Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources

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ScienceDirect Advances in Space Research xxx (2016) xxx–xxx www.elsevier.com/locate/asr

Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources Reza Ghoddousi-Fard ⇑, Franc¸ois Lahaye Canadian Geodetic Survey, Natural Resources Canada, Canada Received 15 January 2016; received in revised form 19 February 2016; accepted 20 February 2016

Abstract Single frequency code and single frequency code and phase GPS precise point positioning scenarios using various ionospheric sources are evaluated by assessing their performances with respect to dual frequency solutions. These include Canadian regional and global vertical total electron content (VTEC) maps produced by Natural Resources Canada and different International GNSS Service (IGS) coordination or analysis centres. Furthermore, two of the most commonly used single layer ionospheric mapping functions applied for conversion of VTEC to slant TEC are evaluated with each and every one of the ionospheric VTEC sources. Results show that the quality of code only solutions depends on ionospheric activity level, and the TEC map and mapping function selected. Code and phase single frequency solutions are also improved when assisted with an external ionosphere source. Crown copyright Ó 2016 Published by Elsevier Ltd. on behalf of COSPAR. All rights reserved.

Keywords: GPS; Ionosphere; TEC

1. Introduction Accurate modeling of the ionosphere is required for reliable precise positioning (see e.g. Stankov and Jakowski, 2007). Ionospheric perturbations are characterised by rapid changes of the ionisation that can induce spatial gradients of total electron content (TEC) substantially larger than during non-disturbed periods (see e.g. Vo and Foster, 2001; Stankov et al., 2006; Jakowski et al., 2008; Stankov et al., 2009). Several regional and global ionosphere maps developed by different centers provide 2 dimensional (2D) state of TEC at a single layer shell at elevation of assumed peak electron density. These maps can assist with singlefrequency GNSS positioning and serve applications such as space weather studies. Even though efforts to develop enhanced models that account for electron density inhomogeneities have been carried out, the common ⇑ Corresponding author.

E-mail address: [email protected] (R. Ghoddousi-Fard).

practice in GNSS positioning still is based on modeling of slant ionospheric delays from an external vertical TEC (VTEC) map combined with a mapping function usually based on a single layer model. Over the past 2 decades a number of organizations have developed 2D global and regional maps of VTEC. A number of International GNSS Service (IGS) Associate Analysis Centers have continuously contributed to the IGS combined VTEC maps. These include the Center for Orbit Determination in Europe (CODE), European Space Agency (ESA), Jet Propulsion Laboratory (JPL) and Technical University of Catalonia (UPC) (Herna´ndez-Pajares et al., 2009). The Canadian Geodetic Survey of Natural Resources Canada (EMR) has also developed a number of ionospheric products (Ghoddousi-Fard, 2014) and has recently resumed submission of its global VTEC maps to IGS data centers. IGS combined as well as each centerspecific global maps can be accessed in IONosphere map Exchange (IONEX) format (Feltens and Schaer, 1998) through IGS global data centers. Furthermore, a

http://dx.doi.org/10.1016/j.asr.2016.02.017 0273-1177/Crown copyright Ó 2016 Published by Elsevier Ltd. on behalf of COSPAR. All rights reserved.

Please cite this article in press as: Ghoddousi-Fard, R., Lahaye, F. Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.02.017

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number of centers have developed regional maps to serve various applications such as regional space weather studies and navigation. Ghoddousi-Fard et al. (2011) developed regional VTEC maps covering Canada and adjacent regions based on spherical cap harmonics (SCH) that can be compared with data from other sensors to understand the evolution of processes in the ionosphere. Independent computations by each analysis center based on different approaches, data sources, number of stations and satellite constellations affect the time-specific VTEC values of the ionospheric grid points. Considering the primary application of VTEC maps is to correct ionospheric delay on single frequency GNSS measurements, different positioning solutions are expected to result from using different VTEC maps. In other words, VTEC maps may be evaluated by analyzing the quality of single frequency positioning solutions assisted with such maps. However, several other factors may affect GNSS single frequency positioning results. Therefore, such analysis should not be considered a way to assess the accuracy of VTEC maps but rather a means to evaluate the quality of single frequency positioning one might expect when using such VTEC products. Since TEC values are required along the path between satellite and receiver, a conversion of VTEC values (extracted from maps) to the satellite to receiver line of sight is required. Furthermore, differential code biases (DCB) must be applied to single-frequency pseudorange observations to account for differences between the references for the satellite clocks and P1 code measurements. Nevertheless, slant to vertical TEC conversion and DCB estimation are the main in-situ elements affecting most of the current VTEC map production approaches themselves. Even though approaches to mitigate the mismodelling caused by single layer model assumptions have been implemented, such as voxel basis functions or a two layer model (Juan et al., 1997; Herna´ndez-Pajares et al., 1997, 1999), the users of such VTEC maps are still limited by the mapping function and DCB values applied in their GNSS processing software. Wienia (2008) evaluated global VTEC maps using phase-adjusted pseudo range single frequency observations and a precise point positioning (PPP) method. Le et al. (2009) compared regional and global maps and studied positioning accuracy using such maps over Europe. There has not been a comprehensive study on the performance of existing VTEC maps and mapping functions in the position domain. This paper evaluates several single frequency positioning solutions using a selection of publicly available global VTEC maps, Canadian regional maps and published mapping functions under various ionospheric activity levels as compared with dual-frequency ionosphere-free PPP solutions. A review of commonly used ionospheric mapping functions is followed by introducing evaluation studies and their results. Last are the summary and conclusions.

