Interactive comment on “ Validation and application of optimal ionospheric shell height model for single-site TEC estimation ” by Jiaqi Zhao and Chen Zhou

We thank referee #1 for careful reading and valuable comments on the manuscript. Accordingly, we have modified the text. All the modifications and changes are shown in the revised manuscript "manuscript-version2" in red font. Our responses to the referee’s comments are listed below and the file "Response to RC1". The two files are in supplement. Response to Referee #1 COMMENTS TO THE AUTHOR: Referee #1: Interactive comment on “Validation and

recall that under some conditions they even obtained unphysical negative shell height).

Reply:
We thank for your important remark.We agree with that the shell height should correspond to the height of the ionosphere barycentre by an ionospheric point of view.While for accurate TEC and DCB estimation, because of VTEC model error and mapping function error and so on, optimal shell height is different with ionosphere barycentre.Actually for different VTEC model, the optimal shell height is also different.Lu et al. (2017) did the similar work by using another ionospheric shell height estimation method.In our manuscript, the optimal shell height is also affected by the accuracy of reference values of DCB.The optimal shell height is more like a modification of the mapping function for the selected VTEC model, and have relationship with solar activity.We believe that optimal shell height and ionosphere barycentre could be closer with the improvement of the VTEC model and mapping function.

Reference
Lu W, Ma G, Wang X, Wan Q, Li J (2017) Evaluation of ionospheric height assumption for single station GPS-TEC derivation.Advances in Space Research 60(2):286-294 It is not clear why the technique proposed for ionospheric shell height estimation cannot be implemented to isolated GNSS receivers not belonging to IGS stations (line 107).

Reply:
We thank for your carefully reading and helpful comment.The optimal ionospheric shell height is calculated from IGS DCB values.DCB is normally not released by non-IGS stations, which means ionospheric shell height cannot be calculated by using this method.However, if we could get the long-term observations and reference values of DCB from non-IGS station, this technique could also work.We have deleted this mistake.Please see page 6 line 107-109 in the revised manuscript.
A Fourier model of the shell height is constructed for GOLD and PTBB for a complete solar cycle between 2003 and 2013.This model does not include any input regarding solar activity.It is well known that the current solar cycle is considerably less strong than the previous one.The ionosphere development has also been substantially lower.Thus it is also expected the optimum shell height should follow a different pattern.A discussion on this point is essential for the correct understanding of this work.

Reply:
We thank for your constructive suggestion.We total agree with the reviewer that solar activity is the dominant factor for ionospheric variability.However, other factors such as atmospheric variability and human activity can also cause ionospheric disturbance.In this study, we do not consider all physical factors explicitly.However, we try to include all the factors by utilizing empirical modeling with data.The Fourier model is a preliminary result.Evaluations on different models will be investigated and compared in the following work.line 52: the work of Sardón et al. (1994) was not oriented towards real-time ionospheric VTEC, but to develop a technique of prediction of DCBs under adverse conditions (antispoofing, ionospheric disturbances).

Reply:
We appreciate the reviewer for this helpful comment.We have accordingly made the revision.Please see Lines 52-54 in the revised manuscript.line 77: specify that the Nava et al. (2007) technique uses multiple stations to obtain a "coinciding pierce point".

Reply:
We thank for the reviewer for providing this suggestion.We have accordingly made the revision.Please see Lines 77 in the revised manuscript.line 125-126: the polynomial model is referred to Lanyi and Roth (1988).However the expression used in this article does not correspond to the one used by those authors.

Reply:
We appreciate the reviewer for pointing out this mistake.We have replaced this reference with (Wild, 1994;Komjathy, 1997).Please see Lines 128-129 in the revised manuscript.
Line 132: does the regional center of the model correspond to the location of the receiver?

Reply:
Yes.The regional center of the model is the location of the receiver.line 134: it is not clear why 9 VTEC models are applied per day.It should be specified that a VTEC model is generated over 3 hours of time.

Reply:
We have accordingly made the revision.Please see Lines 137-139 in the revised manuscript.
line 166: I suggest to indicate explicitly that the 40/L corresponds to a period of 100 days.

Reply:
We thank for this helpful suggestion.We have accordingly write it explicitly.Please see Lines 170-171 in the revised manuscript.line 178: why only stations providing P1 code measurements of pseudorange were used?Will the result be significantly different if any station would have been selected regardless of the measured code?

