Accuracy of GPS total electron content: GPS receiver bias temperature dependence


Corresponding author: A. Coster, MIT Haystack Observatory, Westford, MA 01886, USA. (


[1] Having an accurate method to estimate and remove ionospheric effects is a major issue for low-frequency radio astronomy arrays, as the ionosphere is one of their largest error terms. One way to estimate the ionosphere is to measure total electron content (TEC) using dual frequency global positioning system (GPS) signals. This technique uses the dispersive nature of the ionosphere, as both group and phase velocities are (to first order) dependent on the inverse square of the frequency and on TEC. Using these properties, TEC can be measured to a high degree of accuracy by computing the delay difference between signals at GPS's two frequencies (L1 = 1575.42 and L2 = 1227.6 MHz). Unfortunately, effects other than ionospheric dispersion also introduce differential delay differences. These additional differences, called biases, can be separated into those introduced by the satellite and those by the receiver. Receiver biases show the most significant variations, sometimes over intervals of hours. Changing temperature conditions at the receiver antenna, along the cable, or in the internal receiver hardware are thought to be responsible for some of these variations. We report here on an investigation of the temperature dependence of the GPS receiver bias. Our results show that for our particular receiver, antenna, and cable set-up, a temperature-dependent bias is clearly evident, and that this temperature dependence varies from receiver to receiver. When the receiver bias temperature dependence is removed, a noise level of 1–3 TEC units still remains in the bias estimation.

1 Introduction

[2] The ionosphere is a dispersive medium, and taking advantage of this property enables the ionosphere's total electron content (TEC) to be estimated from the differential delay between the two global positioning system (GPS) frequencies (L1 = 1575.42 and L2 = 1227.6 MHz). Unfortunately, other effects in the system, called biases, can introduce delays between the two frequencies that must be removed to resolve the TEC. The biases can further be separated into those introduced by the satellite and those introduced by the receiver.

[3] This article presents the results of an investigation into GPS receiver bias dependence on temperature. Receiver and satellite biases can be significant and, if not removed correctly, can corrupt the GPS TEC measurements. It has been shown that the GPS satellite biases are fairly stable day to day [Sardon et al., 1994; Wilson et al., 1999], whereas the GPS receiver biases can have significant variation, sometimes over intervals of hours. The National Aeronautics and Space Administration Jet Propulsion Laboratory (JPL) delivers estimates of the 10 day average of the individual satellite biases to the U.S. Air Force. According to Komjathy [2012], for each satellite, the 10 day average differs from the individual day values by 0.1–0.3 ns.

[4] This study was motivated by the science requirements of the Murchison Widefield Array (MWA), a new low-frequency array being built in Western Australia. GPS measurements can be used to validate the MWA's fine-scale regional measurements of differential TEC and to facilitate the MWA's measurements of the magnetic field vector within the solar wind. This last measurement—that of the magnetic field within the solar wind—will provide critical new information to aid in the prediction of the impact of coronal mass ejections on the Earth's ionosphere and magnetosphere. However, to successfully make this measurement, the Faraday rotation due to the ionosphere must be removed with an extremely high degree of accuracy. Having fully calibrated absolute GPS TEC measurements will greatly aid this process. Standard errors for absolute GPS TEC estimation are estimated by the community to be ~3 TECU (1 TECU = 1016 electrons/m2), although the differential GPS TEC measurement is far more accurate (~0.02 TECU). The MWA project would like to have absolute TEC values from GPS at the 0.1–0.2 TECU level, an extremely demanding goal.

[5] Slant TEC estimates are obtained with GPS observations using the following equation:

display math(1)

[6] In equation (1), STECi,k represents the slant TEC and has the units of TECU; the subscript k represents the satellite transmitting the GPS signals, and the subscript i represents the receiver. the constant 2.85 is the conversion factor from ns to TECU and has the units of TECU/ns; DL1 and DL2 represent the total measured range delay in ns from the L1 and L2 frequencies, respectively. Bk and Bi (BkL1–BkL2 and BiL1–BiL2) are differential delays in ns due to the satellite and receiver, respectively. Finally, ε represents other error sources, again in ns, which are assumed to be small after averaging. In standard practice, the relative TEC measurement of the carrier phase data, which has a noise that is two orders of magnitude smaller than the pseudo-range data, is used to “smooth” the TEC measurement from the pseudo-range data [Komjathy, 1997; Gao and Liu, 2002]. As the carrier phase measurement of TEC is only a relative measurement of TEC, it does not provide additional information on the size of the receiver bias. It is worth noting that part of the GPS modernization includes a third civilian frequency, L5 = 1176.45 MHz [Kaplan, 2006], which will allow for additional estimates of the TEC using the various frequency combinations at each time sample. The advantages for the atmospheric science community for GPS modernization are described in Van Dierendonck and Coster [2001].