2. Single layer model ionospheric mapping functions Commonly used 2D ionospheric models for GNSS positioning applications, including IGS TEC maps, require a mapping function (obliquity factor) which is used for conversion between vertical and slant TEC at ionospheric pierce point (IPP). A number of mapping functions have been developed and published in the literature over the past decades. In this section a selection of those are reviewed prior to investigating their effect on single-frequency GPS positioning. The simplest yet most commonly used ionospheric mapping function is based on the single layer model on a spherical earth. Using the geometric relation between zenith angle at user’s location (z) and zenith angle at the pierce point (z0 ) on a spherical Earth with radius R and shell height of H, one can write:
 R  sinðzÞ ð1Þ z0 ¼ asin RþH Hence the single layer mapping function (hereafter referred to as standard mapping function) is written as: m¼

1 cosðz0 Þ

ð2Þ

Hobiger et al. (2007) showed that neglecting the Earth’s oblateness in the single layer mapping function has an impact of less than 0.16% in slant TEC values even at observation elevation angles down to 10°. They concluded that the difference between the spherical and ellipsoidal approaches in mapping functions and pierce point locations are negligible relative to other error sources. Schaer (1999) proposed a modified single layer mapping function using a reduced zenith angle i.e.: m¼

1     R cos asin RþH opt  sinða  zÞ

ð3Þ

Best fit of Eq. (3) to the extended slab model (Coster et al., 1992) was achieved at H opt ¼ 506:7 km and a ¼ 0:9782 (Beutler et al., 2007). The corresponding mapping function, which is widely used, is hereafter referred to as modified standard mapping function. Clynch et al. (1989) proposed a mapping function in the form of a polynomial obtained from least-square fit to TEC ratios when assuming a homogeneous electron density shell between altitudes of 200 and 600 km. This mapping function is written as follow: m ¼ a0 þ a1  x 2 þ a2  x 4 þ a3  x 6

ð4Þ

where x ¼ 2zp a0 ¼ 1:0206; a1 ¼ 0:4663; a2 ¼ 3:5055; a3 ¼ 1:8415 Smith et al. (2008) studied geometric and numerical errors as a result of using a shell model and concluded that such errors can reach up to 14% on days of quiet

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Fig. 1. Ionospheric mapping functions. stn: standard, mstn: modified standard, Clynch: Clynch et al. (1989), Smith: Smith et al. (2008), brdc: broadcast, Klob: Klobuchar (1975).

Fig. 2. Difference between either of modified standard (mstnD), Clynch et al. (1989) (ClynchD), Smith et al. (2008) (SmithD), Broadcast (brdcD) and Klobuchar (1975) (KlobD) mapping functions and standard mapping function at VTEC of 15 TECU on f 1 frequency.

ionosphere activity. They proposed a mapping function based on a modified single layer model as follow: m¼

1

1 

P 100

 cosðz0 Þ

ð5Þ

where P ¼ A þ B  H    A ¼ atan z0  180  A0  A 1  A 2  A3 p      B ¼ atan z0  180  B0  B 1  B 2  B3 p A0 ¼ 64:4297; A1 ¼ 0:0942437; A2 ¼ 1:39436; A3 ¼ 19:6357 B0 ¼ 64:3659; B1 ¼ 0:104974; B2 ¼ 1:41152; B3 ¼ 0:0463341

The mapping function used in the GPS broadcast ionospheric model (IS-GPS-200, 2013) is written as: m ¼ 1 þ 16ð0:53  EÞ3

ð6Þ

where E is elevation angle between the user and satellite in semi-circles. The broadcast mapping function provides very similar results to original formulation by Klobuchar (1975) written as:


96  e m¼1þ2 90

3 ð7Þ

Please cite this article in press as: Ghoddousi-Fard, R., Lahaye, F. Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.02.017

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Fig. 3. (a) Standard mapping function at shell heights of 350 and 450 km. (b) Difference between standard mapping functions on Fig. 3a at VTEC of 15 TECU on f 1 .