Reply:
We thank for this comment.CODE also provides the DCB of P1-C1, but only for satellites.And we are not sure whether the receiver DCB is C1-P2 bias in CODE DCB file, for the station providing C1 code but no P1 code.So we use the DCB of P1-P2 for reference.Accordingly, we just select stations with P1 code.
On figure 2 an anomaly appears at the end of 2010, where a gap (or values outside the vertical axis limit ?) appears on the estimated shell heights.In this article there is not a discussion about this strange behavior, but in the previous article (Zhao and Zhou 2018), figure 3 shows that all stations have simultaneously anomalous DCBs during a few months.I suggest to make a deeper investigation on why this happen, but clearly these DCBs values are not reliable.Some hypotheses: an error in CODE processing chains; an error in the receivers firmware that affect the time estimate; some error at GPS system level.The impact to the results of this article concern the Fourier model to represent the whole solar cycle behavior of the shell height, but should not affect the station comparisons of 2014.

Reply:
We appreciate the reviewer for raising this important comment.We totally agree with the reviewer that the anomaly at the end of 2010 could be an error in CODE processing procedures.We have followed the reviewer's suggestion that discussion on the data gap have been added in the revised manuscript.Please see Line 217-219.variability, what is the benefit of using a Fourier model up to order 40?A much lower order could provide comparable results.On the other hand, the fast variability is not achievable with this model.-both stations show the limits of the proposed approach: the distributions on the right panels present each a missing tail, suggesting that the imposed shell height limits are not adequate.For GOLD station we could expect shell heights higher than 1000 km and for PTBB shell heights lower than 100 km, which are unphysical, because outside the ionosphere Reply: We thank for this constructive suggestion.We set the elevation cut-off angle as 15° and 30°，Figure 2 shows their results.When elevation mask angle is set as 30°, the spreading is larger, compare to 15°.But their optimal ionospheric shell heights have similar fluctuation frequencies.When other stations around apply the model, their elevation mask angles must to be same with the reference station.it is a good prediction or not.I think a more explicit discussion of the whole validation approach is needed Reply: We appreciate the reviewer for raising this comment.We have followed the suggestion.Please see Line 172-176.The difference between DCB released by CODE and DCB calculated using the predicted optimal ionospheric shell heights are plotted in red dots.The difference between DCB released by CODE and DCB calculated using the fixed ionospheric shell height are plotted in black dots.In both Figure 4 and Figure 5, the general distribution and mean value of red dots is smaller than that of black dots, which means the DCB estimation is improved by using the predicted optimal ionospheric shell heights.

Technical corrections
line 108: I think "it is intuitional and practical" should read "it is intuitive and practical" Reply:

General comments
This manuscript presents a method to determine the optimal ionospheric shell height (= effective height of an assumed thin ionospheric shell) based on TEC measurements and DCB code biases.The method is a further development and validation of a method developed by the same authors.
In my opinion, this is a useful contribution to the scientific community, particularly since in many applications it has become common practice to "just assume" a fixed ionospheric height (of often about 350 km) without thinking.As criticism, one might argue that physical inputs, such as ionosondes, radar measurements, or solar activity data, have not been used in this method.However, I think that the approach of this paper can be considered just one approach, and is useful to be compared to other approaches which may use other information.Besides, the optimal method to determine the effective ionospheric height may depend on the application, which makes it useful to try different methods.
The paper is mostly clear and well written.The authors have clearly well taken into account the comments made by the two earlier reviewers.I have only a few more questions, see here below.Furthermore, I have made editorial comments to improve the English in the annotated manuscript, onward from page 3 of this pdf-file. Replay: We thank the referee for the encouraging evaluation on our study.

