includes a scalar OVM as well as a FGM for vector data. The project's Web site [Bock et al., 2000] states that the satellite was validated to provide scalar magnetometer readings with accuracy better than 0.5 nT, with a resolution about 10 pT. See Reigber et al.  for satellite and instrument details.
 SAC-C, by the Argentine Commission on Space Activities, has a circular orbit at 702 km altitude and 98.2 degrees inclination. Data are available from 23 January 2001 through 4 December 2004. It carries a scalar helium magnetometer as well as a FGM for vector readings. We retrieved scalar magnetic field values at 1 Hz resolution for the entire available time range. For more details, see Colomb et al. .
 As of 2002, accurately oriented vector data could not be obtained from SAC-C, because attitude data from the star imager on board were not available. It is not clear whether this has since been rectified. The data were conditioned in a similar manner to Ørsted; disturbances from magnetic torquer coils remain, but should not exceed 1–2 nT. Overall, most measurements have accuracy within 3 nT [Tøffner-Clausen, 2002].
 Some data from Ørsted, CHAMP, and SAC-C was used in the creation of the IGRF models. However, the data used were substantially restricted. For instance, data from Ørsted were only used at nighttime, with the sun at least 5° below the horizon, with Kp at most 1+ at the time of observation and Kp at most 2 for the preceding two hours, and with change of the ring current index Dst at most 1 nT/hour. Polar data was rejected if the interplanetary magnetic field (IMF) was over 3 nT. One data point per minute was used from the remaining set [Olsen et al., 2005]. Additionally, an attempt was made to ignore the short-wavelength crustal field by subtracting the CM4 model [Sabaka et al., 2004] of the same. Only roughly 0.21% of the available Ørsted data, 0.14% of the available CHAMP data, and 0.21% of the available SAC-C data were used in preparing the IGRF-11 model.
4. Data Processing Methods
 We focused on validation of the IGRF by analyzing the scalar field values from UARS, Ørsted, CHAMP, and SAC-C. These satellites have copious data (for every second, with gaps ranging in size from minutes to weeks) for dates from 15 March 1991 through the present; we examined data through May 2010. Unlike some other satellites carrying magnetometers, the data product for these satellites express field values in the standard units of nT, and the satellite position is reported in the easily handled Earth-Centerd Earth-Fixed (ECEF) coordinate frame, with polar coordinates. The data have also been well conditioned with regard to electromagnetic interference from other instruments on board the satellites, time and location drift, and magnetometer calibration. All these factors made them a natural first choice for analysis.
 Desired analysis consisted of using the reported time and location for each reading to compute the IGRF-predicted field strength, comparing it to the recorded reading, maintaining a mean misfit, mean absolute misfit, and root-mean-square (rms) misfit, each in terms first of nT and secondly as a percentage of the magnitude of the IGRF. The algorithm also keeps a list of the maximum absolute or relative differences with their time and location. The standard deviation of the misfits are then calculated. We will report results primarily as root mean square relative misfit.
 Analysis functions were implemented in MATLAB using the MatlabMPI parallel programming toolkit [Kepner, 2002]. A parallelized IGRF implementation must evaluate associated Legendre polynomials at a range of orders and degrees at once. An algorithm for this was found by Zhang and Jin , obtained online (http://ceta.mit.edu/ceta/comp_spec_func, copyright Zhang, Jin) and used with permission.
 A considerable amount of spurious data were present in some of the data sets. The following five filters are used to eliminate questionable data:
 1. Time filter. We discarded any results with recorded times from outside the range of 8 September 1991 through 3 July 2010. For example, some data had recorded times in 1980. These data were not used in our analysis. A total of 10,240 data points were discarded for this reason, or 0.001% of the total.
 2. Altitude filter. Data with recorded orbital altitudes below 280 km or above 1200 km were discarded, because this is outside all of our satellites' orbits. 400,420 data points were discarded for this, or 0.051% of the total.
 3. Field magnitude filter. Any recorded magnetic field strengths below 4,000 nT or above 64,000 nT were discarded. 9,040 data points were discarded, or 0.001% of the total.
