Radio Science

An ionospheric error model for time difference of arrival applications

Authors


Abstract

[1] The geolocation accuracy of satellite-borne time difference of arrival systems is limited by the uncompensated differential delay experienced by each pair of satellite-to-ground paths. We have measured the ionospheric component of this differential delay using two-frequency GPS receivers at seven ground sites; one in the high-latitude region, four at midlatitudes, and two in the equatorial region. The measurements were expressed as differential total electron content (ΔTEC) so that they could be used at frequencies other than the 1.6 and 1.2 GHz GPS values and be readily compared with ionospheric models. Three models were used to account for the measured ΔTEC values: a climatological model (RIBG) and two modified versions of Jet Propulsion Laboratory's GIM model (a postanalysis and a near-real-time implementation using a subset of reporting stations). All three models reduced the ΔTEC differences significantly. The postanalysis GIM model performed better than the other two in the tails of the residual error distribution, but there was no clear difference among the three in about 75% of the cases. However, when the data were binned according to local time and angle between the two rays, both GIM models were more accurate than RIBG between about 0600 and 1900 LT at angles greater than about 50°.

1. Introduction

[2] At elevation angles greater than about 10° the dominant environmental source of geolocation errors for single-frequency satellite-based time difference of arrival (TDOA) systems like GPS is the uncompensated differential delay induced by the ionosphere along each pair of satellite-to-ground paths. It is difficult to estimate the magnitude of these errors accurately, however, because they depend on the poorly known ionospheric correlations between the two rays. For the same reason, it is difficult to evaluate how effective any particular ionospheric model might be in reducing the errors [Klobuchar, 1975, 1987, 1996; Feess and Stephens, 1987; Jorgensen, 1989; Klobuchar and Doherty, 1990; Greenspan et al., 1991]. This experiment was designed to directly measure the differential ionospheric errors and to evaluate the effectiveness of three global ionospheric models being considered for operational use in correcting them.

[3] We made a series of measurements in the first three months of 2000 using two-frequency GPS receivers. Since the frequency of L1 is 1.575 GHz and that of L2 is 1.228 GHz, L2 will be delayed more than L1 as the two rays pass through the ionosphere. If the transmitter and receiver biases are known, this differential delay can be cast in terms of the total electron content (TEC) along the (assumed common) ray path [Klobuchar, 1996]. The difference in the simultaneously measured slant path TEC values from two different GPS satellites to the same receiver is what we call a ΔTEC measurement; at frequencies above a few hundred megahertz this is directly related to the ionospheric TDOA error.

[4] The measurements were then compared with three ionospheric models. We derived the empirical distribution functions of the residual errors when the data were sorted by both local time at the receiver and by the ground angle between the two ray paths.

[5] Finally, we found an analytical expression for the ΔTEC error that bounded the 70th and 95th percentiles of the cumulative probability function for the measurements themselves as well as for the residual errors when the ionospheric errors were “corrected” by using the models.

2. Description of the Experiment

2.1. Ionospheric Models

[6] We compared the measured ΔTEC values with the values predicted by three ionospheric models: RIBG and two versions of Jet Propulsion Laboratory's (JPL's) GIM.

2.1.1. RIBG.

[7] This is a climatological model developed by Reilly [1993, 1991] and Reilly et al. [1991] which combines three separate models with an efficient ray trace procedure. The underlying models are ionospheric conductivity and electron density (ICED) [Tascione et al., 1988], for the main ionospheric layers, the topside model of Bent et al. [1975], and the plasmasphere model of Gallagher et al. [1988]. The RIBG version used in this study was SFP1996.1, dated February 1996. This was the latest U.S. government version of RIBG that was provided to us for testing. However, more recent versions of RIBG have since been developed [Reilly and Singh, 1997] which use GPS data to determine separate driving parameters for different properties on the electron density profiles.

