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Improved GPT2w (IGPT2w) model for site specific zenith tropospheric delay estimation in China
Journal of Atmospheric and Solar–Terrestrial Physics 198 (2020) 105202
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Research Paper
Improved GPT2w (IGPT2w) model for site specific zenith tropospheric delay estimation in China Zheng Du a, Qingzhi Zhao a, *, Wanqiang Yao a, Yibin Yao b a b
College of Geomatics, Xi’an University of Science and Technology, Xi’an, 710054, China School of Geodesy and Geomatics, Wuhan University, Wuhan, 430079, China
A R T I C L E I N F O
A B S T R A C T
Keywords: GNSS ZTD residual GPT2w IGPT2w Periodic signals
Despite being a systematic error source in global navigation satellite system (GNSS) positioning and navigation, tropospheric delay is a key parameter in GNSS meteorology. Therefore, deriving a zenith tropospheric delay (ZTD) value as accurately as possible from an empirical model without using auxiliary data is a prerequisite for the high-precision application of GNSS positioning and navigation. To reach the goal above, this paper proposes an improved model to estimate the ZTD based on the Global Pressure and Temperature 2 wet (GPT2w) model, which is called the improved GPT2w (IGPT2w) model. The GPT2w-derived ZTD is first calculated as the initial value of the IGPT2w model, and the time series of ZTD residual can thus be obtained between the GNSS- and GPT2w-derived ZTDs over GNSS stations. Analysis of the long time series variation of ZTD residuals using the multichannel singular spectrum analysis method reveals evident periodic signals. The Lomb–Scargle method is then used to determine the specific values of these periodic signals, and different periods are identified at various GNSS stations. Therefore, a ZTD residual model that considers annual, semi-annual, and seasonal periods is established. The IGPT2w model, in which the ZTD value is obtained by combining the estimated ZTD residual and the GPT2w-derived ZTD, can be acquired. A total of 188 GNSS stations in China throughout 2015 to 2017 are selected to validate the IGPT2w model. In the model, the GNSS-derived ZTD is obtained using the GAMIT/ GLOBK software, and the accuracy is validated using radiosonde data with root mean square and bias of 1.9 and 0.1 cm, respectively, at 33 collocated stations in China. Statistical results reveal that the accuracy of the IGPT2wderived ZTD is improved by 13.7% compared with that of the GPT2w-derived ZTD when the GNSS-derived ZTD is regarded as the reference. Such result indicates that the proposed IGPT2w model outperforms the GPT2w model in China.
1. Introduction Tropospheric delay is a major error source in satellite navigation and positioning but an important parameter in global navigation satellite system (GNSS) meteorology. Therefore, this error should be carefully processed in such applications as GNSS positioning and navigation (Jin et al., 2010). Tropospheric delay is generated when radio signals penetrate the troposphere with values of approximately 25 m at low elevation angles and 2.5 m in the vertical direction (Chen et al., 2012). The zenith tropospheric delay (ZTD) is an average value projected by the value of slant path delay with different azimuth and elevation angles into the vertical direction using a mapping function. ZTD comprises zenith hydrostatic delay (ZHD) and zenith wet delay (ZWD). ZHD can be precisely modeled using the observed surface pressure. Meanwhile,
describing ZWD is difficult because it is highly affected by tempora l–spatial variations in atmospheric water vapor. Therefore, the ZTD model cannot be indirectly described by combining ZHD and ZWD models. Nevertheless, certain empirical models directly estimate the ZTD, thereby improving the accuracy of GNSS navigation and positioning. In general, three types of ZTD models, including very-long-baseline interferometry, real-time kinematic, and differential global positioning system, are commonly used for tropospheric corrections in GNSS (Ahn et al., 2006; Song et al., 2016; Kim et al., 2017; Yao et al., 2019). The first type involves established models based on real-time GNSS-derived ZTDs in regional networks. Tao (2008) estimated ZWD parameters using real-time orbit and clock products provided by the Jet Propulsion Lab oratory (JPL) based on precise point positioning (PPP) technique. Ye
* Corresponding author. E-mail address: [email protected] (Q. Zhao). https://doi.org/10.1016/j.jastp.2020.105202 Received 30 November 2019; Received in revised form 31 December 2019; Accepted 2 January 2020 Available online 28 January 2020 1364-6826/© 2020 Elsevier Ltd. All rights reserved.
