An accurate and efficient algorithm for Faraday rotation corrections for spaceborne microwave radiometers

Transcription

1 RADIO SCIENCE, VOL. 46,, doi: /2010rs004509, 2011 An accurate and efficient algorithm for Faraday rotation corrections for spaceborne microwave radiometers Malkiat Singh 1 and Michael H. Bettenhausen 2 Received 27 August 2010; revised 6 May 2011; accepted 23 May 2011; published 25 August [1] Faraday rotation changes the polarization plane of linearly polarized microwaves which propagate through the ionosphere. To correct for ionospheric polarization error, it is necessary to have electron density profiles on a global scale that represent the ionosphere in real time. We use raytrace through the combined models of ionospheric conductivity and electron density (ICED), Bent, and Gallagher models (RIBG model) to specify the ionospheric conditions by ingesting the GPS data from observing stations that are as close as possible to the observation time and location of the space system for which the corrections are required. To accurately calculate Faraday rotation corrections, we also utilize the raytrace utility of the RIBG model instead of the normal shell model assumption for the ionosphere. We use WindSat data, which exhibits a wide range of orientations of the 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 new three shell model and compared with the raytrace method for computation time and accuracy. We have compared the Faraday results obtained with climatological (International Reference Ionosphere and RIBG) and physics based (Global Assimilation of Ionospheric Measurements) ionospheric models. We also study the impact of limitations in the availability 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 for spaceborne microwave radiometers, Radio Sci., 46,, doi: /2010rs Introduction 1 Computational Physics Inc., Springfield, Virginia, USA. 2 Remote Sensing Branch, Remote Sensing Division, Naval Research Laboratory, Washington, D.C., USA. Copyright 2011 by the American Geophysical Union /11/2010RS [2] The ionosphere influences the propagation of radio signals that traverse through or are reflected by it. Ionospheric effects are important for a number of applications including satellite communications, GPS single frequency navigation, HF over the horizon radar, satellite altimetry and space based radar. Ionospheric effects include absorption, refraction, signal group delay, phase advance, and Faraday rotation of the polarization vector. All of these effects are inversely proportional to the square of the frequency of the signal. The effects of Faraday rotation on measurements from spaceborne microwave radiometers such as the Special Sensor Microwave/Imager (SSM/I) [Hollinger et al., 1990], and the TRMM Microwave Imager (TMI; are generally neglected because the effects are considered to be small at microwave frequencies. However, recent studies for microwave radiometers for soil moisture and salinity measurements [Skou et al., 1999; LeVine and Abraham, 2002] and ocean surface wind direction measurements [Meissner and Wentz, 2006] have shown that Faraday rotation effects can introduce significant errors. Faraday rotation can be important for soil moisture and salinity measurements due to the use of lower frequency (L band) measurements. Wind direction retrievals from passive microwave measurements as demonstrated by WindSat utilize all four Stokes components [Bettenhausen et al., 2006]. The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation and their relationship with brightness temperature are defined by Meissner and Wentz [2006]. Bettenhausen et al. [2006] have shown that the third Stokes measurements must be accurate to about 0.1 K (equivalent to 0.08 Faraday rotation angle at 10.7 GHz) to support wind direction retrievals because the wind direction signal is small. The Faraday rotation effect can be written as e 3 Faraday Rotation ðradiansþ ¼ 8 2 m 2 c" 0 f 2 Z hs 0 B k N e ds; where e and m are electron charge and mass respectively, " 0 is the electric permittivity of free space, B is the magnetic field vector (Tesla), k is the wave vector along the raypath, N e is electron density (el/m 3 ), f is the frequency (Hz), and B k is the dot product of B and k vectors. The integral is up to satellite height. Equation (1) can also be written as Faraday Rotation ðradiansþ ¼ 2: f 2 Z hs 0 B k N e ds; ð1þ ð2þ 1of16

