an accurate and efficient algorithm for faraday rotation corrections for spaceborne microwave...

An accurate and efficient algorithm for Faraday rotationcorrections for spaceborne microwave radiometers

Malkiat Singh1 and Michael H. Bettenhausen2

Received 27 August 2010; revised 6 May 2011; accepted 23 May 2011; published 25 August 2011.

[1] Faraday rotation changes the polarization plane of linearly polarized microwaveswhich propagate through the ionosphere. To correct for ionospheric polarization error, it isnecessary to have electron density profiles on a global scale that represent the ionospherein real time. We use raytrace through the combined models of ionospheric conductivityand electron density (ICED), Bent, and Gallagher models (RIBG model) to specifythe ionospheric conditions by ingesting the GPS data from observing stations that are asclose as possible to the observation time and location of the space system for which thecorrections are required. To accurately calculate Faraday rotation corrections, we alsoutilize the raytrace utility of the RIBG model instead of the normal shell model assumptionfor the ionosphere. We use WindSat data, which exhibits a wide range of orientations ofthe raypath and a high data rate of observations, to provide a realistic data set for analysis.The standard single‐shell models at 350 and 400 km are studied along with a newthree‐shell model and compared with the raytrace method for computation time andaccuracy. We have compared the Faraday results obtained with climatological (InternationalReference Ionosphere and RIBG) and physics‐based (Global Assimilation of IonosphericMeasurements) ionospheric models. We also study the impact of limitations in theavailability of GPS data on the accuracy of the Faraday rotation calculations.

Citation: Singh, M., and M. H. Bettenhausen (2011), An accurate and efficient algorithm for Faraday rotation corrections forspaceborne microwave radiometers, Radio Sci., 46, RS4008, doi:10.1029/2010RS004509.

1. Introduction

[2] The ionosphere influences the propagation of radiosignals that traverse through or are reflected by it. Iono-spheric effects are important for a number of applicationsincluding satellite communications, GPS single frequencynavigation, HF over‐the‐horizon radar, satellite altimetryand space‐based radar. Ionospheric effects include absorp-tion, refraction, signal group delay, phase advance, andFaraday rotation of the polarization vector. All of theseeffects are inversely proportional to the square of the fre-quency of the signal. The effects of Faraday rotation onmeasurements from spaceborne microwave radiometerssuch as the Special Sensor Microwave/Imager (SSM/I)[Hollinger et al., 1990], and the TRMM Microwave Imager(TMI; http://trmm.gsfc.nasa.gov/) are generally neglectedbecause the effects are considered to be small at microwavefrequencies. However, recent studies for microwave radio-meters for soil moisture and salinity measurements [Skouet al., 1999; LeVine and Abraham, 2002] and ocean sur-face wind direction measurements [Meissner and Wentz,2006] have shown that Faraday rotation effects can intro-

duce significant errors. Faraday rotation can be importantfor soil moisture and salinity measurements due to the use oflower frequency (L band) measurements. Wind directionretrievals from passive microwave measurements as dem-onstrated by WindSat utilize all four Stokes components[Bettenhausen et al., 2006]. The Stokes parameters are a setof values that describe the polarization state of electro-magnetic radiation and their relationship with brightnesstemperature are defined by Meissner and Wentz [2006].Bettenhausen et al. [2006] have shown that the third Stokesmeasurements must be accurate to about 0.1 K (equivalentto 0.08° Faraday rotation angle at 10.7 GHz) to supportwind direction retrievals because the wind direction signal issmall. The Faraday rotation effect can be written as

Faraday Rotation radiansð Þ ¼ e3

8�2m2c"0 f 2

Zhs

0

B � k Ne ds; ð1Þ

where e and m are electron charge and mass respectively, "0is the electric permittivity of free space, B is the magneticfield vector (Tesla), k is the wave vector along the raypath,Ne is electron density (el/m3), f is the frequency (Hz), andB · k is the dot product of B and k vectors. The integral is upto satellite height. Equation (1) can also be written as

Faraday Rotation radiansð Þ ¼ 2:3646� 104

f 2

Zhs

0

B � k Ne ds; ð2Þ

1Computational Physics Inc., Springfield, Virginia, USA.2Remote Sensing Branch, Remote Sensing Division, Naval Research

Laboratory, Washington, D.C., USA.

Copyright 2011 by the American Geophysical Union.0048‐6604/11/2010RS004509

RADIO SCIENCE, VOL. 46, RS4008, doi:10.1029/2010RS004509, 2011

RS4008 1 of 16

http://dx.doi.org/10.1029/2010RS004509

[3] In addition to the frequency, two other geophysicalquantities which determine the Faraday rotation effect arethe geomagnetic field and the total numbers of electronsalong the raypath. Even though these two geophysicalparameters are measured continuously at specific locations(there are a number of geomagnetic observatories both athigh–latitudes and the equator for the magnetic field andionosonde and GPS receiver stations for the ionosphere) onthe globe, we have to rely on models for both magnetic fieldand ionospheric densities due to the temporal and spatialcoverage of the space systems. Observations are normallyused to update the models to represent real time conditions.[4] For the magnetic field, we use the International Geo-