[7] In the early days of GPS, the community was uncertain how to estimate either set of biases. However, it was quickly recognized that the delay introduced by satellite and receiver biases was constant as a function of elevation, whereas the delay introduced by the ionosphere changed. Techniques applying this property to estimate the biases use a mapping function to convert each line of sight TEC estimate into a zenith TEC estimate. The standard mapping function is given by Mannucci et al. [1993], although several others have been defined in the literature [Komjathy, 1997].

[8] Work at the National Aeronautics and Space Administration JPL e.g., Lanyi and Roth, 1988 led to initial algorithms for bias estimation. Other methods were developed using single receivers [Coco et al., 1991; Gaposchkin and Coster, 1993]. More powerful techniques based on the global network of receivers [Wilson and Mannucci, 1993; Sardon et al., 1994] then followed.

[9] The estimation of satellite and receiver biases remains an area of significant and active investigation within the ionospheric community. New and enhanced techniques have been recently developed that estimate receiver differential biases for all available GPS stations (typically around 1000 sites) on a daily basis [Komjathy et al., 2005; Rideout and Coster, 2006]. The research community needs more efficient and improved estimation algorithms to properly perform process and quality checks on the large amount of GPS data currently available on a daily basis.

[10] This article looks at one possible error in the standard practices of determining receiver biases. Specifically, the most common procedure within the community involves applying a single value for the receiver bias, and another one for each individual satellite bias, over a 24 h time frame. For the satellite biases, most groups rely on those estimated by the international Global Navigation Satellite Systems (GNSS) organization (IGS) or those of IGS contributing members, such as JPL. These values are available online at Crustal Dynamics Data Information System (CDDIS). The IGS satellite biases are a combination of the different research centers’ products, and therefore the accuracy of the IGS biases may be higher than those derived at JPL [Komjathy, 2012]. Receiver biases, on the other hand, are typically estimated by the individual groups processing TEC. Typically, the estimate of the receiver bias is made over a full 24 h period. This practice averages over any temporal changes in temperature. In the literature, there have been hints of issues with temperature [Ciraolo et al., 2007]. For example, the thermal influence on time and frequency transfer has been studied [Rieck et al. 2003]. The National Radio Astronomy Observatory has also described phase stability measurements versus temperature for several coaxial cable types [Norrod, 2003]. It is assumed that the temperature dependence in the L1–L2 receiver bias is due to a combination of the following: (1) the hardware in the preamplifier of the antenna, (2) the cable connecting the antenna, and (3) the receiver hardware itself. Receivers that measure the biases due to the receiver hardware have been built [Dyrud et al., 2008], but they neglect to account for the receiver bias due to items 1 and 2. Our goal is modest. Our aim is to demonstrate that temperature changes, on a time scales less than 24 h, will impact estimated receiver biases, and this, in turn, will affect the absolute TEC that can be obtained with GPS.

1.1 Description of Experiments

[11] For these experiments, we have used three Scintillation Network Decision Aid (SCINDA) GPS receivers located at the MWA site, −26.703° latitude and 116.657° longitude, and a fourth SCINDA GPS receiver at the MIT Haystack Observatory Optics Facility, 42.612° latitude and −71.485° longitude. The SCINDA is a network of ground-based receivers that monitor scintillations at the UHF and L-Band frequencies caused by electron density irregularities in the equatorial ionosphere [Groves et al., 1997]. The four SCINDA receivers are all NovAtel dual frequency GISTM receivers running the specialized software described in Carrano and Groves [2006]. The GPS receivers at the MWA site were provided by the Air Force Research Laboratory, while the GPS receiver at the MIT Haystack Optics building is on loan from the University of Calgary.

[12] The temperature data were collected with EasyLog USB data loggers, which collect temperature, dew point, and humidity data. Temperature data were verified by comparing the measurements to the National Oceanic and Atmospheric Administration's measurements in Westford, Massachusetts, the town where the MIT Haystack Optics building is located. These temperature loggers have a stated manufacturer's accuracy of ±0.5 °C (±1 °F) and a repeatability of ±0.1 °C (±0.2 °F) over the range −35 °C to 80 °C (−31 °F to 176 °F). In our sampling, the temperature is measured to the 0.5 °C level.

