Investigating GNSS PPP–RTK with external ionospheric constraints

Abstract

Real-Time Kinematic Precise Point Positioning (PPP–RTK) is inextricably linked to external ionospheric information. The PPP–RTK performances vary much with the accuracy of ionospheric information, which is derived from different network scales, given different prior variances, and obtained under different disturbed ionospheric conditions. This study investigates the relationships between the PPP–RTK performances, in terms of precision and convergence time, and the accuracy of external ionospheric information. The statistical results show that The Time to First Fix (TTFF) for the PPP–RTK constrained by Global Ionosphere Map (PPP–RTK-GIM) is about 8–10 min, improved by 20%–50% as compared with that for PPP Ambiguity Resolution (PPP-AR) whose TTFF is about 13–16 min. Additionally, the TTFF of PPP–RTK is 4.4 min, 5.2 min, and 6.8 min, respectively, when constrained by the external ionospheric information derived from different network scales, e.g. small-, medium-, and large-scale networks, respectively. To analyze the influences of the optimal prior variances of external ionospheric delay on the PPP–RTK results, the errors of 0.5 Total Electron Content Unit (TECU), 1 TECU, 3 TECU, and 5 TECU are added to the initial ionospheric delays, respectively. The corresponding convergence time of PPP–RTK is less than 1 min, about 3, 5, and 6 min, respectively. After adding the errors, the ionospheric information with a small variance leads to a long convergence time and that with a larger variance leads to the same convergence time as that of PPP-AR. Only when an optimal prior variance is determined for the ionospheric delay in PPP–RTK model, the convergence time for PPP–RTK can be shorten greatly. The impact of Travelling Ionospheric Disturbance (TID) on the PPP–RTK performances is further studied with simulation. It is found that the TIDs increase the errors of ionospheric corrections, thus affecting the convergence time, positioning accuracy, and reliability of PPP–RTK.

Introduction

Precise Point Positioning (PPP) can achieve a positioning accuracy of better than 10 cm within 30 min by applying satellite precise orbit and clock products. Much work has been done to reduce the convergence time of PPP, such as using multi-constellation Global Navigation Satellite System (GNSS) data (Guo et al., 2017; Li et al., 2019a, 2019b; Li et al., 2015a; Li et al., 2019c; Lou et al., 2016), performing integer ambiguity resolution (Collins et al., 2008; Ge et al., 2008; Hu et al., 2019; Laurichesse et al., 2009; Li & Zhang, 2012; Li et al., 2016; Liu et al., 2017), and adding priori atmospheric information (de Oliveira et al., 2017). However, it still needs more than 15 min to achieve the solutions with centimeter-level accuracy due to the ambiguity of carrier observations. For the Real-Time Kinematic (RTK) technique, though the ambiguity can be quickly fixed, it cannot be applied by the users far away from the reference stations. PPP–RTK is the integer ambiguity resolution enabled PPP by adding the priori atmospheric information from a local reference network, which has a short convergence time (Wübbena et al., 2005; Li & Zhang et al. 2011; de Oliveira et al., 2017; Zhang et al., 2018). It extends the PPP concept by providing different kinds of corrections to users, such as satellite orbit, clock, and atmospheric delay corrections as well as satellite phase and code biases. These corrections, when accurately provided, enable regional or even global PPP–RTK users to recover the integer nature of ambiguities, thus improving the positioning accuracy and convergence behavior (Geng et al., 2010; Geng & Shi, 2017; et al., ; 2019; Liu et al., 2017; Li et al., 2019a; Li et al., 2019b; Li et al., 2019c; Zha et al., 2021).