2 [3] In addition to the frequency, two other geophysical quantities which determine the Faraday rotation effect are the geomagnetic field and the total numbers of electrons along the raypath. Even though these two geophysical parameters are measured continuously at specific locations (there are a number of geomagnetic observatories both at high latitudes and the equator for the magnetic field and ionosonde and GPS receiver stations for the ionosphere) on the globe, we have to rely on models for both magnetic field and ionospheric densities due to the temporal and spatial coverage of the space systems. Observations are normally used to update the models to represent real time conditions. [4] For the magnetic field, we use the International Geomagnetic Reference Field (IGRF) model (available at that provides an empirical representation of earth s magnetic field. The model was developed by the International Association of Geomagnetism and Aeronomy (IAGA) after incorporating the ground as well as space based observations. The current IGRF model is used in almost all magnetic intensity calculations without any other competing global geomagnetic field model. For any given day, the model provides the geomagnetic field components as a function of altitude. In addition to values that are provided by the model, there are also small day to day variations in earth s magnetic field. These variations are measured at a number of observatories around the globe. The index that quantifies the fluctuations in the horizontal component of magnetic field is known as Kp and is available in almost real time at se/rwc/kp/. The index (Kp) can range between 0 and 9 with 1 being calm and 5 or more indicating disturbed or storm conditions. The fluctuations in the horizontal components of magnetic field for the quiet, disturbed, and storm conditions (Kp index < 5, 5 8, and >8, respectively) are in the range of 0 120, , and >500 nt (nanotesla), respectively. The normal magnetic field components vary from 30,000 to 60,000 nt and in terms of percent variations, the three conditions (quiet, disturbed and stormy) can be as large as 0.04, 1.67 and >1.67 percent of the minimum value (normally near the magnetic equator). To study the frequency of occurrence of disturbed or storm conditions, we have analyzed the Kp data for one full year (2003) which represents a moderate year in the solar cycle. The results of the analysis shows that quiet, disturbed, and storm conditions occur , and 0.27 percent of the time, respectively. The modeled magnetic field values can be corrected for day to day variations if the accuracy requirements are of that order. [5] The electron density of the ionosphere is a function of solar activity, magnetic latitude, and local time. It is also affected by the dynamics and interaction of plasma with the magnetic field. The ionosphere can of course be represented by models. There are a number of models described in the literature that are designed with specific applications in mind. The common climatological models for general ionospheric applications in the literature are International Reference Ionosphere (IRI) [Bilitza and Reinisch, 2008] and RIBG [Reilly and Singh, 1997, 2001, 2004; Singh and Reilly, 2006]. The IRI model can provide electron density, electron temperature, ion temperature and ion composition, total electron content (TEC), and spread F probability. For given location, time, date and the solar conditions, this model provides the electron density, electron temperature, ion temperature, and ion composition in the altitude range from 50 km to 2000 km and TEC up to altitudes of 2000 km. The IRI and the other climatological models are based on similar databases but in recent years these models have developed the capability of ingesting the observations to update the models. The IRI model can be updated with bottom side ionosonde and in situ electron density data but lacks the ability to ingest the abundantly available GPS observations 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 Naval Research Laboratory during the 1990s and later upgraded by Geoloc Corporation [Reilly and Singh, 1997]. This model combines three ionospheric models for different altitude ranges: the Ionospheric Conductivity and Electron Density [Tascione et al., 1988] model that was developed at NOAA and 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 from 3000 km up to GPS altitude. The models are joined in such a way that there are no discontinuities at the joints as discussed in more detail by Reilly and Singh [1997]. On the observation side of ionospheric physics, there are hundreds of dual frequency (L1 = MHz and L2 = MHz) GPS receivers that operate continuously around the globe and the data is collected by a number of networks for easy access. The differential path delay term from the two frequencies for GPS satellites has components from ionospheric delay (proportional to TEC), the troposphere, and the biases for both the GPS satellites and receivers. Any technique to infer the ionospheric term from GPS observations has to solve simultaneously for the biases of the GPS receivers which may also be time dependent. A technique that can ingest the GPS data and simultaneously solve for the specification of the ionosphere and the satellite and receiver biases has been described in detail by Reilly and Singh [1997]. This technique employs discrete inverse theory s methodology [Menke, 1989] and is used to solve for the driving parameters of the RIBG model and the biases of the GPS receivers and satellites. In this technique, the GPS data is fed into the RIBG model to calculate the driving effective sunspot number that will replicate the observed ionospheric effect in the GPS data. The outputs of this methodology are essentially effective sunspot number which is the driver for the RIBG model and the biases. To test the validity of this RIBG technique for the analysis of GPS data, the ionospheric profiles have been compared with independent observations from a large number of instruments, e.g., the incoherent radar data, world campaign from different 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 been developed for specific applications [Mandrake et al., 2005]. The GAIM (Global Assimilation of Ionospheric Measurements) model, on the other hand, is based on a physicsbased ionosphere plasmasphere model with the capability of assimilating ionospheric measurements from a variety of instruments. The GAIM model has the capability to assim- 2of16