[13] In all of the cases discussed in this article, the length of the cable connecting the receiver and the antenna was primarily outside (>70% of length of cable was outside). We consider only two temperatures, one outside the building, and one inside the building. For the experiment at the Westford site, data were collected for over a hundred days using two temperature sensors and a single GPS receiver. In this experiment, the goal was to determine and remove the temperature dependence on the receiver bias. The second experiment at the MWA site in Western Australia involved a comparison of TEC estimates from three collocated GPS receivers over a single 24 h period. The primary goal of this experiment was to observe if the temperature dependence on receiver bias estimation varies from receiver to receiver. We will first describe experiments at the MIT Haystack Observatory where data were collected from days 15 to 150 of 2010. This data set is used for an initial attempt at modeling and removing the temperature dependence on receiver bias estimation. The remaining residual bias is examined for dependence on other parameters that may need to be considered in future modeling efforts.

1.2 Experiment at MIT Haystack Observatory

[14] In our first experiment, temperature data were collected with two temperature loggers that were collocated with the single GPS SCINDA receiver that ran continuously at the Optics Facility of the MIT Haystack Observatory in Westford, Massachusetts. One of these temperature loggers was located inside the building near the receiver and the other was located outside the building on the pole of the antenna. It is important to note that the antenna was located on a black top roof, where temperatures are expected to exceed the local ambient temperature.

1.2.1 Receiver Bias Estimation

[15] To test our hypothesis of temperature dependence on receiver bias estimation, daily receiver bias values were estimated for the nighttime period from 12:00 a.m. to 2:00 a.m. when the ionosphere is expected to be fairly stable. The receiver bias estimation procedure is based on a modified version of the bias determination procedure described by Rideout and Coster [2006]. GPS data from several other receivers in the New England area were used in this estimation procedure to strengthen the results. The bias estimation procedure uses the minimum scallop estimation approach described in Rideout and Coster [2006]. The minimum scalloping method was developed by P. Doherty (private communication) and depends on the assumption that when the receiver bias is correctly accounted for; on average, there should be no correlation between elevation and vertical TEC.

[16] In practice, the algorithm loops through a series of receiver bias estimates for a given receiver. For a given receiver bias, we bin and median filter vertical TEC values by elevation angle and determine the flatness of the resulting median TEC versus elevation-angle data. The receiver bias that gives the flattest value of TEC versus elevation angle is the minimum scallop receiver bias. We generally choose data around local midnight (in this case from 12 a.m. to 2 a.m. local time) where there should be less inhomogeneity in the TEC. This time period is valid for midlatitude locations. For equatorial locations, a time period shifted later, for example, 2 a.m. to 4 a.m. local time, would be more appropriate due to the possible presence of nighttime equatorial irregularities. The sites analyzed here were all midlatitude.

[17] The error in the individual receiver bias can then be reduced further using a second method, called “least squares,” and described by Rideout and Coster [2006]. The least squares method gives a differential receiver bias between a group of receivers that view overlapping sections of the sky. Only one absolute receiver bias measurement is then needed to calculate all the receiver biases for the group. This absolute receiver bias is calculated by averaging the independent minimum scallop estimates of receiver bias for each receiver in the group. This reduces the error in absolute receiver bias by the square root of the number of receivers in the group. It is important to note, however, that any remaining error in receiver bias is correlated across all the data from that group, and cannot be reduced by further averaging.

[18] The main modification to the MIT Automated Processing of GPS software involves an improvement in the handling of errors used for the TEC processing, described here for completeness. Errors are now tracked throughout the software, and random and correlated receiver bias errors described above are handled separately. This allows for optimal estimation of binned measurements using weighted averages and allows error values to be calculated independently for each binned measurement. The bin-to-bin variability in the TEC measurements was greatly reduced using this approach.

[19] The random error is associated with fitting the code data to phase data. For each arc, defined as a length of phase data without any possible phase breaks, a distribution function of the difference of code to phase data fitting is generated. The standard deviation of this distribution function is calculated and divided by the square root of the number of points. This random error is then weighted by an elevation factor associated with the line of sight to vertical conversion. Finally, the data are binned, and the resultant random error is the weighted mean of the binned errors. Figure 1 shows the receiver bias estimates using this technique as a function of day of year.

Figure 1.

Estimated receiver biases for receiver in Westford, Massachusetts, versus day of year, 2010.

1.2.2 Temperature Study Results

[20] For each day, an average temperature for the nighttime period from 12:00 a.m. to 2:00 a.m. was computed. Figure 2 shows the outdoor and indoor temperature data plotted as a function of corresponding bias value from days 15 to 150 of 2010 (January 2010 to June 2010). The linear lines associated with the indoor temperature results reflect the 0.5 sampling of the data. This is not apparent in the measured outdoor temperatures, although they were measured with the same temperature resolution because the range of temperatures observed is much greater. It is important to note that the temperatures and bias values shown in this plot are from the 12:00 a.m. to 2:00 a.m. local time period. We did investigate calculating the bias value during different time periods during the day; however, our estimates of the biases during the daytime were noisier in part because of the minimum scalloping technique we used to estimate the bias. In this article, the temperature bias will be demonstrated using the nighttime values.