Among all kinds of corrections, the precise ionospheric delay is the main bottleneck limiting the fast and successful ambiguity resolution for PPP–RTK (Hernández-Pajares et al., 2011; Jakowski et al., 2008). Consequently, the precise determination of the ionospheric delay along a specific Line-of-Sight (LoS) and its correction are of great significance for reducing the convergence time of PPP–RTK. Ionospheric delays used for augmenting PPP with Ambiguity Resolution (PPP-AR) are mainly derived with the following two ways. The first one is the two-dimensional vertical thin-shell TEC (Total Electron Content) model established at a specific altitude by using mathematical algorithms, such as spherical harmonics function (Schaer, 1999), spherical cap harmonics function (Haines, 1988; Li et al., 2015b), B-splines, and trigonometric B-splines (Mautz et al., 2005; Schmidt et al., 2008) and so on. Existing studies indicated that benefiting from the vertical ionospheric corrections, the convergence time of PPP–RTK can be reduced (Banville et al., 2014; Psychas et al., 2018). One main problem with the estimated TEC models is the mapping function error. To improve the accuracy of vertical ionospheric TEC model, the ionosphere is divided into many three-dimensional voxels of the same size. Hernández-Pajares et al. (1999) first presented GNSS-based data-driven tomographic models. Since then, many scholars further improved the ionosphere tomographic models by different methods (Wen et al., 2015; Wen et al., 2007, 2008; Zheng et al., 2017, 2018, 2020, 2021). Olivares-Pulido et al. (2019) presented a 4D tomographic ionospheric model to support PPP–RTK, achieving an accuracy better than 10 cm in the horizontal direction within about 10 min. The second one is to interpolate the ionospheric delays with high accuracy derived from networks. It has been demonstrated that PPP–RTK can achieve mm-level positioning accuracy in the horizontal direction by using ionospheric corrections derived from a small-scale network (about dozens of kilometers) (Teunissen et al., 2010; Zhang et al., 2011). Also, the convergence behavior can be significantly improved with TEC in the slant and vertical directions when leveraging a regional network and Global Ionospheric Maps (GIMs) (Xiang et al., 2020). In addition, Wang et al. (2017) found that ten seconds were required to make most of the horizontal positioning errors smaller than 10 cm by using 1 Hz data when the network corrections are provided, such as the satellite clocks, the satellite phase biases, and the ionospheric delays.

Previous studies showed that the convergence time of PPP–RTK can be reduced when constrained by external ionospheric information. However, the performances of PPP–RTK constrained by the ionospheric information, which is derived from different scale networks with different accuracy levels by giving different prior variances, and derived under different ionospheric conditions, are not fully studied. This study addresses the aforementioned problems. The performances of PPP–RTK constrained by different accuracy of ionospheric information for different situations are discussed and analyzed. Finally, summary and conclusions are given.

Methodology

In this section, the algorithms of the PPP–RTK server model, the estimation of ionospheric delay based on carrier phase observations and the PPP–RTK user model are introduced.

PPP–RTK server model

The difference between the observed and calculated observation equations for un-differenced and un-combined at k-th epoch can be described as:

$$\left\{ \begin{aligned} {\text{E}}(\Delta P_{r,i}^{s} (k)) &= {\mathbf{c}}_{r}^{s} (k) \cdot {\Delta }{\mathbf{x}}_{r} (k) + c \cdot [{\text{d}}t_{r} (k) - {\text{d}}t^{s} (k)] + g_{r}^{s} \cdot T_{r}^{s} (k) \\ & \quad + \mu_{i} \cdot I_{r,i}^{s} (k) + B_{r,i}^{{}} - B_{i}^{s} \\ {\text{E}}(\Delta \Phi_{r,i}^{s} (k))& = {\mathbf{c}}_{r}^{s} (k) \cdot {\Delta }{\mathbf{x}}_{r} (k) + c \cdot [{\text{d}}t_{r} (k) - {\text{d}}t^{s} (k)] + g_{r}^{s} \cdot T_{r}^{s} (k) \\ & \quad - \mu_{i} \cdot I_{r,i}^{s} (k) + b_{r,i}^{{}} - b_{i}^{s} + \lambda_{i} \cdot N_{r,i}^{s} \\ \end{aligned} \right.$$

