Introduction

Taking advantage of the dispersive nature of the ionospheric refractivity, global navigation satellite system (GNSS) can make remarkable contributions to monitoring spatial and temporal structure and variability of the ionosphere by measuring the total electron content (TEC) of the ionosphere (Brunini et al. 2005). On the other hand, it is well-known that ionospheric delay accounts for one of the largest and most variable sources of error in GNSS applications (Feltens and Schaer 1998; Krankowski et al. 2005). For the single-frequency (SF) GNSS receivers, one important way to mitigate the ionospheric error is to introduce external ionospheric TEC information or models (Wang et al. 2016a; Li et al. 2018a). Therefore, GNSS retrieval of ionospheric TEC is and remains to be an important issue in the precise positioning and timing of applications (Mannucci et al. 1998; Hernandez-Pajares et al. 1999).

Following the success of the GPS and the GLONASS system, the Chinese BeiDou navigations satellite system (BDS) began providing continuous positioning, navigation and timing services in the Asia-Pacific area at the end of 2012 (Montenbruck and Steigenberger 2013). By the end of 2016, the regional BDS (BDS-2) comprised a total of 6 geostationary orbit (GEO) satellites, 6 inclined geosynchronous orbit (IGSO) satellites and 4 medium earth orbit (MEO) satellites. On March 30, 2015, China began to build the 3rd generation BeiDou system (BDS-3), which will offer a fully global navigation service by 2020. At the end of August 2018, seventeen BDS-3 in-orbit validation satellites had been launched (http://mgex.igs.org/IGS_MGEX_Status_BDS.php). The multi-GNSS experiment (MGEX) project undertaken by the international GNSS service (IGS) provides an excellent opportunity to track all available GNSS signals and to conduct tracking data analysis (Montenbruck et al. 2014). The MGEX network and the signals transmitted by the current BDS satellites make it possible to apply multi-constellation and multi-frequency observations to precise ionospheric TEC estimation.

Over decades, considerable effort has been put into developing methods for GNSS-based ionosphere TEC retrieval (Komjathy et al. 2005). The commonly used ionospheric TEC determination methods can be classified into two main categories: (1) the dual-frequency (DF) approach and (2) the SF approach.

As one of the simplest and effective DF methods, the carrier-to-code leveling (CCL) method has been widely used for retrieving ionospheric observables based on geometry-free (GF) linear combinations of GNSS pseudorange and carrier phase measurements (Hernandez-Pajares et al. 2011; Zhang 2016). The ionospheric observables obtained from the CCL technique can be interpreted as a combination of the slant TEC (STEC) along the satellite-receiver line-of-sight, the satellite differential code biases (SDCBs), and the receiver DCBs (Li et al. 2017a). According to the study of Ciraolo et al. (2007), there is a systematic arc-dependent error called leveling error which is induced by code-delay noise and multi-path effects in the CCL-based ionospheric observable (Brunini and Azpilicueta 2009). With the goal of attaining more accurate TEC measurement retrieval, Zhang et al. (2012) proposed a DF method for extracting ionospheric observables from raw observations using precise point positioning (PPP). Considering that the PPP-derived ionospheric observables have an identical form to their counterparts obtained from leveling the GF GNSS carrier phase to code but are less affected by the extracting error, this PPP-based method has also been widely used in GNSS-based ionospheric studies (Yuan et al. 2015; Liu et al. 2016).

Compared with the more common DF approach, an attractive advantage of the SF method is that it can work with mass-market SF receivers, which are much more cost-effective than geodetic-grade DF receivers. Several studies have already been performed regarding SF GNSS retrieval of TEC (Yuan and Ou 2001; Wu et al. 2006; Schueler and Oladipo 2013), and most of the literature focuses on the code-minus-carrier (CMC) combination method. Schueler and Oladipo (2014) found that the CMC-based SF method is suitable for ionosphere monitoring for mid- and high-latitude sites; however, the precision level of CMC-based TEC estimate is bounded by the code-delay noise and multi-path effects. Zhang et al. (2017) proposed a SF PPP approach that enables the joint estimation of vertical TEC (VTEC) and SDCBs, and found that the SF approach performs well when applied to GPS L1 data collected by a single receiver. However, only GPS was considered in their study. Compared to other GNSS systems, BDS is the only navigation system employing GEO, IGSO, MEO satellites in a common constellation. According to the studies of Shu et al. (2016) and Wang et al. (2016b), both the precision and stability of BDS SDCB, estimated together with ionospheric model coefficients from BDS data, are lower compared with those of GPS and Galileo. The reasons of this phenomenon are related to the sparsely distributed tracking stations of BDS and the quality of BDS observables. Therefore, it is worthwhile to study how to obtain high-precision ionospheric TEC using the BDS data on just one frequency.