Fig. 4. IPP location at station MAUI during March 17, 2015: (a) VTEC values as extracted from CODE final VTEC maps; (b) difference between slant TEC using standard and modified mapping functions.

where e is the elevation angle of the satellite in degrees at the user’s location. Klobuchar (1975) derived this equation as an approximation (within 2% from 5° to 90° elevation angle) of standard mapping function (1) with H ¼ 350 km and R ¼ 6370 km. Fig. 1 shows the values of all above-mentioned mapping functions (at shell height of 350 km, where applicable) over all elevation angles. Fig. 2 represents L1 ionospheric delay differences relative to the standard mapping function for a VTEC of 15 TECU. One can notice from Figs. 1 and 2 that all studied mapping functions provide almost the same results at elevation angles above 50°. However at low elevation angles, the studied mapping functions can cause slant

delay variations up to several decimetres even at a moderate VTEC value of 15 TECU. Changing shell height will also systematically affect single-layer model mapping functions. Fig. 3a shows standard mapping function calculated based on two commonly used shell height of 350 and 450 km. Fig. 3b shows the L1 ionospheric delay difference between the two mapping functions of Fig. 3a when applied to a VTEC of 15 TECU, reaching again several decimeters. Fig. 4a shows VTEC values at IPPs observed at 5-min interval from station MAUI (see Fig. 6 for location of this station) during 24 h on March 17, 2015 (as extracted from CODE final VTEC maps). Fig. 4b shows the difference

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between using the standard (with shell height of 450 km) and modified mapping functions on the calculated slant TEC. It is noted that slant TEC values from the standard mapping function are systematically larger (with daily mean of +2.2 TECU and up to more than 15 TECU at elevation angles down to 10 degrees) than those from the modified mapping function. For the purpose of this paper the modified standard and standard mapping functions (with shell height consistent with the selected VTEC map), which are two of the most commonly used mapping functions, are used to evaluate the impact of obliquity mapping on single frequency positioning.

(March 17–19, 2015) ionospheric conditions. One should note that Kp index is a global indicator of the Earth’s geomagnetic field disturbances and should suffice for the purpose of this paper for characterization of global ionospheric activity level as sensed by single frequency GNSS measurements. In addition to P1 pseudorange-only single frequency solutions introduced later, an enhanced single frequency approach using both pseudorange and carrier-phase is also evaluated and will be introduced in the next section. Then the effect of different mapping functions will be evaluated. Finally, overall performance of all solutions over a number of stations is presented.

3. Evaluation data sets

4. Single frequency code and phase (L1P1) GPS PPP performance

Three periods with different ionospheric activity levels are selected, each covering 3 days. The selected periods and their ionospheric activity levels as characterized by Kp index are shown in Fig. 5. Various PPP processing scenarios at 10 stations shown in Fig. 6 are carried out during these 3 selected periods covering very quiet (July 18–20, 2015), slightly disturbed (May 10–12, 2014) and stormy

As an alternative to traditional single frequency pseudorange point positioning with an external ionospheric model, improved results can be achieved using a combination of single frequency phase and pseudorange observations (e.g. L1 and P1). First order ionospheric effects can be removed from such observables given the equal but

50 40 30 20 10

Daily sum of 3-hourly Kp index value Daily max 3-hourly Kp index value

0

Fig. 5. Daily sum and max of 3-hourly Kp index during studied periods.

Fig. 6. Location of studied stations.