(1)

where ${\text{E}}( \cdot )$ denotes the expectation operator; $P_{r,i}^{s} (k)$ and $\Phi_{r,i}^{s} (k)$ are the code and phase observations, respectively, from receiver r to satellite s at epoch k for frequency i (i = 1,2); the 3 × 1 vector $\Delta {\mathbf{x}}_{r} (k)$ denotes the receiver’s position increment at epoch k; the 3 × 1 vector ${\mathbf{c}}_{r}^{s} (k)$ indicates the unit vector from receiver to satellite; c denotes the speed of light in vacuum; ${\text{d}}t_{r}$ and ${\text{d}}t^{s}$ denote the receiver and satellite clock errors, respectively; $T_{r}^{s} (k)$ denotes the zenith non-hydrostatic tropospheric delays at epoch k, while the hydrostatic tropospheric delays can be corrected by an empirical model; $g_{r}^{s}$ denotes the non-hydrostatic tropospheric mapping function; $B_{r,i}^{{}}$ and $B_{i}^{s}$ are the receiver and satellite code hardware biases, respectively; $b_{r,i}^{{}}$ and $b_{i}^{s}$ are the receiver and satellite carrier phase hardware biases, respectively; $\mu_{i} = f_{1}^{2} /f_{i}^{2}$ is the conversion coefficient of ionospheric delay for frequency i; $I_{r,i}^{s} (k)$ is the first order ionospheric delay for frequency i along a line-of-sight at epoch k; $\lambda_{i} = c/f_{i}$ indicates the phase wavelength for frequency i; $N_{r,i}^{s}$ is the phase integer ambiguity for frequency i.

Equation (1) is a rank deficiency system due to the linear dependency of some columns of the design matrix. To eliminate the rank deficiency, Odijk et al. (2016) proposed the null space of the design matrix and chose a minimum constraint set, i.e., S-basis constraint. Based on the S-basis, the types of rank deficiency and their S-basis constraints can be found in Table 1. It should be noted that this choice of S-basis holds for the Code Division Multiple Access (CDMA) signals, and different choice of S-basis should be applied for the Frequency Division Multiple Access (FDMA) signals (Zhang et al., 2021).

Table 1 The rank deficiencies information, including involved parameters, sizes, and S-basis for PPP–RTK network

In Table 1, p and q denote the reference receiver and satellite, respectively. In addition, $B_{{{\text{IF}}}}^{s}$, $B_{{{\text{GF}}}}^{s}$, $B_{{r,{\text{IF}}}}^{{}}$ and $B_{{r,{\text{GF}}}}^{{}}$ can be described as:

$$\left\{ \begin{aligned} B_{{{\text{IF}}}}^{s} &= \frac{{\mu_{2} B_{1}^{S} - \mu_{1} B_{2}^{S} }}{{\mu_{2} - \mu_{1} }} \hfill \\ B_{{{\text{GF}}}}^{s} & = \frac{{B_{2}^{S} - B_{1}^{S} }}{{\mu_{2} - \mu_{1} }} \hfill \\ D_{{{\text{DCB}}}}^{S} & = B_{2}^{S} - B_{1}^{S} \hfill \\ B_{{{\text{IF}}}}^{s} & = \frac{{\mu_{2} B_{r,1} - \mu_{1} B_{r,2} }}{{\mu_{2} - \mu_{1} }} \hfill \\ B_{{{\text{GF}}}}^{s} & = \frac{{B_{r,2} - B_{r,1} }}{{\mu_{2} - \mu_{1} }} \hfill \\ D_{{{\text{DCB}}}}^{r} & = B_{r,2} - B_{r,1} \hfill \\ \end{aligned} \right.$$

(2)

When the rank deficiencies in Eq. (1) are eliminated based on the S-basis listed in Table 1, the full-rank un-differenced and un-combined PPP–RTK network code and phase observations at epoch k can be expressed as:

$$\left\{ \begin{aligned} {\text{E}}(\Delta P_{r,i}^{s} (k))& = c \cdot [\overline{{{\text{d}}t}}_{r \ne p} (k) - \overline{{{\text{d}}t}}^{s} (k)] + g_{r}^{s} \cdot \overline{T}_{r \ne p}^{s} (k) \\ & \quad + \mu_{i} \cdot \overline{I}_{r,i}^{s} (k) + \overline{B}_{r \ne p,i > 2}^{{}} - \overline{B}_{i>2}^{s} \\ {\text{E}}(\Delta \Phi_{r,i}^{s} (k)) & = c \cdot [\overline{{{\text{d}}t}}_{r \ne p} (k) - \overline{{{\text{d}}t}}^{s} (k)] + g_{r}^{s} \cdot \overline{T}_{r \ne p}^{s} (k) \\ & \quad- \mu_{i} \cdot \overline{I}_{r,i}^{s} (k) + \overline{b}_{r \ne p,i > 2}^{{}} - \overline{b}_{i}^{s} + \lambda_{i} \cdot N_{r \ne p,i}^{s \ne q} \\ \end{aligned} \right.$$

(3)

The formulations of the estimable parameters in Eq. (3) can be found in Table 2.