The first contribution of this work is thus to modify the methodology presented by Zhang et al. (2017) in such a way that the SDCBs are not estimated together with the ionospheric model coefficients any longer, but fixed them to the very precise values available from externally sources. In addition, when performing the ionospheric TEC accuracy validation in Zhang et al. (2017), there is no other external reference TEC data used except for the customary CCL-based TEC estimates, implying that only intra-technique comparison has been conducted. With this in mind, we consider more studies on the issue of performance analysis of SF-based ionospheric TEC by means of inter-technique comparison, that is, by comparing them with external reference TEC data and using different ionospheric modeling methods. This is the other contribution of the present work.

In this study, we propose a new TEC estimation procedure based on SF BDS data. The next section presents the proposed method to determine ionospheric TEC based on SF BDS data, the following section describes the experimental data and processing strategies. Then we show the preliminary experimental results. In the end, we present conclusions about the results. It should be noted that we hereafter refer to the ionospheric TEC determination approach using the CCL method based on the BDS B1 + B2 data as the DF approach.

Methodology

The SF-based algorithm for estimating the ionospheric TEC is described in detail in this section. Generally, it consists of two sequential steps: first, extract the ionospheric observables from the original SF BDS code and carrier phase observations; second, separate the ionospheric TEC from the ionospheric observables by establishing an ionospheric model and fixing SDCBs to very precise values provided externally.

Extraction of ionospheric observables from SF BDS measurements

The mathematical expression for BDS code and carrier phase, which is observable in units of length, can be expressed as follows (Leick et al. 2015; Li et al. 2017b):

$$\left\{ \begin{gathered} P_{{r,j}}^{s}(k)=\rho _{r}^{s}(k)+d{t_r}(k) - d{t^s}(k)+T_{r}^{s}(k)+{\mu _j} \cdot I_{{r,1}}^{s}(k)+{b_{r,j}} - b_{j}^{s}+{\varepsilon _p}(k) \hfill \\ \phi _{{r,j}}^{s}(k)=\rho _{r}^{s}(k)+d{t_r}(k) - d{t^s}(k)+T_{r}^{s}(k) - {\mu _j} \cdot I_{{r,1}}^{s}(k)+N_{{r,j}}^{s}+{\varepsilon _\phi }(k) \hfill \\ \end{gathered} \right.$$
(1)

where \(P_{{r,j}}^{s}(k)\) and \(\phi _{{r,j}}^{s}(k)\) denote the code and carrier phase observations from satellite \(s\) to receiver \(r\) on frequency band \(j\) at epoch \(k\); \(\rho _{r}^{s}(k)\) is the geometric vacuum distance from the satellite at the epoch of transmission to reception at the receiver; \(d{t_r}(k)\) and \(d{t^s}(k)\) refer to the receiver and satellite clock errors; \(T_{r}^{s}(k)\) is the slant tropospheric delay; \(I_{{r,1}}^{s}\) is the ionospheric delay on the first frequency; \({\mu _j}=\frac{{f_{1}^{2}}}{{f_{j}^{2}}}\) is the frequency-dependent factor that is used to convert ionospheric delay from the first frequency to other frequencies; \({b_{r,j}}\) and \(b_{j}^{s}\) are the receiver and satellite code hardware delays; \(N_{{r,j}}^{s}\) is the carrier phase ambiguity in length absorbing receiver and satellite phase instrumental delays and initial phase biases; \({\varepsilon _{\text{p}}}(k)\) and \({\varepsilon _\phi }(k)\) are the combinations of the observational noise and multipath in code and carrier phase observations.