3 ilate the TEC measurements from GPS receivers, bottomside electron profiles from ionosonde, line of sight ultraviolet (UV) emissions, in situ electron density measurements from satellites and TEC measurements from tomography experiments. The complete list of observations that can be assimilated by the GAIM model is discussed in detail by Schunk et al. [2004]. The physics based model used by the GAIM model is the Utah State University s time dependent ionosphere model [Sojka, 1989], which is also known as Ionosphere Forecast Model (IFM). The main ionospheric data assimilation methodology used by this forecast model is 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, tiraweb.nrl.navy.mil). The output from the NRL site is the nmf2, hmf2, electron density profiles and TEC on a latitude and longitude grid for both the GAIM model and the IFM model. The outputs from the latter model are referred to as the 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 provide meaningful comparisons of Faraday rotation angle computation time and accuracy for different methods. An assessment of the impact of Faraday rotation on WindSat measurements is given by Meissner and Wentz [2006]. Our analysis includes a new technique that considers the ionosphere consisting of three shells at three altitudes instead of a single shell. The analysis shows that three shells can provide accuracy similar to that of a raytrace technique with computation time similar to that of a single shell model. The ionospheric model used in the algorithm is updated by ingesting the GPS data from a receiver that is close to the radiometer measurements. The implications and limitations of the distances between the radiometer measurement location and GPS receiver location are also discussed. 2. Data Analysis [9] The WindSat instrument used for the present analysis is in sun synchronous orbit at around 850 km altitude and crosses the equator at 18:00 and 06:00 local time (LT). WindSat is a multifrequency polarimetric microwave radiometer that operates in discrete bands at 6.8, 10.7, 18.7, 23.8, and 37.0 GHz. WindSat has a 6 foot spinning offset reflector antenna with nominal earth incidence angles ranging from 49.9 to 55.3 for different channels. The 10.7, 18.7, and 37.0 GHz channels are fully polarimetric. Our analysis focuses on the measurements at 10.7 and 18.7 GHz because, as discussed by Meissner and Wentz [2006], the Faraday rotation effects for WindSat are most important for the third Stokes measurements at those frequencies. According to equation (2), Faraday rotation essentially depends on two terms: B k and N e. The first term depends on the magnitude of the magnetic field intensity and the angle between the magnetic field and the wave vector. WindSat measurements are taken in both the forward and aft directions relative to the velocity vector of the satellite. This scanning geometry results in a large azimuthal variation of the wave vector which when combined with the magnetic field variation can produce very interesting variations in B k for a complete satellite orbit. The second term is the ionospheric term which we shall compute using the RIBG model. The RIBG model will be updated with the GPS data from a station that is close to the WindSat observations in both time and location. The two equatorial crossing times also make an interesting combination because the ionospheric contents at these local times are markedly different. Even though the ionosphere is in the decay stage at the 18:00 LT crossing, the variations in Faraday rotation have components from the ionosphere and the B k term. The equatorial crossing at 06:00 LT, on the other hand, occurs when the ionospheric content is very close to the minimum and variations in Faraday rotation are mostly due to the B k term. [10] Each term in the integral for the calculation of Faraday rotation is a function of height. Calculating the product at each height and summing it up to the satellite height can become a complex and time consuming process. To minimize this complexity and the computing time, it has been customary in the literature to assume that the ionosphere is just a shell at either 350 or 400 km (also known as the ionospheric pierce point). The magnetic field and the wave vector are also calculated at the pierce point. The resultant product at the pierce point is multiplied by the obliquity factor (sec c) to calculate Faraday rotation. We shall instead use the actual raytrace capability of the RIBG model to calculate Faraday rotation. To illustrate the process for Faraday rotation calculations for the WindSat orbit with full raytrace, we first choose the ascending part of the orbit on 24 September 2003 that is shown on the left of Figure 1. For the ionospheric specifications for this orbit time period, we analyze the GPS data from the station that is closest to this particular orbit. The closest station with GPS data available for that day is the Bermuda station. Although at present there are many more stations that are operating close to this orbit location, in 2003, Bermuda was the only GPS station reporting data near the orbit in time and location. The GPS data was analyzed for two hours (16:00 to 18:00 LT) to derive the effective sunspot number and solve for the biases. The two hours of data contains enough information to complete the analysis and this time period is close to the orbit crossing time at the equator. The value for the effective sunspot number is 78.5 which means that the observed local ionosphere can be reproduced using this sunspot number in the RIBG model. The published sunspot number for this date by NOAA s National Geophysical Data Center is 64 and for the subsequent day (25 September 2003) the published number is 77. The published sunspot number is one number for the whole globe for one day whereas effective sunspot number is more local in nature for the time period of the GPS data analysis and the location around the receiver itself. [11] To calculate the Faraday rotation for the 24 September 2003 orbit, we run the RIBG model with an effective sunspot number of 78.5 at 22:00 UT (equatorial crossing local time). Electron density profiles are created at each 10 km interval from 80 to 500 km, and at each 50 km interval from 500 km to the satellite altitude. Although for this particular example the ionospheric density profiles are created at one universal time, ionospheric profiles can be calculated periodically when processing satellite data. Similarly the magnetic field components (Bx, By, and Bz) are calculated from the IGRF10 model at the same height intervals as those of 3of16

