GPS Solutions

, Volume 17, Issue 3, pp 337–346

# Application of the TaiWan Ionospheric Model to single-frequency ionospheric delay corrections for GPS positioning

• Lung-Chih Tsai
• Joz Wu
• Chao-Han Liu
Original Article

## Abstract

The performance of a three-dimensional ionospheric electron density model derived from FormoSat3/COSMIC GPS Radio Occultation measurements, called the TaiWan Ionosphere Model (TWIM), in removing the ionospheric delays in single-frequency pseudorange observations is presented. Positioning results using TWIM have been compared with positioning results using other ionospheric models, such as the Klobuchar (KLOB) and the global ionospheric model (GIM). C/A code pseudoranges have been observed at three International GPS Service reference stations that are representative of mid-latitude (BOR1 and IRKJ) and low-latitude (TWTF) regions of the ionosphere. The observations took place during 27 geomagnetically quiet days from April 2010 to October 2011. We perform separate solutions using the TWIM, KLOB, GIM ionospheric models and carry out a solution applying no ionospheric correction at all. We compute the daily mean horizontal errors (DMEAN) and the daily RMS (DRMS) for these solutions with respect to the published reference station coordinates. It has demonstrated that TEC maps generate using the TWIM exhibit a detailed structure of the ionosphere, particularly at low-latitude region, whereas the Klobuchar and the GIM only provide the basic diurnal and geographic features of the ionosphere. Also, it is shown that even for lower satellite elevations, the TWIM provides better positioning than the Klobuchar and GIM models. Specifically, using TWIM, the difference of the uncorrected solution (no ionospheric correction), and the other solutions, relative to the uncorrected solution, is 45 % for the mean horizontal error (DMEAN) and 42 % for the horizontal root-mean-square error (DRMS). Using Klobuchar and GIM, the percent for DMEAN only reaches to about 12 % and 3 %, while the values for the DRMS are only 12 and 4 %, respectively. In the vertical direction, all models have a percentage of about 99 and 70 % for the mean vertical error (VMEAN) and vertical root-mean-square error (VRMS), respectively. These percentages show the greater impact of TWIM on the ionospheric correction compared to the other models. In at least 40 % of the observed days and across all stations, TWIM has the smallest DMEAN, VMEAN, DRMS, and VRMS daily values. These values reach 100 % at station TWTF. This shows the overall performance of TWIM is better than the Klobuchar and GIM.

### Keywords

Single-frequency GPS positioning Ionospheric model Ionospheric delay

## Introduction

Degradation of the accuracy of the derived coordinates using global positioning system (GPS) can be attributed to the following sources: errors in satellite ephemeris, errors by satellite clocks, atmospheric errors, receiver errors, multipath interference, and position dilution of precision (PDOP) (Spencer et al. 2003). Since 2000, when the selective availability in GPS has been removed, the ionosphere has become the main error source in single-frequency GPS positioning (Camargo et al. 2000; Ovstedal 2002). This is due to the ability of the ionosphere to impact an incoming radio wave. The behavior of the ionosphere primarily depends on the local time, season, solar activity, viewing direction, location of the receiver, and the earth’s magnetic field (Klobuchar 1987; Camargo et al 2000). These can in turn cause different effects to the accuracy of GPS positioning. First-order ionospheric delay can be practically removed in the GPS signals using dual-frequency GPS receivers, which uses both L1 (1,575.42 MHz) and L2 (1,227.60 MHz) bands. However, these devices are more expensive and thus are generally not available for most users.

A number of ionospheric models have been developed for single-frequency GPS positioning. One of the first models was developed by Klobuchar (1987). He developed an operational ionospheric model to be broadcast by GPS satellites to provide the user corrections of approximately 50 % RMS of the ionospheric range error (Klobuchar 1987). Ovstedal (2002) used the precise satellite orbits and satellite clock corrections and the global ionospheric model (GIM) supplied by International GPS Service (IGS) in absolute GPS positioning. He was able to demonstrate a sub-meter epoch-to-epoch accuracy in the horizontal and approximately 1-meter accuracy in the vertical. One of the most recent models is called the Multi Instrument Data Analysis System (MIDAS). Developed for regional applications, Allain and Mitchell (2009) used data from dual-frequency receivers distributed across Europe. They compared their results with other models such as Klobuchar and the International Reference Ionosphere (IRI). They showed that using MIDAS, and precise satellite orbit data from IGS, the average position is within 1.5 m while using Klobuchar and IRI, the average position is 4 and 3 m respectively.

