We recently proposed a method to establish an optimal ionospheric shell height model based on the international GNSS service (IGS) station data and the differential code bias (DCB) provided by the Center for Orbit Determination in Europe (CODE) during the time from 2003 to 2013. This method is very promising for DCB and accurate total electron content (TEC) estimation by comparing to the traditional fixed shell height method. However, this method is basically feasible only for IGS stations. In this study, we investigate how to apply the optimal ionospheric shell height derived from IGS station to non-IGS stations or isolated GNSS receivers. The intuitive and practical method to estimate TEC of non-IGS stations is based on optimal ionospheric shell height derived from nearby IGS stations. To validate this method, we selected two dense networks of IGS stations located in regions in the US and Europe. Two optimal ionospheric shell height models are established by two reference stations, namely GOLD and PTBB, which are located at the approximate center of two selected regions. The predicted daily optimal ionospheric shell heights by the two models are applied to other IGS stations around these two reference stations. Daily DCBs are calculated according to these two optimal shell heights and compared to respective DCBs released by CODE. The validation results of this method are as follows. (1) Optimal ionospheric shell height calculated by IGS stations can be applied to its nearby non-IGS stations or isolated GNSS receivers for accurate TEC estimation. (2) As the distance away from the reference IGS station becomes larger, the DCB estimation error becomes larger. The relation between the DCB estimation error and the distance is generally linear.

Dual-frequency GPS signal propagation is affected effectively by ionospheric dispersive characteristics. Taking advantage of this property, ionospheric total electron content (TEC) along the path of signal can be estimated by differencing the pseudorange or carrier phase observations from dual-frequency GPS signals. Carrier phase leveling or smoothing of code measurement is widely adopted to improve the precision of absolute TEC observations (Mannucci et al., 1998; Horvath and Crozier, 2007). In general, it is considered that the derived TEC in carrier phase leveling or smoothing technique consists of slant TEC (STEC), the combination differential code bias (DCB) of satellite and receiver, multipath effects and noise. The DCB is usually considered as the main error source and could be as large as several TEC units (TECu) (Lanyi and Roth, 1988; Warnant, 1997).

For TEC and DCB estimations, mapping functions with a single-layer model (SLM)
assumption have been intensively studied for many years. Sovers and
Fanselow (1987) firstly simplified the ionosphere to a spherical shell. They
set the bottom and the top side of the ionospheric shell as

The ionospheric shell height is considered to be the most important parameter for a mapping function, and the shell height is typically set to a fixed value between 350 and 450 km (Lanyi and Roth, 1988; Mannucci et al., 1998). Birch et al. (2002) proposed an inverse method to estimate the shell height by using simultaneous VTEC and STEC observations and suggested the shell height is preferably a value between 600 and 1200 km. Nava et al. (2007) utilized multiple stations to obtain a shell height estimation method by minimizing the mapping function errors; this method is referred as the “coinciding pierce point” technique. Their results indicated that the suitable shell heights for the midlatitude is 400 and 500 km during the geomagnetic undisturbed conditions and disturbed conditions, respectively. In the case of the low latitude, the shell height at about 400 km is suitable for both quiet and disturbed geomagnetic conditions. Jiang et al. (2018) applied this technique to estimate the optimal shell height for different latitude bands. In their case, the optimal layer height is about 350 km for the entire globe. Brunini et al. (2011) studied the influence of the shell height by using an empirical model of the ionosphere and pointed out that a unique shell height for whole region does not exist. Li et al. (2018) applied a new determination method of the shell height based on the combined international GNSS service (IGS) Global Ionospheric Maps and the two methods mentioned above to the Chinese region and indicated that the optimal shell height in China ranges from 450 to 550 km. Wang et al. (2016) studied the shell height for a grid-based algorithm by analyzing goodness of fit for STEC. Lu et al. (2017) applied this method to different VTEC models and investigated the optimal shell heights at solar maximum and at solar minimum.