magnetic Reference Field (IGRF) model (available at http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html) that providesan empirical representation of earth’s magnetic field. Themodel was developed by the International Association ofGeomagnetism and Aeronomy (IAGA) after incorporatingthe ground as well as space‐based observations. The currentIGRF model is used in almost all magnetic intensity cal-culations without any other competing global geomagneticfield model. For any given day, the model provides thegeomagnetic field components as a function of altitude. Inaddition to values that are provided by the model, there arealso small day to day variations in earth’s magnetic field.These variations are measured at a number of observatoriesaround the globe. The index that quantifies the fluctuationsin the horizontal component of magnetic field is known asKp and is available in almost real time at http://rwc.lund.irf.se/rwc/kp/. The index (Kp) can range between 0 and 9 with1 being calm and 5 or more indicating disturbed or stormconditions. The fluctuations in the horizontal components ofmagnetic field for the quiet, disturbed, and storm conditions(Kp index < 5, 5–8, and >8, respectively) are in the range of0–120, 120–500, and >500 nT (nanoTesla), respectively.The normal magnetic field components vary from 30,000 to60,000 nT and in terms of percent variations, the threeconditions (quiet, disturbed and stormy) can be as large as0.04, 1.67 and >1.67 percent of the minimum value (nor-mally near the magnetic equator). To study the frequency ofoccurrence of disturbed or storm conditions, we have ana-lyzed the Kp data for one full year (2003) which represents amoderate year in the solar cycle. The results of the analysisshows that quiet, disturbed, and storm conditions occur93.048, 6.678 and 0.27 percent of the time, respectively.The modeled magnetic field values can be corrected for dayto day variations if the accuracy requirements are of thatorder.[5] The electron density of the ionosphere is a function of

solar activity, magnetic latitude, and local time. It is alsoaffected by the dynamics and interaction of plasma with themagnetic field. The ionosphere can of course be representedby models. There are a number of models described in theliterature that are designed with specific applications inmind. The common climatological models for generalionospheric applications in the literature are InternationalReference Ionosphere (IRI) [Bilitza and Reinisch, 2008] andRIBG [Reilly and Singh, 1997, 2001, 2004; Singh andReilly, 2006]. The IRI model can provide electron density,electron temperature, ion temperature and ion composition,total electron content (TEC), and spread‐F probability. Forgiven location, time, date and the solar conditions, this

model provides the electron density, electron temperature,ion temperature, and ion composition in the altitude rangefrom 50 km to 2000 km and TEC up to altitudes of2000 km. The IRI and the other climatological models arebased on similar databases but in recent years these modelshave developed the capability of ingesting the observationsto update the models. The IRI model can be updated withbottom‐side ionosonde and in situ electron density data butlacks the ability to ingest the abundantly available GPSobservations because of its upper altitude limit of 2000 km.[6] The RIBG (raytrace through the combined models of

ionospheric conductivity and electron density (ICED), Bent,and Gallagher models) model was developed at NavalResearch Laboratory during the 1990s and later upgraded byGeoloc Corporation [Reilly and Singh, 1997]. This modelcombines three ionospheric models for different altituderanges: the Ionospheric Conductivity and Electron Density[Tascione et al., 1988] model that was developed at NOAAand is used to provide the electron density up to 1000 km,the Bent et al. [1975] that is used from 1000 to 3000 km,and the Gallagher et al. [1988] model that is used from3000 km up to GPS altitude. The models are joined in such away that there are no discontinuities at the joints as discussedin more detail by Reilly and Singh [1997]. On the obser-vation side of ionospheric physics, there are hundreds ofdual frequency (L1 = 1575.42 MHz and L2 = 1227.60 MHz)GPS receivers that operate continuously around the globeand the data is collected by a number of networks for easyaccess. The differential path delay term from the two fre-quencies for GPS satellites has components from iono-spheric delay (proportional to TEC), the troposphere, andthe biases for both the GPS satellites and receivers. Anytechnique to infer the ionospheric term from GPS observa-tions has to solve simultaneously for the biases of the GPSreceivers which may also be time dependent. A techniquethat can ingest the GPS data and simultaneously solve forthe specification of the ionosphere and the satellite andreceiver biases has been described in detail by Reilly andSingh [1997]. This technique employs discrete inversetheory’s methodology [Menke, 1989] and is used to solvefor the driving parameters of the RIBG model and the biasesof the GPS receivers and satellites. In this technique, theGPS data is fed into the RIBG model to calculate the drivingeffective sunspot number that will replicate the observedionospheric effect in the GPS data. The outputs of thismethodology are essentially effective sunspot number whichis the driver for the RIBG model and the biases. To test thevalidity of this RIBG technique for the analysis of GPS data,the ionospheric profiles have been compared with inde-pendent observations from a large number of instruments,e.g., the incoherent radar data, world campaign from dif-ferent instruments, and precision location for GPS solutions[Reilly and Singh, 2001, 2004; Singh and Reilly, 2006].[7] IRI and RIBG are updated climatological models.

There are also a number of other ionospheric models (e.g.,GIM, U.S.‐TEC, MIDAS, and EMAD) that have beendeveloped for specific applications [Mandrake et al., 2005].The GAIM (Global Assimilation of Ionospheric Measure-ments) model, on the other hand, is based on a physics‐based ionosphere‐plasmasphere model with the capability ofassimilating ionospheric measurements from a variety ofinstruments. The GAIM model has the capability to assim-