We present ionospheric delay correction for single-frequency GPS pseudoranges using a numerical and phenomenological model, called the TaiWan Ionospheric Model (TWIM). Its performance is compared to other ionospheric models such as Klobuchar and GIM. It is hoped that this will provide single-frequency GPS users an alternative ionospheric model that would give higher positioning accuracy than most models used today.

## Ionospheric corrections

Given the total electron content (TEC) along the signal path, ionospheric delay (dion) in the pseudorange can be determined as:
$$d_{\text{ion}} = \frac{{40.3{\text{TEC}}}}{{f^{2} }}$$
(1)

Numerically, one TEC is equivalent to about 16 cm of delay in the L1 pseudorange. Generally, ionospheric effects usually amount to about 30 m of error in the pseudorange measurements, depending on the elevation angle of the satellite. Moreover, among the errors that contribute to the pseudorange measurements, the ionospheric error is the most variable and difficult to compensate since the ionosphere is very dynamic and ionospheric radio propagation is dependent on the frequency of the radio wave.

In this study, the TWIM, which is a three-dimensional ionospheric electron density (ne) model, is used to calculate ionospheric delay for GPS single-frequency receivers (Tsai et al 2009). It is a numerical and phenomenological model of global ionospheric electron density that is constructed from monthly weighted and half-hourly radio occultation (RO) measurements from Formosat3/COSMIC GPS. The half-hour vertical ne profiles are derived from 30-day data set from the day of observation with much weight (Gaussian-shaped weighting function) given to days closer to the day of observation. For example, ne profiles for day 254 use RO observations from day 224 to 254. Additionally, further interpolation is done within the half-hour profiles. The 30-day data provide enough RO observations to produce global electron density profiles at a 30-min temporal resolution with a 3° spatial resolution.

Each layer (F2, F1, E, or D) is characterized by a Chapman-type function, which is described by its peak electron density (nemax), peak density height (hm), and scale height H. Thus, these parameters can be used to obtain the electron density ne at a specific longitude (θ), latitude (λ) and height (h) using the least-squares error fitting of the observed profile to the Chapman functions:
$$n_{e} \left( {\theta ,\lambda ,h} \right) = \sum\limits_{i = 1}^{n} {n_{e\max } \left( {\theta ,\lambda } \right)} \exp \left\{ {\frac{1}{2}\left[ {1 - \frac{{h - h_{m} \left( {\theta ,\lambda } \right)}}{{H\left( {\theta ,\lambda } \right)}} - \exp \left( { - \frac{{h - h_{m} \left( {\theta ,\lambda } \right)}}{{H\left( {\theta ,\lambda } \right)}}} \right)} \right]} \right\}$$
(2)
Each i represents a physical layer of F2, F1, E, or D. Surface spherical harmonics are applied to map the derived Chapman layer parameters in geodetic coordinates. Thus, the TWIM is three dimensional. In addition, all of these layers can occur during the daytime. The F1 and D layers decay at night and can be hidden within the other layers, but the F1 and D-layer parameters are still derivable at all times by least-squares error fitting. The TWIM does not account for electron density in the plasmasphere.

It should also be noted that because every electron density estimates by the TWIM uses a 30-day data set, any disturbances in the ionosphere, such as storms, in a given day will be masked by the other non-disturbed days within the 30-day period. Moreover, the Formosat3/COSMIC cannot provide dense enough RO observations to observe disturbed Ne profiles when magnetic storm happens. During these events, the retrieved Ne profiles are removed because the signal-to-noise ratio values are very low. Therefore, such events may not be observed. However, this data set is enough to model the ionosphere during geomagnetically quiet days.

The ne profiles are used to calculate the point-to-point slant TEC (STEC) between the receiver and each GPS satellite (Tsai et al 2009). These are used to determine the ionospheric delay on the L1 frequency. The corrections made using TWIM are compared to two of the most commonly used ionosphere models for single-frequency positioning. Both vertical TECs generated by Klobuchar and GIM, which are two-dimensional ionosphere models, are translated to slant delay using elevation-dependent mapping functions.