[8] The version of RIBG we tested was driven by two environmental proxies, the sunspot number and an auroral activity index. We used the hourly Ionospheric Activity Index (IACTIN) provided by the Air Force Weather Agency's (former Air Force Space Command's) 55th Space Weather Squadron (SWxS) for the sunspot number [Secan and Wilkinson, 1997]. IACTIN is an equivalent sunspot number developed to work with the 1988 URSI foF2 model [Rush et al., 1989], which is used by RIBG. The index is based on measurements of foF2 from different locations worldwide (but mostly the continental United States). As described by Secan and Wilkinson [1997, p. 1718], IACTIN, which they refer to as an effective sunspot number (SSNe), “is defined as the SSN value that, when input to the URSI foF2 model, gives a weighted zero-mean difference between the observed and modeled foF2 values.” The weighting depends on latitude and local time.

[9] The 55th SWxS generates several varieties of hourly IACTIN, the differences depending on the number of hours (ranging from 6 to 24) of foF2 data used to produce the index (ionospheric memory). We used an ionospheric memory of 24 hours as recommended by K. Scro (private communication, 2000). That is, data from the hour of interest and data from the previous 23 hours were used to calculate the IACTIN. We used this version of the hourly IACTIN because it would better allow RIBG to track variations in the ionosphere, while not being as prone to errors when the data it is based on have errors, or when some of the data are missing (especially since we are comparing the RIBG climatological model to frequently updated data-driven approaches such as the JPL GIM models; see the description below).

[10] During the data collection phase of the experiment the IACTIN ranged from 76 to 170, with an average value of 121. During the test periods, geomagnetic activity ranged from low to moderate. We attempted to use the 55th SWxS's auroral activity index, but it was much more variable than our experimental data, so we held the index constant at a moderate value of 4.

2.1.2. Interpolated DICE.

[11] This is a shell model of the ionosphere based on the near-real-time TEC measurements made by a 50-station subset of the worldwide network of the GPS receivers used in JPL's GIM ionospheric model. [Mannucci et al., 1998] We refer to this model as DICE to distinguish it from the publicly available GIM model. The model and the software to calculate the slant path TEC values from it were supplied to us by JPL. It was updated at 15-min intervals throughout the data collection phase. Because it is purely a fit to measured data, it requires no other inputs to drive it. “Interpolated” means that the value at any given time is evaluated by linearly interpolating between the closest updated model before and after the desired time. This is the way it should be used if data are being analyzed well after the time when they were collected.

2.1.3. Extrapolated DICE.

[12] When it is necessary to analyze data in near real time, it is not possible to wait for the next DICE update to use in interpolating to the time of interest. The “extrapolated” DICE model is meant to simulate that situation. Here we assumed that it would take 50 min to receive and process all the data from the 50-station network of GPS receivers and a new model would then be disseminated to users every hour. Thus the user would be working with data that were between 50 and 110 min old. The “extrapolated” model is the most recent model the user would have available, updated only because of the Earth's rotation.

2.2. Ground Truth Receiver Locations

[13] We used data from seven different GPS receiver sites (termed “validation stations” because we were attempting to validate the usefulness of the models). One was in the high-latitude region, four were at midlatitudes, and two were equatorial. Table 1 lists their locations, and Figure 1 plots their positions graphically. While we realized that the number of validation stations (especially at the low and high latitudes) is limited, the number and selection of their locations resulted from the combined constraints of the need to test particular situations (such as near and far from stations used to drive the DICE models) and the need for quality data with high availability (based on International GPS Service ratings) not used to drive the DICE models. Three of the midlatitude stations (JPLM, NICO, and URUM) were selected to test different longitude regions reasonably close (∼500 km) to the DICE driver stations. One station (KERG) was purposely chosen because it was far away from the driver stations. The two equatorial stations (RIOP and BAKO) were initially selected to be near the DICE driver sites, but during the test periods the driver sites near RIOP were available less then 50% of the time. Hence RIOP represents a station distant from driver data, while BAKO represents a station near driver data. The one high-latitude station (TIXI) was originally going to be combined with the midlatitude stations, but the results were different enough to warrant its own bin. The high latitudes also fall under the constraint of needing high-quality data, with high availability from stations not used to drive the DICE maps.

Figure 1.

Locations of the seven validation stations.