Figure 2.

Nighttime receiver bias versus outdoor temperature (top) or indoor temperature (bottom) for the time period days 15–150 of 2010. Data collected at the MIT Haystack Observatory Optics building in Westford, Massachusetts.

[21] The Pearson correlation coefficient (r), which represents the linear relationship between two variables, is determined for both the outdoor temperature versus bias (top plot) and the indoor temperature versus bias (bottom plot). Other fits to the data (such as fitting to a quadratic) were no more significant, so we continued with the linear model. For a statistical sample size greater than 100, as is the case here with 131 points, an r value greater that 0.254 or less than −0.254 is rated significant. Because the Pearson correlation coefficient, or r value, is equal to −0.289 for the outdoor temperature versus bias value versus −0.253 for the indoor temperature versus bias, we decided to first remove the correlation observed between the receiver bias and the outdoor temperature. Recall that the receiver bias dependence on temperature is likely due to three factors: the antenna, the cable, and the receiver hardware. As we described earlier, along with the antenna, greater than 70% of the length of the cable in this experiment was located outside. A more detailed initial modeling of receiver biases should take these three factors into account.

[22] Also observed is an apparent break in the indoor temperature data versus bias (this can be observed visually at approximately 24 °C). To remove this correlation, a linear trend was computed for the outdoor temperature versus receiver bias (shown in the top plot of Figure 3) and removed from the data. The residuals are shown in the bottom plot of Figure 3.

Figure 3.

The top plot shows the computed linear trend in the receiver bias versus outdoor temperature data set. The bottom plot shows the receiver bias residuals versus temperature.

[23] Figure 4 shows the resulting receiver bias residuals versus the indoor temperature. In this plot, a clear break can still be observed in the data at 24 °C (75 °F). It is assumed that data below 24 °C are indicative of time periods where the heater in the optics building was having difficulty maintaining a constant temperature. The breakpoint likely shows the dependence of receiver electronics with respect to the temperature. For this data set, the data were divided into two different groups corresponding to the residuals associated with the data less than and greater than or equal to 24 °C. Linear fits were computed for each of these groups. These fits and the corresponding Pearson r values are shown in Figures 5 and 6. For the data corresponding to temperatures greater than 24 °C, the Pearson r value is fairly high (r = 0.398).

Figure 4.

Receiver bias residuals versus indoor temperature for days 15–150 of 2010.

Figure 5.

The computed linear trend in the receiver bias residual versus indoor temperature greater than or equal to 24°C.

Figure 6.

The computed linear trend in the receiver bias residual versus indoor temperature less than 24°C.

[24] The Rieck et al. [2003] study also measured two different linear regimes in the temperature dependence, similar to that observed in our experiment. They observed a slightly different breakpoint than the one we observed (at 24 °C). Their study differed from ours in that they only looked at temperature effects on the L1 (1575.42 MHz) frequency of the GPS signal, and not the delay difference between the L1 and the L2 (1227.6 MHz) frequencies as was done in our study. For the low noise amplifier of the Ashtech Choke Ring antenna that they were using, they measured a temperature phase delay relation of −0.1 ps/K above 20 °C (68 °F) and a +0.17 ps/K below 15 °C (59 °F).

[25] Once these linear trends have been estimated and removed from the data, the final residuals are shown in Figure 7. Clearly, a 2–3 TEC fluctuation level remains in biases versus day of year plot. To test if additional factors that should be accounted for in our bias estimation techniques, several atmospheric parameters were examined to see if any additional correlations could be found. The atmospheric parameters selected to study for correlations included the Ap, the Kp, and the daily and 90 day averaged F10.7 cm solar flux. These parameters are important in measuring solar activity (daily and 90 day averaged F10.7 cm solar flux) and the Earth's magnetic and electric field disturbances (the Kp and the Ap). The only parameter that strongly stood out as highly correlated with our residuals was the daily F10.7 cm solar flux, which is a measurement of solar radio emissions at 2800 MHz. This correlation can be observed in Figure 7 where a scaled daily F10.7 cm value is overplotted in red onto the final residual receiver bias. The correlation factor between these two data sets is 0.64.

Figure 7.

Final residual receiver bias following removal of both indoor and outdoor temperature trends (shown in black). The scaled daily F10.7 measurements are overplotted in red.