Generally, the precise satellite clock products provided by IGS are generated using the ionospheric-free (IF) combination. Therefore, the published IGS satellite clock offsets \({\text{d}}t_{I}^{s}\) absorb the IF combination of satellite code hardware delays as follows (Li et al. 2012; Zhang et al. 2017):

$${\text{d}}t_{I}^{s}(k)={\text{d}}t_{{}}^{s}(k)+\frac{{{\mu _2}}}{{{\mu _2} - 1}}b_{1}^{s} - \frac{1}{{{\mu _2} - 1}}b_{2}^{s}$$
(2)

which should be carefully considered when using raw observations since satellite clock offsets are fixed to the precise products in PPP processing. Using the a priori known receiver coordinate and satellite coordinate and considering (2), the code and carrier phase observation equation on the first frequency can be expressed as:

$$\left\{ \begin{aligned} \Delta P_{{r,1}}^{s}(k) & =P_{{r,1}}^{s}(k) - \rho _{r}^{s}(k)+{\text{d}}t_{I}^{s}(k) - T_{{r,0}}^{s}(k) \\ & =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{t_r}(k)+{b_{r,1}}+I_{{r,1}}^{s}(k)+\frac{1}{{{\mu _2} - 1}}(b_{1}^{s} - b_{2}^{s})+{\varepsilon _{\text{p}}}(k) \\ \Delta \phi _{{r,1}}^{s}(k) & =\phi _{{r,1}}^{s}(k) - \rho _{r}^{s}(k)+{\text{d}}t_{I}^{s}(k) - T_{{r,0}}^{s}(k) \\ & =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{t_r}(k) - I_{{r,1}}^{s}(k)+N_{{r,1}}^{s}+\frac{{{\mu _2}}}{{{\mu _2} - 1}}b_{1}^{s} - \frac{1}{{{\mu _2} - 1}}b_{2}^{s}+{\varepsilon _\phi }(k) \\ \end{aligned} \right.$$
(3)

where \(\Delta P_{{r,1}}^{s}(k)\) and \(\Delta \phi _{{r,1}}^{s}(k)\) denote the observed-minus-calculated code and carrier phase observations; \(\rho _{r}^{s}(k)\) is the geometric distance computed from a priori known receiver coordinate and satellite coordinate; \(T_{{r,0}}^{s}(k)\) is the approximate slant tropospheric delay computed using an empirical model; \(ZT{D_r}(k)\) is the zenith tropospheric delay at the receiver, and \(m_{r}^{s}(k)\) is the corresponding mapping function.

Note that (3) represents a rank-deficient system and does not have a unique solution. This issue can be eliminated using a re-parameterization process (Odijk et al. 2016), which is given as follows:

$$\left\{ \begin{aligned} \Delta P_{{r,1}}^{s}(k) & =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+\left( {{\text{d}}{t_r}(k)+{b_{r,1}}} \right)+\left( {I_{{r,1}}^{s}(k)+\frac{1}{{{\mu _2} - 1}}(b_{1}^{s} - b_{2}^{s})} \right)+{\varepsilon _{\text{p}}}(k) \\ & =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\hat {t}}_r}(k)+\hat {I}_{{r,1}}^{s}(k)+{\varepsilon _p}(k) \\ \Delta \phi _{{r,1}}^{s}(k) & =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+\left( {{\text{d}}{t_r}(k)+{b_{r,1}}} \right) - \left( {I_{{r,1}}^{s}(k)+\frac{1}{{{\mu _2} - 1}}(b_{1}^{s} - b_{2}^{s})} \right) \\ & \quad +\left( {N_{{r,1}}^{s} - {b_{r,1}}+\frac{1}{{{\mu _2} - 1}}(b_{1}^{s} - b_{2}^{s})+\frac{{{\mu _2}}}{{{\mu _2} - 1}}b_{1}^{s} - \frac{1}{{{\mu _2} - 1}}b_{2}^{s}} \right)+{\varepsilon _\phi }(k) \\ & {\kern 1pt} =m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\hat {t}}_r}(k) - \hat {I}_{{r,1}}^{s}(k)+\hat {N}_{{r,1}}^{s}+{\varepsilon _\phi }(k) \\ \end{aligned} \right.$$
(4)

