Introduction

The ionospheric delay primarily depends on several variables such as the time of the day, season, solar activity, location of the receiver, and the earth’s magnetic field (Klobuchar 1987; Camargo et al. 2000). In middle latitude areas, where the ionosphere is generally well behaved, the absolute delay in signal is about 2–5 m (Spencer et al. 2003). The delay can be greater in the polar and equatorial regions, where the ionosphere is characterized to have high electron density and high temporal and spatial gradients. Irregularities in the ionosphere that cause rapid fluctuations in the amplitude and phase (called scintillation) of GPS signals could also cause degraded performance of GPS receivers (Misra and Enge 2012; Dubey et al. 2006).

In differential positioning (DGPS), the underlying principle is that two receivers (test and reference) close to each other will experience the same ionospheric delay. For short baselines, the effect of the ionosphere on positioning is effectively removed, whereas for longer baseline, the effect generally increases. An increase in horizontal electron density gradient due to enhanced ionization especially during increased solar activity can also affect differential positioning. Therefore, it is important to provide an ionospheric model that is accurate in order to reduce the dependence of positioning with density gradients.

We present the differential ionospheric delay correction for single-frequency code-based DGPS positioning using a numerical and phenomenological model called the TaiWan Ionospheric Model (TWIM) (Tsai et al. 2009). Its performance compared with other ionospheric models, such as the Klobuchar model and global ionospheric maps, will also be presented. This study is motivated by our assumption that TWIM would be available for regular single-frequency users in the future.

Single-frequency code-based differential GPS

In DGPS, stationary or moving receivers use the measurements from fixed reference receivers at a known location to remove the common errors through differencing, among which are the tropospheric delay, and satellite clock and orbital errors. Errors that are unique to each receiver such as receiver measurement noises, receiver clock error, and multipath cannot be removed using this technique.

The pseudorange between the reference receiver (r) and satellite (s) is given by

$$P_{r}^{s} = \rho_{r}^{s} + c\left( {{\text{d}t}_{r} - {\text{d}t}^{s} } \right) + \text{d}_{\text{rel}}^{s} + {\text{d}t}_{\text{GD}}^{s} + \text{d}_{{r,{\text{trop}}}}^{s} + \text{d}_{{r,{\text{ion}}}}^{s} + \varepsilon$$
(1)

where P s r is the pseudorange at L1 frequency and ρ s r is the geometric range between r and s. While dt r is the reference receiver clock bias, dt s is the satellite clock bias, d srel is the error due to relativistic effects, dt sGD is the satellite timing group delay or instrument delay, d s r,trop is the tropospheric delay, and d s r,ion is the ionospheric delay. Other un-modeled errors such as receiver instrumental delay, multipath, and measurement noise are included in ε. The same equation follows for the test (rover) receiver u and the same satellite s:

$$P_{u}^{s} = \rho_{u}^{s} + c\left( {{\text{d}t}_{u} - {\text{d}t}^{s} } \right) + \text{d}_{\text{rel}}^{s} + {\text{d}t}_{\text{GD}}^{s} + \text{d}_{{u,{\text{trop}}}}^{s} + \text{d}_{{u,{\text{ion}}}}^{s} + \varepsilon$$
(2)

In order to estimate the geometric range ρ s u between u and s, Eq. (1) is subtracted from (2) to yield

$$\rho_{u}^{s} + {c\text{d}t}_{ur} = P_{ur}^{s} + \rho_{r}^{s} - \text{d}_{{{ur},{\text{trop}}}}^{s} - \text{d}_{{{ur},{\text{ion}}}}^{s}$$
(3)

The pseudorange difference P s ur is available from the observation of the test and reference receivers, while ρ s r can be easily calculated from the known positions of r and s. The differential tropospheric delay d s ur,trop is obtained through a tropospheric model, while the differential ionospheric delay d s ur,ion is determined using different ionospheric models. The remaining are four unknowns, that is, the X, Y, Z coordinates imbedded in ρ s u and the difference cdt ur between the receiver clock errors of u and r are calculated through the standard least squares estimation.