 4. Coordinate rate-of-change filter. Some results, particularly from UARS, exhibit a jump in the reported latitude, longitude, or altitude, after which the value jumps back to its previous neighborhood. To account for this, we took a rolling mean and standard deviation of the first time derivative of each coordinate, using a six minute window, along arcs of data between any time gaps of ten seconds or more. Any data point which had a normal 1 s time gap on both sides, yet had a change in one coordinate more than four standard deviations away from the mean change, were removed. 416,057 data points were discarded for this, or 0.053% of the total.
 5. Isolated point removal. For measured values which differed from the IGRF by 1300 nT or more, we examined the median difference of the preceding and following five data points, the mean difference of the preceding and following 100 data points, and the time gap on either side. If both before and after the point, either the median and mean difference were both at least 800 nT less than at the point or the time gap was at least four minutes, then it was removed as an isolated point. These values were selected to remove only a small proportion of outliers, while leaving large but potentially legitimate differences. 1,708 isolated points were discarded.
 Data exceeding the first two boundary parameters were entirely from CHAMP, while data exceeding the field magnitude boundaries were roughly half from CHAMP and half from UARS. It should be understood that different projects are in different stages of data conditioning and filtering, and that the CHAMP data continue to undergo revision and review.
 Upon application of the above filters, some data were still found to exhibit abnormalities. For the most part, these fell into six categories of ‘discrepancy event’:
 1. Apparent magnetometer instrument errors on CHAMP, during which field strengths begin to take random extreme values. Two such events were noted, and a total of 28,487 data points were removed (.004% of the total.)
 2. Magnetometer reset processes after resuming function. On CHAMP these are characterized by abnormally high values, unchanging for several seconds, which gradually jump downward in a “staircase” effect. On CHAMP, SAC-C, or UARS another apparent reset process heads immediately downward from an abnormally high peak. 21 such events were noted, and a total of 1189 data points were removed.
 3. Incorrect recorded positions would result in wildly inaccurate IGRF values. 20 events of this type, totalling 26,630 data points (.003% of the total) were removed.
 4. “Gaps”: These events, on Ørsted, CHAMP, and UARS, are associated with a loss of instrument function. When recording resumes, a new time is not recorded, and data from some unknown time later is appended to the existing “packet” as if arriving at the normal 1 Hz rate. At some point, a new packet is initiated with a correct time. The result appears as a discontinuity where measured data extends continuously from one side, but model data extends continuously from the other. This kind of error was noticed and discussed by Lowes et al. . 20 such events were noted, and a total of 249 data points were removed.
 5. “Swoop”: These discrepancy events involve a continuous, natural-appearing deviation of recorded values from the IGRF, notable only in the large (greater than 10,000 nT) variance attained, and the sharpness of the transition. As seen in Figure 1, the event seems naturalistic on a local scale, but anomalistic on a larger scale. 2 of these events were noted, both from CHAMP; one occurred at an altitude of 345 km, latitude roughly 10° north, longitude 90° east; the other at 400 km altitude, 50° south, and 20° east. These data were left in the calculations, since they do not appear to be an artifact.
5. Results and Analysis
5.1. Contour Maps
Figure 2 shows the relative RMS misfit of measured values from the IGRF, from all four satellites, on global maps. The maps have 3 degree resolution for both latitude and longitude. The misfit from both the 10th and 11th generation IGRF is shown; with the amount of data included, these are identical. The “specks” of higher misfit visible over the Pacific, as well as the higher-misfit area over South America, are due to UARS, as can be seen in Figure 3. The South Atlantic Anomaly contributes to the general higher misfit over South America [De Santis and Qamili, 2010], but the response of UARS is much greater than that of the other satellites. A map is also shown with the misfit from the 11th generation in quiet geomagnetic conditions, corresponding to a planetary Kp index at most 2. The discrepancy seen around South America is lessened here. Finally, the number of data points for each position is shown. The patterns in amount of data are due to the different orbital coverage of each satellite.