where identifiers with \(\wedge\) on the top denote the re-parameterized parameters of the model,

$$\left\{ \begin{gathered} \widehat{d}t_{r} (k) = dt_{r} (k) + b_{{r,1}} \hfill \\ \widehat{I}_{{r,1}}^{s} (k) = I_{{r,1}}^{s} (k) - \frac{1}{{\mu _{2} - 1}} \cdot DCB^{s} \hfill \\ \hat{N}_{{r,1}}^{s} = N_{{r,1}}^{s} - b_{{r,1}} + \frac{{\mu _{2} + 1}}{{\mu _{2} - 1}} \cdot b_{1}^{s} - \frac{2}{{\mu _{2} - 1}} \cdot b_{2}^{s} \hfill \\ \end{gathered} \right.$$
(5)

with \({\text{DC}}{{\text{B}}^{\text{s}}}=b_{2}^{s} - b_{1}^{s}\) being the satellite DCB.

We can find that (4) still has a rank deficiency of one size, the re-parameterized parameters \({\text{d}}{\hat {t}_r}(k)\), \(\hat {I}_{{r,1}}^{s}(k)\) and \(\hat {N}_{{r,1}}^{s}\) are still not individually estimable. To cope with this problem, we re-parameterize the clock as the changes of the original receiver clocks relative to the first epoch, and estimate such receiver clock offsets re-parameterized as white noise. Consequently, the resulting full-rank observation equations can be expressed as follows:

$$\left\{ \begin{gathered} \Delta P_{{r,1}}^{s}(k)=m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\bar {t}}_r}(k)+\bar {I}_{{r,1}}^{s}(k)+{\varepsilon _{\text{p}}}(k) \hfill \\ \Delta \phi _{{r,1}}^{s}(k)=m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\bar {t}}_r}(k) - \bar {I}_{{r,1}}^{s}(k)+\bar {N}_{{r,1}}^{s}+{\varepsilon _\phi }(k) \hfill \\ \end{gathered} \right.$$
(6)

where

$$\left\{ \begin{aligned} {\text{d}}{{\bar {t}}_r}(k) & ={\text{d}}{{\hat {t}}_r}(k) - {\text{d}}{{\hat {t}}_r}(1)={\text{d}}{t_r}(k) - {\text{d}}{t_r}(1) \\ \bar {I}_{{r,1}}^{s}(k) & =\hat {I}_{{r,1}}^{s}(k)+{\text{d}}{{\hat {t}}_r}(1)=I_{{r,1}}^{s}(k) - \frac{1}{{{\mu _2} - 1}} \cdot {\text{DC}}{{\text{B}}^s}+{\text{d}}{{\hat {t}}_r}(1) \\ \bar {N}_{{r,1}}^{s} & =\hat {N}_{{r,1}}^{s}+2{\text{d}}{{\hat {t}}_r}(1) \\ \end{aligned} \right.$$
(7)

with \({\text{d}}{\bar {t}_r}(k)\), \(\bar {I}_{{r,1}}^{s}(k)\) and \(\bar {N}_{{r,1}}^{s}\) being the estimable receiver clocks, ionospheric observables and ambiguities, respectively.

Applying the same deviation process to the second frequency, the full-rank observation equations can be expressed as:

$$\left\{ \begin{gathered} \Delta P_{{r,2}}^{s}(k)=m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\bar {t}}_r}(k)+\bar {I}_{{r,2}}^{s}(k)+{\varepsilon _{\text{p}}}(k) \hfill \\ \Delta \phi _{{r,2}}^{s}(k)=m_{r}^{s}(k) \cdot {\text{ZT}}{{\text{D}}_r}(k)+{\text{d}}{{\bar {t}}_r}(k) - \bar {I}_{{r,2}}^{s}(k)+\bar {N}_{{r,2}}^{s}+{\varepsilon _\phi }(k) \hfill \\ \end{gathered} \right.$$
(8)