In the recent study by Zhao and Zhou (2018), a method to establish an optimal ionospheric shell height model for single-station VTEC estimation has been proposed. This method calculates the optimal ionospheric shell height in order to minimize the difference between the estimated DCB and the DCB released by the Center for Orbit Determination in Europe (CODE). Five optimal ionospheric shell height models were established by the proposed method based on the data of five IGS stations at different latitudes and the corresponding DCBs provided by CODE during 2003 to 2013. For the five selected IGS stations, the results have shown that the optimal ionospheric shell height models improve the accuracies of DCB and TEC estimation compared to a fixed ionospheric shell height of 400 km in a statistical sense. We also found that the optimal ionospheric shell height shows 11- and 1-year periods and is correlated to the solar activity, which indicated the connection of the optimal shell height with ionospheric physics.

While the proposed optimal ionospheric shell height model is promising for
DCB and TEC estimation, this method also can be implemented to isolated GNSS
receivers not belonging to IGS stations, if we can get the long-term
observations and reference values of DCB from the isolated GNSS receivers.
By considering the spatial correlation of ionospheric electron density, it
is intuitive and practical to adopt the optimal ionospheric shell height of
a nearby IGS station to the non-IGS stations. So whether an optimal
ionospheric shell height model can improve the TEC

The purpose of this study is to investigate the feasibility of applying the optimal ionospheric shell height model derived from IGS station to nearby non-IGS GNSS receivers for accurate TEC and DCB estimation. By selecting two different regions in the US (Region I) and Europe (Region II) with dense IGS stations, we calculate the daily DCBs of 2014 by using the optimal ionospheric shell heights derived from data from 2003 to 2013 of two central stations in two regions. We also try to find the DCB estimation error and its relation to the distance away from the central reference station.

In Zhao and Zhou (2018), we proposed a concept of optimal ionospheric shell height for accurate TEC and DCB estimation. Based on daily data of a single site, this approach searches for a daily optimal ionospheric shell height, which minimizes the difference between the DCBs calculated by the VTEC model for a single site and reference values of DCB. For a single site, its long-term daily optimal ionospheric shell heights can be estimated and then modeled. In our case, the polynomial model (Wild, 1994; Komjathy, 1997) is applied to estimate satellite and receiver DCBs, and the DCBs provided by CODE are used as the reference.

In the polynomial model, the VTEC is considered as a Taylor series expansion
in latitude and solar hour angle, which is expressed as follows:

Based on the thin shell approximation, the observation equation can be
written as follows:

To estimate DCBs, the method above requires a definite thin shell height
value. Conversely, if we get the daily solutions of DCBs, the optimal
ionospheric shell height can be estimated. The optimal ionospheric shell
height is assumed to be between 100 and 1000 km and is defined as the shell
height with the minimum difference between DCB

After the method above is applied to 11-year data, the estimated optimal
ionospheric shell heights can be modeled by a Fourier series, which is
expressed as follows:

This model can be applied to neighboring stations' DCB estimations. Instead of fixed shell height, this model provides a predicted optimal ionospheric shell height. Note that, while in the establishment and application of the model, the VTEC model, mapping function and elevation cut-off angle are constant, all of them affect the optimal ionospheric shell height.

The previous section introduced a method to establish a daily optimal ionospheric shell height model based on a single site with reference values of DCBs. To analyze the improvement of DCB estimation by this model for the reference station and other neighboring stations, we present two experiments to evaluate and validate this method by using IGS stations located in the US and Europe. To ensure the accuracy and consistency of DCB, we only select IGS stations with pseudorange measurements of P1 code, and whose receiver DCBs have been published by CODE.

Figure 1 shows the location and distribution of the selected IGS stations in two regions. Table 1 shows the information of the geographical location, distance to reference station in each region and receiver types of all stations. Based on the RINEX data of the GOLD station in Region I and the PTBB station in Region II during the period of 2003–2013, two separate optimal ionospheric shell height models for each region are established by the aforementioned method. Then the model is applied to estimate DCB in 2014 for all the other stations in each region. Note that the reference stations GOLD and PTBB are marked with black triangles in the figure. The other neighboring stations are located in different orientations of GOLD and PTBB with different distances, which range from 136 to 1159 km for region I and range from 190 to 1712 km for region II. In the table, the receiver type is corresponding to 2003–2014 for GOLD and PTBB, and 2014 for the other stations. In region I, the receiver type of GOLD was changed once in September 2011. The five selected stations used four receiver types in 2014; TABV and PIE1 had the same receiver type. In region II, there are 9 receiver types for the 16 stations. The receiver type of PTBB has changed twice in 2006.