The atmospheric errors (tropospheric and ionospheric) are highly dependent to elevation angles. Thus, these errors can be minimized by setting elevation masks or cutoff angles. These angles do not allow data coming from satellites at low elevation where they are subject to more biases as compared to higher elevation angles. Moreover, setting an elevation mask can also minimize other elevation-dependent biases, such as multipath effects. In this study, elevation masks are set at 10°.

## Method

Single-frequency GPS data are recorded during the most geomagnetically quiet days per month from April 2010 to October 2011—a total of 27 days— by three IGS reference stations (BOR1, IRKJ, and TWTF) at a 2-min sampling interval. Tables 1 and 2 show the details of each station and the geomagnetically quiet days used in this study. These days were chosen since TWIM is a quiet time model.
Table 1

Details of the three IGS stations used in this study

Code

Location

Longitude

Latitude

Height (m)

Local time (h)

BOR1

Boroweic, Poland

52°16′37′′

17°04′24′′

124.9

+2

IRKJ

Irkutsk, Russia

52°13′08′′

104°18′58′′

502.1

+7

TWTF

Taoyuan, Taiwan

24°57′13′′

121°09′52′′

201.5

+8

Table 2

List of days used in this study with their minimum, maximum 3-h Kp index, and daily total Kp index

Date

DOY

Minimum 3-h Kp index

Maximum 3-h Kp index

Total Kp index

2010 (14 days)

04/26/2010

116

0

7

23

05/23/2010

143

0

3

7

05/24/2010

144

0

7

20

06/12/2010

163

3

10

33

07/10/2010

191

0

3

17

07/17/2010

198

0

3

20

08/22/2010

234

0

3

17

08/30/2010

242

0

3

13

09/11/2010

254

0

3

10

09/12/2010

255

0

7

20

10/02/2010

275

0

0

0

11/06/2010

310

0

7

20

11/26/2010

330

0

10

20

12/10/2010

344

0

3

10

2011 (13 days)

01/30/2011

30

0

3

13

02/03/2011

34

0

7

23

03/15/2011

74

0

3

13

04/26/2011

116

0

7

30

05/20/2011

140

0

7

33

06/29/2011

180

0

7

33

07/27/2011

208

0

13

47

08/18/2011

230

0

10

37

08/31/2011

243

0

7

20

09/19/2011

262

0

10

37

09/23/2011

266

0

7

33

10/28/2011

301

0

3

3

10/29/2011

302

0

3

3

GPS raw data in RINEX format (ftp://igscb.jpl.nasa.gov/igscb/data/format/rinex211.txt) are used. The RINEX files include the pseudorange and carrier phase observations and the satellite navigation message for each GPS satellite. Satellite positions are calculated using the GPS satellite broadcast ephemeris, that is only the data contained in the RINEX files are used. The receiver positions are calculated using L1 C/A code pseudoranges corrected for satellite clock bias, relativistic clock bias, timing group delay, and tropospheric error. A base solution is carried out in which the ionospheric delay is not applied; this solution is referred to as uncorrected (UNC) solution. Three additional solutions are carried out per epoch and station but now correcting the ionospheric delay using the Klobuchar (KLOB), GIM and TWIM models respectively.

The errors in the horizontal (east and north) and vertical are calculated by differencing the computed position and the known reference position. The horizontal error D is calculated as
$$D = \sqrt {E^{2} + N^{2} }$$
(3)
where E and N are the differences in the east and north direction between both solutions. The daily mean and the root-mean square (RMS) values of these differences are calculated per station and labeled as EMEAN, NMEAN, ERMS, and NRMS. To represent the daily horizontal position accuracy, we compute the daily mean of (3) denoted by DMEAN. We also compute the distance RMS, labeled DRMS, which contains 65–69 % probability of observations being within a horizontal circle with radius DRMS and center at the reference position, as (Conley et al. 2006):
$${\text{DRMS}} = \sqrt {{\text{ERMS}}^{2} + {\text{NRMS}}^{2} }$$
(4)
The daily mean error along the vertical (VMEAN) and its RMS error (VRMS) are used to describe the vertical accuracy of each observation. The difference of uncorrected and corrected data, relative to the uncorrected data, is then expressed in percent. This percentage expresses the impact of the ionospheric model and is given by.
$$\% {\text{diff}} = \frac{{{\text{UNC}} - {\text{ION}}}}{\text{UNC}} \times 100\;\%$$
(5)
where UNC is the value obtained using no ionospheric models and ION is value obtained when using an ionospheric model. Eq. (5) is applied to mean and RMS values. Positive percentage shows improvement in the observations while negative percentage pertains to measurements whose errors are worse than the uncorrected data.