Table 2 The estimable parameters of un-differenced and un-combined PPP–RTK and corresponding formulations

As is shown in Table 2, the ionospheric parameters are biased by Geometry-Free (GF) receiver and satellite code biases. These biases are also contained in the interpolated ionospheric delays that are provided to users. As a result, the interpolated user ionospheric delay for different satellites will contain different combinations of receiver code biases when the different receivers of a network observe different satellites, resulting in system biases for PPP–RTK users of the network. It is worth noting that the system biases can be eliminated by single differencing between satellites when the ionospheric delays are used as constraints in PPP–RTK.

It should be mentioned that Eq. (3) ignores the spatial correlation of ionospheric delays in the network. The slant ionospheric delays derived from the network receivers for the same satellite are approximately equal if the distances between the receivers are a few hundred kilometers (Odijk, 2002). Accordingly, the redundant observation equation, i.e., the single-differenced ionospheric code observations described in Eq. (4), can be added to Eq. (1).

$$I_{p}^{s} (k) - I_{r \ne p}^{s} (k) = 0,{\kern 1pt} W = S^{ - 1}$$

(4)

where W and S denote the weight and variance–covariance matrix of the single-differenced ionospheric code observations between receivers.

Once the single differenced ionospheric code observations between receivers are added to Eq. (1), it will not be a rank deficiency system. After eliminating the rank deficiency, the full rank un-differenced and un-combined PPP–RTK can be formulated as:

$$\left\{ \begin{aligned} {\text{E}}(\Delta P_{r,i}^{s} (k))& = c \cdot [\overline{{{\text{d}}t}}_{r \ne p} (k) - \overline{{{\text{d}}t}}^{s} (k)] + g_{r}^{s} \cdot \overline{T}_{r \ne p}^{s} (k) \\ & \quad + \mu_{i} \cdot \overline{I}_{r,i}^{s} (k) + \overline{B}_{r \ne p,i> 2}^{{}} - \overline{B}_{i>2}^{s} + \frac{{\mu_{i} }}{{\mu_{2} - \mu_{1} }}\overline{B}_{{r \ne p,{\text{DCB}}}} \\ {\text{E}}(\Delta \Phi_{r,i}^{s} (k)) & = c \cdot [\overline{{{\text{d}}t}}_{r \ne p} (k) - \overline{{{\text{d}}t}}^{s} (k)] + g_{r}^{s} \cdot \overline{T}_{r \ne p}^{s} (k) \\ & \quad- \mu_{i} \cdot \overline{I}_{r,i}^{s} (k) + \overline{b}_{r \ne p,i> 2}^{{}} - \overline{b}_{i}^{s} + \lambda_{i} \cdot N_{r \ne p,i}^{s \ne q} \\ \overline{I}_{r}^{s} - \overline{I}_{r \ne p}^{s} &= 0 \\ \end{aligned} \right.$$

(5)

where the estimable forms of the biased receiver code biases, phase biases, and ionospheric delays are listed in Table 3.

Table 3 The estimable formulation for the un-differenced and un-combined PPP–RTK sever model

It can be seen from Table 3 that the estimable ionospheric parameters of the network reference receiver include GF receiver code biases. Therefore, the interpolated ionospheric delays provided to users contain the same receiver Differential Code Bias (DCB), which is eventually absorbed by the estimated receiver code and phase bias of users.