Total electron content

The receiver-to-satellite TEC is estimated using three different models: Klobuchar model (KLB), the global ionospheric maps (GIM), and the TWIM. KLB is a continuous, operational ionospheric model with eight coefficients broadcasted by each GPS satellite. It is based on a cosine representation of the hourly variation of the total electron content where the daytime ionosphere is a half-cosine function, while the nighttime ionosphere is a horizontal (constant) function. This is used to provide the user corrections of approximately 50 % RMS of the ionospheric range error (Klobuchar 1987). GIM provided by the International GNSS Service (IGS) is a discrete vertical TEC (VTEC) map with a spatial resolution of 2.5° in latitude and 5° in longitude. Each global map is updated every 2 h. It uses the TEC data from hundreds of ground-based GPS/GLONASS stations worldwide to produce global ionospheric TEC maps (Schaer 1999; Hernández-Pajares et al. 2009). Both KLB and GIM approximate the ionosphere as a thin layer. The line-of-sight TEC or slant TEC (STEC) is derived from the VTEC at the ionospheric pierce point (IPP), which is assumed to be at 350 km, multiplied by an obliquity factor (m) that is satellite elevation (α) dependent (Ovstedal 2002):

$${\text{STEC}} = {\text{VTEC}} \times m$$
(4)

where

$$m = 1 + 16 \cdot \left( {0.53 - \alpha } \right)^{3}$$
(5)

On the other hand, TWIM is a three-dimensional ionospheric electron density (n e ) model based on Formosat3/COSMIC GPS radio occultation (RO) measurements (Tsai et al. 2009). In this model, a Chapman-type function is used to characterize each layer i (F2, F1, E, or D) of the ionosphere:

$$n_{e} \left( {\theta ,\lambda ,h} \right) = \sum\limits_{i = 1}^{n} {n_{e\hbox{max} } \left( {\theta ,\lambda } \right)} \exp \left\{ {\frac{1}{2}\left[ {1 - \frac{{h - h_{m} \left( {\theta ,\lambda } \right)}}{{H\left( {\theta ,\lambda } \right)}} - \exp \left( { - \frac{{h - h_{m} \left( {\theta ,\lambda } \right)}}{{H\left( {\theta ,\lambda } \right)}}} \right)} \right]} \right\}$$
(6)

where n emax is the peak electron density, h m is the peak density height, and H is the scale height. This is used to obtain the n e at a specific longitude θ, latitude λ, and height h using the least squares error fitting of the observed profile to the Chapman functions.

Surface spherical harmonics are applied to map the derived Chapman layer parameters in geodetic coordinates, which gives TWIM its three-dimensional spatial feature. 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. However, TWIM does not account for electron density in the plasmasphere. Since TWIM is in three dimensions, the STEC can be calculated as

$${\text{STEC}} = \int\limits_{r}^{s} {n_{e} (l){\text{d}l}}$$
(7)

where l is the line of sight from the r to s (Misra and Enge 2012).

It should also be noted that because every electron density estimated by the TWIM uses a 30-day data set, any disturbances in the ionosphere, such as geomagnetic storms, in a given day will be masked by the other non-disturbed days within the 30-day period. Moreover, the Formosat3/COSMIC cannot only provide about 1–4 RO points per 10° × 10° grid for every half-hour model, which is not dense enough to observe disturbed n e profiles when geomagnetic storm or other small-scale phenomena happen. During these events, the retrieved n e 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 sufficient to model the ionosphere during geomagnetically quiet days.

Given the STEC along the signal path between r and s, the ionospheric delay d ion in the pseudorange can be determined as

$$\text{d}_{{r,{\text{ion}}}}^{s} = \frac{{40.3 \cdot {\text{STEC}}}}{{f^{2} }}$$
(8)

where f is the frequency of the signal. Numerically, one TEC unit (TECU) is equivalent to about 16 cm of delay in the L1 (f = 1,575.42 MHz) pseudorange.