## Results and discussion

This section demonstrates the performance of TWIM in GPS positioning. The results, which include the total electron content maps, the horizontal and vertical positional errors and RMS values, are compared to the Klobuchar and GIM models at different elevation angles, time, and geographical locations. The percent occurrence of the days that yielded lowest errors for all days of observations is also presented.

### Total electron content map

Nine global VTEC maps for Klobuchar, GIM, and TWIM at epochs 0300UT, 0400UT, and 0500UT on August 22, 2010 (DOY 234) are shown in Fig. 1, which is chosen arbitrarily. The test sites are shown in red triangles. The basic diurnal and geographic features of the ionosphere are shown in all nine maps, that is, peak density during day time and in the low-latitude regions while the minimum density is featured during night time and in the mid-latitude regions. We can also observe the westward shift of the peak density area as the time moves forward. The maximum VTEC value shown on the GIM maps is the greatest among the three models while the Klobuchar maps displayed the least peak density that is shifted to higher latitudes. The TWIM, on the other hand, provides a more detailed VTEC maps as compared with the other two models. For example, the well-known equatorial ionization anomaly (EIA) is clearly shown in the TWIM VTEC maps in the daytime sector. Furthermore, the TWIM can provide three-dimensional electron densities while Klobuchar and GIM only provide two-dimensional (longitudinal and latitudinal) VTEC models.

The calculated 24-h ionospheric slant delays in meters using Klobuchar, GIM, and TWIM on DOY 234 for the three test sites are shown in Fig. 2. The white and gray areas indicate local daytime (0600–1800LT) and nighttime (1800–0600LT). The figure clearly shows the typical diurnal variation of ionosphere where electron density is higher during daytime than in night time. Moreover, the mid-latitude stations (BOR1 and IRKJ) produce low ionospheric delay while the low-latitude station (TWTF) provides high ionospheric slant delays during daytime. This corresponds to the high density provided by the EIA. Meanwhile, all stations yield approximately equal ionospheric slant delays during nighttime. Slant delays calculated from the Klobuchar model are consistent for all stations, showing the half cosine-shaped daytime and constant nighttime electron densities. The GIM-derived slant delays, on the other hand, are closer to the TWIM-derived slant delays with the TWIM providing a wider range of values than GIM. However, GIM produce larger ionospheric delay than TWIM. This is because GIM have the largest peak electron density as previously described in Fig. 1.

Figure 3 that shows the horizontal errors D (Eq. 3) and the vertical errors using the three models KLB, GIM, and TWIM for the three stations and epochs 0300UT, 0400UT, and 0500UT for DOY 234. As shown in this figure, the TWIM generally provides the smallest error for the stations in both horizontal and vertical direction. Meanwhile, Klobuchar provide largest error in both horizontal and vertical directions. This indirectly shows that TWIM generally provides a better estimation of the slant TEC around the GPS receivers.

Different elevation masks have been applied to test which elevation angle will yield better results. The resulting DRMS and VRMS (Eq. 4) at different elevation masks (0°, 5°, 10°, 15°, and 20°) for DOY 234 for different ionospheric models are shown in Fig. 4. The positioning accuracy has improved when low elevation masks (5° and 10°) are applied. However, as the elevation mask is set higher (15° and 20°), the positioning seems to degrade in quality. This is because at larger elevations masks, more satellites are ignored, which will result in larger dilution of precision values and worse positioning quality. It indicates that TWIM performs better than the other models for observations at low elevations and demonstrates the ability of TWIM to provide good electron density profiles. The figure shows that even without applying any elevation masks, TWIM still produces better results than the other models.