Figure 3 shows the relative RMS misfit of measured values from the IGRF 11th generation, from each satellite, on global maps. The maps have 3 degree resolution for both latitude and longitude. Note that the map for UARS has a different color scale, so that all the maps have visible variation.
5.2. Misfit by Altitude
Figures 4 and 5 show the relative RMS misfit recorded values and IGRF-11 for each kilometer of altitude attained by the satellites. Altitude here is above the reference radius, 6,371.2 km from the center of the Earth. The satellites have very different orbital profiles, so the number of data points (black dots, with axis on top) provide important context. Triangles show the RMS misfit from the IGRF-10; circles show the RMS misfit from the IGRF-11. The plots titled “polar” use data from 80° to 85° north latitude; “midlatitude” is from 30° to 35° north latitude; and “equatorial” is from 0° to 5° north latitude. UARS had no data in the polar region.
 A discrepancy between CHAMP data and the IGRF-10 at lower altitudes is particularly visible in the polar region. This has been substantially improved in the IGRF-11. Keep in mind that only CHAMP has an orbit below 500 km altitude, and only UARS orbited between 500 and 600 km. SAC-C has a very narrow altitude range, contained within the altitude range of Ørsted.
5.3. Misfit by Kp Index
 Kp is an index representing the global level of geomagnetic activity. It varies between 0 and 9, with high levels indicating more geomagnetically active conditions, and is computed every 3 hours by averaging readings from geomagnetic observatories around the world (the “p” stands for “planetary.”) The relative RMS misfit by Kp index is shown in Figure 6. As might be expected, the deviation of field strengths from model predictions increases during more geomagnetically active periods.
 The error bars indicate standard deviations of the relative RMS misfits about their mean. Triangles show the misfit from the IGRF-10, and circles show the misfit from the IGRF-11; they are substantially indistinguishable in this case. Black dots show the number of data points, with axis on the right. UARS is displayed because it has the largest misfits by far. CHAMP is displayed because it is the only satellite at altitudes below 500 km, where the effects of the magnetic field on signal propagation are amplified by higher electron densities and greater magnetic field magnitudes. The spike seen in relative misfits for CHAMP for Kp 6+ should not be given too much emphasis, in the context of the low number of data points available. Ørsted, not shown, has very well behaved misfits positively correlated with Kp, climbing from 0.1% misfit at Kp 0 to about 0.8% misfit at Kp 9. SAC-C, not shown, has the same general pattern, with somewhat more noise.
5.4. Misfit Over Time
Figure 7 shows the relative RMS misfit from the IGRF-11 over each four week's time. Time axis labels are GPS week numbers, which started on 5 January 1980. The vertical lines in the background appear every five years (in 1995, 2000, 2005, and 2010) at the first week of the year. Activity levels have changed over time, and the peaks of activity seen in 2003 and 2004 correspond to disturbed geomagnetic conditions during that period. Recall that all data before 15 March 1999 (week 1000) is from UARS only.
5.5. Diurnal Pattern of Misfit
 The local time of day at the satellite's position forms a useful measure of the satellite's orientation relative to the Sun. Solar effects can have a large impact on geomagnetic phenomena. The IGRF-11 displays a dramatically improved agreement with CHAMP here. Figure 8 shows the relative RMS misfit from the IGRF-10 and IGRF-11 by local time of day. The plots of Figure 8 titled “Low Kp” correspond to quiet geomagnetic conditions, with Kp values at most 2. The other plots include all data. The peaks in the number of data points for “All satellites” around 630 and 1350 min, or 10:30 and 22:30, are due to SAC-C which has a sun-synchronized orbit at these times of day.
 The mean absolute difference from the IGRF-11 using all three satellites' measurements was found to be 69.6 nT, a value less than 0.20% of the average field strength of 35,146 nT, with a standard deviation of 116.7 nT. The RMS misfit from the IGRF-11 was 135.9 nT, 0.39% of the average field strength, with a standard deviation of 195.6 nT. The mean absolute difference relative to the model (i.e. the mean of ∣m − r∣/r, where m is a measured B field magnitude and r is the IGRF-11 predicted value) was 0.22%, with a standard deviation of 0.39%. The relative RMS misfit was 0.45%, with a standard deviation of 0.62%.
