New leveling and bias estimation algorithms for processing COSMIC/FORMOSAT-3 data for slant total electron content measurements



[1] Ionospheric modeling can be improved by the inclusion of occultation data between satellites and GPS transmitters. COSMIC/FORMOSAT-3 provides a large data set of such occultations. In order to utilize these absolute measurements in an assimilative model, the data must be carefully processed; the level of the ionospheric combination and the differential biases must be accurately determined. The COSMIC GPS receivers operate in a high multipath environment; a phase leveling algorithm, utilizing the information in the multipath, improves the leveling errors by at least 0.4 total electron content units. Receiver biases are then computed from the leveled data by making some simplifying assumptions about the structure of the ionosphere and plasmasphere. This processing algorithm provides occultation measurements as slant total electron content to an accuracy of 3.55 total electron content units.

1. Introduction

[2] Ionospheric remote sensing is in a rapid growth phase driven by an abundance of ground and space-based GPS receivers, new UV remote sensing satellites and the advent of data assimilation techniques for space weather. The success of the GPS/MET experiment in 1995 inspired a number of follow-on radio occultation missions for profiling the atmosphere and ionosphere. These include the Argentine Satelite de Aplicanciones Cientificas-C (SAC-C) [e.g., Hajj et al., 2004a, 2004b], the U.S.-funded Ionospheric Occultation Experiment (IOX) [e.g., Straus, 2007], and Germany's Challenging Minisatellite Payload (CHAMP) [e.g., Jakowski et al., 2007]. The joint U.S./Taiwan Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC;, a constellation of six satellites, nominally provides up to 3000 ionospheric occultations per day. The COSMIC 6-satellite constellation was launched in April 2006 and observed final orbits in November 2007. COSMIC now provides an unprecedented global coverage of GPS occultation measurements (between 1400 and 2400 good soundings per day as of March 2011, reduced from the nominal number by degradation of the sensors), each of which yields electron density information with 1–3 km vertical resolution. Calibrated measurements of ionospheric delay (which is directly proportional to total electron content or TEC) suitable for input into assimilation models are currently made available in near real time (NRT) from COSMIC with a latency of 30–120 min. Similarly, NRT total electron content (TEC) data are available from two of the worldwide NRT networks of ground GPS receivers (∼75 5 min sites and 125 hourly sites, operated by JPL and others). The combined ground and space-based GPS data sets provide a new opportunity to more accurately specify the three-dimensional ionospheric density with a time lag of only 15–120 min. With the addition of the vertically resolved occultation data, the retrieved profile shapes represent the hour-to-hour ionospheric weather much more accurately [Komjathy et al., 2010]. We have now begun integrating COSMIC-derived TEC measurements with ground-based GPS TEC data and assimilating these data into models such as the JPL/USC Global Assimilative Ionospheric Model (GAIM) [Hajj et al., 2004a, 2004b; Hajj and Romans, 1998; Mandrake et al., 2005] so that three-dimensional global electron density structures and ionospheric drivers can be estimated. Recently the COSMIC GPS measurements along with ground-based GPS measurements have been assimilated into JPL/USC GAIM for a study of ionospheric storm [Pi et al., 2009] revealing identifiable features of equatorial anomaly enhancements.

[3] Over the course of the past 15 years, we have used the Global Ionospheric Mapping (GIM) software developed at the Jet Propulsion Laboratory [Mannucci et al., 1998] to compute high precision slant ionospheric delay by removing the satellite and receiver differential biases from ionospheric observables using ground-based GPS receivers. In this paper, we describe a new and improved algorithm to estimate ionospheric observables using spaceborne GPS observations from COSMIC satellites. These high-precision spaceborne GPS observations can be combined with ground-based GPS observations to serve as a backbone to our GAIM estimating the global 3-D electron density field.

2. Extracting Slant TEC From GPS Observations

[4] Dual-frequency GPS observations consist of pseudorange and carrier phase measurements, at the two GPS frequencies f1 = 1.5754 GHz and f2 = 1.2276 GHz. The corresponding pseudoranges, P1 and P2 are given by

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The associated carrier phases, L1 and L2 are given by

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Equations (1) and (2) are composed of the following terms: ρ is the geometric range, dTdt is the clock error differential, equation image are the frequency dependent ionosphere delay, equation image is the troposphere delay, equation image are the satellite interfrequency biases, equation image are the receiver interfrequency biases, mpx are the frequency dependent multipath error and εequation image are the noise terms. Addionally, λi are the wavelengths of the two frequencies and