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et al. (2008) highlighted the feasibility of this method, and the accurate ZTD was estimated using the PPP technique. Generation strategies for real-time and near-real-time tropospheric products were respectively developed using Kalman filtering and backward smoothing based on the PPP technique by Dou�sa et al. (2018). The second type comprises ZTD models based on single/multi-meteorological parameters obtained from the ground, and the commonly used models include Saastamoinen (1972), Hopfield (1969), and Ifadis (1986) models. The ZTD accuracy estimated by these models ranges from the decimeter-to centi meter-level under different conditions (Yao et al., 2019). For example, the ZTD accuracy estimated by the Hopfield model is low at stations with high elevation because this model is subject to elevation. By contrast, the Saastamoinen model is slightly affected by station height. The third type includes ZTD empirical models by which the ZTD parameter can be estimated without any auxiliary information. Several examples of series models include the University of New Brunswick (Collins and Langley, 1997, 1998), global pressure and temperature (Bohm et al., 2007; Lagler et al., 2013; Bohm et al., 2015), IGGtrop (Li et al., 2012, 2015), TropGrid (Krueger et al., 2004; Schüler, 2014), global ZTD (GZTD) (Yao et al., 2013, 2016) and Global Tropospheric Model (GTrop) (Sun et al., 2019) series models. Currently, the ZTD accuracy estimated by empirical models is approximately 4 cm on a global scale (Yao et al., 2015). Although the accuracy of ZTD derived from empirical models has significantly improved, further increasing the accuracy of obtaining the ZTD parameter remains a research point in GNSS positioning and nav igation and GNSS meteorology, especially for regional areas. To achieve the aforementioned goal, this study proposes an improved GPT2w (IGPT2w) model for estimating ZTD parameters to further improve the quality of the GPT2w model in China. In the IGPT2w model, the GPT2wderived ZTD is considered the initial value, and the time series of ZTD residual is modeled considering the periodic signals. Therefore, the final ZTD parameter of IGPT2w is obtained by combining the GPT2w-derived ZTD and the ZTD residual estimated from the established residual model. This paper is organized as follows. Section 2 introduces the GPT2w model and the retrieval of the ZTD parameter. Section 3 de scribes the experimental description and establishment of the IGPT2w model. Section 4 presents the analysis and validation of the IGPT2w model. Finally, Section 5 provides the conclusion.
horizontal direction. P ¼ P0 ⋅e
gm ⋅dM Rg ⋅Tv ðh
h0 Þ
Tv ¼ T0 ð1 þ 0:6077⋅QÞ T ¼ T0 þ dT*dh
(1)
e0 ¼ Q*P0 =ð0:622 þ 0:378*QÞ=100 e ¼ e0 ðP*100=P0 Þλþ1 where P0 , T0 , and e0 are the pressure, temperature, and water vapor pressure, respectively, at the grid point; P, T, and e are the pressure, temperature, and water vapor pressure, respectively, at the target location; gm is gravity and has a value of 9.80665 m/s2; dM is the molar mass of dry air with a value of 28.965 � 10 3 kg/mol; Rg is the universal gas constant with a value of 8.3143 J/K/mol; Tv is the virtual temper ature (Kelvin); Q is the specific humidity; dT is the temperature lapse rate (degree/km);λ is the water vapor decrease factor; h and h0 represent station height and grid height, respectively. 2.2. Retrieval of ZTD Only the nine meteorological parameters can be obtained from the GPT2w model. Therefore, the ZTD derived from GPT2w must be calculated. ZTD comprises ZHD and ZWD. In this study, ZHD and ZWD are calculated separately based on corresponding formulas and then added together to obtain the ZTD. The ZHD parameter is calculated as follows (Davis et al., 1985): ZHD ¼
1
0:0022768⋅P 0:00266⋅cos 2 ϕ 0:00028⋅h
(2)
where P is the surface pressure (hPa), h is the geoid height of the station (km), and ϕ is the latitude of station (radian). ZWD can be calculated based on the formula proposed by Askne and Nordius (1987): � � k3 Rd 0 ZWD ¼ 10 6 k 2 þ es Tm ðλ þ 1Þgm � (3) k’2 ¼ k2 k1 Mw Md
2. GPT2w model and retrieval of ZTD parameter
Rd ¼ R=Md
2.1. Description of the GPT2w model
where R is the molar gas constant with a value of 8.314 J/(mol K); Mw and Md are the molar masses of water and dry air with values of 18.0152 and 28.9644 g/mol, respectively; k1 , k2 , and k3 are refractivity formula constants with values of 77.604 � 0.014 K/mbar, 64.79 � 0.08 K/mbar, and (3.776 � 0.004) � 105 K2 /mbar, respectively; Rd denotes the spe cific gas constant for the dry constituents; gm is the gravity acceleration at the mass center of the vertical column of the atmosphere; Tm is the weighted mean temperature; and λ is the water vapor reduction factor. According to the above formulas, only four of the nine meteorological parameters derived from GPT2w (P, e, Tm , and λ) are required to calculate the ZTD in this work.