### Positioning results

Figure 5 shows a sample diurnal variation of the east, north and vertical errors at TWTF station for DOY 234. The effect of the ionosphere in GPS positioning is clearly shown in the vertical error where errors of uncorrected solutions reach about 15 m in daytime and approach zero in nighttime. Moreover, improvements made by different models in the positioning are most evident in the vertical (height) direction. This is because the ionospheric delays are all directed above the receiver. The ionospheric effect in the horizontal is present in all directions (east–west and north–south). In this case, parts of the errors in one direction, say east, can be canceled by the errors in the opposite direction (west).

The performance of TWIM in the horizontal and vertical directions for the stations for DOY 234 can be stated as follows: TWIM exhibits the smallest mean horizontal error (DMEAN) of 1.05, 0.99, and 1.18 m for stations BOR1, IRKJ, and TWTF, corresponding to percentage (Eq. 5) of 10, 9, and 29 %. The DRMS using TWIM is also the smallest among the models with values of 1.28, 1.11, and 1.35 m for the same station sequence, corresponding to percentage of 7, 8, and 33 %. This means that TWTF has exhibited the greatest positioning improvement compared to the other two stations. Regarding the vertical direction, Klobuchar provides the smallest mean vertical error (VMEAN) (−0.15 m) at station BOR1 (95 %), but TWIM has given the best results in stations IRKJ and TWTF at −0.44 m (78 %) and −0.18 m (96 %), respectively. TWIM also provides sub-meter mean vertical errors for all stations. The VRMS are smallest using TWIM with values at 1.65, 1.87, and 2.18 m for stations BOR1, IRKJ, and TWTF, respectively. This corresponds to percentages of 54, 37, and 60 %. For this example, the generally trend of increasing accuracy (based on DRMS and VRMS) is GIM, Klobuchar, and TWIM for station BOR1 while for stations IRKJ and TWTF, the trend is Klobuchar, GIM, and TWIM.

Figure 6 shows DMEAN and VMEAN observed at station TWTF for all days of observations and using all ionospheric models. The DMEAN ranges 1.24–5.67 m using Klobuchar, 1.19–5.96 m using GIM, and 1.08–4.32 m using TWIM. The VMEAN varies from 3.16 to 10.06 m, −1.82 to 1.64 m, −1.43 to 1.60 m, and −3.29 to 1.01 m using Klobuchar, GIM, and TWIM, respectively. Using TWIM, 48 % of the days observed have DMEAN of 1.50 m or less while 70 % of the observed days have VMEAN of 1.00 m or less. Negative VMEAN corresponds to observations below the reference point. The DRMS and VRMS results for all days of observation using all ionospheric models are shown in Fig. 7. The DRMS using Klobuchar, GIM, and TWIM vary from 1.49 to 6.73 m, 1.42 to 7.14 m, and 1.33 to 5.07 m, respectively. Using TWIM, 16 (59 %) of the days have DRMS of 2.0 m or less. The VRMS range from 1.98 to 6.12 using Klobuchar, 1.89 to 6.63 m using GIM, and 1.75 to 6.38 using TWIM. TWIM provides the smallest DMEAN and DRMS for all observations, whereas it provides the smallest VMEAN and VRMS for 11, and 15 days, respectively.
The corresponding percentages of the DMEAN and VMEAN for the difference of uncorrected solution and ionospheric corrected solution at station TWTF for all days of observation are shown in Fig. 8. The values demonstrate the amount of corrections made by each model to the uncorrected solutions. The percentage of DMEAN ranges from 0.56 to 12.33 %, 0.16 to 2.56 %, and 8.70 to 45.07 % using Klobuchar, GIM, and TWIM, respectively. This shows the good performance of TWIM in correcting positioning in the horizontal direction. On the other hand, the percentage of VMEAN using Klobuchar, GIM, and TWIM varies 55.98–99.44 %, 71.74–99.48 %, and 46.71–99.05 %, respectively. Figure 9 shows the percentage of DRMS and VRMS for all days of observations at TWTF station. Using Klobuchar, GIM, and TWIM, the percentage of DRMS ranges from 0.14 to 11.89 %, 0.23 to 3.67 %, and 9.21 to 42.32 %, respectively, while the percentage of VRMS varies from 44.95 to 70.72 %, 46.70 to 69.00 %, and 36.17 to 71.54 %, respectively.