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bs;U1 is the satellite instrumental delays on U1 for satellite S, br;U1 is the receiver instrumental delays on U1 for receiver r; and eP 1 and eU1 are multipath and noise on P 1 and U1 respectively.

opposite sign of ionospheric effects on carrier-phase and pseudorange measurements (phase advance and code delay) as one may realise from the pseudorange and carrier-phase observation equations on f 1 frequency as follow: P 1 ¼ q þ c:ðdT  dtÞ þ I 1 þ T þ bs;P 1 þ br;P 1 þ eP 1 U1 ¼ q þ c:ðdT  dtÞ þ k1 N 1  I 1 þ T þ b

s;U1

ð8Þ

þ br;U1

Hence L1P1 observable can be derived as:

þ eU1

ð9Þ

where: P 1 and U1 are pseudorange and carrier phase measurements respectively, q is the geometric range between the satellite and receiver, c is the vacuum speed of light, dT and dt are the offsets of the receiver and satellite clocks from GPS time respectively, I 1 is the ionospheric delay on f 1 , T is the tropospheric delay, k1 is the wavelength of the f 1 , N 1 is the carrier phase ambiguity on f 1 , bs;P 1 is the satellite instrumental delays on P 1 for satellite S, br;P 1 is the receiver instrumental delays on P 1 for receiver r,

(a)

L1P 1 ¼

P 1 þ U1 : 2

ð10Þ

One may note that while the L1P1 combination is ionosphere-free (to the first order) and should have less noise compared to P 1 , it contains an initial phase ambiguity of the carrier which should be estimated. Therefore pseudorange measurements must be included for proper estimation of the ambiguities, which are themselves affected by ionospheric delays. Our L1P1 PPP solutions use both the P 1 pseudoranges and L1P1 combination in parameter solutions similar to dual frequency code and carrier-phase PPP solutions. Quality of pseudorange measurements affects the ambiguity resolution and hence the solution convergence time. Therefore external ionospheric information plays a crucial role in proper ambiguity resolution. Fig. 7 shows forward L1P1 single epoch solutions with standard mapping

Foward single epoch lp11 soluon at ALGO, March 17, 2015

4.5 3.5 lat

lon

h

2.5 1.5

m

0.5 -0.5 -1.5 0

2

4

6

8

UT hour

10

12

Forwad single epoch lp1cod1 soluon at ALGO, Match 17, 2015

(b) 4.5 3.5

lat

lon

h

2.5 1.5 m

0.5 -0.5 -1.5 0

2

4

6

UT hour

8

10

12

Fig. 7. Forward L1P1 single epoch solutions with standard mapping function at station ALGO during first 12 h of March 17, 2015: (a) no external ionosphere source to correct P 1 ; (b) CODE (codg) ionosphere map used to correct P 1 .

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Forward single epoch lp11 soluon at MAUI, March 17, 2015

(a) 20 15 lat

lon

h

10 5

m

0 -5 -10 0

2

4

6

8

10

12

UT hour

Forward single epoch lp1cod1 soluon at MAUI, March 17, 2015

(b)

20 lat

15

lon

h

10 m

5 0 -5 -10 0

2

4

6

8

10

12

UT hour

Fig. 8. Forward L1P1 single epoch solutions with standard mapping function at station MAUI during first 12 h of March 17, 2015: (a) no external ionosphere source to correct P 1 ; (b) CODE (codg) ionosphere map used to correct P 1 .

24 hours mean posion difference between 2freq iono-free and L1P1 at NRC1

L1P1 std of 24 hours kinemac posion esmates at NRC1

0.45

0.15

0.4 0.1

0.35 0.3

0.05

0.25

m

Lat 0

m

Lat

0.2

Lon

Lon

0.15

H

H

0.1

-0.05

0.05 Jul-18-2015 Jul-19-2015 Jul-20-2015

Mar-17-2015 Mar-18-2015 Mar-19-2015

(b)

May-10-2014 May-11-2014 May-12-2014

Jul-18-2015 Jul-19-2015 Jul-20-2015

Mar-17-2015 Mar-18-2015 Mar-19-2015

(a)

0 May-10-2014 May-11-2014 May-12-2014

-0.1

Fig. 9. 24 h L1P1 kinematic (single-epoch) PPP at station NRC1: (a) mean position difference from 2freq iono-free; (b) standard deviation.

function at mid-latitude station ALGO during first 12 h of stormy day of Match 17, 2015 with and without

ionospheric correction for P 1 . Improved convergence time can be noted in Fig. 7b as compared with Fig. 7a.

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R. Ghoddousi-Fard, F. Lahaye / Advances in Space Research xxx (2016) xxx–xxx Dual-freq. iono-free std of 24 hours kinemc posion esmates at NRC1

Jul-20-2015

Jul-19-2015

Jul-18-2015

Mar-19-2015

Mar-18-2015

Mar-17-2015

May-12-2014

Lon May-11-2014

Lat

0 May-10-2014

m

0.06 0.04 0.02

H

Fig. 10. Standard deviation of 24 h dual frequency ionospheric-free kinematic PPP position estimates at NRC1.