Table 1. GPS Validation Stations
NameLatitude, °NLongitude, °EAltitude, km
High Latitude
TIXI71.6128.90.05
Midlatitude
JPLM34.2241.80.42
KERG−49.470.30.07
NICO35.133.40.16
URUM43.687.60.86
Equatorial
BAKO−6.5106.80.16
RIOP−1.7281.32.82

2.3. Data Collection and Reduction

[14] The data collection period consisted of three 9-day spans in January, February, and March 2000. The data from all the available GPS satellites in view of each receiver site (with elevation angles above 10°) were transmitted to JPL for reduction to TEC measurements by subtraction of JPL-determined hardware differential delay biases from the differential delay data. The GPS ephemerides and the TECs were then sent to us for further analysis.

[15] The JPL slant path TEC values have an error of 2–3 TEC units (TECU) (B. Wilson, private communication, 2001). While there are numerous other organizations providing TEC data of similar accuracy, we went to JPL for our TEC processing because one of the possible outcomes of this experiment is the expansion of a near-real-time network of GPS TEC receivers that JPL currently runs for the Air Force.

[16] The TEC data typically consisted of measurements at 1 min intervals, 24 hours per day, for 9 days in each span. Some receiver sites (notably RIOP and NICO) had equipment problems for a few days during the 27 day collection period, but most of the sites were operational during the whole time.

[17] We then converted the TEC data into ΔTEC data by subtracting the smallest TEC measurement at each site from all the others measured at that same time. (This can be done without loss of generality, since the subsequent processing assumes that we have sampled only the nonnegative portion of a zero-mean distribution.)

2.4. Available Data

[18] Table 2 presents the number of ΔTEC measurements that were available from each of the seven receiver sites. The second column refers to the measurements which were used with the uncorrected data and RIBG and interpolated DICE models; the third column shows the smaller number of measurements available for the extrapolated DICE model. (The DICE models were supplied during the data collection periods only, so during the first 50 min of each 9 day collection there was no DICE model available.)

Table 2. Available ΔTEC Measurements
NameTotal Number“Extrapolated” Number
High Latitude
TIXI225,563224,519
Midlatitude
JPLM177,612176,933
KERG162,064161,261
NICO136,902136,404
URUM235,038234,031
Equatorial
BAKO148,592147,969
RIOP60,40360,153

3. Data Analysis

3.1. Probability Distributions

[19] We examined the empirical probability distributions of the ΔTEC residuals after subtracting the predictions of each of the three ionospheric models from the measurements and compared them to the uncorrected ΔTEC distribution. (Although the measured ΔTECs are defined to be nonnegative, the difference between the measured and the model ΔTECs can have either sign.) The data were edited to remove points that differed from the sample mean by more than 4 times the sample standard deviation. Figures 24 display these distributions for the combined results of the two equatorial stations, the four midlatitude stations, and the single high-latitude station. The stepped curves are the measured distributions, and the smooth curves are the best-fitting Gaussian distributions.

Figure 2.

Empirical probability distributions for the combined residuals of the two equatorial stations. The abscissa is in TEC units.

Figure 3.

Empirical probability distributions for the combined residuals of the four midlatitude stations. The abscissa is in TEC units.

Figure 4.

Empirical probability distributions for the residuals of the single high-latitude station. The abscissa is in TEC units.

[20] None of these empirical distributions is consistent with a Gaussian distribution, so it is not possible to estimate confidence limits as simple multiples of the standard deviation. Instead, we calculated the empirical cumulative distribution function for each of the cases from the data values themselves. The cumulative distribution functions are shown in Figures 57, where the numbers in the body of each figure are the values of ΔTEC (in TEC units), where the probability of not exceeding that value is the percentage indicated. These empirical distributions clearly show not only that all three models are very effective in reducing the ΔTEC errors but also that there is no clear distinction between them until the probability is above about 70–80%.

Figure 5.

Cumulative probability distributions for the combined residuals of the two equatorial stations. The abscissa is in TEC units.

Figure 6.

Cumulative probability distributions for the combined residuals of the four midlatitude stations. The abscissa is in TEC units.

Figure 7.

Cumulative probability distributions for the residuals of the single high-latitude station. The abscissa is in TEC units.