SINGH AND BETTENHAUSEN: FARADAY ROTATION RS4008RS4008

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ilate the TEC measurements from GPS receivers, bottom‐side electron profiles from ionosonde, line‐of‐sight ultravi-olet (UV) emissions, in situ electron density measurementsfrom satellites and TEC measurements from tomographyexperiments. The complete list of observations that can beassimilated by the GAIM model is discussed in detail bySchunk et al. [2004]. The physics‐based model used by theGAIM model is the Utah State University’s time‐dependentionosphere model [Sojka, 1989], which is also known asIonosphere Forecast Model (IFM). The main ionosphericdata assimilation methodology used by this forecast modelis a Kalman filter and is detailed by Scherliess et al. [2004].The GAIM model is run at Air Force Weather Agency(AFWA) and at Naval Research Laboratory (NRL, http://tiraweb.nrl.navy.mil). The output from the NRL site is thenmF2, hmf2, electron density profiles and TEC on a latitudeand longitude grid for both the GAIM model and the IFMmodel. The outputs from the latter model are referred to asthe background parameters.[8] For the purpose of this paper, we will use the orbit,

measurement frequencies, and scan geometry of the Wind-Sat polarimetric radiometer [Gaiser, 2004] to providemeaningful comparisons of Faraday rotation angle compu-tation time and accuracy for different methods. An assess-ment of the impact of Faraday rotation on WindSatmeasurements is given by Meissner and Wentz [2006]. Ouranalysis includes a new technique that considers the iono-sphere consisting of three shells at three altitudes instead ofa single shell. The analysis shows that three shells canprovide accuracy similar to that of a raytrace technique withcomputation time similar to that of a single‐shell model. Theionospheric model used in the algorithm is updated byingesting the GPS data from a receiver that is close to theradiometer measurements. The implications and limitationsof the distances between the radiometer measurementlocation and GPS receiver location are also discussed.

2. Data Analysis

[9] The WindSat instrument used for the present analysisis in sun‐synchronous orbit at around 850 km altitude andcrosses the equator at 18:00 and 06:00 local time (LT).WindSat is a multifrequency polarimetric microwave radi-ometer that operates in discrete bands at 6.8, 10.7, 18.7,23.8, and 37.0 GHz. WindSat has a 6 foot spinning offsetreflector antenna with nominal earth incidence anglesranging from 49.9° to 55.3° for different channels. The 10.7,18.7, and 37.0 GHz channels are fully polarimetric. Ouranalysis focuses on the measurements at 10.7 and 18.7 GHzbecause, as discussed by Meissner and Wentz [2006], theFaraday rotation effects for WindSat are most importantfor the third Stokes measurements at those frequencies.According to equation (2), Faraday rotation essentiallydepends on two terms: B · k and Ne. The first term dependson the magnitude of the magnetic field intensity and theangle between the magnetic field and the wave vector.WindSat measurements are taken in both the forward and aftdirections relative to the velocity vector of the satellite. Thisscanning geometry results in a large azimuthal variation ofthe wave vector which when combined with the magneticfield variation can produce very interesting variations inB · k for a complete satellite orbit. The second term is the

ionospheric term which we shall compute using the RIBGmodel. The RIBG model will be updated with the GPS datafrom a station that is close to the WindSat observations inboth time and location. The two equatorial crossing timesalso make an interesting combination because the iono-spheric contents at these local times are markedly different.Even though the ionosphere is in the decay stage at the18:00 LT crossing, the variations in Faraday rotation havecomponents from the ionosphere and the B · k term. Theequatorial crossing at 06:00 LT, on the other hand, occurswhen the ionospheric content is very close to the minimumand variations in Faraday rotation are mostly due to the B · kterm.[10] Each term in the integral for the calculation of Far-

aday rotation is a function of height. Calculating the productat each height and summing it up to the satellite height canbecome a complex and time consuming process. To mini-mize this complexity and the computing time, it has beencustomary in the literature to assume that the ionosphere isjust a shell at either 350 or 400 km (also known as theionospheric pierce point). The magnetic field and the wavevector are also calculated at the pierce point. The resultantproduct at the pierce point is multiplied by the obliquityfactor (sec c) to calculate Faraday rotation. We shall insteaduse the actual raytrace capability of the RIBG model tocalculate Faraday rotation. To illustrate the process forFaraday rotation calculations for the WindSat orbit with fullraytrace, we first choose the ascending part of the orbit on24 September 2003 that is shown on the left of Figure 1. Forthe ionospheric specifications for this orbit time period, weanalyze the GPS data from the station that is closest to thisparticular orbit. The closest station with GPS data availablefor that day is the Bermuda station. Although at presentthere are many more stations that are operating close to thisorbit location, in 2003, Bermuda was the only GPS stationreporting data near the orbit in time and location. The GPSdata was analyzed for two hours (16:00 to 18:00 LT) toderive the effective sunspot number and solve for the biases.The two hours of data contains enough information tocomplete the analysis and this time period is close to theorbit crossing time at the equator. The value for the effectivesunspot number is 78.5 which means that the observed localionosphere can be reproduced using this sunspot number inthe RIBG model. The published sunspot number for thisdate by NOAA’s National Geophysical Data Center is 64and for the subsequent day (25 September 2003) the pub-lished number is 77. The published sunspot number is onenumber for the whole globe for one day whereas effectivesunspot number is more local in nature for the time period ofthe GPS data analysis and the location around the receiveritself.[11] To calculate the Faraday rotation for the 24 September

2003 orbit, we run the RIBG model with an effective sun-spot number of 78.5 at 22:00 UT (equatorial crossing localtime). Electron density profiles are created at each 10 kminterval from 80 to 500 km, and at each 50 km interval from500 km to the satellite altitude. Although for this particularexample the ionospheric density profiles are created at oneuniversal time, ionospheric profiles can be calculated peri-odically when processing satellite data. Similarly the mag-netic field components (Bx, By, and Bz) are calculated fromthe IGRF10 model at the same height intervals as those of


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the density profile. The wave normal components are alsocalculated at all the height intervals by assuming a straightline raypath, i.e., no bending due to the ionosphere which isa valid assumption for the microwave frequency rangebetween the satellite and the earth surface. This is done foreach scan of the antenna beam up to the satellite altitudeusing Simpson’s rule for the integration. The result for a

10.7 GHz frequency channel is shown in Figure 2 for theascending part of the orbit. The results in Figure 2 showlarge variations in the Faraday rotation values and thesevariations are more due to the dot product of magnetic fieldwith the wave vector than the changes in total electroncontent along the raypath. To illustrate this point moreclearly, we choose the data at two points close to 20° lati-

Figure 2. Faraday rotation angle for the 10.7 GHz frequency channel of WindSat satellite for ascendingorbit in the Figure 1. The effective sunspot was calculated from the Bermuda GPS data for 24 September2003 and the ionosphere was created by RIBG model at 21:00 UT (18:00 LT equatorial crossing time).The Faraday rotation angle was calculated with full raytrace technique of RIBG model. The stars show thevariations in Faraday rotation angle during a single sweep in a forward scan.