TWIM has already been used in several applications. Macalalad et al. (2013) derived absolute ionospheric delay corrections from TWIM to remove the ionospheric delays in single-frequency pseudorange observations for stand-alone GPS positioning. Also, Tsai et al. (2010, 2014a, b) used the TWIM to perform three-dimensional and continuous radio wave ray tracing simulation to identify the radio wave source position (in this case, ionosondes) from angle of arrival measurements. They also used TWIM to provide maximum usable frequency or MUF for radio frequency transmissions. Since TWIM is a numerical and parameterized model, it can be used for ionospheric electron density predictions for real-time modeling (Tsai et al. 2014a, b).

Method

GPS L1 pseudorange (C/A) observations are obtained in RINEX format from nine GPS stations continuously operated by the National Land Surveying and Mapping Center of Taiwan’s Ministry of the Interior (http://www.nlsc.gov.tw/) shown in Fig. 1, with a 30-second sampling rate. These stations form baselines that range from 19 to 340 km and generally cover the northern crest of the equatorial ionization anomaly (EIA) region. For a baseline, one station is the reference station, while another is the station. Here, the baseline vector is defined as the displacement vector from the reference station to the test station. For each of the 72 possible baselines, 70 days are selected during a period of solar minimum from 2008 to 2010. The days used only include two of the quietest days for each month from February 2008 to December 2010 with daily average 3-h Kp index of only up to 0.875 only, as shown in Fig. 2. The maximum monthly mean sunspot number for this period was only 25 while the F10.7 cm solar flux for each day of observation ranged from 65.6 to 92.2, which is well within the period solar minimum of Solar Cycle 24.

Fig. 1
figure 1

Nine permanent continuously operating GPS stations around Taiwan maintained by the National Land Surveying and Mapping Center of Taiwan Ministry of the Interior

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Fig. 2
figure 2

Solar and geomagnetic activity indices during the days of observation from 2008 to 2010 (70 days total). Red-dashed lines show the monthly mean sunspot number; blue solid lines and black bars are the daily average solar flux F10.7 cm and Kp index, respectively

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GPS satellite positions are calculated using the broadcast ephemeris. A satellite elevation mask of 10 degrees is applied, and an elevation-dependent weighting function is used in the least squares adjustment as described in Jin and de Jong (1996). The tropospheric delay is assumed to follow the modified Hopfield model (Goad and Goodman 1974). Epoch-by-epoch positions of the test station are computed for a period of 24 h using single-frequency single-differenced pseudoranges for each day and baseline. No carrier-phase smoothing is applied since this study is limited to the use of C/A observation only. A base solution, referred to as uncorrected (UNC) solution, has been carried out per epoch, and baseline with no differential ionospheric delay correction is applied. Three additional solutions are performed which corrected for the differential ionospheric delay using the KLB, GIM, and TWIM models.

The east, north, and vertical positioning errors (E, N, and V) for every epoch are determined by subtracting its computed position of the test station from its known position. The hourly, daily, and seasonal mean values of E and N are calculated and represented as EMEAN and NMEAN, respectively. To represent the horizontal position accuracy, the hourly, daily, and seasonal horizontal mean (DMEAN) is computed as

$${\text{DMEAN}} = \sqrt {{\text{EMEAN}}^{2} + {\text{NMEAN}}^{2} }$$
(9)

The daily mean errors along the vertical (VMEAN) is used to describe the vertical accuracy of each observation. This study is limited to the calculation of mean values only because for code-based DGPS, the measurement noise increases greatly to the point where it dominates the mean values and becomes the main contributor to the root-mean-square values. Moreover, getting the mean values is sufficient since static positioning is the main application of this study.