Precise ionospheric delay estimation

As is mentioned above, the convergence time can be reduced by applying the constraint of ionospheric information, which can be derived from GF combination. In recent decades, three methods were used for the extraction of ionospheric observables, such as the Carrier-to-Code Levelling (CCL) method (Ciraolo et al., 2007), the un-differenced and un-combined PPP (UD-PPP) method (Zhang et al., 2012), and the zero-difference integer ambiguity (PPP-Fixed) method (Ren et al., 2020). The CCL method only considers the noise of a specific satellite in a continuous arc, leading to large errors between different satellites when the reference of each satellite is different. The UD-PPP method uses external constraints and performs the adjustment of the selected network by the least-squares method. The estimated ambiguity can be the best for the satellites tracked by a certain receiver, and the differences between different satellites can be effectively reduced. In addition to the advantages of the UD-PPP, the PPP-Fixed fully utilizes the constraints of receivers for the whole network. Furthermore, ionospheric observables extracted from carrier phase observations by using the PPP-Fixed method have higher accuracy than those extracted with CCL and UD-PPP. The PPP-Fixed method is applied for ionospheric delay extraction. The processing flow chart of this method is plotted in Fig. 1 and its detailed processing steps can be found in (Ren et al., 2020).

Fig. 1

Flow chart of ionospheric delay estimated by using the PPP-Fixed method

PPP–RTK user model

The receiver of a user can be theoretically regarded as a part of network. Therefore, the rank deficiencies and corresponding S-basis for users are nearly the same as those of network receivers. Unlikely, the positions of users need to be estimated and the $\overline{{{\text{d}}t}}^{s}$, $\overline{B}_{i > 2}^{s}$, and $\overline{b}_{i}^{s}$ need to be corrected as well. Therefore, the PPP–RTK user model is given as follows:

$$\left\{\begin{aligned} & E(\Delta {P}_{r,i}^{s}(k)+c\cdot {\overline{\mathrm{d}t}}^{s}(k)+{\overline{\mathrm{d}t}}_{j>2}^{S}(k))\\ &={c}_{u}^{S}(k)\cdot \Delta {x}_{u}(k)+c\cdot {\widehat{\mathrm{d}t}}_{u}(k)+{g}_{u}^{s}\cdot {\widehat{T}}_{u}^{s}(k)+{\mu }_{i}\cdot {\widehat{I}}_{u}^{s}(k)+{\overline{B}}_{u,i>2}+{\mu }_{i}\cdot {\widehat{B}}_{u,\mathrm{GF}}\\ & E(\Delta {\Phi }_{r,i}^{s}(k)+c\cdot {\overline{\mathrm{d}t}}^{s}(k)+{b}_{j>2}^{S}(k))\\ &={c}_{u}^{S}(k)\cdot \Delta {x}_{u}(k)+c\cdot {\widehat{\mathrm{d}t}}_{u}(k)+{g}_{u}^{s}\cdot {\widehat{T}}_{u}^{s}(k)-{\mu }_{i}\cdot {\widehat{I}}_{u}^{s}(k)+{\widehat{b}}_{u,i>2}+{\lambda }_{i}\cdot {N}_{u,i}^{s\ne q}\\ & {\overline{I}}_{\mathrm{Interpolate}}^{s}={\overline{I}}_{r\ne p}^{s}\end{aligned}\right.$$

(6)

where the formulations of the estimable parameter are listed in Table 4. When the ionospheric delays provided to users are interpolated by network ionospheric delays, Eq. (6) represents the PPP–RTK user model, and $\widehat{B}_{{u,{\text{GF}}}}$ needs to be estimated. It should be noted that the ionospheric delay corrections contribute to the rapid convergence at the beginning significantly when a new satellite is observed, or outages occur.

Table 4 The estimable parameters for a PPP–RTK user

Results and discussion

In this section we first describe the performances of the PPP–RTK constrained by the external ionospheric information derived from different types of networks. Then we analyze the dependencies of convergence time on the prior variances of the ionospheric delays. Finally, a Travelling Ionospheric Disturbance (TID) is simulated and its impacts on the PPP–RTK performances are presented. For PPP–RTK test, the data collected by about 190 GNSS stations from 2020/04/01 to 2020/04/08 are applied. The distribution of GNSS stations, which are mainly located in the Chinese mainland, is plotted in Fig. 2.