Figure 1. The representative ascending and descending orbits for WindSat satellite. The equatorialcrossing local times of orbits are 18:00 and 06:00 LT, respectively. The inclination for the satellite is81.3°, and altitude varies from 830 to 850 km.


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tude and marked with an asterisk on the forward scan por-tion of the antenna. The Faraday rotation varies from 0.123to 0.2045 degrees (a factor of 1.6626) between the twopoints. The variation in the total electron portion is only1 TEC unit (from 45.70 to 46.71 TEC units), and theobliquity factors (sec c) for the two points are 1.4490 and1.4558. The obliquity factor is essentially related to theraypath length. The B · k term for these two points variesfrom 15680 to 25105 nT (factor of 1.595). These variationstaken separately show that 95% of the variation in Faradayrotation is due to the variation of the B · k factor within onescan. The results also show the bifurcation in the northernhemisphere that can be associated with the forward and theaft portions of the scan.[12] The above analysis in Figure 2 is for the ascending

portion of the orbit that has an 18:00 LT equatorial crossingtime so that the ionospheric electron density is early in thedecaying stage. The other half of the orbit (right side ofFigure 1) is the descending portion with a 06:00 LT equa-torial crossing. The Faraday rotation calculation process isrepeated for the descending portion of the orbit with theresults shown in Figure 3. Almost all of the variation is fromthe B · k term since, as we mentioned earlier, the iono-spheric electron density at 6am local time is close to theminimum possible for the day. Another feature that is dif-ferent from the ascending orbit is that we can distinctly seethe forward and aft portion of the orbit. The maximumFaraday rotation values for the descending portion of theorbit are at high latitudes due to higher values of the mag-netic field and the wave normal product. Both the ascendingand descending orbit analysis show that the magnetic field

term plays an important role in the determination of theFaraday rotation correction.[13] The time required to compute Faraday rotation for

half of the orbit for one channel is about 14 s using a2.4 GHz Intel Pentium D computer. For WindSat this cal-culation should be completed for five distinct propagationpaths covering 10 channels (6.8, 10.7 and 18.7 GHz verticaland horizontal polarizations and 10.7 and 18.7 GHz ±45 degree polarizations). The required computational timewould exceed the time required for geolocation of the mea-surements; so, it is desirable to reduce the computationalrequirements.[14] One common method to save computing time is the

use of a shell model for the ionosphere instead of the ray-trace method. The shell model assumes that the entire ion-osphere is in the form of a shell at either 300 or 400 km(ionospheric pierce point, IPP). In the published literature,authors have used different shell heights: 300 km [Yueh,2000], 350 km [Bishop et al., 2009; LeVine and Abraham,2002], and 400 km [Komjathy et al., 2005; Mannucci et al.,1998]. To calculate the Faraday rotation, the vertical totalelectron content at the pierce point is scaled to the satelliteby the obliquity factor (sec c) and the B · k term is alsocalculated at the IPP. We calculate the total electron contentfrom RIBG up to the satellite altitude on a latitude/longitudegrid with the effective sunspot number for the observationtime. The results from the shell methodology are shown inFigures 4a and 4b for the shells at 400 and 350 km, respec-tively, as percent variation from the raytrace method for theascending portion of the orbit. The values for the percentvariation that are plotted are derived using the following:

Figure 3. Faraday rotation angle for the descending portion of the orbit in Figure 1 for the 10.7 GHzfrequency channel. The equatorial crossing time is at 06:00 LT when ionosphere is almost nonexistent.

Faraday rotation with raytraceð Þ � Faraday rotation assuming ionosphere as shell at IPPð ÞFaraday rotation with raytraceð Þ � 100:


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[15] The differences between the shell model and raytracecan vary from 7% to 12% in the equatorial latitudes forshells at 350 and 400 km, respectively. The difference canrange up to 15% at high latitudes although this is primarilybecause the absolute rotation is lower. The computation time

using the shell model is reduced from 14 s to 4 s which isabout 28% of the time taken by the raytrace method.[16] The shell model reduces the computation time but at

the expense of reduced accuracy. We have expanded theshell concept further by adding multiple shells instead of the

Figure 4a. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as single shell at 400 km. The results are for the ascending portion ofthe orbit shown in Figure 1.

Figure 4b. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as single shell at 350 km for the ascending portion of the orbit shownin Figure 1.