The GPT2w model is an empirical model that does not use auxiliary information and considers the annual and semi-annual amplitudes of certain meteorological parameters, such as pressure, temperature, and water vapor pressure. This model was established based on the monthly mean pressure level data derived from the fourth product of European Center for Medium-Range Weather Forecasts (ECMWF ERA-Interim) reanalysis data with a horizontal resolution of 1� � 1� throughout 2001 to 2010, which can be downloaded from (http://ggosatm.hg. tuwien.ac.at/DELAY/SOURCE/GPT2w/MATLAB/gpt2_1w.m). Only the ellipsoidal coordinates (latitude, longitude, and height) of GNSS stations and the corresponding modified Julian data are required as the input information because the GPT2w model is a blinded model. The nine outputs are as follows: hydrostatic and wet mapping function co efficients, temperature in degrees Celsius and its lapse rate in degrees per km, mean temperature of the water vapor in Kelvin, pressure in hPa, water vapor pressure in hPa, water vapor decrease factor, and geoid €hm et al., 2015). The specific procedure of GPT2w undulation in m (Bo for estimating the corresponding parameters is performed as follows. (a) The four grid points surrounding the input location are first determined. (b) The corresponding parameters at the desired height are calculated using the data of the four grid points at the given grid height based on Eq. (1). (c) The parameters from the four grid points are interpolated to the target location using the bilinear interpolation method in the
3. Improved GPT2w (IGPT2w) model for ZTD estimation 3.1. Experiment description A total of 188 GNSS stations in China are selected from 2014 to 2017 for the experiment. The GNSS observations are processed using the GAMIT/GLOBK (Ver. 10.4) software, and the elevation cutoff mask of 10� is selected in this research. The tropospheric parameters of ZTD and gradients in the east/west and north/south directions are both estimated with an interval of 1 h. The specific configurations of GAMIT software have been presented in Table 1. Additionally, the accuracy of the GNSSderived ZTD is compared with that obtained from the radiosonde data at 2
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height less than 100 m between the GNSS and radiosonde stations are regarded as a criterion for selecting the collocated stations. Therefore, 33 collocated stations in China are chosen, and Fig. 2 provides the root mean square (RMS) and bias of the ZTD differences derived from GNSS and radiosonde. The long-term ZTD residuals between GNSS-derived and radiosonde-derived ZTD are first obtained at collocated stations and then calculated the RMS value of each station. It can be observed from Fig. 2 (a) that the RMS value of the ZTD difference derived from the GNSS observation ranges from 0 cm to 3 cm when compared with radiosonde data. The average RMS and bias are approximately 1.9 and 0.1 cm, respectively, for the 33 selected stations from 2015 to 2017.