Fig. 2

Distribution of GNSS stations

The parameter estimation is performed by the least square method for the network, while the Kalman filter and kinematic mode are applied for users. The satellite with the maximum elevation angle for a specific epoch is selected as a reference. The satellite positions and satellite clock errors are derived from the precise satellite orbit and clock products provided by the International GNSS Service (IGS). The receiver positions of the network are fixed to their known values. The code and phase biases of receivers and satellites are treated as time-invariant parameters. Other kinds of error sources in positioning, such as the solid earth tide, the ocean tide, the relativistic effects, as well as satellite receiver Phase Center Offset (PCO) and Phase Center Variation (PCV), are corrected by corresponding models. The ambiguity term of the reference station is fixed by estimating the Un-calibrated Phase Delays (UPD). In this study, the UPDs are estimated by using the method introduced by Li & Zhang (2012). The ambiguities of users are resolved by using the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method (Teunissen, 1995). Other main processing strategies are listed in Table 5.

Table 5 The detailed processing setups for PPP–RTK

PPP–RTK test for different network scales

In PPP–RTK processing, the Kalman filter is reinitialized every hour.

Figure 3 presents the distribution of GNSS stations and validation stations, which are applied for the regional network test of the PPP–RTK constrained by interpolated ionospheric delays.

Fig. 3

Distribution of GNSS stations and validation stations for a regional network

Figure 4 illustrates the performance comparisons of the PPP-AR and the PPP–RTK constrained by Global Ionosphere Map (PPP–RTK-GIM). As we can see, The Time to First Fix (TTFF) of PPP–RTK-GIM for different IGS stations is improved by 20%-50% as compared with that of PPP-AR.

Fig. 4

Statistical results of PPP-AR and PPP–RTK constrained by GIM

To better understand how the ionospheric delays derived from different network scales affect the performances of a PPP–RTK user, the small-scale (about 300 km), medium-scale (about 500 km) and large-scale (800 km) networks are designed and shown in Fig. 5. The number of reference stations is about 80, 40, and 30 for small-scale, medium-scale, and large-scale networks, respectively. Both small-scale and medium-scale networks consider the spatial variations of the ionosphere and the distribution of GNSS stations, while the distances between the reference stations for small-scale network are smaller than those for medium-scale network. The large-scale network contains less reference stations that can nearly cover China, leading to the longest distances between reference stations. Hence, the large-scale network considers neither the spatial variations of the ionosphere nor the distribution of available reference stations.

Fig. 5

The designed networks with point spacing of a 300, b 500, and c 800 km

Figure 6 compares the statistical results of TTFF and fixing rate for the PPP–RTK constrained by the ionospheric delays interpolated from small-scale, medium-scale, and large-scale networks, denoted as PPP–RTK-300, PPP–RTK-500, and PPP–RTK-800, respectively. From Fig. 6 we can see that PPP–RTK-300 performs best, followed by PPP–RTK-500, and then PPP–RTK-800. This is due to the fact that the spatial variation of the ionosphere depends much on the distance between stations. The TTFF is about 4.4, 5.2, and 6.8 min for the PPP–RTK-300, PPP–RTK-500, and PPP–RTK-800, respectively. And their corresponding fixing rate is 97%, 96%, and 93%, respectively.

Fig. 6

Statistical results of the PPP–RTK constrained by the ionospheric delays interpolated from different scales of network

The relationship between convergence time and different prior variances

The above results indicate that the TTFF can be reduced for the PPP–RTK constrained by ionospheric delay. However, due to the large number of the stations and long data length in the test, we could not determine the optimum variance to test them one by one. To figure out the relationship between the variance of the given ionospheric delay corrections and the convergence time of PPP–RTK, we added the random errors to the provided ionospheric delay corrections as follows:

$$\tilde{I}_{{r,{\text{bias}}}}^{s} { = }\tilde{I}_{r}^{s} + \sigma$$

(7)

where $\sigma$ denotes the random error. In order to avoid the added random error close to zero, the random error should meet the following condition:

$$0.5 \times S_{{{\text{Max}}}}^{{{\text{Error}}}} \le \left| \sigma \right| \le S_{{{\text{Max}}}}^{{{\text{Error}}}}$$

(8)

where $S_{{{\text{Max}}}}^{{{\text{Error}}}}$ expresses the threshold value of the added error.