Results and discussions

This section demonstrates the performance of TWIM in single-frequency code-based single-differenced DGPS positioning. Its results include the dependence of daily mean error on baseline length, seasonal variation, relation to solar activity and dependence on spatial TEC gradients and a comparison to the Klobuchar and GIM models.

Dependence of daily mean error to baseline vector

For example, consider CHGO as the reference station and the other eight stations are used as test stations to form eight baseline vectors. Figure 3 shows the daily mean error value for UNC for each baseline as a function of the baseline vector along the east, north, and vertical directions for DOY 266 of 2009. The daily horizontal mean error as a function of horizontal baseline is also shown. The daily mean error correlated very well to the baseline vector along the east (R = 0.93) and north (R = 0.99). This means that if the reference station is located east or north of the test station, the mean error along that direction is also positive, and vice versa. The error–baseline relationship along the horizontal is due to the presence the horizontal ionospheric gradients. Therefore, the aim is to decrease this dependence of positioning error to the baseline vector by imposing actual ionospheric model to estimate these errors. An ideal correction should produce a horizontal slope, while a positive slope indicates under correction and negative slope indicates an overcorrection. A parameter, called the error–baseline slope, is defined as the slope of the best-line fit of each mean error versus baseline graph. This term is used as a measure of the performance of each ionosphere model in mitigating the differential ionospheric delay for a 24-hour survey, where a smaller error–baseline slope means better correction. For error–baseline slopes of the order of 10−8, 10−7, and 10−6, the expected 24-hour mean error along the horizontal would approximate to about 0.2, 0.3, and 0.5 m, respectively, for a baseline of up to 300 km. In Fig. 3, the error–baseline slope along the east and north directions are 6.05 × 10−7 and 1.72 × 10−6, respectively. Overall the horizontal error–baseline slope is 9.59 × 10−7 (R = 0.811), while the vertical error–baseline slope is −7.11 × 10−4 (R = −0.488).

Fig. 3
figure 3

East (top-left), north (top-right), horizontal (bottom-left), and vertical (bottom-right) mean error versus baseline vector components for all possible test–reference station pairs, using CHGO as the test station, for DOY 266 of 2009

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Figure 4 shows the daily mean errors against the east baseline vector for UNC using KLB, GIM, and TWIM as source of differential ionospheric corrections for all 72 possible baseline combinations on DOY 266 of 2009. Here, each receiver has been considered as a test and reference station. The reduction of the slope after using ionospheric models is evident along the horizontal component (row 3). This implies that for a 24-hour survey, the dependence of the mean error with baseline is due to the influence of the ionosphere. Along the horizontal, the TWIM made the best correction in both the east and the north directions by reducing the slope along the east by 99 %, north by 65 %, and horizontal by 73 %. GIM reduced the slope along the east, north, and horizontal by 81, 58, and 65 %, respectively. This makes GIM the second best-performing model. KLB overcorrects by 49 % in the east direction as indicated by the change of sign of the error–baseline slope. In the north and horizontal direction, KLB reduced the slope by 53 and 58 %, respectively. This makes KLB the least-performing model. Along the vertical, the reversed relationship between the mean vertical error and the height differences (vertical baseline) is evident and unaffected by the application of ionospheric models. This suggests that the ionosphere may not be the main source of this relationship. Perhaps it is due to the troposphere since this delay also depends on the height of the receiver.

Fig. 4
figure 4

Scatter plot of the mean error versus baseline vector without differential ionospheric correction UNC (col 1) and with differential ionospheric corrections using KLB (col 2), GIM (col 3), and TWIM (col 4) for east (row 1), north (row 2), horizontal (row 3), and vertical (row 4) directions for DOY 266 of 2009 and for all possible baselines

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The effect of the ionosphere is more evident along the horizontal than the vertical for DGPS positioning, which is contrary to the effect of ionosphere in stand-alone positioning where the effect of ionospheric correction is most evident in the vertical direction rather than in the horizontal direction (Macalalad et al. 2013). This is because in DGPS positioning, only the largely horizontal ionospheric gradients are considered and not the absolute ionospheric delay as employed in stand-alone positioning. Therefore, the subsequent discussions will focus on the performance of the different ionospheric models in DGPS positioning along the horizontal and its east and north components.