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usual single shell. We tested this concept with multipleshells at different heights and have chosen to use three shellsat altitudes of 200, 400, and 650 km. Our tests showed thattwo shells did not provide sufficient accuracy and that fouror more shells did not significantly improve the accuracyover that achieved with the three‐shell model. For the three‐shell model technique, we calculate values of TEC thatrepresent the electron content in the ionosphere at each ofthe three‐shell heights. The shell at 200 km contains theelectron content of the ionosphere up to 300 km, the shell at400 km contains electron content from 300 to 500 km, andthe shell at 650 km contains the electron content above500 km to satellite altitude. B · k is also evaluated at eachshell height. The latter part is multiplied with correspondingoblique total electron content to obtain three components ofFaraday rotation that are summed up for the total rotation.The result is shown in Figure 4c as the difference betweenthe raytrace and the three‐shell model on percentage basisand the errors in three‐shell model are less than 3%. Com-putation times for the raytrace, single‐shell, and three‐shellmodels are 14, 4, and 5.5 s, respectively.[17] Next we repeat the same analysis for the single‐shell

model versus the three‐shell model for the descendingportion of the orbit that has almost no ionospheric variation.The analysis for the descending orbit for the models withsingle shells at 350 km, 400 km, and for the three‐shellmodel is performed in the same way as for the ascendingorbit. The results are displayed in Figures 5a–5c in terms ofthe absolute difference instead of percent basis becausesome of the actual values are near zero. The comparisonsbetween the different shell models show that the three‐shellmodel again provides better accuracy than the single‐shellmodels.

[18] The analysis that we have discussed above is for theday 24 September 2003 when the solar activity as measuredby sunspot number was in the moderate range. Furthermore,the satellite equatorial crossing times were either 18:00 or06:00 LT when electron densities have declined from thepeak or are near the daily minimum, respectively. To test thethree‐shell technique at extreme ionospheric conditions, wetest the three‐shell model using the electron density profilecalculated for a sunspot number of 150 and the local time of13:30. This is the time when the electron density peaks as afunction of local time. Such solar conditions occur only atthe peak of the solar cycle (within about a one year intervalof the 11 year solar cycle). Faraday rotation results for theextreme conditions for the ascending orbit are shown inFigure 6. The calculations for Figure 6 were done with theraytrace method and the comparison of the Faraday rotationwith the moderate solar conditions used for Figure 2 resultsshows that the rotation is increased by about a factor of two.The overall behavior of the results is similar to that shown inFigure 2 with a notable difference in the equatorial anomalyregion. The strength and width of the equatorial anomaly isa function of sunspot number and local time. The differ-ences between results for raytrace and results for the shellmodels are shown in Figures 7a–7c. The results are derivedby the same method that was used for the results shown inFigure 4. The percent differences for the above three sce-narios are 15%, 11%, <5% respectively. These values arevery similar to those of Figures 4 except for the differencesat southern high latitudes. We also can see some peculiarmultiple bifurcation‐like behavior in the figures for thesouthern hemisphere. This may be the result of ionosphericstructure at high latitudes or possible break down of themodel at extreme conditions. The data tables that are used in

Figure 4c. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as three shells at 200, 400, and 650 km for the ascending portion of theorbit shown in Figure 1.


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the ionospheric models are based on ground observations.The data tables are at sunspot numbers of 0 and 100. Themodels may not work well for the extreme conditions oneither side. We have seen at the extreme low solar activity,the effective sunspot number values have to be negative torecreate the TEC data from GPS observations [Reilly andSingh, 1997]. The conditions that we are depicting in thisexample can happen for only a few days in an 11 year solar

cycle, but the example demonstrates the accuracy of thethree‐shell model over the single‐shell model.

3. Error Analysis

[19] In this section we discuss the absolute accuracy ofthe methodology for the Faraday rotation calculations asdescribed in the preceding section of this paper. The inac-

Figure 5a. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphereand the ionosphere considered as single at 400 km for the descending portion of the orbit shown inFigure 1.

Figure 5b. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphereand the ionosphere considered as single at 350 km for the descending portion of the orbit shown inFigure 1.


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curacies in the calculation of the Faraday rotation angle forspaceborne microwave radiometers can be due to the mag-netic field or ionospheric modeling errors or errors in theintegration of their product. The geomagnetic field andionospheric modeling errors can be reduced by using geo-magnetic measurements and GPS reference station data that

are close in time to the observations. We update the mag-netic field model monthly. The day to day variability in themagnetic field which amounts to much less than 1% asdiscussed in the introduction part can be corrected depend-ing upon the accuracy constraints.

Figure 5c. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphereand the ionosphere considered as three shells at 200, 400, and 650 km for the descending portion of theorbit shown in Figure 1.

Figure 6. Faraday rotation angle for the ascending portion of Figure 1 for 10.7 GHz frequency channelwith raytrace technique. The ionospheric conditions represent extreme values with sunspot number of150 and at 13:30 LT when ionosphere is at its peak.


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[20] There are also errors in the ionospheric portion of theequation due to inaccuracies of the models. The resultsshown in the previous section were generated using theupdated RIBG model. As discussed earlier in the introduc-tion, there also exist a number of other ionospheric modelsthat can be used for similar analyses. In this section, wecompare the Faraday rotation calculated using IRI andGAIM ionospheric models with the results obtained usingthe RIBG model. The IRI model can be run on the Internetor the Fortran source code can be compiled and run on a

personal computer. The GAIM model requires considerablehardware power and at present is run either at AFWA orNRL. For the RIBG and IRI models, the latitude/longitudegrid is adjustable but for the global GAIM the grid is fixedwith the longitude grid is set at 15° and a variable latitudegrid of 4.667° for low and middle latitudes (<±67°) and 3°for high latitudes. It may be possible in the future to runthe GAIM model for regional and local ionosphere, we willuse global GAIM output which is currently available fromNRL site.

Figure 7a. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as single shell at 400 km as related to Figure 6 conditions.

Figure 7b. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as single shell at 350 km as related to Figure 6 conditions.


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Figure 8a. Faraday rotation angle for ionospheric generated by three models ((top) IRI, (middle) RIBG,and (bottom) GAIM) for the ascending orbit of Figure 1 for the day 26 September 2003 and 18:00 LTequatorial crossings time.