Table 1 Some configurations of GAMIT software for GNSS processing. Term
Value
Reference frame
Troposphere
Satellite quality center correction Satellite altitude angle Tide model
Inertial coordinate system Axial precession Nutation Prior zenith tropospheric delay Interval Mapping function Bock I Bock II/IIA Bock IIR
J2000.0 Ifadis, 1986 Ifadis, 1986 0.5m hourly HGMF、DGMF x,y,z: 0.2100, 0.0000, 0.8540m x,y,z: 0.2790, 0.0000, 1.2300m x,y,z: 0.0000, 0.0000, 0.0000m 10� IERS2003 model; FES2004 model and Polar tide model
3.2. Time series analysis of ZTD residual After the ZTD values derived from GPT2w and GNSS are calculated over the 188 GNSS stations, the time series of the ZTD residual between GPT2w and GNSS can be further obtained. Examples of time series of ZTD residuals between GPT2w and GNSS at XJML, XJSS, and XJKE stations throughout 2015–2017 are presented in Fig. 3. In this figure, the existence of periodic signals is possibly observed in the time series of
the collocated stations. The distribution of GNSS and radiosonde stations is presented in Fig. 1. Herein, the horizontal distances in latitudinal and longitudinal directions less than 30 km, and the vertical difference in
Fig. 1. Geographical distribution of GNSS and radiosonde stations selected in China over throughout 2015–2017.
Fig. 2. Comparison of RMS and bias of ZTD difference between GNSS and radiosonde throughout 2015–2017 at 33 collocated stations in China. 3
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Fig. 3. Time series of ZTD difference between GNSS and GPT2w at stations XJML, XJSS, and XJKE throughout 2015–2017.
Fig. 4. First RC (mm) corresponding to the 10 largest eigenvalues when MSSA method is applied to the time series of ZTD residual at XJKE throughout 2015–2017.
ZTD residuals for the three selected GNSS stations. At present, spectrum analysis is widely used to discuss the periodic characteristics of ZTD (Jin et al., 2009; Li et al., 2012; Bałdysz et al., 2015). The multi-channel singular spectrum analysis (MSSA) method, which is an extension of singular spectrum analysis (SSA), is first introduced to verify the exis tence of periodic signals in the time series of ZTD residuals (Oropeza and Sacchi, 2011). The SSA method is designed to extract useful information from a noisy time series, whereas its multivariate form MSSA is effective in determining the periodic variations of time series (Zhang et al., 2017). The theory of MSSA can be summarized as follows: if the time series of ZTD residual includes n complex exponentials, then the corresponding Hankel matrix of the time series is a matrix of rank n. The missing data and noise error will increase the rank of the Hankel matrix, and this issue can be resolved only by rank reduction (Hua, 1992). Fig. 4 shows the first reconstructed components (RC) (mm) corresponding to the 10 largest eigenvalues when the MSSA method is applied to the time series of the ZTD residual at the XJKE station. The figure depicts the evident existence of periodic signals in the time series of the ZTD residual; semi-annual, seasonal, and annual periods appeared in RC1-2, RC4-5,
and RC6, respectively. However, the specific values of periodic signals cannot be determined by the MSSA method, and this incapability is the defect of this method. Therefore, the Lomb–Scargle periodogram method is used to deter mine the specific values of periodic signals of ZTD residuals between GNSS and GPT2w at the selected GNSS stations. The periodogram method introduced in this paper, which can handle time series with gaps or uneven sampling intervals (Zhao et al., 2019a), is similar to Fourier spectrum analysis. Fig. 5 presents the periodic variations for XJML, XJSS, and XJKE stations using the Lomb–Scargle periodogram method throughout 2015–2017. Findings show the following: only the annual period exists at XIML station; the annual and semi-annual periods exist at XJSS station; and the annual, semi-annual, and seasonal periods exist at XJKE station. Such results further verify the existence of some periods in the time series of the ZTD residuals. Additionally, the time series of the ZTD residuals at different GNSS stations in China have different periodic signals. Additionally, the Lomb–Scargle periodogram method is used to analyze the time series of ZTD residuals at the 188 selected GNSS 4
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Fig. 5. Periodic variations for XJML, XJSS, and XJKE stations obtained using the Lomb–Scargle periodogram method, in which the first to third columns represent the annual, semi-annual, and seasonal periods.
stations have three types of periods. Additionally, it can be observed from Fig. 6 that the distribution of GNSS stations with different types of periods for ZTD residuals exhibits certain regional features. Such phe nomenon is mainly related to the distribution of atmospheric water vapor (Zhao et al., 2019b).