Seasonal variation

Comparison of seasonal mean error–baseline slopes for winter (DOY 1–79 and DOY 356–365), spring (DOY 80–172), summer (DOY 173–265), and autumn (DOY 266–355) from 2008 to 2010 is shown in Fig. 5. The overall horizontal error for UNC generally follows the usual VTEC seasonal variation where it is relatively lower in the winter and summer months compared with their succeeding equinoxes (spring and autumn). East errors are generally half the errors along the north direction, which suggests that the electron density gradients along the north are about half the amount of density gradients in the east direction. This result is fairly reasonable for an area within the EIA region. This variation is effectively reduced by all the models by at least 45 % when application of differential ionospheric delay is done.

Fig. 5
figure 5

East (top), north (middle), and horizontal (bottom) seasonal error–baseline slope (× 10−6) using different models from 2008 to 2010

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Along the east direction, GIM generally provides the best corrections. It is denoted as the decrease in the seasonal mean error–baseline slope by at least 89 %, except for autumn where it overcorrected by 5 %. TWIM provides the second best correction along east direction though it slightly overestimates the corrections by as much as 40 % in autumn. It surpassed GIM in summer where it only overcorrected by 1 %. KLB delivers the least correction where it overestimates the differential correction by at least 44 % for all seasons.

TWIM provides the best correction along the north direction during winter and spring with 97 and 89 % correction, respectively. GIM provides best correction in autumn with 98 % correction. For summer, all models are equally poor where they overcorrect by 55 % for GIM, 59 % for TWIM, and 73 % for KLB.

Overall, GIM provides the best corrections along the horizontal direction followed by TWIM and KLB. GIM corrects most effectively in spring with 91 % correction. It is least effective during summer with 76 % correction, but still provides good results compared to KLB. TWIM best performs in summer (79 % correction) even slightly outdoing GIM, but it is not as successful as GIM during autumn with only 68 % correction. Nevertheless, TWIM is still better than KLB for all seasons. KLB, generally the least-performing model, still provides correction of only 53, 67, 55, and 44 % during winter, spring, summer, and autumn, respectively.

Relationship with solar activity

The hourly IPPs for KMNM-SICH and CHGO-FUSI for 0600UT of DOY 310 of 2010 are shown in Fig. 6. It shows the typical coverage of IPPs for every hour. As expected, the difference in the IPPs for the reference and test stations generally follows the relative positions of the each station. Thus, for a mainly eastward baseline such as KMNM-SICH, their simultaneous IPP pairs are also along the east. For this configuration, the differential positioning will primarily depend on the east TEC gradients. While for CHGO-FUSI, where the baseline is generally along the north, their IPPs pairs are along the north as well. Therefore, the positioning results for this alignment will also be north dependent.

Fig. 6
figure 6

Typical hourly IPPs for KMNM-SICH (left) and CHGO-FUSI (right). The IPPs are for DOY 310 of 2010 at 0600UT. Red dots refer to the IPPs of the reference station, and blue dots are for test station

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Skone and Shrestha (2002) reported that significant degradation in single-frequency differential positioning accuracies can be attributed to the enhancement of electron density gradients. This is observed by comparing results from a low (June 1999) and high (May 2000) solar activity dates for a 430-km baseline in the equatorial South American region. Figures 7 and 8 show the half-hourly diurnal variation of the east, north, and horizontal positioning errors for baselines KMNM-SICH and CHGO-FUSI for DOY 189 of 2008, where the solar flux F10.7 cm is only 65.6, the lowest for the whole dataset. The errors for both baselines for DOY 310 of 2010 are shown in Figs. 9 and 10. This day has the highest solar activity whose F10.7 cm value is 92.2. Each curve shows the hourly mean for every 30 min for UNC, KLB, GIM, and TWIM, which is the same timing as in modeling TWIM. The small variation in the positioning error during low solar activity is amplified when solar activity is increased. The improvement in positional error, by applying of ionospheric correction, is only evident in the east direction for KMNM-SICH, while it is only apparent in the north direction for CHGO-FUSI. For low solar activity, the corrections made by the models do not differ as much compared to the corrections made during higher solar activity. However, the application of such models fail to completely remove these error because the models may not take into account the gradients caused by larger solar activity. This will be discussed in the next section.