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The ionosphere combinations,

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are dominated by the ionospheric contributions in the domain where the noise and carrier multipath contributions are low. The Δ's are, by convention, the 2 minus 1 channels. For connected arcs of data, the ambiguity terms, due to differences in cycles, are constant. Additionally, the multipath on the carrier phase is generally 2 orders of magnitude smaller than those on the pseudorange measurements [Hofmann-Wellenhof, 2001]. With these assumptions, the equations for PI and LI become

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Solving for equation image requires the determination of the ambiguity factor and the satellite and receiver differential biases, equation image and equation image respectively. These two processes are referred to as phase leveling technique and satellite and receiver bias estimation. Once dion,L1 is determined, the TEC is simply a scaling factor of dion,L1, i.e.,

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where 1 TECU = 1016 e/m2.

[5] An example of the PI combination compared to a (leveled) equation imageI combination for several arcs of data is shown in Figure 1. It can be seen that the PI combination (green) contains much larger noise term than the equation imageI combination (red).

Figure 1.

Example of PI combination versus (leveled) equation imageI combination for many arcs of data for site in Australia on 16 November 2010.

3. Leveling Algorithm

[6] The leveling algorithm is designed to use the unbiased level of the PI code ionospheric observable to set the level of the LI phase ionospheric observable. This effectively adjusts the leveled LI combination to provide a low-noise measure of the ionospheric delay content plus some biases (to be discussed in section 4). To correct the level of the LI combination, a constant value 〈N〉 must be determined to characterize the weighted mean of PILI, i.e.,

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where index i is over all measurements in a connected arc. A method of determining the wi is needed. The leveled LI combination is then

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[7] Using equation (3), the PILI combination yields

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Since the biases and the ambiguity terms are assumed to be constant over a phase connected arc, the distribution around these constant values is determined by ΔmpP, and the two noise terms. If we assume that the noise terms are negligible compared to the multipath, then PILI provides the constant terms needed for the leveling plus additional multipath distribution. The optimal choice of wi is then determined by the distribution of ΔmpP.

[8] The COSMIC spacecraft configuration introduces significantly more multipath error than ground based GPS stations as shown by Hwang et al. [2010]. As can be seen in the observation equations, equation (4), the pseudorange multipath contribution increases the variance of the PI combination; the PI combination is distributed around the ionospheric delay content by the multipath distribution. Therefore, it is possible to minimize the variance of the PILI combinations using the multipath to develop a weight. Since we have no a priori knowledge of the distribution of the multipath, the method of maximum entropy [Cover and Thomas, 1991] suggests we should take only the mean and variance of the multipath and use a normal distribution function as the weighting function. This approach introduces the least amount of self-correlation between the weighted average and the sample of the distribution. Consider now the two multipath combinations (not to be confused with the mpx terms)

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where a, b, c and d are frequency dependent combinations, for the given f1 and f2 these are

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It was shown by Estey and Meertens [1999] that these combinations can be expressed as

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where equation imagei are combinations of the carrier multipath, equation image, and the noise, equation image. Therefore the difference

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It has also been shown [Hofmann-Wellenhof, 2001] that the noise and carrier phase multipath are 2 orders of magnitude smaller than the pseudorange multipath. For each connected arc the ambiguities are constant, so equation (11) is a distribution of the differential pseudorange multipath. Using this distribution to derive the Gaussian statistics of the weighting function the width of the differential multipath contribution to be used during the weighting. This implies that points in the arc that have large deviation from the mean of the differential multipath contribution will not have a large influence on the calculation of 〈N〉. This is what is desired to help reduce the variance in the weighted PILI calculation. The weighting function is then the normal distribution of the difference in the multipath combinations with mean and variance

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The wi for each measurement is calculated as

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[9] Using this we find

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where 〈·〉μ,σ2 represents the weighted average of · using the weighting function described in equation (6) with weights defined by equation (13). Thus

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This can be converted to slant TEC (STEC) by

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Determination of B, which is dominated by the satellite and station differential biases, is discussed in section 4.