Figure 7c. Percent difference in Faraday rotation angles between full raytrace through the ionosphereand the ionosphere considered as three shells at 200, 400, and 650 km as related to Figure 6 conditions.


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[21] The analysis presented in the previous section wasbased on the updated version of RIBG data for one day(24 September 2003) with the effective sunspot numberbased on GPS data from the Bermuda station (the stationclosest to the orbit). For this particular day GAIM modelresults are available on the NRL site for the IFM modelonly while both GAIM and IFM outputs are available for26 September 2003. Therefore we will compare the threemodels on 26 September 2003 for the ascending portion oforbit. The effective sunspot number for 26 September 2003for the RIBG after ingestion of Bermuda GPS data is 76.8and the Faraday rotation results from RIBG ionosphere areshown in Figure 8a (middle).The corresponding Faradayrotation results with ionosphere generated by IRI are shownin Figure 8a (top). Similar results for Faraday rotation usingthe ionosphere generated by the GAIM model for the sameday and time are shown Figure 8a (bottom).[22] The Faraday rotation results for both the IRI and

RIBG models (Figure 8a) show very similar behavior as afunction of latitude except that the IRI model results areconsistently lower. A comparison between the results fromthe RIBG and the GAIM shows that the maximum valuesfor Faraday rotation angles are very close to each other.However, there is a distinct difference between the twomodels at equatorial latitudes. The equatorial anomaly ismuch more pronounced on the two sides of equator for boththe RIBG and the IRI models while the anomaly on thesouth side of the equator is less prominent for the GAIMmodel. Figure 8b shows the differences in Faraday rotationbetween the different models in more detail. This plot is thesubtraction of Faraday rotations results for the IRI andGAIM models from the RIBG results. The differencebetween the RIBG and the IRI results show that the RIBG ishigher by 0.01°–0.03° at middle and high latitudes but thedifference can be as high a 0.04° at low latitudes. Thesunspot number used by IRI for this particular day is 59.1and the effective sunspot number inferred by RIBG using

the GPS reference station is 76.8. The Faraday rotationdifference is primarily due to the difference in the sunspotnumber used by the two models. The sunspot numbersused by the various ionospheric models and published bythe National Geophysical Data Center for the two days(24 September 2003 and 26 September 2003) are shown inTable 1. The difference between the RIBG and GAIMmodels shows that at low latitude (−20°–15°), the GAIMresults are consistently higher. The RIBG model lacks theingestion from any equatorial GPS station for this particularexample while GAIM assimilates the data from equatoriallatitudes. The Faraday rotation results from the three modelsare different at high latitudes especially in the northernhemisphere. No ionospheric data observed at high latitudesis used for the update of either RIBG or GAIM modelswhich means that the climatological and physics‐basedmodels are inherently different at high latitudes.[23] Meissner and Wentz [2006] have calculated the

Faraday rotation angle with the ionosphere that was gener-ated by the IRI model and concluded that rotation error ofgreater than >0.15° will add appreciable error to the thirdStoke parameter and subsequently degrade the accuracy ofwind vector retrieval. For this particular orbit, this thresholdis reached 0.01 percent of the time with the IRI model, 11.44percent of the time for the GAIM model and 8.53 percent forthe RIBG model.[24] These results are representative of the ionosphere

during moderate solar activity conditions and the equatorial

Figure 8b. Faraday rotation angle difference between different ionospheric models (IRI, RIBG, andGAIM) from Figure 8a results.

Table 1. Sunspots for 2 Days Used by Different Models

Model/Sunspot 24 September 2003 26 September 2003

NOAA 64 77RIBG 78.5 76.8IRI 59.2 59.1IFM 74.3 71.8


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crossing time of 18:00 LT, when the electron content of theionosphere has decayed from its maximum of the day.Faraday rotation effects will be greater during times withhigh solar activity. The same will be true when the electroncontent is at its peak for the day. To study the local timeeffect on the Faraday rotation results, we calculate the TECfrom the GAIM model for the same day at 2pm local time.The threshold limit for the degradation as set by Meissnerand Wentz [2006] will be reached 16.4% of the timeinstead of 11.4% at 6pm. The effect of extreme ionosphericconditions on this threshold can be calculated from Figure 6

results and the 0.15° threshold is exceeded 76.35% of thetime under such conditions.[25] To minimize the error from the ionospheric models,

the assimilating models should have better representation ofthe existing ionosphere than the nonassimilating models. Inthe assimilating category, the GAIM model has the capa-bility of ingesting a large variety of ionospheric observa-tions on the global scale. The RIBG model assimilates onlythe GPS data and also just one GPS receiver at a time buthas the flexibility of selecting the GPS stations of choice fordata ingestion.

Figure 9a. The GPS stations locations that are used in the study of accuracy degradation versus thedistance.

Figure 9b. Differences of Faraday rotation angles between pairs of stations versus the distance betweenthe pair. The vertical bars show the range of differences between each pair of stations over the completedata set in the orbit. Also shown are the mean, median, and 3 sigma values for each pair of stations.