Fig. 7
figure 7

Hourly mean error for KMNM-SICH for east (top), north (middle), and horizontal (bottom) direction for DOY 189 of 2008 (F10.7 cm = 65.6). The results show the positioning error for UNC (black), KLB (blue), GIM (red), and TWIM (green)

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Fig. 8
figure 8

Hourly mean error for CHGO-SICH for east (top), north (middle), and horizontal (bottom) directions for DOY 189 of 2008 (F10.7 cm = 65.6)

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Fig. 9
figure 9

Hourly mean error for KMNM-SICH for east (top), north (middle), and horizontal (bottom) directions for DOY 310 of 2008 (F10.7 cm = 92.2)

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Fig. 10
figure 10

Hourly mean error for CHGO-SICH for east (top), north (middle), and horizontal (bottom) directions for DOY 310 of 2008 (F10.7 cm = 92.2)

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The relationship between the east, north, and horizontal error–baseline slopes with solar flux F10.7 cm for UNC, KLB, GIM, and TWIM is shown in Fig. 11. For UNC, the error–baseline slope tends to increase as the solar flux intensity increases especially for north direction. But upon application of ionospheric corrections, this relationship tends to decrease. This dependence of the slope with solar activity is generally reduced by GIM more than KLB and TWIM along the east direction. KLB performs worst along the east direction due to the negative offset even though KLB seems to have the lowest slope compared with GIM and TWIM. TWIM performs similarly with GIM especially for low solar flux. Along the north direction, GIM and TWIM perform equally well as they reduce the dependence of northeast error to solar activity by about 60 %, while KLB only reduces it by 10 %. Their performances mostly differ for higher solar flux number. Overall, GIM provides best solution for all solar activity by reducing its dependence on solar flux, followed by TWIM and KLB. This order is similar to the daily and seasonal variation results, which suggests that the general performance of the three models with respect to their ability estimates differential ionospheric corrections used for differential GPS positioning.

Fig. 11
figure 11

Scatter plot of east (row 1), north (row 2), and horizontal (row 3) error–baseline versus daily solar flux (F10.7 cm) from 2008 to 2010. The results also show the best-fit line for UNC (col 1), KLB (col 2), GIM (col 3), and TWIM (col 4)

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Diurnal analysis: dependence on spatial TEC gradients

As mentioned above, the TEC spatial gradients are the main contributors to the positioning errors along the horizontal for DGPS. As seen in Figs. 7 and 8, the correction results made by the models do not seem to diverge from each other along the east direction, while in the north direction, significant improvements are observed. This suggests that the eastward ionospheric gradients for all the models do not differ as much as in the north direction. For this example, GIM and TWIM provide very similar results though GIM is slightly better than TWIM. KLB, on the other hand, provides the least correction.