Figure 7 presents the convergence time for different errors of less than 0.5 TECU, 1 TECU, 3 TECU and 5 TECU that are added to the network-derived ionospheric delays with optimum variances. As is shown in Fig. 7, all the positioning accuracy better than 10 cm can be achieved within 7 min. Although the variance can properly describe the accuracy of ionospheric delay, the convergence time increases with the increase of added ionospheric errors. The convergence time is about 3, 5, and 6 min, respectively, when the corresponding added error is about 1 TECU, 3 TECU, and 5 TECU with optimum variance. Meanwhile, the convergence time is less than 1 min when the added error of ionospheric delay is smaller than 0.5 TECU with optimum variance.

Fig. 7

The convergence time of the ionospheric delay corrections with an error of a smaller than 0.5 TECU, b 1 TECU, c 3 TECU, and d 5 TECU as optimum variance, respectively

Figure 8 illustrates the convergence time of PPP-AR and that of the PPP–RTK constrained by ionospheric delay corrections with an error of 3 TECU for different variances. It shows that the variances of ionospheric delay corrections affect the convergence time significantly. On the one hand, a very small variance of the ionospheric delay with an error of 3 TECU leads to the lower fixing rate and a longer convergence time compared with those of PPP-AR. Meanwhile, the stability and reliability of the positioning errors during the un-convergence period are not ideal. On the other hand, when the variance of ionospheric delay correction is too large, the convergence time of the PPP–RTK constrained by ionospheric delay corrections is nearly the same as that of PPP-AR, but the positioning accuracy during the un-convergence period is better than that of PPP-AR. When the variance matches the accuracy of ionospheric delay corrections, the convergence time of the PPP–RTK constrained by ionospheric delay corrections can be reduced significantly compared with that of PPP-AR.

Fig. 8

The convergence time of a PPP-AR, and the ionospheric delay corrections with an error of 3 TECU while b the variance is very small (< 1 TECU), c the variance is the optimum (= 3 TECU), and d the variance is large (> 5TECU)

Figure 9 describes the convergence time for the different errors of ionospheric delay corrections with different variances. As shown in Fig. 9, the convergence time varies similarly for the different errors of ionospheric delay corrections. Generally, the convergence time is long when the variance is very small, while the convergence time is almost the same as that of PPP-AR when the variance is large. The convergence time will be reduced significantly if the variance can best match the accuracy of the ionospheric delay corrections when compared with that of PPP-AR. In terms of the ionospheric delay corrections without errors, the convergence time will be the same as that of PPP-AR if the variance is large enough, and the convergence time will be very short if the variance is very small.

Fig. 9

The convergence time of the PPP–RTK constrained by external ionospheric delay information with an error of a 0, b 1 TECU, c 3 TECU, and d 5 TECU for different variances

The performance of PPP–RTK during the periods of ionospheric disturbances

Travelling Ionospheric Disturbance (TID) is the ionospheric density fluctuation that propagates as a wave through the ionosphere at a wide range of velocities and frequencies (Belehaki et al., 2020; Saito et al., 1998; Tsugawa et al., 2004, 2007). The high occurrence rate of TIDs and complicated variety of their characteristics regarding their velocity, propagation direction, and amplitude impact the operation of ground-based infrastructures, especially real-time kinematic services, and radio communication (Hernández-Pajares et al., 2006, 2012). In this study, a TID is simulated and its impacts on the convergence time of the PPP–RTK constrained by external ionospheric delay information are analyzed. For this purpose, the simulated TID can be expressed as follows:

$$S = A_{0} + A_{1} \cdot \cos (2 \cdot {\uppi } \cdot f_{0} \cdot t + \varphi_{0} )$$

(9)

where $S$ indicates the simulated results; $A_{0}$ expresses the direct component in TECU; $A_{1}$ is the amplitude of the TID in TECU; $f_{0}$ is the frequency of the TID in Hz; t is the number of epoch; $\varphi_{0}$ denotes the initial phase of the TID with an unit of rad. Suppose that $A_{0} = 0$, $A_{1} = 1$, and $f_{0} = 0.2\;{\text{mHz}}$, the simulated TIDs with different initial phases are shown in Fig. 10.