Geographical location of the selected IGS stations in the US region (Region I) and Europe region (Region II). The black triangle in each plot is the reference station.

Information for the stations.

Statistical results of mean and RMS of average

Figure 2 shows the estimated daily optimal ionospheric shell height of
GOLD and PTBB during the period from 2003 to 2013. The left panel shows the
variation in the daily optimal ionospheric shell height and the fitting
result by Eq. (

Variation in the daily optimal ionospheric shell height (black) and the fitting result (red).

Figure 3 shows the amplitude spectra of the daily optimal ionospheric
shell height of the two reference stations estimated by the Lomb–Scargle
analysis (Lomb, 1976; Scargle, 1982). As can be found in Fig. 3, the peaks
correspond to 11-year, 1-year, 6-month and 4-month cycles. The amplitudes of
11- and 1-year cycles are more evident than other periods in both
stations. As mentioned earlier, 0.01 per day is about the maximum frequency
of Eq. (

Lomb–Scargle spectra of the daily optimal ionospheric shell height.

We establish two optimal ionospheric shell height models for each region
from the 40th-order Fourier series based on the 11-year data of GOLD and
PTBB. To investigate the availability zone of the optimal ionospheric shell
height model, we apply the models to the stations of each region as shown in
Fig. 1 and Table 1. Based on the predicted daily optimal ionospheric shell
heights in 2014 calculated by the model at GOLD or PTBB, each station is
applied to estimate DCB separately in 2014 using Eqs. (

The results of this comparison are shown in Fig. 4. The panels for the
stations are arranged by their distances to reference stations, and this is also
applied to Table 2; from the top panels to the bottom panels, the distance
of the corresponding station to the reference station gradually increases.
The left and right panels show the daily differences and the histograms of
the statistical results in 2014, respectively. For all of the stations, the
daily average differences of DCBs calculated using the optimal ionospheric
shell height model are reduced compared to those using the fixed ionospheric
shell height. For GOLD and TABV, the improvement is substantial; the daily
average

Comparisons of the average

Comparisons of the average

Table 2 shows the quantitative statistical results of average

Figures 6 and 7 show the relation between the statistical results
of average

Relation of the accuracy for DCB estimation with the distance to GOLD. The red lines are the linear fitting results.

Relation of the accuracy for DCB estimation with the distance to PTBB. The red lines are the linear fitting results.

In this study, we implement and validate a method to transfer the optimal
ionospheric shell height derived for IGS stations to non-IGS stations or
isolated GNSS receivers. We establish two optimal ionospheric shell height
models by the 40th-order Fourier series based on the data of IGS stations
GOLD and PTBB in two separate regions. These two models are applied to the
stations in each region, where the distance to GOLD ranges from 136 to 1159 km and the distance to PTBB ranges from 190 to 1712 km. The main findings
are summarized as follows:

The optimal ionospheric shell height model improves the accuracy of DCB estimation compared to the fixed shell height for all of the stations in a statistical sense. These results indicate the feasibility of applying the optimal ionospheric shell height derived from IGS stations to other neighboring stations. The IGS stations can calculate and predict the daily optimal ionospheric shell height and then release this value to the nearby non-IGS stations or isolated GNSS receivers.

For other stations in each region, the error of DCB by the optimal
ionospheric shell height increases linearly with the distance to the
reference station GOLD or PTBB. For the mean and the RMS of the daily
average

Due to a requirement of this experiment, we only analyze two regions in midlatitude because of the insufficiency of long-term P1 data. We also ignore the orientation of isolated GPS receivers to the reference station.

This study is based on data services provided by the IGS
(International GNSS Service) and CODE (the Center for Orbit Determination in
Europe). The data from the IGS stations can be downloaded from

JZ and CZ designed the experiments and JZ carried them out. JZ developed the model code and performed the simulations. JZ and CZ prepared the paper.

The authors declare that they have no conflict of interest.

This study is based on data services provided by the IGS (International GNSS Service) and CODE (the Center for Orbit Determination in Europe). This work is supported by the National Natural Science Foundation of China (NSFC grants 41574146 and 41774162). Thanks also to Dalia Buresova and Max van de Kamp and two anonymous referees.

This paper was edited by Dalia Buresova and reviewed by Max van de Kamp and two anonymous referees.