Therefore, it is important to differentiate the TEC spatial gradient for each model used in this study. The study region is limited to approximately ± 20° around Taiwan (5°N ~ 40°N and 100°E ~ 140°N) where the IPPs are usually located, as shown in Fig. 6. Figure 12 shows the regional VTEC maps at 0600UT (1400UT, peak ionization at the center of the region) during DOY 189 of 2008 and DOY 310 of 2010 for KLB, GIM, and TWIM. KLB shows a much simpler ionosphere where the typical morphology of the EIA is not present. GIM and TWIM exhibit a more realistic representation of the ionosphere where EIA peaks are clearly seen. Also, the general VTEC values are higher for increased solar activity. An initial impression suggests that most of the TEC gradients follow the north direction and only small gradients are present along the east direction and most of these are due to the EIA. On DOY 189 of 2008, the regional VTEC ranges (max–min) are about 3 TECU for KLB, 11 TECU for GIM, and 10 TECU for TWIM at 0600UT. In DOY 310 of 2010, these gradients increase to 9 TECU, 38 TECU, and 20 TECU, for KLB, GIM, and TWIM, respectively. This shows that the GIM estimates the largest gradients, followed by TWIM and KLB. This is greatly reduced during nighttime where there is minimal ionization in the ionosphere. In fact, the TEC gradient for this region during this time only vary from 0 TECU for KLB to only 6 TECU for GIM and TWIM.

Fig. 12
figure 12

Regional VTEC maps using KLB (col 1), GIM (col 2), and TWIM (col 3) during for DOY 189 of 2008 (row 1) and DOY 310 of 2010 (row 2) at 0600UT. Contour lines at 1 TECU interval. Purple dots are occultation points used in modeling the TWIM

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Considering the IPPs for each baseline pair, the hourly STEC and ionospheric delay difference between these points can be determined. The hourly root-mean-square (RMS) differential ionospheric delay for DOY 189 of 2008 and DOY 310 of 2010 for baselines KMNM-SICH and CHGO-FUSI is shown in Figs. 13 and 14, respectively. The RMS value of the differential ionospheric delay can be used to describe the amount of gradients present for each hour. For KMNM-SICH baseline, the RMS differential ionospheric delay for all models and for both days is very similar which is reflected by the amount of correction in the positioning error, as shown in Figs. 7–10. For DOY 189 of 2008, the RMS differential delay for all models for CHGO-FUSI is similar since this is during a day of low solar activity. However, this parameter is increased for high solar activity like in DOY 310 of 2010. The difference between the models of this baseline and day is also reflected with the positioning results (also in Figs. 7–10). This shows that taking into account, the gradients along the east do not contribute to the positioning results as much as in the north direction where much of the gradients are present. Nighttime gradient does not seem to differ from day to day, which produces minimal improvements during this time. Thus, the RMS of the differential ionospheric delay can be used to measure the ability of a model to estimate the corrections that it can provide, that is, if the RMS value approaches the UNC positional error, the model would perform better.

Fig. 13
figure 13

Hourly RMS differential ionospheric delay for baselines KMNM-SICH on DOY 189 of 2008 (top) and DOY 310 of 2010 (bottom)

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Fig. 14
figure 14

Hourly RMS differential ionospheric delay for baselines CHGO-FUSI on DOY 189 of 2008 (top) and DOY 310 of 2010 (bottom)

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Figure 15 shows the scatter plot of hourly RMS ionospheric delay using the three models as compared to the hourly UNC horizontal mean error for all days of observation (2008–2010) for baselines KMNM-SICH and CHGO-FUSI. The diagonal lines show the delay-to-error ratio between the two parameters where ratios closer to 1.00 means better gradient estimates, hence, better correction. Red marks indicate daytime period, while blue marks refer to nighttime. The nighttime plots show low ratio, which indicates that during this time, the differential ionospheric delay has little effect in improving the UNC solution. The daytime plot indicates better effect of the differential ionospheric delay estimates to positioning error. It shows that an eastward baseline (KMNM-SICH) has low delay-to-error ratio and they are similar for all three models. This implies that along the east direction, there is little difference between the estimated gradients of the three models and their corresponding correction, which translates to similarity in positioning performance for all models. On the other hand, there are significant differences between the overall delay-to-error ratios for a northward baseline (CHGO-FUSI) where GIM having the largest ratio and KLB has the lowest. This means that on an hourly basis, GIM has the best estimates of hourly gradients with ratio of at least 0.25 for large horizontal errors. This is followed closely by TWIM with ratio close to 0.20, while KLB only have <0.10. This shows that GIM and TWIM estimate the horizontal gradients considerably better than the standard KLB that translates to better positioning performance.