Specific comments
Equation ( 1): Shouldn't the equation also include a term "TV(φ0,S0)" ?Otherwise, the equation seems to say that VTEC=0 at the regional center.Or is TV(φ,S) supposed to mean: the difference in VTEC between a station and the regional center? Reply: We thank the referee for this comment.
The term "TV(φ0,S0)" is included in equation ( 1).According to equation (1), This VTEC model fits the VTEC over a period of time, time is an input parameter.In Equation ( 5): • You are applying this optimization over 11 years, and therefore only for the two reference stations, right?Reply: Yes, Equation ( 5) are applied to estimate the optimal ionospheic shell heights over 11 years, based on the data of the two reference stations.Then two models are established by fitting the estimated optimal ionospheic shell heights.And other stations are used to verify whether the two models can be applied to nearby stations.Equation ( 5 Reply: We thank the referee for this comment. The matrix φ contains (φ-φ 0 ) i (S-S 0 ) j f(z).
Please see Line 167-168 in the revised manuscript.
• And presumably, θ contains only 1s and 0s, right? Reply: Yes, for example, the i-th STEC corresponds to the j-th satellite, then θ(i,j)=1, and Please see Line 168-169 in the revised manuscript.
• What does the matrix E contain?It cannot be the model coefficients from equation (1), because you are applying this only for the reference stations, so φ -φ0 and S -S0 are 0.So is E the vector of TV(φ,S) values as in equation ( 2)? (= VTEC, in the reference stations) Reply: We thank the referee for this valuable comment.It's our mistake.We do not make it clear.In the two cases, all of the DCB estimations are based on single site.
Only the two reference stations are applied to establish two optimal shell height models using equation ( 1)-( 6).Each station is applied to estimate DCB separately in 2014 using equation ( 1)-( 4), same as the general single site DCB estimation (one station, one VTEC model), except the shell heights are provided by the optimal shell height models.Their VTEC models (i.e.TV(φ,S)) are independent of each other, and the VTEC model coefficients (Eij) are unknown.
So the matrix E contains Eij.
"φ0 and S0 denote φ and S at regional center" (line 137-138 in the manuscripts), "the region" here means the cover region of IPP for single station in one day, not the regions in figure 1.So "regional center" means the center of the cover region of IPP.
And for different station, φ0 and S0 are different.φ and S are the location of IPP, they change all the time.
For example, the steps of estimating the DCB of TABV in 2014-1-1: Frist, the optimal shell height model based on GOLD (the reference station near TABV) provides the predicted shell height in 2014-1-1, and h in equation ( 3) is set as the predicted shell height.
Then the data of TABV in 2014-1-1 and h are substituted into equation ( 1)-( 4).The data of TABV contains raw STEC (T OS PRN ), IPP location (φ,S), the center of IPP (φ 0 ,S 0 , you can simplify set them as the location of TABV), and elevation angel (El).So only the VTEC model coefficients (Eij) and DCB are unknown.
Finally, the VTEC model coefficients (E ij ) and DCB are estimated by least square method.Equation ( 1)-( 4) can be written as: We have modified, please see Line 137-138 in the revised manuscript.
• If so, where do you get these VTEC values from?From equation (2) using your optimal shell height model? Reply: where, E and DCB is unknown, and can be estimated by least square method.
• Do you mean with the "polynomial model": equation (1)?If so, how do you find the coefficients? Reply: Yes, "polynomial model" is expressed as equation ( 1), the coefficients are unknown, and are estimated with DCB.See the reply above.
• Or do you mean that you assume that the optimal ionospheric height at the stations is equal to that at the reference station?If so, what do you mean with the "polynomial model"?And when do you use equation (1)? Reply: Yes, the two optimal ionospheric shell height models provide shell height to the stations nearby."polynomial model" means the polynomial VTEC model which is expressed as equation (1).Equation ( 1) is used in DCB estimation and the optimal ionospheric shell height estimation.
Line 348-350: "For different region, the error at 0 km (i.e. the error for the reference station) is different, which should be also considered."This is a little vague; can you explain more what you mean to say with this?For instance, something like: "The quality of the DCB estimations depends also on the quality of the optimal shell height model at the reference stations themselves, which may also not be equally good in all areas." Reply: We thank the referee for this helpful comment.
Yes, the quality of the DCB estimations also depends on the quality of the optimal shell height model at the reference stations themselves.Figure 6 and Figure 7 show the error of DCB by the optimal ionospheric shell height increases linearly with the distance to the reference station GOLD or PTBB.So the quality of the DCB estimations depends on the slope, the distance and the error at 0 km.
We have modified this part, please see line 351-353 in the manuscripts.