Table 2 Number of GNSS stations and the corresponding periods, where A, B, C, and D represent annual, semi-annual, and seasonal periods with 120 and 90 days, respectively. Type
Number
AþBþCþD BþCþD AþCþD CþD AþC AþBþD AþBþC BþC BþD AþB AþD A B C
27 25 25 23 21 16 11 8 2 8 5 6 8 3
3.3. Establishment of ZTD residual model Owing to the existence of some periodic signals in the time series of ZTD residuals between GNSS and GPT2w at GNSS stations, fitting those values is possible by establishing the corresponding periodic model. Herein, a periodic model of the time series of ZTD residuals of 188 GNSS stations is established considering the annual, semi-annual, and seasonal periods and the average of ZTD residuals. Finally, the periodic model of ZTD residuals can be determined as a combination of four periodic items and the average term, which is expressed as follows: � � � � doy doy 2π ϕ1 þ A2 cos 4π ϕ2 þ … ΔZTD ¼ A0 þ A1 cos 365:25 365:25 � � � � doy doy A3 cos 6π ϕ3 þ A4 cos 8π ϕ4 365:25 365:25 (4)
stations in China, and the statistical results further reveal that different stations show various periodic signals. Table 2 shows the number of GNSS stations and the corresponding periods. Fig. 6 provides the first six distributions of GNSS stations with the maximum number of period. Table 1 reveals that the time series of ZTD residuals at most GNSS sta tions has at least two types of periods, and more than 50% of GNSS
where ΔZTD refers to the ZTD residual; A0 is the average of ZTD re sidual; A1 A4 represent the annual, semi-annual, and seasonal periods of 120 and 90 days, respectively, while ϕ1 ϕ4 refer to their 5
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Fig. 6. First six distributions of GNSS stations with the maximum number of period.
corresponding phases. These unknown parameters are estimated by the least square method in this study. Therefore, the final IGPT2w model can be obtained by combining the initial value of ZTD derived from the GPT2w model and the ZTD residual estimated by the fitted model in Eq. (4).
derived/GTrop-derived ZTD. Such result indicates the good perfor mance of the established IGPT2w model, and a good consistency existed between IGPT2w and Gtrop models. In addition, Table 3 gives the sta tistical result of ZTD accuracy based on different models in China. It can be concluded that IGPT2w model has the highest accuracy in China, Gtrop model is the second, and GPT2w model is the worst. The accuracy of the IGPT2w model proposed has been improved by approximately 10% when compared with GTrop-ZTD model in China.
4. Validation of IGPT2w model for ZTD estimation The quality of the established IGPT2w model is the key to evaluating its further usage. Therefore, the fitted ZTD residual model and the ZTD time series estimated by the IGPT2w model are first compared at three GNSS stations. Afterward, the accuracy of the established IGPT2w model and its improvement rate are further analyzed. In this work, two indexes are introduced to evaluate the established IGPT2w model, namely, RMS and mean absolute error (MAE), which are computed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m �2 1 X RMS ¼ ZTDmodel ZTDGNSS i i m i¼1 (5) m � � 1 X model GNSS �ZTD MAE ¼ ZTDi � i m i¼1
4.2. Validation of internal and external accuracies of IGPT2w model The above analysis presents only the result of the IGPT2w model at the three selected GNSS stations and thus cannot reflect the total accu racy of the IGPT2w model in China. Therefore, the validation of internal and external accuracies of the IGPT2w model is further analyzed to evaluate its performance in entire China. In this section, only the ZTD data derived from GNSS and the GPT2w model throughout 2015–2016 are used to establish the IGPT2w model, while the ZTD data for the year 2017 are used to perform the external validation at 188 GNSS stations. Fig. 8 shows the internal/external validation of the IGPT2w model (first and second rows, respectively); the first and second columns are the RMS and MAE, respectively. The values of RMS and MAE decrease from the southeast to the northwest of China, and this pattern corresponds to the distribution of atmospheric water vapor in the country (Zhao et al., 2019b). The maximum values of RMS and MAE appear in the southeast of China for the internal and external validations. In addition, Table 4 provides the statistical results regarding the internal/external accuracies of GPT2w and IGPT2w models. The accuracy of the ZTD derived from the IGPT2w model (approximately 2.2 cm) is improvement to that from the GPT2w model (approximately 2.5 cm) when the GNSS-derived ZTD is considered to be the reference.