4 Figure 1. The representative ascending and descending orbits for WindSat satellite. The equatorial crossing local times of orbits are 18:00 and 06:00 LT, respectively. The inclination for the satellite is 81.3, and altitude varies from 830 to 850 km. the density profile. The wave normal components are also calculated at all the height intervals by assuming a straight line raypath, i.e., no bending due to the ionosphere which is a valid assumption for the microwave frequency range between the satellite and the earth surface. This is done for each scan of the antenna beam up to the satellite altitude using Simpson s rule for the integration. The result for a 10.7 GHz frequency channel is shown in Figure 2 for the ascending part of the orbit. The results in Figure 2 show large variations in the Faraday rotation values and these variations are more due to the dot product of magnetic field with the wave vector than the changes in total electron content along the raypath. To illustrate this point more clearly, 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 ascending orbit in the Figure 1. The effective sunspot was calculated from the Bermuda GPS data for 24 September 2003 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 the variations in Faraday rotation angle during a single sweep in a forward scan. 4of16

5 Figure 3. Faraday rotation angle for the descending portion of the orbit in Figure 1 for the 10.7 GHz frequency channel. The equatorial crossing time is at 06:00 LT when ionosphere is almost nonexistent. tude and marked with an asterisk on the forward scan portion of the antenna. The Faraday rotation varies from to degrees (a factor of ) between the two points. The variation in the total electron portion is only 1 TEC unit (from to TEC units), and the obliquity factors (sec c) for the two points are and The obliquity factor is essentially related to the raypath length. The B k term for these two points varies from to nt (factor of 1.595). These variations taken separately show that 95% of the variation in Faraday rotation is due to the variation of the B k factor within one scan. The results also show the bifurcation in the northern hemisphere that can be associated with the forward and the aft 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 crossing time so that the ionospheric electron density is early in the decaying stage. The other half of the orbit (right side of Figure 1) is the descending portion with a 06:00 LT equatorial crossing. The Faraday rotation calculation process is repeated for the descending portion of the orbit with the results shown in Figure 3. Almost all of the variation is from the B k term since, as we mentioned earlier, the ionospheric electron density at 6am local time is close to the minimum possible for the day. Another feature that is different from the ascending orbit is that we can distinctly see the forward and aft portion of the orbit. The maximum Faraday rotation values for the descending portion of the orbit are at high latitudes due to higher values of the magnetic field and the wave normal product. Both the ascending and descending orbit analysis show that the magnetic field term plays an important role in the determination of the Faraday rotation correction. [13] The time required to compute Faraday rotation for half of the orbit for one channel is about 14 s using a 2.4 GHz Intel Pentium D computer. For WindSat this calculation should be completed for five distinct propagation paths covering 10 channels (6.8, 10.7 and 18.7 GHz vertical and horizontal polarizations and 10.7 and 18.7 GHz ± 45 degree polarizations). The required computational time would exceed the time required for geolocation of the measurements; so, it is desirable to reduce the computational requirements. [14] One common method to save computing time is the use of a shell model for the ionosphere instead of the raytrace method. The shell model assumes that the entire ionosphere 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 total electron content at the pierce point is scaled to the satellite by the obliquity factor (sec c) and the B k term is also calculated at the IPP. We calculate the total electron content from RIBG up to the satellite altitude on a latitude/longitude grid with the effective sunspot number for the observation time. The results from the shell methodology are shown in Figures 4a and 4b for the shells at 400 and 350 km, respectively, as percent variation from the raytrace method for the ascending portion of the orbit. The values for the percent variation that are plotted are derived using the following: Faraday rotation ðwith raytraceþ Faraday rotation ðassuming ionosphere as shell at IPPÞ 100: Faraday rotation ðwith raytraceþ 5of16