where

$$\left\{ \begin{gathered} {\text{d}}{{\hat {t}}_r}(1)={\text{d}}{t_r}(1)+{b_{r,2}} \hfill \\ {\text{d}}{{\bar {t}}_r}(k)={\text{d}}{t_r}(k) - {\text{d}}{t_r}(1) \hfill \\ \bar {I}_{{r,2}}^{s}(k)=I_{{r,2}}^{s}(k) - \frac{{{\mu _2}}}{{{\mu _2} - 1}} \cdot {\text{DC}}{{\text{B}}^s}+{\text{d}}{{\hat {t}}_r}(1) \hfill \\ \bar {N}_{{r,2}}^{s}=N_{{r,2}}^{s} - {b_{r,2}}+\frac{{2{\mu _2}}}{{{\mu _2} - 1}} \cdot b_{1}^{s} - \frac{{{\mu _2}+1}}{{{\mu _2} - 1}} \cdot b_{2}^{s}+2{\text{d}}{{\hat {t}}_r}(1) \hfill \\ \Delta P_{{r,2}}^{s}(k)=P_{{r,2}}^{s}(k) - \rho _{r}^{s}(k)+{\text{d}}t_{I}^{s}(k) - T_{{r,0}}^{s}(k) \hfill \\ \Delta \phi _{{r,2}}^{s}(k)=\phi _{{r,2}}^{s}(k) - \rho _{r}^{s}(k)+{\text{d}}t_{I}^{s}(k) - T_{{r,0}}^{s}(k) \hfill \\ \end{gathered} \right.$$
(9)

with \(\bar {I}_{{r,2}}^{s}(k)\) and \(\bar {N}_{{r,2}}^{s}\) being the estimable ionospheric observables and ambiguities on the second frequency, respectively.

Apparently from (7) and (9), the estimable ionospheric observables \(\bar {I}_{{r,1}}^{s}\) and \(\bar {I}_{{r,2}}^{s}\) from SF BDS measurements are biased by the SDCB, the combination of the receiver clock at the first epoch and the receiver hardware delay on the corresponding frequency.

Ionospheric TEC modeling

Once the ionospheric observables are obtained, the absolute ionospheric TEC can be isolated from the ionospheric observables in two different ways: (1) the ionospheric model parameters are estimated in conjunction with SDCB parameters and biased receiver clock parameters; and (2) the ionospheric model parameters and biased receiver clock parameters are simultaneously estimated by constraining the SDCB to a priori known values, such as the multi-GNSS DCB products provided by the German aerospace center (DLR) or the Chinese academy of sciences (CAS) (available at: ftp://igs.ign.fr/pub/igs/products/mgex/dcb/) (Li et al. 2018b). In this study, we use the adjusted spherical harmonic (ASH) expansion based on a thin-layer approximation to model variation of ionospheric VTEC (Liu et al. 2018), which reads,

$$\left\{ \begin{gathered} {\text{VTEC}}(\varphi ',\lambda ')=\sum\limits_{{n=0}}^{4} {\sum\limits_{{m=0}}^{{{n_{}}}} {\tilde {P}_{{nm}}^{{}}} } (\sin \varphi ') \cdot \left( {{{\tilde {E}}_{nm}}\cos (m\lambda ')+{{\tilde {F}}_{nm}}\sin (m\lambda ')} \right) \hfill \\ \varphi '=\pi /2 - \frac{{\pi \left( {\pi /2 - \cos {\varphi _C}} \right)}}{{{\theta _{\hbox{max} }}}} \hfill \\ {\varphi _C}=\arccos [\sin \varphi \sin {\varphi _N}+cos\varphi cos{\varphi _N}\cos ({\lambda _N} - \lambda )] \hfill \\ \lambda '=\arcsin (\frac{{\sin (\lambda - {\lambda _N})}}{{\sin {\varphi _C}}}\cos \varphi ) \hfill \\ \end{gathered} \right.$$
(10)