Editorial comments
Please see the comments in red, annotated in the copy of the manuscript, below. Reply:

Introduction
Dual-frequency GPS signal propagation is affected effectively by ionospheric dispersive characteristics.Taking advantage of this property, ionospheric 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/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/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 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 h-35 and h+75 km, where h is taken to be 350 km above the surface of the earth and allowed to be adjusted.In this model, the electron density was evenly distributed in the vertical direction.Based on this model, Sardόn et al. (1994) introduced the Kalman filter method for real-time ionospheric VTEC estimation, which can also be a promising prediction of DCBs under adverse conditions (antispoofing, ionospheric disturbances).Klobuchar (1987) assumed that STEC equals VTEC multiplied by the approximation of the standard geometric mapping function at the mean vertical height of 350 km along the path of STEC.Lanyi and Roth (1988)  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 preferred to be 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 mid-latitude is 400 km 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 IGS GIMs 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 with regards to minimize |ΔDCB| by comparing to the DCB released by 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 the time 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-year 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/DCB estimation of nearby stations needs to be verify.
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/DCB estimation.By selecting two different regions in U.S.
and Europe with dense IGS stations, we calculate the daily DCBs of 2014 by using the optimal ionospheric shell heights derived from data from 2003-2013 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.

Method
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 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: 00 00 ( , ) ( ) ( ) Based on the thin shell approximation, the observation equation can be written as: ( , ) ( , ) ( ) where PRN os T is slant TEC calculated by carrier phase smoothing, the superscript PRN denotes GPS satellite.
PRN DCB denotes the combination of GPS satellite and receiver DCB.z denotes the zenith angle of IPP.According to Lanyi and Roth (1988), the standard geometric mapping function () fz is expressed as follows: where Re denotes the earth's radius, El denotes the elevation angle, and h denotes the thin ionospheric shell height.Note that h also affects the location of the IPP.
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 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: where k is the order of Fourier series and is set to 40, This model can be applied to neighboring stations' DCB estimation.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.

Experiment and Results
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 region in U.S. 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 presents the location and distribution of the selected IGS stations in two regions.Table 1 presents     At the end of 2010, a gap appears, for the DCB provided by CODE is simultaneously anomalous for both stations (Zhao and Zhou, 2018), and the data during this period are abandoned.
Fig. 2 Variation of the daily optimal ionospheric shell height (black) and the fitting result (red) Figure 3 presents 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 Figure 3, the peaks correspond to 11-year, 1-year, 6-month and 4-month cycles.The amplitudes of 11-year and 1-year cycles are more evident than other periods in both two stations.As mentioned earlier, 0.01 per day is about the maximum frequency of (6).Higher frequencies would not be useful because of their small amplitudes.This result shows that the optimal ionospheric shell height of GOLD and PTBB is periodic, and the 40th-order of Fourier series is suitable for modelling its variation.
Fig. 3 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 Figure 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 equation ( 1)-( 4).The difference of DCBs in all stations in each region calculated using the optimal ionospheric shell height model at the reference stations and DCBs provided by CODE is then compared to the difference of DCBs calculated using a fixed ionospheric shell height (400 km) and DCBs released by CODE.
The results of this comparison are shown in Figure 4.The panels for the stations are arranged by their distances to reference station, 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 ΔDCBs is close to zero.For the other stations, the median daily average ΔDCB is negative, but smaller in absolute value than using the fixed shell height.This result shows the improvement of the model seems to be related with the distance to GOLD.Data gaps on the figure correspond to days when data from that station are not available.Figure 5 is the same format as Figure 4, and presents the results of Region II.Comparing to the results of fixed ionospheric height, Figure 5 also indicates that the ΔDCB calculated using the optimal ionospheric shell heights at PTBB is on average smaller than that calculated using fixed ionospheric shell height.Both Figure 4 and Figure 5 present that the accuracy of DCB estimation can be improved using optimal ionospheric heights from reference stations.Table 2 presents the quantitative statistical results of average ΔDCB in 2014.For all the stations in each region, the mean values and the root mean squares (RMS) using the optimal ionospheric shell height model are smaller than those using the fixed ionospheric height.For Region I, the improvements of GOLD and TABV are the most significant.Their mean values are reduced to 0.12 and 0.08 TECu, respectively; the root mean squares are reduced by 4.43 and 4.33 TECu, respectively.
For Region II, the improvement for DCB estimation are the most obvious for WTZZ, with mean value of ΔDCB decreases from 2.34 to 0.02.We could note that TABV and WTZZ station are quite close to the reference stations in each region.