[26] We suspect that this observed correlation is related to the mapping function used in our receiver bias estimation. The mapping function we used is described in Rideout and Coster [2006] and depends only on the elevation and has no built-in term for a changing ionospheric shell height or any azimuthal dependence. Our approach was based on the assumption that there is little variability in the ionospheric shell height in the 2 h window after midnight when all the receiver biases are estimated in the MIT Automated Processing of GPS software. Nevertheless, we suspect that the ionospheric shell height does have some day-to-day variability that is not captured in the MITGPS mapping function, and this additional error source could perhaps account for some of the unmodeled errors observed. Another possibility is that some of this error can be attributed to remaining errors in the satellite biases, which are assumed to be correct.

1.3 Experiment at MWA Site

[27] On 25 April 2008, three SCINDA GPS receivers were located at the MWA site, separated by approximately 10–20 m. Figure 8 shows the exact placement of these receivers at the MWA site.

Figure 8.

Exact location of the MW1, MW2, and MW3 receivers at the MWA site.

[28] Shown in Figure 9 are the line-of-sight (LOS) differential TEC measurements between these three receivers as a function of time of day. The LOS differential TEC values are used here because no additional errors are introduced by use of a mapping function. The different colors represent the TEC measured to the different individual GPS satellites observed. Because the receivers were so closely spaced, the difference between their individual TEC measurements should be zero for the full 24 h period observed. If a mistake was made in one of their receiver bias estimations, and the receiver bias was constant for the 24 h period, the difference should be a constant. The top plot in Figure 9 shows the differential LOS TEC measurements between receivers MW2 and MW1, the middle plot shows the differential LOS TEC between receivers MW1 and MW3, and the bottom plot shows the differential LOS TEC between MW2 and MW3. As all of receivers were within 10–20 m of each other, the differential LOS TEC values should be either constant as a function of time of day, or at the very least, relatively constant. What is observed, however, are larger values in the differential TEC during the daytime and smaller values during the night. This diurnal structure is largest in the bottom plot, which shows the differential LOS TEC values between MW2 and MW3, and smallest in the middle plot, which shows differential LOS TEC between MW1 and MW3. One would surmise from this that the receiver MW2 has a receiver bias temperature dependence that differs significantly from that of MW1 and MW3 and that the MW2 bias at least is not constant for the full 24 h period. Although the exact temperature excursion between day and night on 27 April 2008 is not known for the MWA site, it is known from historical records that in Learmonth, Western Australia, the low temperature for 27 April 2008 was 20 °C (68 °F) and the high temperature was 32 °C (90 °F). Based on the observations in Figure 9, we are suggesting that the observed large diurnal changes in the differential TEC are due to a receiver bias temperature dependence that differs between the receivers MW1, MW2, and MW3. In other words, the temperature dependence for each receiver is unique and thus requires that the receiver bias temperature dependence be estimated for each individual receiver.

Figure 9.

Differential line of sight TEC between three different SCINDA receivers at the MWA site plotted as a function of time of day.

2 Summary

[29] There are four main conclusions from this research.

  1. A clear temperature dependence on the GPS receiver bias is evident.
  2. This temperature dependence appears to include both indoor and outdoor temperatures.
  3. Other factors that are not now accounted for, such as the daily F10.7 cm flux, appear to be important in receiver bias estimation.
  4. GPS receiver biases appear to differ between individual receivers, and each receiver must be treated independently.

[30] With respect to our initial reason for undertaking this investigation—which was the question how accurately can the absolute TEC be determined using GPS for use in the low-frequency array calibration—we think that a 2–3 TEC scatter remains in our bias estimation process even after the temperature effect is removed (see Figure 7). This number does not represent the relative TEC accuracy, which is based on the differential phase measurements of TEC. The relative TEC can be estimated on the order of 0.01 TEC units. The 2–3 TEC scatter remaining in our observed bias estimation is likely due to the inaccuracies of the model used for the ionospheric mapping function. As was mentioned earlier, our mapping function does not account for any changes in the daily ionospheric scale height. The error in our mapping function also includes the lack of a valid plasmaspheric model. [Mazzella, 2012] Recently, Carrano et al. [2009] developed a Kalman filter model to estimate plasmaspheric TEC. If a better understanding of how to model the temperature effect in the bias estimation can be obtained, it is possible that an extended version of the Kalman filter for bias determination will lead to more accurate estimates of the bias. Clearly more experimental and modeling studies are warranted.


[31] The authors gratefully acknowledge support from the National Science Foundation to Siena College (grant no. ATM-0753840) and to MIT Haystack Observatory (grant no. AGS-1023098).