3.2. Local Time and Separation Dependence

[21] Although the postanalysis model (interpolated DICE) is better than either the real-time model (extrapolated DICE) or the climatological model (RIBG) above about the 60% level, the improvement was not as great as had been expected. Since there were over 200,000 data points in each latitude region, we could investigate the reasons for this by gridding the data in the two dimensions that most influence the magnitude of the ΔTEC residual: the local time at each station and the angle θ between the two rays. The grid was set up to be every 2 hours in local time and every 10° in angle. In order to obtain a statistically valid result the 4σ editing criterion was applied independently at each grid point, and only points with at least 50 samples (after editing) were retained. The results indicated that many of the data were obtained when the angle between the rays was less than 90° and about half of the data occurred during the night and evening hours. Both of these conditions resulted in a relatively large number of small residuals in all the models since the uncorrected ΔTECs were themselves small.

3.3. Analytical Models

[22] Since the accuracy of the three ionospheric models varies as a function of both local time and the angle between the rays, the most concise way of characterizing the results is an analytical representation. When the residuals were plotted versus angle, they showed a strong linear trend with long-period fluctuations. Accordingly, we used a linear term and two sine terms as basis functions for our representation (one with a 180° period and the other with a 90° period). Since the residuals have to be zero when θ is zero, cosine terms were ruled out. The constraint on local time was lenient: We required only that the functions have a 24 hour periodicity. We used a Fourier expansion to fit the time residuals and retained only the most significant terms (24, 12, and 8 hour periods).

[23] We define the quantities:

M

number of points in angle separation;

N

number of points in local time;

T(P)

an M × N matrix of ΔTEC values which are associated with the P percentile point of the cumulative probability distribution;

F(θ)

an M × 3 matrix describing the dependence of T(P) on the angle θ (degrees) between the rays;

G(τ)

a 7 × N matrix describing the dependence of T(P) on the local time τ (hours);

C

a 3 × 7 matrix of least squares coefficients.

Then

equation image

[24] We found that the data could be accurately represented by using the following functions (where the subscript i refers to the row of the matrix and j refers to the column):

equation image
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equation image
equation image
equation image
equation image

[25] We then calculated the coefficient matrix C for both the 70% and the 95% levels for the residuals in each of the four models (including the “model,” which was just the uncorrected measurements). Given our choice of basis functions, we used an unconstrained unweighted least squares fit to determine the C matrix; this treated all the data equally. Since we had so much data, it was unlikely that outliers would have significantly altered the numerical values of the coefficients. Figures 813 display the results. From left to right, then top to bottom, each figure shows the levels for the uncorrected data, then the residual levels after RIBG, extrapolated DICE, and interpolated DICE corrections. The values of C are given in Table A111.

Figure 8.

Contours of constant ΔTEC at the 70% level for the combined residuals of the two equatorial stations. The contour interval is 5 TECU.

Figure 9.

Contours of constant ΔTEC at the 95% level for the combined residuals of the two equatorial stations. The contour interval is 5 TECU.

Figure 10.

Contours of constant ΔTEC at the 70% level for the combined residuals of the four midlatitude stations. The contour interval is 3 TECU.

Figure 11.

Contours of constant ΔTEC at the 95% level for the combined residuals of the four midlatitude stations. The contour interval is 6 TECU.

Figure 12.

Contours of constant ΔTEC at the 70% level for the residuals of the single high-latitude station. The contour interval is 3 TECU.

Figure 13.

Contours of constant ΔTEC at the 95% level for the residuals of the single high-latitude station. The contour interval is 6 TECU.

3.4. Discussion of Results

[27] For equatorial latitudes, at the 70% level, all four models exhibit large residuals well above 15 TECU over much of the domain (Figure 8). The largest errors are between 1200 and 1800 LT, but the errors remain significant from 0800 LT through local noon to well past local midnight. The result of applying any of the three ionospheric models is to shift the time of the peak error to 1800–2400 LT. This is the postsunset region, where complex physics drives strong gradients (irregularities, etc.), and is not modeled well. The region of lowest errors also expands to encompass most of the postmidnight and presunrise region. This is when the ionosphere is quiet and the TEC is the least, so the modeling errors tend to be smaller. In one case the errors after “correction” are larger than the measured ΔTECs themselves. At the 95% level (Figure 9,) the morphology is similar, but the magnitudes have increased by a factor of 2. In general, the models do a poor job of reproducing the ΔTECs in the equatorial regions. This is most likely due to the limited amount of equatorial data used in either the averaging of the data into the model climatologies for RIBG or the interpolation between driver stations in the case of the DICE models. In the case of RIBG, other sources of error such as using a global index to select the climatology in the equatorial region could be potentially significant. The errors in the DICE models, such as an error due to mapping from slant to vertical or the assumption of a constant shell height, are much smaller in comparison.