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[26] The assimilating ionospheric model requires theobserved ionospheric data to ingest. The most abundantionospheric data that can be assimilated or ingested intomodels is the GPS data which is readily available on aglobal scale from a number of data centers in near real time.The availability of this GPS data has improved rapidly overthe last decade; this trend is continuing. For the Americas,the CORS network (http://www.ngs.noaa.gov/cors) com-piles data from a few hundred stations that cover NorthAmerica, Hawaii, Guam, and the Caribbean, and isexpanding into South America. The GPS data from thisnetwork is available on an hourly basis on the same day andis typically available to users within two hours of theobservation time. The European network of GPS receiversnot only covers most of Europe (http://www.epncb.oma.be/_dataproducts/data_access/) but also has data from islandsin the Atlantic Ocean and the North Sea. The Australiannetwork covers Australia with stations on the coastal areas(http://www.ga.gov.au/earth‐monitoring.html) and aroundthe entire continent. Even though data from more and moreGPS stations is becoming available there are still gaps inthe global coverage of the ionospheric observations. For

example, at the present time there is no coverage in theSouth Atlantic Ocean. Data gaps may also occur becausesome stations may be down temporarily or due to problemswith the data delivery system. In the presence of data gaps,models are obtained by either interpolation or with the filtersfrom the ionosphere generated in the other parts of theglobe.[27] Another error can also arise if there is large distance

between the ingested data source point and the ionosphereapplication point. To include different scenarios underwhich ionospheric data may be unavailable at the applica-tion point, we have conducted a study of varying distancesfrom reference GPS stations versus accuracy degradation.We take the GPS data from one station and predict theionosphere at a second station and compare with the iono-sphere predicted by the GPS receiver of the second station.Figure 9a represents the stations that we have included in thecase study. The latitude/longitude difference between thestations can be more than 40 degrees and physical distancesbetween pairs of stations can vary from as little as 900 km toa maximum of 6000 km. We have analyzed the GPS datathat was available for 24 September for three years (2001,

Figure 10. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphereand the ionosphere considered as single shell at 400 km, single shell at 350 km, and three‐shell model.


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2003, and 2007). These three years represent high, moderateand low solar activity levels. The GPS data was available foronly three stations for the year 2001 while for the other twoyears it was available from six stations. We calculate theFaraday rotation with the sunspot number based on differenteffective sunspot numbers at two stations and associate itwith the distance between the stations. Figure 9b shows theresulting differences in Faraday rotation in degrees versusdistance at 10.7 GHz for the ascending part of the orbit inFigure 1.The effective sunspot number was determined fromthe GPS data for all the stations between 1900 to 2100 UTfor the respective days. The vertical bars show the range ofdifferences between each pair of stations over the completedata set in the orbit. The plot also shows the mean, median,and 3 sigma values of the differences for all stations. Theseresults show that we can expect accuracy in Faraday rotationto be better than 0.005° as long as we have GPS station datawithin 1500 km of the orbit based on the small sample ofdata that we have analyzed.[28] Another source of error in the Faraday rotation can be

due to the method used to integrate the product of themagnetic field and the electron content along the propaga-tion path. The results of Figures 4a–4c can also be repre-sented in terms difference in degrees instead of percentoccurrences and are shown in Figure 10. These differencesin Faraday rotation angle can be as high as 0.02°, 0.01°, and0.002° for single‐shell models at 400 km and 350 km andthe three‐shell model respectively. We can also computeerrors for extreme ionospheric conditions by combining thepercent occurrences of Figures 7a–7c with Figure 6 Faradayrotation values. The errors for extreme ionospheric condi-tions can be as high as 0.041°, 0.024°, and 0.009° for the400 km shell, the 350 km shell, and three‐shell modelsrespectively. The microwave analyses done by other authorsas mentioned in this paper have used a single shell at dif-ferent heights from 300 to 400 km and our analysis canquantify the errors. Bettenhausen et al. [2006] have alsoshown from WindSat data analysis that modeling and cali-bration errors for the third and fourth Stokes parameters areabout 0.1 K for brightness temperatures which translates into 0.08° at 10.7 GHz. The combination of Faraday rotationerrors due to ionospheric model errors approximated by thedifferences shown in Figure 8b with the single‐shell modelerrors shown above is of the same order. Use of the three‐shell model along with an assimilating ionospheric modelcan reduce the errors significantly. The full raytrace methodcould be used to further reduce the errors when there are nocomputational time constraints.

4. Conclusions

[29] We have studied Faraday rotation for microwaveradiometry using WindSat as an example but the analysismethod can be applied to other missions. The Faradayrotation term although small at microwave frequencies canstill affect the accuracy of the final products. We haveindentified the errors in the calculations for Faraday rotationthat relate to the contribution from the ionospheric term. Theerrors can arise from the magnetic field component, iono-spheric modeling, distance between the reference source fordata ingestion and the application point, and assuming theionosphere as a single shell. We have discussed in the paper

methods to reduce these errors. For the magnetic fieldcomponent, the magnetic field model should be updated atleast once a month. The assimilating ionospheric models cangenerate better ionosphere than nonassimilating models. TheGAIM model has the advantage of ingesting the ionosphericdata from a large number of instruments while the RIBGmodel can choose the best GPS station location. The iono-spheric corrections using a GPS station within 1500 km ofthe orbit provide Faraday rotation accuracies to better than0.005° at 10.7 GHz. For the distances greater than 1500 km,this analysis can be used to infer the error amount. Single‐shell models have been used for microwave applications withpierce heights ranging from 300 km [Yueh, 2000] to 400 km[Le Vine and Abraham, 2002]. Ionospheric researchers havealso used single shells at either 350 km [Bishop et al., 2009]or at 400 km [Komjathy et al., 2005; Mannucci et al. 1998].We have shown that the three‐shell approach provides muchbetter accuracy than the single‐shell models with similarcomputation time.

ReferencesBent, R. B., S. K. Llewellyn, G. Nesterczuk, and P. E. Schmid (1975), Thedevelopment of a highly‐successful worldwide empirical ionosphericmodel and its use in certain aspects of space communications and world-wide total electron content investigations, in Effect of the Ionosphere onSpace Systems andCommunications, edited by J.M.Goodman, pp. 13–28,Nav. Res. Lab., Washington, D. C.