4. Bias Estimation

[10] One of the features of the GIM data processing system is the ability to estimate the satellite and receiver differential biases, Bij, from equation (16) by taking cross correlations between many different ground based GPS measurements. Each ground station has multiple satellite links which allow an algorithm to be developed for determining the bias between station i and receiver j in a relative sense. By fixing one of the relative satellite biases to be 0, the complete deconvolution of equation image and equation image is performed. This, of course, is done under the assumption that for some weighting function, g we can assume

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where the subscript g indicates a weighted mean using g as the weighting function. For an unbiased distribution, g, this is true for an unbiased noise (e.g., Δequation imageL). For the ground GPS receivers, it is also safe to assume that the multipath is small, thus the weighted mean of zero is also a fair assumption. Another assumption that leads to the same conclusion is that the global average multipath over all satellites and receivers is zero. Under these assumptions, for weighting function g, GIM provides

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where the equation image represents the relative bias where each has been shifted in opposite directions by some constant value (this is the absolute bias of the GPS satellite differential bias fixed to 0).

[11] For the space platform, such as COSMIC, GIM is not suitable to estimate receiver biases. Instead, we can use the relative satellite biases produced by GIM and generate a relative COSMIC receiver bias. Thus we effectively compute

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Unfortunately, the multipath in the COSMIC environment is larger than the ground case. Because of this we cannot make the same assumption about the unbiased distribution and therefore, cannot neglect the weighted mean. This will differ from arc to arc and introduces larger variances in the bias estimation for spaceborne GPS data.

[12] The GIM bias estimation relies on cross correlation between multiple sites measuring the same ionosphere. For the space-based GPS data, this data overlap is not available and an alternate method must be employed. One could use a background model, such as the International Reference Ionosphere (IRI) [Bilitza and Reinisch, 2008], but these models are not able to capture the valuable space weather conditions that may be present. Instead, we employ a simple but effective approach. Since COSMIC is orbiting at around 800 km, above the F region peak, we expect upward looking measurements to be passing through a minimal amount of the ionosphere. If we also restrict the measurements to be in the high-latitude regions, the contribution from the plasmasphere is expected to be very small. Under the assumption that the high-latitude and high elevation angle measurements pass through a minimal amount of TEC, these measurements can be used to derive a differential bias for the receiver. This is done by tabulating the smallest measurement of an arc that passes through the filtered region. Over the course of the day the measurements are averaged. From this average a small component due to the small ionosphere and plasmasphere contributions are subtracted. This average minus estimated contribution is the daily bias for the receiver. A recent study by Yizengaw et al. [2008] showed that the plasmasphere contribution to the total ground based TEC in high-latitude regions was about 25%. Figure 2 shows a histogram of the total plasmasphere contribution for all COSMIC measurements that were upward looking (>70°) and above 60 degrees latitude for all 6 COSMIC satellites between 1 January 2009 and 31 March 2009. The plasmasphere contribution was computed by integrating the line-of-sight TEC through the Sheeley-Tu [Sheeley et al., 2001; Tu et al., 2006] plasmasphere model. The total contribution appears to be at the 0.1 TECU level. It can also be seen from Figure 3 that the daily variation of the mean is minimal and the 1-sigma variation each day is similarly of order 0.1 TECU. In our simplistic model, it is then assumed that the background TEC between COSMIC and the GPS satellite in the upward looking, high-latitude measurements, is 0.5 TECU, with errors of order 0.1 TECU. This estimate is likely to be slightly large by a few tenths of TECU and is intended to compensate for the fact that there is a double bulge in Figure 2 in which the plasmasphere is sometime larger by 50% or more.

Figure 2.

Value of the plasmasphere contribution to upward looking (>70°) high-latitude (>60°) line-of-sight measurements from 1 January 2009 to 31 March 2009 for all six COSMIC satellites.

Figure 3.

The daily variations of the plasmasphere contributions for 2009. The line represents the mean of the plasmasphere contribution, and the error bars represent the 1-sigma range of the plasmasphere contributions.

[13] Applying this process a daily estimate of the receiver bias, equation image can be computed. To eliminate potential data errors and variabilities, the actual bias used is a 10 day running average of the daily estimates. It is also possible to have entire days which do not meet the filtering criterion. For these missing days, the global bias average (for all days) is used in place of the daily estimate in the 10 day average. Figure 4 shows the 10 day average biases for three of the COSMIC satellites between 1 and 24 October 2010. The histogram in Figure 4 shows the number of arcs of data that satisfy the filtering criteria. The variabilities for satellites 4 and 6 appear to be of the order of 1 TECU but the estimated value seems to be stable. Satellite 1 shows a bimodal behavior. This is suspected of being due to a real discrete change in the hardware bias.

Figure 4.

The daily variation in October 2010 of the bias estimates for antenna 0 of COSMIC satellite number 4 (red), antenna 1 of COSMIC 1 (black), and antenna 1 of COSMIC 6 (blue).