24 hours mean posion difference between 2freq iono-free and P1 only with igsg IONEX input with standard mapping funcon at NRC1

24 hours mean posion difference between 2freq iono-free and P1 only with igsg IONEX input with modified mapping funcon at NRC1

1

1

0.8

0.8

0.6

0.6

0.4

0.4

m 0.2

Lat

0

Lon

m

H

-0.2

Lat

0

Lon H

-0.2

-0.4

Jul-18-2015 Jul-19-2015 Jul-20-2015

(b)

Mar-17-2015 Mar-18-2015 Mar-19-2015

May-10-2014 May-11-2014 May-12-2014

Jul-18-2015 Jul-19-2015 Jul-20-2015

Mar-17-2015 Mar-18-2015 Mar-19-2015

-0.4 May-10-2014 May-11-2014 May-12-2014

(a)

0.2

Fig. 11. Mean position difference of 24 h P1 only kinematic PPP assisted with igsg TEC maps using: (a) standard mapping function, (b) modified mapping function; from 2freq iono-free solution at station NRC1.

P1 only with igsg IONEX input with standard mapping funcon, std of 24 hours posion esmates

Jul-20-2015

Jul-18-2015

Jul-19-2015

Mar-19-2015

Mar-18-2015

H Mar-17-2015

(b)

Lon

May-12-2014

Jul-18-2015 Jul-19-2015 Jul-20-2015

Mar-17-2015 Mar-18-2015 Mar-19-2015

H

Lat

May-10-2014

Lon

May-10-2014 May-11-2014 May-12-2014

(a)

Lat

1.4 1.2 1 0.8 m 0.6 0.4 0.2 0 May-11-2014

1.4 1.2 1 0.8 m 0.6 0.4 0.2 0

P1 only with igsg IONEX input with modified mapping funcon, std of 24 hours posion esmates

Fig. 12. Standard deviation of 24 h P1 only kinematic PPP at NRC1 assisted with igsg TEC maps using: (a) standard mapping function; (b) modified mapping function.

Similarly one can see the effect of external ionosphere map in equatorial station MAUI during the same period in Fig. 8 (note different vertical scale compared to Fig. 7). One may note from Fig. 8a that it took more than 6 h for the solution to be converged whereas with ionosphere corrected P 1 the convergence time has decreased to about 0.5 h as can be noted in Fig. 8b. While reliable ionosphere information enhances the ambiguity resolution, for the purpose of daily based overall

statistical comparison carried out in the rest of this paper, backward solutions which are resulted after ambiguity convergence are considered to avoid inclusion of convergence period solutions in statistics. Fig. 9 shows performance of backward L1P1 PPP GPS solutions with no external ionospheric correction for station NRC1 as compared with dual frequency ionosphere-free solutions during the 3 evaluation periods. For the studied periods and station, L1P1 daily mean

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Table 1 Single frequency PPP processing scenarios carried out in this paper.

1 2

n = 1, 2 (1: using standard mapping function; 2: using modified mapping function). Yearly averages refer to values derived from one year of monthly DCB values as provided by CODE.

solutions are within about 10 cm in horizontal components and within about 15 cm in height component from those of the corresponding dual-frequency solutions. Standard deviation of single-epoch position estimates following ambiguity convergence reached about 0.4 m in height component. However, as expected there is no apparent dependency of L1P1 backward solutions to ionospheric activity level as characterized by Kp index. As an indicator of precision of the dual frequency solution, the standard deviation (repeatability) of the solutions can be seen in Fig. 10. 5. Effect of mapping function on single frequency pseudorange only (P1) solutions using external ionosphere map In order to study the effect of different mapping functions on single frequency positioning with external ionospheric maps, single-epoch P1-only solutions were estimated for station NRC1 using the same VTEC map but two different mapping functions. Fig. 11a and b show the mean position differences of single-epoch solutions obtained over 24 h (with respect to dual frequency ionosphere free solution) using IGS VTEC maps (igsg), applying standard and modified mapping functions respectively. It is noted that change of mapping functions significantly affected P1-only position results, especially in height. However standard deviation of positioning components are not much affected as a result of changing the mapping function as can be compared in Fig. 12a and b. 6. Evaluation of single frequency PPP solutions assisted with various ionospheric maps Several single frequency GPS PPP scenarios, respectively using L1P1 combination and P1-only, with various