The remaining errors that have not been accounted for by the ionospheric delays can be attributed to the uncertainties in the models, both in troposphere and ionosphere. Moreover, errors brought about by the accuracy of the broadcast ephemeris can also be a source of error since the precise ephemeris of the IGS was not used.

Table 3 summarizes the percentage of occurrence of Klobuchar, GIM, and TWIM with the smallest mean horizontal and vertical errors, DRMS, and VRMS for stations BOR1, IRKJ, and TWTF. As mentioned above, all observed days show that TWIM provided the smallest DMEAN and DRMS at station TWTF. Meanwhile, 41 % (11 days) and 56 % (15 days) of the observed days provide the least VMEAN and VRMS, respectively. On the other hand, 26 % (7 days) and 30 % (8 days) of the observed days the Klobuchar model has smallest VMEAN and VRMS, respectively. GIM provides the smallest VMEAN and VRMS in 33 % (9 days) and 15 % (4 days) of the days of observation. Thirteen days or 48 % using TWIM have 2.5 m of VRMS or less. In stations BOR1 and IRKJ, the TWIM also provides the most occurrences of smallest DMEAN, VMEAN, DRMS, and DRMS with at least 50 % occurrence, except for VMEAN at station BOR1.
Table 3

Percentage of occurrence of ionospheric model with smallest DMEAN, VMEAN, DRMS, VRMS

Percentage of occurrence

BOR1

IRKJ

TWTF

KLB (%)

GIM (%)

TWIM (%)

KLB (%)

GIM (%)

TWIM (%)

KLB (%)

GIM (%)

TWIM (%)

DMEAN

26

0

74

48

0

52

0

0

100

VMEAN

37

33

30

22

19

59

26

33

41

DRMS

26

0

74

22

15

63

0

0

100

VRMS

4

46

50

4

30

67

30

15

56

The performance of TWIM across all stations and all days of observation are generally better than the other models expect for the mean vertical error at BOR1. For all stations, the general order of increasing accuracy (based on the RMS errors) is GIM, Klobuchar, and TWIM in the horizontal while it is Klobuchar, GIM, and TWIM along the vertical. In addition, the TWIM seems to provide a good ionospheric model at low latitude (EIA region) as shown in the results at station TWTF especially in the horizontal. The accuracy of TWIM can also be attributed to the number of occultation use in the modeling as well as the distribution of the occultation points at each region. That is, the denser the occultation points in a specific region the better the estimate of electron density the TWIM will provide. However, such evaluation is beyond the scope of the study.

## Conclusions and future work

GPS satellite-receiver slant TEC at three stations (BOR1, IRKJ, and TWTF) were determined for 27 days with geomagnetically quiet conditions during a period of 2 years using the TWIM model and compared with commonly used models such as Klobuchar and GIM. All models have exhibited typical diurnal characteristics of the ionosphere. However, the TEC maps using TWIM provide a more detailed representation of the current state of the ionosphere as compared to using Klobuchar and GIM ionospheric maps, especially in the low-latitude region, which resulted to better positioning at this region. The TWIM can provide three-dimensional ionospheric electron density values and improve horizontal and vertical positioning significantly compared to Klobuchar and GIM.

Best possible positioning is achieved using a 10° elevation mask. The solutions were able to remove the effect of very low elevation angles but still able to maintain enough satellites to produce good DOPs and positioning results. In addition, the TWIM provides better positioning even for lower elevation masks as compared with the other models. This indicates the high quality of TWIM in producing three-dimensional ionospheric electron density maps over a wider range of satellite elevation.

It is shown that the TWIM is for all days of observation and across all stations generally better than the other models. The average order of increasing accuracy is GIM, Klobuchar, and TWIM along the horizontal, and it is Klobuchar, GIM, and TWIM along the vertical. Judging from the overall result at station TWTF, the TWIM seems to provide good ionospheric model at low latitude (EIA region) particularly in the horizontal.

In the future research, positioning can be improved by using precise GPS orbit information provided by IGS and better tropospheric models. The performance of the TWIM in GPS positioning with respect to the number of occultation points should also be studied. Lastly, the performance of the TWIM in GPS positioning at various geomagnetic and solar activities can be explored in order to establish the applicability of TWIM in different space weather conditions.

### References

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