[28] For midlatitudes at the 70% level (Figure 10) the measurements exhibit large residuals (greater than 15 units) centered between 0600 and 1800 LT, which increases with greater separation angles with a maximum centered at 1400 LT. This corresponds to the peak daytime ionosphere. When the models are applied, the residuals decrease dramatically. For RIBG the region of greater than 15 units shrinks to 0900–1800 LT for separation angles greater than 100°. Since the separation angle is related to the correlation length, as the separation angle increases the mismodeling of the TEC along the two paths is less likely to cancel out. For the DICE models, the greater than 15 unit TEC contour essentially disappears from the plots. At the 95% level (Figure 11) the plots mirror the 70% level plots, except the magnitudes of the residuals are about a factor of 2 greater.

[29] For high latitudes, at the 70% level (Figure 12), the region of large ΔTECs shifts to 0900–2100 LT with a peak at 1500 LT, again during the peak of the ionospheric electron content. When the models are applied, the region of greater than 15 units shrinks. For RIBG, only above 110° separation angle do errors this large remain. For both DICE models the residuals are greater than 15 units only between 0200 and 0400 LT and above 120°. At the 95% level (Figure 13) a similar structure is observed, but the magnitudes of the residuals are 2–3 times greater.

4. Conclusions

[30] We have analyzed approximately 106 ΔTEC measurements in order to construct an empirical error model for the ionospheric delays experienced by TDOA geolocation systems. ΔTEC measurements (rather than just TEC measurements) were needed because the correlation between the delays experienced by any two satellite-to-ground rays was not known. Even if it had been, the correlation of the ionospheric model errors for the two rays is not known.

[31] The observed ΔTEC values are often large enough to seriously degrade the accuracy of GPS-derived geolocations. The uncorrected path differential exceeds 100 TECU five percent of the time during the evening hours in the equatorial region. This is equivalent to a TDOA error of over 50 ns at L1. Large errors can also occur at midlatitudes, shifted toward midday instead of the evening and at somewhat larger angles between the two rays. Since the TDOA errors scale approximately as f2 down to several hundred megahertz, the errors are even larger for systems using lower frequencies.

[32] We used three models to attempt to account for these differential ionospheric delays: one based on climatology using IACTIN as the driving parameter, one based only on global TEC measurements made between 50 and 110 min before the time of interest, and the same model used in an “after the fact” manner, where measurements were available both before and after the time of interest. The models performed equally well in about 75% of the cases, but the data-driven models were significantly more accurate than the climatological model in regions of local time and angular separation where the measured ΔTEC was the largest. In both the high-latitude and the midlatitude regions the DICE models reduced the largest residuals by a factor of 3–5; the climatological model reduced them by a factor of about 2. All the models were much less successful in the equatorial region.

[33] Any given level of the cumulative probability function of the ΔTEC values, either as they were measured or after the predictions of one of the ionospheric models had been subtracted, can be represented by simple functions of local time and separation angle between the rays. For local time these were just trigonometric functions with a fundamental periodicity of 24 hours. For separation angle we used a linear term and two sine terms. These functions can be used with the least squares coefficients given in Table A1 to improve the error analysis of GPS and any other single-frequency TDOA geolocation system.

Acknowledgments

[34] This analysis could not have been done without the essential contributions of Anthony Mannucci, Lawrence Sparks, Brian Wilson, and others of JPL's Ionospheric and Atmospheric Remote Sensing Group under Task Order 15137. They collected the GPS data and reduced them to TEC measurements; they also provided the DICE ionospheric model fields. This work was performed under U. S. Air Force contract F4701-00-C-009.

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