where ZTDmodel refers to the ZTD value derived from the GPT2w or IGPT2w model; ZTDGNSS is the ZTD value derived from GNSS observa tions, which is estimated using GAMIT software and considered as the reference in the comparison; and m is the total number of ZTD pairs. 4.1. Analysis of fitted ZTD residual model To analyzed the fitted ZTD residual model at different stations, an example of ZTD residuals is given in three selected stations randomly in Fig. 7. The ZTD residual model is first presented at stations XJML, XJSS, and XJKE in this section (first column of Fig. 7). Findings show that the fitted model can effectively describe time series variation of ZTD re siduals. In addition, the ZTD time series derived from GNSS, GPT2w, GTrop and IGPT2w are presented at these stations (second column of Fig. 7). It can be observed from Fig. 7 that The IGPT2w-derived ZTD agrees well with that from GNSS when compared with the GPT2w-
4.3. Improvement rate of IGPT2w for ZTD estimation The improvement rate, which refers to the proportion between the absolute value of RMS difference between IGPT2w and GNSS and the RMS value of GPT2w, is introduced in this section to further assess the performance of the proposed IGPT2w model. Herein, the coefficients of 6
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Fig. 7. ZTD residual model and the ZTD time series derived from GNSS, GPT2w, GTrop and IGPT2w at XJML, XJSS, and XJKE stations throughout 2015–2017; the first column represents the ZTD residual model, and the second column refers to the ZTD time series.
difference between GNSS-GPT2w and GNSS-IGPT2w models are calcu lated. Finally, the improvement rate of the IGPT2w model for ZTD estimation compared with GPT2w can be obtained (Fig. 9); the com parison reveals that the accuracy of estimated ZTD values at 188 GNSS stations is improved to different degrees. Statistical results reveal that the average improvement rate of the 188 stations in China is 12.0%, which indicates the good performance of the proposed IGPT2w model. In addition, the frequency distributions of the RMS values derived from GPT2w, GTrop and IGPT2w at 188 GNSS stations are analyzed (Fig. 10). These distributions show that most GNSS stations derived from the IGPT2w model with RMS values less than 3 cm account for 87.2% of all GNSS stations. Meanwhile, stations derived from the GPT2w model with RMS values less than 3 cm account for only 70.7% of all stations. Findings further reveal that the average RMS/MAE values of the ZTD difference between GNSS-GPT2w and GNSS-IGPT2w at the 188 selected GNSS stations over the entire 2017 (2015–2017) are 2.58/2.03 cm and
Table 3 Statistical accuracy of ZTD estimated by three models in China (Unit: cm). Model
RMS
MAE
GPT2w GTrop IGPT2w
2.58 [1.13,5.80] 2.52 [1.16, 5.67] 2.23 [0.95,5.44]
2.03 [0.88, 4.87] 1.98 [0.92, 4.74] 1.67 [0.69, 4.42]
the ZTD residual model in Eq. (4) are first estimated using the least square method with the data throughout 2015–2016 and adopted to estimate the ZTD residual in the year 2017. The initial ZTD value derived from the GPT2w model for 2017 is then calculated. Finally, the ZTD value derived from the IGPT2w model can be obtained by combining the estimated ZTD residual and the GPT2w-derived ZTD. The GPT2w- and IGPT2w-derived ZTD values are compared with those from 188 GNSS stations over the entire 2017, and the RMS values of the ZTD 7
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Fig. 8. Validations of the IGPT2w model in China at 188 stations, where the first and second rows represent the internal/external validation, respectively while the first and second columns are the RMS and MAE, respectively. Table 4 Statistical result of internal/external accuracies of GPT2w and IGPT2w models in China when compared with the GNSS-derived ZTD (unit: cm). Model
Type Internal
GPT2w IGPT2w
External
RMS
MAE
RMS
MAE
2.61 2.25
2.06 1.72
2.53 2.23
1.98 1.67
Fig. 10. Frequency distributions of RMS derived from GPT2w, GTrop and IGPT2w at 188 GNSS stations in China.