external ionosphere TEC maps and applying the two previously mentioned mapping functions were evaluated. Furthermore, L1P1 combination solutions with P1 corrected using various external ionosphere maps were computed (see in Table 1 solution names of lp1[???]n). As mentioned before, satellite DCBs can have nonnegligible effect on single frequency solutions. Current IONEX format version allows satellite DCB parameters in its header. External maps in the present evaluation study are used with their associated DCB values. Fig. 13 shows difference between GPS satellites DCB values as reported in various IONEX maps on March 17, 2015 as well as yearly averages of CODE-provided values, from those in final IGS maps (igsg). One can note that all presented DCB sets are within about 0:6 ns in this day. Since the average of the DCB differences is absorbed by the clock estimation, only the variability around the mean will have an impact on PPP results. The standard deviations of the DCB differences were found to not exceed 0.22 ns (about 6 cm). In order to further evaluate the effect of different DCB sets on single-frequency solutions, some of the selected scenarios are carried out with two different sets of DCB values (see Table 1, shaded area). In order to compare the performance of PPP solutions, Euclidean distance as defined bellow: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn 2 D¼ ð11Þ ðP i  I i Þ i¼1 is used with n = 1 (height component), n = 2 (horizontal components), and n = 3 (3D components) where P i are the estimated position components (mean over 24 h of single epoch backward solution) from dual frequency ionospheric-free PPP solution and I i are estimated position components using one of the mentioned single frequency PPP scenarios. RMS of these metrics which includes both

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GPS satellite DCB esmates difference from igsg on Mar 17, 2015 0.6 codg(0.060)

0.4

emrg(0.181) 0.2 esag(0.175) ns

0 -0.2 G01 G03 G05 G07 G10 G12 G14 G16 G18 G20 G22 G24 G27 G29 G31 -0.4

igrg(0.104) jplg(0.061) upcg(0.217)

-0.6

yrly. avg.(0.125)

PRN

Fig. 13. GPS satellite DCB difference from igsg. Note that legend naming for values extracted from IONEX files follows first 4 characters of IONEX file names. Values in parenthesis are standard deviations in ns.

bias and variability over all studied stations averaged within each selected period is used to evaluate overall performance of single frequency solutions under various ionospheric conditions. Daily GPS observations from 10 globally distributed stations including 4 in Canada (Fig. 6) are processed with a 10 degree elevation cut-off angle in up to 36 single frequency scenarios (see Table 1) as well as using dual-frequency ionosphere-free over each 3-day period. Figs. 14–16 show RMS of our metrics D (Eq. (11)) from single frequency solutions for 4 Canadian stations during the 3 evaluation periods. Solutions are ordered by

increasing 3D RMS. Figs. 17–19 show RMS of metrics when calculated over all 10 stations using the same ordering scheme. Note that regional ionosphere maps could only be evaluated using Canadian stations. It can be noted that L1P1 solutions (lp1) when assisted with an external ionosphere map provide the best single frequency solution among the studied scenarios. L1P1 without external ionosphere map also provide improved results compared to P1only solutions. Also noted is that changing the source of DCB in regional models (sDn vs. scn and sDd vs. scd) did not cause any significant change in associated single frequency solutions.

4 Canadian staons, May 10-12, 2014

1.2 1 0.8 m

3D

2D

h

0.6 0.4 0.2 lp1cod2 lp1cod1 lp1jpl2 lp1jpl1 lp12 lp1scd2 lp1sDd2 lp1scn2 lp11 lp1sDn2 lp1scd1 lp1sDd1 lp1scn1 lp1sDn1 igs2 igr2 scn2 upc2 sDn2 cod2 jpl2 emr2 esa1 scd2 sDd2 esa2 cod1 igs1 upc1 igr1 emr1 scn1 sDn1 scd1 sDd1 jpl1

0

Fig. 14. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over Canadian stations during May 10–12, 2014.

4 Canadian staons, Mar 17-19, 2015

1.2 1 0.8 m

0.6

3D

2D

h

0.4 0.2 lp1cod2 lp1cod1 lp1jpl2 lp1jpl1 lp12 lp1scn2 lp1sDn2 lp11 lp1scn1 lp1sDn1 lp1scd1 lp1scd2 lp1sDd1 lp1sDd2 cod2 esa2 upc2 igs2 igr2 esa1 emr2 cod1 scn2 sDn2 jpl2 scd2 upc1 sDd2 igs1 igr1 emr1 scn1 sDn1 scd1 sDd1 jpl1

0

Fig. 15. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over Canadian stations during Mar 17–19, 2015.