6 Figure 4a. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as single shell at 400 km. The results are for the ascending portion of the orbit shown in Figure 1. [15] The differences between the shell model and raytrace can vary from 7% to 12% in the equatorial latitudes for shells at 350 and 400 km, respectively. The difference can range up to 15% at high latitudes although this is primarily because the absolute rotation is lower. The computation time using the shell model is reduced from 14 s to 4 s which is about 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 the shell concept further by adding multiple shells instead of the Figure 4b. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as single shell at 350 km for the ascending portion of the orbit shown in Figure 1. 6of16

7 Figure 4c. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as three shells at 200, 400, and 650 km for the ascending portion of the orbit shown in Figure 1. usual single shell. We tested this concept with multiple shells at different heights and have chosen to use three shells at altitudes of 200, 400, and 650 km. Our tests showed that two shells did not provide sufficient accuracy and that four or more shells did not significantly improve the accuracy over that achieved with the three shell model. For the threeshell model technique, we calculate values of TEC that represent the electron content in the ionosphere at each of the three shell heights. The shell at 200 km contains the electron content of the ionosphere up to 300 km, the shell at 400 km contains electron content from 300 to 500 km, and the shell at 650 km contains the electron content above 500 km to satellite altitude. B k is also evaluated at each shell height. The latter part is multiplied with corresponding oblique total electron content to obtain three components of Faraday rotation that are summed up for the total rotation. The result is shown in Figure 4c as the difference between the raytrace and the three shell model on percentage basis and the errors in three shell model are less than 3%. Computation times for the raytrace, single shell, and three shell models 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 descending portion of the orbit that has almost no ionospheric variation. The analysis for the descending orbit for the models with single shells at 350 km, 400 km, and for the three shell model is performed in the same way as for the ascending orbit. The results are displayed in Figures 5a 5c in terms of the absolute difference instead of percent basis because some of the actual values are near zero. The comparisons between the different shell models show that the three shell model again provides better accuracy than the single shell models. [18] The analysis that we have discussed above is for the day 24 September 2003 when the solar activity as measured by sunspot number was in the moderate range. Furthermore, the satellite equatorial crossing times were either 18:00 or 06:00 LT when electron densities have declined from the peak or are near the daily minimum, respectively. To test the three shell technique at extreme ionospheric conditions, we test the three shell model using the electron density profile calculated for a sunspot number of 150 and the local time of 13:30. This is the time when the electron density peaks as a function of local time. Such solar conditions occur only at the peak of the solar cycle (within about a one year interval of the 11 year solar cycle). Faraday rotation results for the extreme conditions for the ascending orbit are shown in Figure 6. The calculations for Figure 6 were done with the raytrace method and the comparison of the Faraday rotation with the moderate solar conditions used for Figure 2 results shows that the rotation is increased by about a factor of two. The overall behavior of the results is similar to that shown in Figure 2 with a notable difference in the equatorial anomaly region. The strength and width of the equatorial anomaly is a function of sunspot number and local time. The differences between results for raytrace and results for the shell models are shown in Figures 7a 7c. The results are derived by the same method that was used for the results shown in Figure 4. The percent differences for the above three scenarios are 15%, 11%, <5% respectively. These values are very similar to those of Figures 4 except for the differences at southern high latitudes. We also can see some peculiar multiple bifurcation like behavior in the figures for the southern hemisphere. This may be the result of ionospheric structure at high latitudes or possible break down of the model at extreme conditions. The data tables that are used in 7of16

8 Figure 5a. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphere and the ionosphere considered as single at 400 km for the descending portion of the orbit shown in Figure 1. the ionospheric models are based on ground observations. The data tables are at sunspot numbers of 0 and 100. The models may not work well for the extreme conditions on either side. We have seen at the extreme low solar activity, the effective sunspot number values have to be negative to recreate the TEC data from GPS observations [Reilly and Singh, 1997]. The conditions that we are depicting in this example can happen for only a few days in an 11 year solar cycle, but the example demonstrates the accuracy of the three shell model over the single shell model. 3. Error Analysis [19] In this section we discuss the absolute accuracy of the methodology for the Faraday rotation calculations as described in the preceding section of this paper. The inac- Figure 5b. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphere and the ionosphere considered as single at 350 km for the descending portion of the orbit shown in Figure 1. 8of16