[14] These bias estimates can be compared to the ones reported by the COSMIC Data Analysis and Archive Center (CDAAC,, 2010). There are not estimates for every day of the month in CDAAC but the monthly averages, the maximum and minimum are presented in Table 1. The last column of Table 1 shows the difference in the bias estimate between CDAAC and the method presented here. The differences seem to agree with the premise that the biases computed here are relative to a particular GPS satellite, whereas the biases in CDAAC are absolute. Given the variation of the biases estimates, the relative level seems to agree well.

Table 1. The Biases Generated by CDAAC for October 2010a
AntennaeAverage (TECU)Maximum (TECU)Minimum (TECU)Δ Average (TECU)
  • a

    The last column shows the difference between the monthly average presented in this work and that from CDAAC. This monthly average represents the level correction between the relative biases generated here and the absolute ones from CDAAC.


5. Discussion

[15] The improved performance of the multipath weighted leveling can be seen in Figure 5. These show the histogram of PIequation imageI after leveling is applied for the data from 21 December 2006. The blue histogram is the new multipath weighted leveling. The red histogram indicates the elevation weighted leveling used in GIM for ground-based GPS data. From the width of the histograms, the new weighting technique levels the LI combination consistently closer to the PI combination than does the elevation weighted approach. This is as expected as the new leveling weights are designed to minimize the variance of this distribution. Table 2 tabulates the RMS difference for each panel of Figure 5. It illustrates a consistent improvement of about 0.5 TECU or more across all satellites.

Figure 5.

Histograms of the difference between PI and LI using the new multipath weighted leveling scheme (New) and the old elevation weighted leveling scheme (Old) for 21 December 2006.

Table 2. RMS Differences of PIequation imageI for 21 December 2006 Using Both the New Multipath Based Weighting Scheme and the Old Elevation Angle Based Weighting Scheme
COSMICMultipath BasedElevation Angle Based
16.1 TECU6.8 TECU
24.0 TECU4.4 TECU
36.4 TECU7.1 TECU
46.5 TECU7.3 TECU
54.2 TECU4.8 TECU
66.5 TECU7.2 TECU

[16] The final result of the leveling/bias removal procedure is now shown in Figure 6. Figure 6 shows the leveled and unbiased data from COSMIC 1 on 22 October 2010. Figure 6 contains all of the connected phase arcs for 1 day. The tight structure shows the leveling algorithm is consistent from arc to arc. The approach of the slant TECU measurement to zero as elevation angle increases shows the biases are also being appropriately computed. We now consider the total error of these slant TECU measurement derived from COSMIC GPS signals. Hwang et al. [2010] have shown that the multipath error for COSMIC is of the order of a few centimeters. Using an upper limit of 4 cm, this is equivalent to 0.4 TECU. From experience with both ground and space based data, we approximate the systematic error of the leveling algorithm to be 2 TECU. We also assume a systematic bias error of 0.5 TECU, based on the systematics of estimating the plasmasphere. This is 100% of the assumed plasmaspheric electron content plus upper ionosphere contribution and represents an upper estimate of the error. The bias algorithm is dependent on the leveling algorithm and the multipath, however. Thus the total error in the bias is in fact given as the sum of the leveling error, multipath error and systematic bias algorithm error. This yields about 2.9 TECU. Additionally, to compute the error of slant TECU measurements, the multipath and leveling error are independent of the bias error. This is due to the 10 day averaging procedure used to compute a daily bias. Thus the square of the total error of a slant TEC measurement is given by the sum of the squares of the multipath, leveling and bias errors. This is estimated to be 3.55 TECU.

Figure 6.

Leveled and bias-corrected day of COSMIC data on 22 October 2010 for COSMIC 1.

6. Conclusion

[17] A new processing system for COSMIC data has been presented. This new system uses the information in the multipath combination to provide a TEC level based on variance reduction of the PIequation imageI combination. This level is more accurate than an alternate weighting scheme, i.e., the elevation angle based weighting scheme used by GIM. Using the correctly leveled phase connected arcs from COSMIC a relative receiver differential bias can be estimated. The process of leveling and bias estimation has an error of order 3.55 TECU.

[18] With a processing system in place, it is now possible to systematically assimilate COSMIC GPS measurements into a model such as JPL/USC GAIM. The authors plan to utilize this tool to systematically include COSMIC measurements on a daily basis. It has been shown that such inclusion leads to improved profile structures and better global data coverage.


[19] This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored by the National Aeronautics and Space Administration.