Please cite this article in press as: Ghoddousi-Fard, R., Lahaye, F. Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.02.017

R. Ghoddousi-Fard, F. Lahaye / Advances in Space Research xxx (2016) xxx–xxx

11

4 Canadian staons, Jul 18-20, 2015

1.2 1 3D

2D

h

0.8 m

0.6

0.4 0.2 lp1cod2 lp1sDn2 lp1scn2 lp1cod1 lp1jpl2 lp12 lp1sDd2 lp1scd2 lp1sDn1 lp1scn1 lp11 lp1jpl1 lp1sDd1 lp1scd1 cod1 esa1 igr2 sDn2 scn2 igs2 scd2 sDd2 jpl2 upc1 emr2 igs1 upc2 emr1 igr1 cod2 esa2 scn1 sDn1 scd1 sDd1 jpl1

0

Fig. 16. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over Canadian stations during Jul 18–20, 2015.

10 global stations, May 10 -12, 2014

1.6 1.4 1.2 1 m

3D

0.8

2D

h

0.6 0.4 0.2 jpl1

igr1

emr2

emr1

esa1

igs1

upc1

esa2

cod1

igr2

upc2

jpl2

cod2

igs2

lp11

lp12

lp1jpl1

lp1cod1

lp1jpl2

lp1cod2

0

Fig. 17. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over global stations during May 10–12, 2014.

10 global stations, Mar 17 -19, 2015

1.6 1.4 1.2

3D

2D

h

1 m 0.8

0.6 0.4 0.2 emr1

emr2

jpl1

igr1

esa1

esa2

igs1

upc1

upc2

igr2

cod1

jpl2

igs2

cod2

lp11

lp12

lp1jpl1

lp1jpl2

lp1cod1

lp1cod2

0

Fig. 18. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over global stations during Mar 17–19, 2015.

Please cite this article in press as: Ghoddousi-Fard, R., Lahaye, F. Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.02.017

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R. Ghoddousi-Fard, F. Lahaye / Advances in Space Research xxx (2016) xxx–xxx

10 global stations, Jul 18, 20- 2015

1.6 1.4 1.2 1 m

3D

0.8

2D

h

0.6 0.4 0.2 jpl1

igr1

igs1

esa1

esa2

upc1

jpl2

emr2

upc2

igr2

emr1

cod2

cod1

igs2

lp11

lp12

lp1jpl1

lp1jpl2

lp1cod1

lp1cod2

0

Fig. 19. RMS of 3D, 2D and h position difference of various single frequency PPP scenarios from dual-frequency iono-free solution over global stations during Jul 18–20, 2015.

One may realize the significant impact of the selected mapping function (specified by the last character of the solution identifier) in P1-only solutions. Even though some exceptions may be noted, in most cases the modified mapping function has provided better results. This is especially noticeable for the jplg VTEC maps produced with a mapping function based on an extended slab model (Mannucci, 2014; personal communications). It should be noted that the modified mapping function is also derived from a slab model fit and provides obliquity factor values that agree to about 0.01, down to 10 degree elevation, compared to the JPL extended slab model mapping function. This shows the importance of using compatible mapping functions for slant to vertical conversions in both VTEC map generation and end-user positioning application. It may also indicate the systematic effect of a mapping function (see e.g. Fig. 4b) on possible inherent biases in VTEC maps. However, effect of several other processing steps leading to generation of a global VTEC map, including spatial and temporal interpolations (see e.g. Wienia, 2008), may surpass the effect of compatibility of selected mapping function. In L1P1 solutions the effect of using different mapping functions is not as significant as for P1 solutions

but still better performance of modified mapping function may be concluded. One may also note the specific effects different VTEC maps and mapping functions have on the horizontal and vertical components of position in Figs. 14–19. It is noted that 3D RMS values are mostly dominated by the height component even though in some cases, especially during the storm period, the horizontal component contribution to 3D RMS is more significant. It can also be noted that changing the mapping function mostly affects the height component. Table 2 summarizes the range of 3D RMS values of the solutions with various ionosphere maps shown in Figs. 14– 19. It is noted that P1-only solutions using any of the selected VTEC maps is degraded during ionospheric storm period (see values for the March 17–19, 2015 period in bold). L1P1 solutions (lp1) with external ionosphere source to correct P1 provided 3D position RMS within a range of about 6–21 cm. While not as significant as for P1-only results, L1P1 solution degradation during storm periods was still apparent. The quality of backward smoothed (ambiguity converged) L1P1 solutions may still be affected by scintillations causing more frequent data gaps or cycle