9 Figure 5c. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphere and the ionosphere considered as three shells at 200, 400, and 650 km for the descending portion of the orbit shown in Figure 1. curacies in the calculation of the Faraday rotation angle for spaceborne microwave radiometers can be due to the magnetic field or ionospheric modeling errors or errors in the integration of their product. The geomagnetic field and ionospheric modeling errors can be reduced by using geomagnetic measurements and GPS reference station data that are close in time to the observations. We update the magnetic field model monthly. The day to day variability in the magnetic field which amounts to much less than 1% as discussed in the introduction part can be corrected depending upon the accuracy constraints. Figure 6. Faraday rotation angle for the ascending portion of Figure 1 for 10.7 GHz frequency channel with raytrace technique. The ionospheric conditions represent extreme values with sunspot number of 150 and at 13:30 LT when ionosphere is at its peak. 9of16

10 Figure 7a. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as single shell at 400 km as related to Figure 6 conditions. [20] There are also errors in the ionospheric portion of the equation due to inaccuracies of the models. The results shown in the previous section were generated using the updated RIBG model. As discussed earlier in the introduction, there also exist a number of other ionospheric models that can be used for similar analyses. In this section, we compare the Faraday rotation calculated using IRI and GAIM ionospheric models with the results obtained using the RIBG model. The IRI model can be run on the Internet or the Fortran source code can be compiled and run on a personal computer. The GAIM model requires considerable hardware power and at present is run either at AFWA or NRL. For the RIBG and IRI models, the latitude/longitude grid is adjustable but for the global GAIM the grid is fixed with the longitude grid is set at 15 and a variable latitude grid of for low and middle latitudes (<±67 ) and 3 for high latitudes. It may be possible in the future to run the GAIM model for regional and local ionosphere, we will use global GAIM output which is currently available from NRL site. Figure 7b. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as single shell at 350 km as related to Figure 6 conditions. 10 of 16

11 Figure 7c. Percent difference in Faraday rotation angles between full raytrace through the ionosphere and the ionosphere considered as three shells at 200, 400, and 650 km as related to Figure 6 conditions. 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 LT equatorial crossings time. 11 of 16

12 Figure 8b. Faraday rotation angle difference between different ionospheric models (IRI, RIBG, and GAIM) from Figure 8a results. [21] The analysis presented in the previous section was based on the updated version of RIBG data for one day (24 September 2003) with the effective sunspot number based on GPS data from the Bermuda station (the station closest to the orbit). For this particular day GAIM model results are available on the NRL site for the IFM model only while both GAIM and IFM outputs are available for 26 September Therefore we will compare the three models on 26 September 2003 for the ascending portion of orbit. The effective sunspot number for 26 September 2003 for the RIBG after ingestion of Bermuda GPS data is 76.8 and the Faraday rotation results from RIBG ionosphere are shown in Figure 8a (middle).the corresponding Faraday rotation results with ionosphere generated by IRI are shown in Figure 8a (top). Similar results for Faraday rotation using the ionosphere generated by the GAIM model for the same day 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 a function of latitude except that the IRI model results are consistently lower. A comparison between the results from the RIBG and the GAIM shows that the maximum values for Faraday rotation angles are very close to each other. However, there is a distinct difference between the two models at equatorial latitudes. The equatorial anomaly is much more pronounced on the two sides of equator for both the RIBG and the IRI models while the anomaly on the south side of the equator is less prominent for the GAIM model. Figure 8b shows the differences in Faraday rotation between the different models in more detail. This plot is the subtraction of Faraday rotations results for the IRI and GAIM models from the RIBG results. The difference between the RIBG and the IRI results show that the RIBG is higher by at middle and high latitudes but the difference can be as high a 0.04 at low latitudes. The sunspot number used by IRI for this particular day is 59.1 and the effective sunspot number inferred by RIBG using the GPS reference station is The Faraday rotation difference is primarily due to the difference in the sunspot number used by the two models. The sunspot numbers used by the various ionospheric models and published by the National Geophysical Data Center for the two days (24 September 2003 and 26 September 2003) are shown in Table 1. The difference between the RIBG and GAIM models shows that at low latitude ( ), the GAIM results are consistently higher. The RIBG model lacks the ingestion from any equatorial GPS station for this particular example while GAIM assimilates the data from equatorial latitudes. The Faraday rotation results from the three models are different at high latitudes especially in the northern hemisphere. No ionospheric data observed at high latitudes is used for the update of either RIBG or GAIM models which means that the climatological and physics based models are inherently different at high latitudes. [23] Meissner and Wentz [2006] have calculated the Faraday rotation angle with the ionosphere that was generated by the IRI model and concluded that rotation error of greater than >0.15 will add appreciable error to the third Stoke parameter and subsequently degrade the accuracy of wind vector retrieval. For this particular orbit, this threshold is reached 0.01 percent of the time with the IRI model, percent of the time for the GAIM model and 8.53 percent for the RIBG model. [24] These results are representative of the ionosphere during moderate solar activity conditions and the equatorial Table 1. Sunspots for 2 Days Used by Different Models Model/Sunspot 24 September September 2003 NOAA RIBG IRI IFM of 16