where \({\text{VTEC}}\) is the ionospheric VTEC at the ionospheric intersecting pierce point (IPP); \(\varphi\) and \(\lambda\) are the geomagnetic latitude and longitude of the IPP; \({\varphi _N}\) and \({\lambda _N}\) are the geomagnetic latitude and longitude of the central point of the selected region; \(\varphi '\) and \(\lambda '\) are the projected latitude and longitude in the sphere cap coordinate system; \({\theta _{\hbox{max} }}\) is the half angle of the sphere cap; \({\tilde {P}_{nm}}(\sin \phi )={N_{nm}} \cdot {P_{nm}}(\sin \phi )\) is the normalized associated Legendre function of degree \(n\) and order \(m\); \({N_{nm}}=\sqrt {\frac{{(n-m)!(2n+1)(2 - {\delta _{0m}})}}{{(n+m)!}}}\) is the normalization function and \(\delta\) is the Kronecker delta; \({P_{nm}}(\sin \phi )\) is the classical, unnormalized Legendre function; \({\tilde {E}_{nm}}\) and \({\tilde {F}_{nm}}\) are the unknown ASH coefficients to be estimated.

The BDS SDCB products for the year of 2017 provided by the DLR and for the year of 2014 provided by the CAS are used to eliminate the impact of SDCB. The biased receiver clock at the first epoch is estimated in conjunction with ionospheric model parameters using the least squares fitting technique. Considering (7), the observation equation can be expressed as follows:

$$\left\{ \begin{gathered} \bar {I}_{{r,1}}^{s} - \frac{1}{{{\mu _2} - 1}}({\text{DC}}{{\text{B}}^s})=\frac{A}{{f_{1}^{2}}} \cdot {\text{MF}}(z) \cdot {\text{VTEC}}+{\text{d}}{{\hat {t}}_r}(1) \hfill \\ P=\frac{1}{{1+{{\cos }^2}E}} \hfill \\ \end{gathered} \right.$$
(11)

where \(A={\text{ }}40.28{\text{ }} \times {\text{ }}{10^{16}}\); \({\text{MF}}(z)={[1 - {\cos ^2}E \cdot R_{e}^{2}/{({R_e}+450)^2}]^{ - 1/2}}\) is the mapping function used to convert the slant to the vertical TEC, denoted by the elevation angle \(E\) of the satellite-linked to the corresponding receiver (Schaer 1999; Brunini and Azpilicueta 2010); \({R_e}\) represents the mean radius of the earth in kilometers; \(P\) is the weight of the ionospheric observable depends on the elevation angle of the satellite. To mitigate the impact of multipath and mapping function errors at low elevations, we only use the data with elevation angles above 10° in the process of ionospheric TEC modeling.

Experimental data and processing strategies

As shown in Fig. 1, a set of BDS data collected by 35 MGEX receivers at a 30 s sampling rate were selected to assess the performance of the SF approach. Two test periods consisting of the months of December 2014 and September 2017 were chosen to sample both high and low solar activity conditions. Table 1 shows the models and strategies used in this study when implementing SF PPP to retrieve the ionospheric observables. The data from the selected stations are processed on a day-by-day basis.

Fig. 1
figure 1

Distribution of the selected MGEX receivers

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Table 1 Models and strategies used in SF PPP
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Preliminary results and analysis

We evaluate the ionospheric estimation accuracy by analyzing the consistency between the ionospheric TEC estimates, determined using both SF and DF methods, and the external ionospheric products. Three representative assessment methods are applied to the ionospheric TEC accuracy validation: (1) a comparison between the VTEC estimates at IPPs of each BDS receiver with the IGS combined final global ionospheric maps (GIMs); (2) a comparison between the GNSS-based TEC estimates with TEC measurements provided by DF JASON altimeters; and (3) a comparison between the variations of STEC estimates with the observed STEC variations along a phase-continuous transmitter–receiver arc collected from BDS receivers. The customary DF CCL method is also used as a reference to validate the performance of the SF-based TEC. It should be noted that the SF data tested in this study were collected from geodetic-grade receivers, the performance of TEC estimated by SF data with low-cost receivers connected to patch-type antennas will be discussed in the future work.

Validation against GIM TEC

The IGS combined final GIM product, which has been acknowledged as one of the most accurate post-processed TEC products with an accuracy of 2–8 TEC units (TECU, 1016 electrons/m2) (see http://www.igs.org/products) (Hernández-Pajares et al. 2009), is used as a reference in the performance assessment of the SF ionospheric TEC determination method.