 All of these are encouraging results in terms of using the IGRF for precise calculation of the higher-order ionosphere error for GPS measurements. However, some extreme values were found. The maximum absolute difference, 23,614 nT, occurred on 12 October 2007 as part of a swoop discrepancy. The largest absolute difference outside of a discrepancy event was 1,332 nT, measured by SAC-C on 18 March 2003. Differences exceeding 1,000 nT were found on only 49 days of data; differences exceed 2,000 nT on only 3 days of data. 2,000 nT is 5.7% of the average field strength. Table 1 summarizes the percentage of data where agreement between the three satellites' measurements and the IGRF-11 model was within given ranges. Notice that 85.73% of overall measurements are within 0.5% of the model field values, while 99.49% are within 2% of the model values.
Table 1. Amount of Data Within Specified Limits of IGRF
3.05 within 1 nT
6.09 within 2 nT
15.15 within 5 nT
29.28 within 10 nT
50.66 within 20 nT
74.19 within 50 nT
81.83 within 100 nT
87.17 within 200 nT
99.59 within 500 nT
100.00 within 1000 nT
2.16 within 0.002%
5.41 within 0.005%
10.80 within 0.01%
21.39 within 0.02%
46.49 within 0.05%
65.08 within 0.1%
77.40 within 0.2%
85.73 within 0.5%
92.80 within 1%
99.49 within 2%
100.00 within 5%
7. Further Work
 Other satellite missions have included magnetometer devices, which could be used to provide validation over a wider range of times, conditions, and locations. Magnetometer data are publicly accessible for NASA missions including MAGSAT and the OGO satellite programs. However, these missions are from the 1960s through 1980s, and it seems that more recent data are of more use in characterizing the current IGRF's value for predicting magnetic field values. We have also obtained data for DoubleStar (Chinese National Space Administration). This satellite is in a highly elliptical orbit which has information relevant to the ionosphere only a small fraction of the time. Potential further sources include the Earth Observation Group Defense Meteorological Satellite Program (DMSP), a set of many satellites in near-polar orbits at 830 km altitude, some of which include SSM magnetometers; the International Space Station, which has had magnetometer experiments on board, such as Lazio; the Mir Space Station, which carried magnetometers such as Sileye and SprutMag until its destruction in March 2001; and a host of other satellites, such as DEMETER, Cosmos-49, Space Technology 5, Dynamics Explorer 2, POGS, FREJA, ASTRID-2, Sunsat, and FEDSAT.
 Comparison of vector values to the model would be useful. Difficulties arise in determining satellite orientation, which must be known with great precision to make a comparison meaningful. Many satellites rely on star imagers for orientation information, which can be difficult to calibrate and prone to error.
 This paper presented satellite measurements of magnetic field magnitude in the ionosphere during the past decade. The measurements were compared with the IGRF model results for a wide range of altitude, latitude, longitude, time, and geomagnetic conditions. Methods of data quality assurance and failure mode determination for scalar magnetometer data were discussed. The results presented in this paper indicate that the IGRF remains highly accurate in the ionosphere, even in geomagnetically active conditions, and over a range of locations and altitudes. Small variations in the geomagnetic field can constitute storms, so these results do not mean that the IGRF is appropriate for all B field simulation purposes. Nonetheless, its use in modeling higher-order GPS error, and other effects, can be pursued with confidence.
 We would like to express our appreciation for the support and assistance given by Peter Stauning, Ørsted Project Scientist, of the Danish Meteorological Institute; Vivien Mende and the GFZ Information Systems and Data Center staff; Peter Reid of the University of Edinburgh; and Abhishek Gupta, of The MathWorks, who responded and confirmed the discovery of a bug in MATLAB during the execution of this research. This project is supported partially by AFOSR grant FA9550-07-0354 and by AFRL grant FA8650-08-D-1451.