13 Figure 9a. distance. The GPS stations locations that are used in the study of accuracy degradation versus the crossing time of 18:00 LT, when the electron content of the ionosphere has decayed from its maximum of the day. Faraday rotation effects will be greater during times with high solar activity. The same will be true when the electron content is at its peak for the day. To study the local time effect on the Faraday rotation results, we calculate the TEC from the GAIM model for the same day at 2pm local time. The threshold limit for the degradation as set by Meissner and Wentz [2006] will be reached 16.4% of the time instead of 11.4% at 6pm. The effect of extreme ionospheric conditions on this threshold can be calculated from Figure 6 results and the 0.15 threshold is exceeded 76.35% of the time under such conditions. [25] To minimize the error from the ionospheric models, the assimilating models should have better representation of the existing ionosphere than the nonassimilating models. In the assimilating category, the GAIM model has the capability of ingesting a large variety of ionospheric observations on the global scale. The RIBG model assimilates only the GPS data and also just one GPS receiver at a time but has the flexibility of selecting the GPS stations of choice for data ingestion. Figure 9b. Differences of Faraday rotation angles between pairs of stations versus the distance between the pair. The vertical bars show the range of differences between each pair of stations over the complete data set in the orbit. Also shown are the mean, median, and 3 sigma values for each pair of stations. 13 of 16

14 Figure 10. Difference in Faraday rotation angles (degrees) between full raytrace through the ionosphere and the ionosphere considered as single shell at 400 km, single shell at 350 km, and three shell model. [26] The assimilating ionospheric model requires the observed ionospheric data to ingest. The most abundant ionospheric data that can be assimilated or ingested into models is the GPS data which is readily available on a global scale from a number of data centers in near real time. The availability of this GPS data has improved rapidly over the last decade; this trend is continuing. For the Americas, the CORS network ( compiles data from a few hundred stations that cover North America, Hawaii, Guam, and the Caribbean, and is expanding into South America. The GPS data from this network is available on an hourly basis on the same day and is typically available to users within two hours of the observation time. The European network of GPS receivers not only covers most of Europe ( be/_dataproducts/data_access/) but also has data from islands in the Atlantic Ocean and the North Sea. The Australian network covers Australia with stations on the coastal areas ( monitoring.html) and around the entire continent. Even though data from more and more GPS stations is becoming available there are still gaps in the global coverage of the ionospheric observations. For example, at the present time there is no coverage in the South Atlantic Ocean. Data gaps may also occur because some stations may be down temporarily or due to problems with the data delivery system. In the presence of data gaps, models are obtained by either interpolation or with the filters from the ionosphere generated in the other parts of the globe. [27] Another error can also arise if there is large distance between the ingested data source point and the ionosphere application point. To include different scenarios under which ionospheric data may be unavailable at the application point, we have conducted a study of varying distances from reference GPS stations versus accuracy degradation. We take the GPS data from one station and predict the ionosphere at a second station and compare with the ionosphere predicted by the GPS receiver of the second station. Figure 9a represents the stations that we have included in the case study. The latitude/longitude difference between the stations can be more than 40 degrees and physical distances between pairs of stations can vary from as little as 900 km to a maximum of 6000 km. We have analyzed the GPS data that was available for 24 September for three years (2001, 14 of 16