Taking stations JFNG and XMIS as an example, we plot in Figs. 2 and 3 the daily time series of absolute VTEC values determined using, respectively, the SF approach based on the BDS B1 data, the DF method based on the BDS B1 + B2 data, and the IGS GIM for IPPs. JFNG is located at mid-latitudes in the northern hemisphere and XMIS is located at low latitudes in the southern hemisphere. The period consists of day of year (DOY) 349–355 in 2014 and 262–268 in 2017. Compared to Fig. 3, Fig. 2 shows a significant decrease in the TEC values in response to the decline of the solar activity level. It is apparent that the time series of VTEC estimates determined by the SF approach and DF method have a similar trend with the GIM TEC, they can all represent the diurnal variation of the ionosphere during both high and low solar activity periods.

Fig. 2
figure 2

Daily time series of VTEC values estimated from BDS B1 + B2 data (in blue), from BDS B1 data (in red), and IGS GIM data (in green) at the sites JFNG and XMIS for the period DOY 349–355 in 2014

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Fig. 3
figure 3

Daily time series of VTEC values estimated from BDS B1 + B2 data (in blue), from BDS B1 data (in red), and IGS GIM data (in green) at the sites JFNG and XMIS for the period DOY 262–268 in 2017

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To further quantify the consistency among the time series of VTEC estimates listed in each panel of Figs. 2 and 3, using the GIM TEC values as a reference, the mean bias and the root mean square (RMS) error of the VTEC estimates determined by the SF and DF methods are presented in Figs. 4 and 5. For the high solar activity year, the mean RMS value of the differences between the VTEC derived from the SF/DF model and GIM are approximately 8.01/10.25 and 7.05/7.22 TECU for the stations JFNG and XMIS, respectively. For the low solar activity year, the mean RMS value of the differences of the VTEC derived from the SF/DF model with respect to GIM-based TEC are approximately 2.66/2.68 and 4.11/3.44 TECU for the stations JFNG and XMIS, respectively.

Fig. 4
figure 4

Daily mean and RMS of the differences between the VTEC estimates relative to GIM at the sites JFNG and XMIS for the period DOY 349–355 in 2014 (unit: TECU)

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Fig. 5
figure 5

Daily mean and RMS of the differences between the VTEC estimates relative to GIM at the sites JFNG and XMIS for the period DOY 262–268 in 2017 (unit: TECU)

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Concerning the statistical results for all selected stations summarized in Table 2, the VTEC time series estimated by the SF method agrees better with the IGS GIM products than those estimated by the DF method for the high solar activity period. For the low solar activity period, the TEC estimated by the DF method agrees better with the IGS GIM product than those estimated by the SF method.

Table 2 Monthly mean and RMS of the differences between the VTEC estimates relative to GIM TEC for all selected receivers (unit: TECU)
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Validation against JASON TEC

The DF ocean altimeter on board the JASON satellite can provide a measurement of the VTEC over its footprint (Orus et al. 2007), with a systematic bias of approximately 2–5 TECU above the real ionospheric TEC values (Jee et al. 2010; Wang et al. 2017). As a reliable TEC data source, the JASON TEC data, for high (December 2014) and low (September 2017) sunspot periods, were also used as a reference to assess the performance of the SF ionospheric TEC estimation method.

For comparison, each panel in Fig. 6 shows the JASON TEC and BDS TEC calculated using the SF and DF methods at the locations and times of the JASON satellite orbits on a randomly selected day from the large number of consistent results. It can be seen that the JASON TEC and BDS-based TEC estimates determined using both the SF and DF methods show a similar variation tendency, again showing that both the SF and DF approaches are able to reproduce the spatial and temporal variations of the local ionosphere. Using the JASON TEC as a reference, we calculated the mean bias and RMS for the SF and DF VTEC time series for the test period and present the results in Fig. 7. Note that since the JASON data for the period DOY 258–273 in 2017 and BDS SDCB data for DOY 341 and 347 in 2014 cannot be downloaded from the main IGS server, statistics for this period are not included in Fig. 7. As summarized in this figure, the mean RMS values are 4.60/4.67 TECU for December 2014 and 2.63/2.34 TECU for September 2017 for the SF/DF methods, respectively, showing that TEC estimates generated by the SF method and the DF method have a potential ability to match the JASON TEC.