Bettenhausen, M. H., C. K. Smith, R. M. Bevilacqua, and N. Y. Wang(2006), P.W, Gaiser, and S. Cox, A nonlinear optimization algorithmfor Windsat wind retrievals, IEEE Trans. Geosci. Remote Sens., 44,597–610, doi:10.1109/TGRS.2005.862504.

Bilitza, D., and B. Reinisch (2008), International Reference Ionosphere2007: Improvements and new parameters, Adv. Space Res., 42, 599–609,doi:10.1016/j.asr.2007.07.048.

Bishop, G. J., J. A. Secan, and S. H. Delay (2009), GPS TEC and the plas-masphere: Some observations and uncertainties, Radio Sci., 44, RS0A26,doi:10.1029/2008RS004037.

Gaiser, P. (2004), The WindSat spaceborne polarimetric microwave radi-ometer: Sensor description and early orbit performance, IEEE Trans.Geosci. Remote Sens., 42, 2347–2361, doi:10.1109/TGRS.2004.836867.

Gallagher, D. L., P. D. Craven, and R. H. Comfort (1988), An empiricalmodel of Earth’s plasmasphere, Adv. Space Res., 8, 15–24, doi:10.1016/0273-1177(88)90258-X.

Hollinger, J. L., L. Pierce, and G. A. Poe (1990), SSM/I instrument evalu-ation, IEEE Trans. Geosci. Remote Sens., 28, 781–790, doi:10.1109/36.58964.

Komjathy, A., L. Sparks, B. D. Wilson, and A. J. Mannucci (2005), Auto-mated daily processing of more than 1000 ground based GPS receiversfor studying intense ionospheric storms, Radio Sci., 40, RS6006,doi:10.1029/2005RS003279.

LeVine, D. M., and S. Abraham (2002), The effect of the ionosphere onremote sensing of sea surface salinity from space: Absorption and emis-sion at L band, IEEE Trans. Geosci. Remote Sens., 40, 771–782,doi:10.1109/TGRS.2002.1006342.

Mandrake, L., B. Wilson, C. Wang, G. Hajj, A. Mannucci, and X. Pi(2005), Daily performance of the JPL/USC global assimilation iono-spheric model, paper presented at 11th International Ionospheric EffectsSymposium, Off. of Nav. Res., Alexandria, Va., 3–5 May.

Mannucci, A., B. Wilson, D. Yuan, C. Ho, U. Lindqwister, and T. Runge(1998), A global mapping technique for GPS derived ionospheric totalelectron content measurements, Radio Sci., 33, 565–582, doi:10.1029/97RS02707.

Meissner, T., and F. J. Wentz (2006), Polarization rotation and the thirdStokes parameter: The effects of spacecraft attitude and Faraday rotation,IEEE Trans. Geosci. Remote Sens., 44, 506–515, doi:10.1109/TGRS.2005.858413.

Menke, W. (1989), Geophysical Data Analysis: Discrete Inverse Theory,Academic, San Diego, Calif.

Reilly, M., and M. Singh (1997), Ionospheric specification from GPSdata and the RIBG ionospheric propagation model, Radio Sci., 32,1671–1679, doi:10.1029/97RS00842.


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Reilly, M., and M. Singh (2001), GPS data analysis near Puerto Ricofor World Day campaigns, Radio Sci., 36, 343–350, doi:10.1029/1999RS002422.

Reilly, M., and M. Singh (2004), Electron density height profiles from GPSreceiver data, Radio Sci., 39, RS1S16, doi:10.1029/2002RS002830.

Scherliess, L., R. W. Schunk, J. J. Sojka, and D. C. Thompson (2004),Development of a physics‐based reduced state Kalman filter for theionosphere, Radio Sci., 39, RS1S04, doi:10.1029/2002RS002797.

Schunk, R. W., et al. (2004), Global Assimilation of Ionospheric Measure-ments (GAIM), Radio Sci., 39, RS1S02, doi:10.1029/2002RS002794.

Singh, M., and M. H. Reilly (2006), Improved positioning by addition ofatmospheric corrections to local area differential GPS, Radio Sci., 41,RS5S29, doi:10.1029/2005RS003339.

Skou, N., B. Laursen, and S. Sobjaerg (1999), Polarimetric radiometer con-figurations: Potential accuracy and sensitivities, IEEE Trans. Geosci.Remote Sens., 37, 2165–2171, doi:10.1109/36.789613.

Sojka, J. J. (1989), Global scale physical model of the F region, Rev.Geophys., 27, 371–403, doi:10.1029/RG027i003p00371.

Tascione, T. F., H. W. Kroehl, R. Creiger, J. W. Freeman, R. A. Wolf, R. W.Spiro, R. V. Hilmer, J. W. Shade, and B. A. Hausman (1988), New iono-spheric andmagnetospheric specificationmodels,Radio Sci., 23, 211–222,doi:10.1029/RS023i003p00211.

Yueh, S. (2000), Estimates of Faraday rotation with passive microwavepolarimetry for microwave remote sensing of earth surfaces, IEEE Trans.Geosci. Remote Sens., 38, 2434–2438, doi:10.1109/36.868900.

M. H. Bettenhausen, Remote Sensing Branch, Remote Sensing Division,Naval Research Laboratory, Code 7223, 4555 Overlook Ave. SW,Washington, D.C. 20375, USA.M. Singh, Computational Physics Inc., 8001 Braddock Rd., Ste. 201,

Springfield, VA 22151, USA. ([email protected])


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an accurate and efficient algorithm for faraday rotation corrections for spaceborne microwave...

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