Table 2 Range of 3D position RMS of difference of single frequency PPP scenarios with various external iono sources and two (see footnote 1 of Table 1) mapping functions (MF) from dual-frequency iono-free solution. Values are in meter. Studied region

Scenario

lp1 with external iono

lp1 with external iono

P1 with external iono

P1 with external iono

Period

1

2

1

2

Canadian

May 10–12, 2014 Mar 17–19, 2015 Jul 18–20, 2015

0.068–0.122 0.079–0.212 0.062–0.091

0.058–0.099 0.069–0.212 0.054–0.071

0.351–0.889 0.526–1.153 0.166–0.637

0.212–0.399 0.371–0.818 0.193–0.407

Global

May 10–12, 2014 Mar 17–19, 2015 Jul 18–20, 2015

0.127–0.149 0.144–0.163 0.094–0.113

0.105–0.121 0.115–0.132 0.083–0.098

0.821–1.263 0.970–1.497 0.457–1.013

0.654–1.015 0.882–1.431 0.398–0.567

MF

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R. Ghoddousi-Fard, F. Lahaye / Advances in Space Research xxx (2016) xxx–xxx

slips that require ambiguity re-initializations (see e.g. Ghoddousi-Fard and Lahaye, 2015). 7. Summary and conclusions Over 2000 single frequency PPP solutions obtained by processing up to 36 scenarios for 10 global stations over 9 days (three 3-day periods) were compared to corresponding dual frequency ionosphere-free PPP solutions. L1P1 as well as P1-only solutions were computed using several regional and global ionosphere maps as well as two of the most commonly used ionospheric mapping functions to perform a comparative evaluation. For the studied periods and stations it is noted that L1P1 using external TEC maps to correct the raw pseudoranges provide best results when compared to dualfrequency ionosphere-free solution. All P1-only solutions assisted with VTEC maps show degradation during ionospheric storm periods, although some VTEC maps are seen to perform better than others. Additionally, the performance degradation is variable over different regions, an indication that the global character of the Kp index may not be representative of local conditions, as well as varying accuracy of TEC maps over different regions. It is also noted that the choice of mapping function can significantly affect the P1-only solution assisted with ionosphere maps, especially in height component. This highlights the importance of using compatible mapping functions for ionospheric VTEC map generation and end-user slant delay computation. Acknowledgements We thank editor and two anonymous reviewers for their comments. IGS and its contributing organizations are thanked for GPS data and products. Kp indices are accessed through World Data Center for Geomagnetism, Kyoto. This is ESS contribution number 20150414. References Beutler, G., Bock, H., Dach, R., Fridez, P., Ga¨de, A., Hugentobler, U., Ja¨ggi, A., Meindl, M., Mervart, L., Prange, L., Schaer, S., Springer, T., Urschl, C., Walser, P., 2007. Bernese GPS Software Version 5.0. Astronomical Institute, University of Bern, Bern, Switzerland. Clynch, J.R., Coco, D.S., Coker, C., Bishop, G.J., 1989. A versatile GPS ionospheric monitor: high latitude measurements of TEC and scintillation. In: Proceedings of ION GPS-89, the 2nd International Technical Meeting of the Satellite Division of The Institute of Navigation, Colorado Springs, CO, 22–27 September. The Institute of Navigation, Washington, pp. 445–450. Coster, A.J., Gaposchkin, E.M., Thonton, L.E., 1992. Real-Time Ionospheric Monitoring System Using the GPS, Technical Report 954. Lincoln Laboratory, Massachusetts Institute of Technology, MA, USA. Feltens, J., Schaer, S., 1998. IGS Products for the ionosphere, IGS position paper. In: Proceedings of the IGS Analysis Centers Workshop. ESOC, Darmstadt, Germany, pp. 225–232, 9–11 February.

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Please cite this article in press as: Ghoddousi-Fard, R., Lahaye, F. Evaluation of single frequency GPS precise point positioning assisted with external ionosphere sources. Adv. Space Res. (2016), http://dx.doi.org/10.1016/j.asr.2016.02.017