Summary
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: 1) The optimal ionospheric shell height model improves the accuracy of DCB estimation comparing 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 station 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.
2) 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 ΔDCBs, in region I, the slopes are about 1.84 and 0.75 TECu per 1000 km; in region II, the slopes are about 0.30 and 0.41 TECu per 1000 km.These results indicate the horizontal spatial correlation of regional ionospheric electron density distribution.For the different region, the error at 0 km (i.e. the error for the reference station) is different, which should be also considered, the quality of the DCB estimations also depends on the quality of the optimal shell height model at the reference stations themselves.
Due to a requirement of this experiment, we only analyze two regions in mid-latitude because of the insufficiency of long-term P1 data.We also ignore the orientation of isolated GPS receivers to the reference station.

Fig. 1
Fig.1 Receiver's DCB released by CODE, IGS and JPL from 2010 to 2011

Fig. 2
Fig.2 the optimal ionospheric shell height with different elevation mask angle at PTBB

Fig. 3
Fig.3 Fourier fitting results of different order for GOLD

Figure 4
Figure 4 and 5 top panels show the difference of the DCBs of 2014 in the reference station with the predictions of the Fourier model.However this model has been presented earlier only in term of shell height.It is therefore difficult to understand if

•
Does the matrix φ contains the function f(z) from equation (3)?
further developed this model into a single thin-layer model, and proposed the standard geometric mapping function and the polynomial model.The single thin-layer model assumed that the ionosphere is simplified by a spherical thin shell with infinitesimal thickness.Clynch et al (1989) proposed a mapping function in the form of a polynomial by assuming a homogeneous electron density shell between altitudes of 200 and 600 km.Mannucci et al (1998) presented an elevation scaling mapping function derived from the extended slab mode.There are also many modified mapping functions according to the standard geometric mapping function.Schaer (1999) proposed the modified standard mapping function using a reduced zenith angle.Rideout and Coster (2006) presented a new mapping function which replaces the influence of the shell height by an adjustment parameter, and set the shell height as 450 km.Smith et al (2008) modified the standard mapping function by using a complex factor.Based on the electron density field derived from the international reference ionosphere (IRI),Zus et al (2017) recently developed an ionospheric mapping function at fixed height of 450 km with dependence on time, location, azimuth angle, elevation angle, and different frequencies.
1) where V T denotes VTEC. and S denote the geographic latitude and the solar hour angle of ionospheric pierce point (IPP), respectively; 0  and 0 S denote  and S at the center of the cover region of IPP in one day.ij E is the model coefficient.m and n denote the orders of the model.A polynomial model fits the VTEC over a period of time.In our case, a VTEC model is generated over 3 hours of time, therefore 8 VTEC models are applied per day.DCB is considered as constant in one day.Since our analysis is based on long-term single site data, we set m and n to 4 and 3, respectively.Huang and Yuan (2014) applied the polynomial model with the same orders to TEC estimation.
PRNDCB and the reference values.This optimization problem can be written as: the daily optimal ionospheric shell height; ref DCB denotes the vector of the reference values of DCBs; s.t. is the abbreviation for subject to;  T=Φ E+θ DCB is the matrix form of all the observation equations in one day; T denotes the vector of os T ; E corresponds to the coefficients of the models, contains ij E ; DCB is the vector of PRN DCB ; Φ is the coefficient matrix of E is the coefficient matrix of DCB , contains only 1s and 0s.E and DCB are unknown.
time, and L is the time span which equals to 4018 days.The maximum frequency of the model is 40/L≈0.01per day, which corresponds to a period of 100 days.By least square method, the model coefficients can be estimated.
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 has been 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 nine receiver types for the sixteen stations.The receiver type of PTBB has changed twice in 2006.

Fig. 1
Fig.1 Geographical location of the selected IGS stations in U.S. region (Region I) and Europe region (Region II).The black triangle in each plot is the reference station.

Figure 2
Figure2presents the estimated daily optimal ionospheric shell height of GOLD

Fig. 4
Fig.4 Comparisons of the average ΔDCB calculated using the predicted optimal

Fig. 5
Fig.5 Comparisons of the average ΔDCB calculated using the predicted optimal

Figure 6 and
Figure 6 and Figure 7 present the relation between the statistical results of

Fig. 6
Fig.6 Relation of the accuracy for DCB estimation with the distance to GOLD.The red lines are the linear fitting results

Table 1
Information for the stations

Table 2
Statistical results of mean (ΔDCB) in 2014