Fig. 6
figure 6

VTEC estimates at each IPP from the BDS B1 + B2 data (in green), from the BDS B1 data (in blue), and from the JASON data (in red) on a day in December 2014 and September 2017. Note that the unit of “number of ground tracks” on the horizontal axis is fifty

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Fig. 7
figure 7

Daily mean and RMS of the differences between the VTEC estimates and the JASON reference data for December 2014 and September 2017 (unit: TECU)

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Validation against differential STEC

The index of differential STEC (dSTEC), which is defined as the difference between the STEC measured at any point along a continuous phase arc and the STEC measured when the satellite is at its highest elevation seen from a given receiver (Hernández-Pajares et al. 2017), has also been used for comparing the performance of the SF method. According to the study of Feltens et al. (2011), the measured dSTEC “truth” can be directly obtained from the GF combination of the raw DF phases when no cycle slips occur over a phase continuous measurement arc, with an accuracy of less than 0.1 TECU.

Figure 8 shows the mean dSTEC errors and the RMS errors of the dSTEC for the SF and DF methods at each randomly selected receiver under different solar activity levels. It is apparent that the mean RMS values are 5.62/5.43 TECU for December 2014 and 3.35/3.02 TECU for September 2017 for the SF/DF methods, respectively, showing that there is no significant difference in the ability to capture both spatial and temporal gradients in the ionosphere between the SF and DF methods.

Fig. 8
figure 8

Mean value of monthly dSTEC-BDS error and the RMS error of the dSTEC-BDS for the SF PPP and DF CCL

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Conclusions

To eliminate the requirement of the customary dual-frequency (DF) approach for code and phase data on two frequencies, a new ionospheric total electron content (TEC) estimation procedure based on single-frequency (SF) BDS data is proposed in this study. The implementation of the proposed method consists of two procedures: first, retrieve ionospheric observables employing SF precise point positioning (PPP) instead of carrier-to-code leveling (CCL) used in the customary DF method; second, isolate the ionospheric TEC from the ionospheric observables by modeling ionospheric TEC using the adjusted spherical harmonic (ASH) expansion and introducing the a priori SDCB estimates provided by the German aerospace center (DLR) and Chinese academy of sciences (CAS).

To assess the performance of the proposed approach, the ionospheric TEC are estimated using a set of BDS measured data collected from receivers of MGEX network for a high (December 2014) and low (September 2017) sunspot period. The accuracy of ionospheric TEC estimated by both the SF and DF methods are validated by comparing them with external TEC data sources. Based on our analysis, we can conclude that the ionospheric VTEC derived using the SF method from the BDS B1 data and using the DF method from the BDS B1 + B2 data are both capable of reproducing the spatial and temporal variations of the ionosphere. Taking the TEC estimates obtained from the IGS final combined global ionospheric map (GIM) product as a reference, the mean RMS of the TEC estimates obtained from the SF approach and DF method are 5.77/6.80 TECU for December 2014 and 3.65/2.48 TECU for September 2017, respectively. The TEC estimates generated by the SF method and the DF method have a comparable ability to match the JASON TEC, with average RMS errors at the level of 4.60/4.67 TECU for December 2014 and 2.63/2.34 TECU for September 2017, respectively. Compared to the BDS-measured differential STEC (dSTEC), the mean RMS errors determined using the SF and DF approaches are 5.62/5.43 TECU for December 2014 and 3.35/3.02 TECU for September 2017, respectively. It can be concluded that the TEC determination performances of the SF PPP and DF CCL approaches are roughly equal.

Overall, the SF approach is a promising way to monitor ionospheric TEC based on SF BDS data. Compared to the customary DF approach, the SF method is more attractive and competitive in cost and has greater potential to be popularized to the mass-market SF receivers that can support BDS code and phase data on only one frequency. It may become possible to use mass-market SF receivers to increase the spatial and temporal resolution of ionospheric information in land areas in the future.