Fig. 10

The simulation results of TID with same direct component, amplitude, and frequency but different initial phases

To study the impacts of TID on PPP–RTK, the TID is firstly simulated by assuming that $A_{0} = 0$, $A_{1} = 1$, $f_{0} = 0.2\;{\text{mHz}}$ and $\varphi_{0} { = }135^{ \circ }$. Then the TID errors are added to the ionospheric delay corrections. The convergence time of the PPP–RTK constrained by ionospheric delay corrections with a best variance and influenced by TID errors for different variances can be found in Fig. 11. As can be seen from Fig. 11, the positioning results of PPP–RTK are poor due to the influence of TID. Moreover, the convergence time and reliability are affected as well, which is mainly related to the prior variance of the ionospheric delay. Particularly, if the same variance is adopted after adding TID errors with that during the durations without the TID errors, the positioning errors will be very large and vary greatly. The positioning accuracy is always larger than 10 cm and there are some situations where the ambiguities are wrongly fixed. In addition, much longer convergence time is needed during the TID period if the prior variance is larger. The positioning errors will be smaller than 10 cm after convergence, but still worse than that without adding TID errors.

Fig. 11

The convergence time of PPP–RTK constrained by ionospheric delay corrections with a no TID, and with a TID of an amplitude of 1 TECU and the variance of b 0.088 m, and c 0.38 m

Furthermore, TIDs are simulated with $A_{0} = 0$, $f_{0} = 0.2\;{\text{mHz}}$ and $\varphi_{0} { = }135^{ \circ }$, but with different amplitudes. The convergence time with the same variances for PPP–RTK during the TID periods are investigated and the results are plotted in Fig. 12. As we can see, the impacts of TIDs increase as the amplitudes of TIDs become larger. Note that the impact of TIDs is more significant at the beginning of convergence for each reinitialized period. When the ambiguity parameter is fixed, the positioning errors are nearly the same as those of the case without TIDs. Meanwhile, the convergence time will be longer for the TIDs with the larger amplitudes.

Fig. 12

The convergence time of the PPP–RTK constrained by external ionospheric information with a no TID, and with TID of the amplitude for b 0.5 TECU, c 1 TECU, and d 2 TECU and the variance of 0.48 m

Conclusions

In this study, the relationship between the accuracy of ionospheric delay corrections and the convergence time is investigated. The PPP–RTK network model, PPP–RTK user model, and the method to extract ionospheric delay are first described. Then, the positioning performances of the PPP–RTK constrained by ionospheric delay with different errors, and with different TID situations are presented. The results of this study are summarized as follows.

• (1) In terms of the TTFF for PPP-AR and PPP–RTK-GIM, the performance of the PPP–RTK constrained by global ionosphere maps can be improved compared with that of PPP-AR. The TTFF is about 8–10 min and 13–16 min for PPP-AR and PPP–RTK-GIM, respectively. The improvement of TTFF for the PPP–RTK-GIM is about 20% ~ 50%.

• (2) The errors of 0.5 TECU, 1 TECU, 3 TECU and 5 TECU are added to the network-derived ionospheric delays, respectively, which are used to constrain PPP–RTK with optimum variances. The results show that all the positioning errors smaller than 10 cm can be achieved within 7 min and the convergence time depends on the accuracy of ionospheric delay corrections. The convergence time is about 3, 5, and 6 min, respectively, when the added errors are about 1, 3, and 5 TECU with optimum variance, while it is about 1 min for the error less than 0.5 TECU. In addition, the convergence time of the PPP–RTK constrained by ionospheric delay with an error of 3 TECU for different variances is investigated. The results indicate that the variance of ionospheric delay affects the convergence time significantly. The convergence time for the ionospheric delay with errors have the similar variation. The convergence time will be very long when the variance is very small, while it will be the same as that of PPP-AR when the variance is very large.

• (3) Finally, the impacts of TID on the convergence time of PPP–RTK are checked by a simulation. The TID can affect the convergence time, positioning accuracy and reliability of PPP–RTK, which are related to the given variances. In addition, the larger variance leads to a longer time of convergence and the impact of TID will be enhanced with the increase of TID amplitude.