15 2003, and 2007). These three years represent high, moderate and low solar activity levels. The GPS data was available for only three stations for the year 2001 while for the other two years it was available from six stations. We calculate the Faraday rotation with the sunspot number based on different effective sunspot numbers at two stations and associate it with the distance between the stations. Figure 9b shows the resulting differences in Faraday rotation in degrees versus distance at 10.7 GHz for the ascending part of the orbit in Figure 1.The effective sunspot number was determined from the GPS data for all the stations between 1900 to 2100 UT for the respective days. The vertical bars show the range of differences between each pair of stations over the complete data set in the orbit. The plot also shows the mean, median, and 3 sigma values of the differences for all stations. These results show that we can expect accuracy in Faraday rotation to be better than as long as we have GPS station data within 1500 km of the orbit based on the small sample of data 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 the magnetic field and the electron content along the propagation path. The results of Figures 4a 4c can also be represented in terms difference in degrees instead of percent occurrences and are shown in Figure 10. These differences in Faraday rotation angle can be as high as 0.02, 0.01, and for single shell models at 400 km and 350 km and the three shell model respectively. We can also compute errors for extreme ionospheric conditions by combining the percent occurrences of Figures 7a 7c with Figure 6 Faraday rotation values. The errors for extreme ionospheric conditions can be as high as 0.041, 0.024, and for the 400 km shell, the 350 km shell, and three shell models respectively. The microwave analyses done by other authors as mentioned in this paper have used a single shell at different heights from 300 to 400 km and our analysis can quantify the errors. Bettenhausen et al. [2006] have also shown from WindSat data analysis that modeling and calibration errors for the third and fourth Stokes parameters are about 0.1 K for brightness temperatures which translates in to 0.08 at 10.7 GHz. The combination of Faraday rotation errors due to ionospheric model errors approximated by the differences shown in Figure 8b with the single shell model errors shown above is of the same order. Use of the threeshell model along with an assimilating ionospheric model can reduce the errors significantly. The full raytrace method could be used to further reduce the errors when there are no computational time constraints. 4. Conclusions [29] We have studied Faraday rotation for microwave radiometry using WindSat as an example but the analysis method can be applied to other missions. The Faraday rotation term although small at microwave frequencies can still affect the accuracy of the final products. We have indentified the errors in the calculations for Faraday rotation that relate to the contribution from the ionospheric term. The errors can arise from the magnetic field component, ionospheric modeling, distance between the reference source for data ingestion and the application point, and assuming the ionosphere as a single shell. We have discussed in the paper methods to reduce these errors. For the magnetic field component, the magnetic field model should be updated at least once a month. The assimilating ionospheric models can generate better ionosphere than nonassimilating models. The GAIM model has the advantage of ingesting the ionospheric data from a large number of instruments while the RIBG model can choose the best GPS station location. The ionospheric corrections using a GPS station within 1500 km of the orbit provide Faraday rotation accuracies to better than at 10.7 GHz. For the distances greater than 1500 km, this analysis can be used to infer the error amount. Singleshell models have been used for microwave applications with pierce heights ranging from 300 km [Yueh, 2000] to 400 km [Le Vine and Abraham, 2002]. Ionospheric researchers have also 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 much better accuracy than the single shell models with similar computation time. References Bent, R. B., S. K. Llewellyn, G. Nesterczuk, and P. E. 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16 Reilly, M., and M. Singh (2001), GPS data analysis near Puerto Rico for World Day campaigns, Radio Sci., 36, , doi: / 1999RS Reilly, M., and M. Singh (2004), Electron density height profiles from GPS receiver data, Radio Sci., 39, RS1S16, doi: /2002rs Scherliess, L., R. W. Schunk, J. J. Sojka, and D. C. Thompson (2004), Development of a physics based reduced state Kalman filter for the ionosphere, Radio Sci., 39, RS1S04, doi: /2002rs Schunk, R. W., et al. (2004), Global Assimilation of Ionospheric Measurements (GAIM), Radio Sci., 39, RS1S02, doi: /2002rs Singh, M., and M. H. Reilly (2006), Improved positioning by addition of atmospheric corrections to local area differential GPS, Radio Sci., 41, RS5S29, doi: /2005rs Skou, N., B. Laursen, and S. Sobjaerg (1999), Polarimetric radiometer configurations: Potential accuracy and sensitivities, IEEE Trans. Geosci. Remote Sens., 37, , doi: / Sojka, J. J. (1989), Global scale physical model of the F region, Rev. Geophys., 27, , doi: /rg027i003p 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 ionospheric and magnetospheric specification models, Radio Sci., 23, , doi: /rs023i003p Yueh, S. (2000), Estimates of Faraday rotation with passive microwave polarimetry for microwave remote sensing of earth surfaces, IEEE Trans. Geosci. Remote Sens., 38, , doi: / M. H. Bettenhausen, Remote Sensing Branch, Remote Sensing Division, Naval Research Laboratory, Code 7223, 4555 Overlook Ave. SW, Washington, D.C , USA. M. Singh, Computational Physics Inc., 8001 Braddock Rd., Ste. 201, Springfield, VA 22151, USA. (smalkiat@yahoo.com) 16 of 16