Fig. 15
figure 15

Scatter plot of hourly RMS differential ionospheric slant delay using KLB (col 1), GIM (col 2), and TWIM (col 3) against hourly UNC horizontal mean error for KMNM-SICH (row 1) and CHGO-FUSI (row 2) baselines for all days of observation from 2008 to 2010. AM = Red, PM = Blue

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Conclusion and future work

TWIM, a quiet-time, three-dimensional electron density model, is used to mitigate the differential ionospheric delay in the L1 pseudorange. Its performance has been evaluated for a total of 70 days of spanning, the most geomagnetically quiet days during a 3-year solar minimum period (2008–2010), and is compared with the performance of KLB and IGS GIM. The pseudorange observations are obtained from nine permanent continuously operating GPS stations distributed across Taiwan, which could form baseline lengths ranging from 19 to 340 km. This covers the northern crest of the EIA. In general, TWIM has a good performance in reducing the differential ionospheric delay especially for long baselines and higher solar activity. It performs better than KLB. TWIM also performs similarly with GIM though GIM has a better performance overall.

Daily mean error along the horizontal reveals a highly positive linear relationship with the baseline vector. That is, when a test station is located to the north (east) with respect to a reference station, its north (east) mean error is positive. The corresponding error–baseline slopes show that GIM generally performs best in reducing the effect of the baseline length in a 24-hour survey. The TWIM performs similarly with GIM and is better compared to KLB. On the other hand, the baseline dependence of positioning error along the vertical is reversed. This indicates that the vertical error brought about by the differences in height (vertical baseline) is not due to the ionosphere.

Seasonal variation shows that GIM provides best correction along the horizontal followed by TWIM and KLB, where the GIM correction is most effective during spring at 91 %. TWIM performs best in summer with 79 % correction compared with GIM’s 76 %. TWIM least performs in autumn with correction of 68 % but still much better than KLB with only 44 %. Across all seasons, KLB performs worst with corrections up to 67 % only.

The error–baseline slope tends to increase with F10.7 cm solar flux. But this dependence is partially reduced with differential ionospheric correction is applied. GIM reduces this dependence the most. TWIM achieved similar results with GIM especially during low solar activity. They even reduce this dependence by about 60 % in the north direction, while KLB only achieved 10 % correction.

Regional TEC analysis showed that GIM and TWIM have more realistic representation of ionospheric gradients than KLB where they both exhibited typical EIA features. In this region, the TEC gradient generally follows the north direction, while lesser gradients appear along the east direction. The associated differential ionospheric slant delay generally dictates the amount of correction that one model can provide to the horizontal error. Overall, GIM has the best differential ionospheric delay-to-horizontal error ratio of more than 0.25, which indicates that it can give the best estimate of hourly ionospheric gradients. This is followed closely by TWIM with about 0.20. KLB only has <0.10 ratio.

A possible reason why GIM performs better than TWIM might be the fact that IGS have three receivers in two locations in Taiwan (TNML, TCMS, and TWTF). Data from these stations would have a significant contribution in producing TEC maps over Taiwan. But we did not include these stations to ensure that our results would be independent on the database used by IGS.

The similar performance of TWIM with GIM shows the quality and applicability of TWIM in mitigating differential ionospheric delays in the C/A code pseudorange caused by electron density gradients in the ionosphere and that it can provide decimeter-to-centimeter accuracy in differential GPS positioning for single-frequency receivers during geomagnetic quiet conditions across all seasons and different levels solar activity even if the baseline length is as far as 300 km. This initial study on the performance of TWIM in DGPS shows that the potential of TWIM as an operational model is great, especially when the follow-on mission of the COSMIC program is operational. The expected increase in number of RO observations from COSMIC2 will definitely produce better and denser modeling of TWIM.