2.23/1.67 cm, respectively. Such a result further indicates the capability of the IGPT2w model to improve the accuracy of the GPT2w model by 13.7% for ZTD estimation in China. 4.4. Accuracy of planar distribution for IGPT2w model under different seasons The two-dimensional (2D) distributions of ZTD residuals between GNSS-GPT2w and GNSS-IGPT2w and the improvement rate of the IGPT2w model compared with the GPT2w model under different sea sons are presented in Fig. 11 to analyze the accuracy of IGPT2w model in entire China under different seasons. The four seasons are divided based on the standard meteorological practice: spring is from March to May, summer is from June to August, autumn is from September to
Fig. 9. Improvement rate of IGPT2w model for ZTD estimation at 188 GNSS stations compared with GPT2w model in China over the entire 2017.
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Fig. 11. 2D distributions of ZTD residuals; the first and second columns refer to the ZTD residuals between GNSS-GPT2w and GNSS-IGPT2w, respectively, and the third column shows the improvement rate of the IGPT2w model compared with the GPT2w model under the four seasons.
November, and winter is from December to February. Fig. 11 shows that the 2D distribution of the ZTD residual between GNSS and IGPT2w models is smaller than that between GNSS and GPT2w models, espe cially in summer and autumn. Additionally, the third column of Fig. 11 shows that the improvement rate of the IGPT2w model is concentrated in summer and autumn. Such result indicates that the established IGPT2w model performs better than the GPT2w model in these seasons, but the capabilities of the GPT2w and IGPT2w models for estimating ZTD in China are similar in spring and winter. The reason for such phenomenon may be related to the distribution of atmospheric water vapor in China. Water vapor fluctuates greatly in summer and autumn, but less in winter and spring (Wong et al., 2015), and the established IGPT2w model can be used to correct those large fluctuations in summer and autumn. Therefore, the IGPT2w model is found better than GPT2w model in summer and autumn. In spring and winter, the water vapor fluctuation is inactive and the ZTD difference is small, therefore, the improvement of IGPT2w model is not clear but it also has some improvement.
5. Conclusion The IGPT2w model is proposed in this study for ZTD estimation in China. The GPT2w-derived ZTD is regarded as the initial value of the IGPT2w model. The MSSA and Lomb–Scargle periodogram methods are introduced to analyze the ZTD residuals between GNSS and GPT2w model, and determine the existence of some periodic items. Afterward, a corresponding ZTD residual model is established by considering the influence of annual, semi-annual, and seasonal periods. Finally, the ZTD of the IGPT2w model can be obtained by combining the initial ZTD value derived from the GPT2w model and the ZTD residual estimated from the established residual model. A total of 188 GNSS stations and 33 collocated radiosonde stations in China throughout 2015–2017 are selected for the experiment. The GNSS observations are processed by the GAMIT/GLOBK software, and the RMS and bias of the GNSS-derived ZTD are 1.9 cm and 0.1 cm, respec tively, when compared with the 33 collocated radiosonde stations. GNSS data of 2015–2016 are used to establish the IGPT2w model and applied for the ZTD estimation for the entire 2017. External validation shows 9
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that the IGPT2w model outperforms the GPT2w model. When the GNSSderived ZTD is regarded as the reference, the average RMS and MAE of the ZTD difference of GNSS-IGPT2w (2.23/1.67 cm) are lower than those of GNSS-GPT2w (2.58/2.03 cm). Furthermore, 2D comparison of the IGPT2w model over entire China reveals that the improvement rate of ZTD estimation mainly appears in summer and autumn when compared with that of the GPT2w model, but the capacities of these models are similar in spring and winter. This finding is due to the relatively large ZTD value in summer and autumn and the failure of only the blinded model (GPT2w) to describe the ZTD variation well. There fore, the ZTD residual model plays a supplementary role in the IGPT2w model, especially for cases with relatively large atmospheric water vapor variations in summer and autumn.
Dou�sa, J., � aclavovic, P., Zhao, L., Ka�cma�rík, M., 2018. New adaptable all-in-one strategy for estimating advanced tropospheric parameters and using real-time orbits and clocks. Rem. Sens. 10 (2), 232. Hopfield, H.S., 1969. Two-quartic tropospheric refractivity profile for correcting satellite data. J. Geophys. Res. 74 (18), 4487–4499. Hua, Y., 1992. Estimating two-dimensional frequencies by matrix enhancement and matrix pencil. IEEE Trans. Signal Process. 40, 2267–2280. Ifadis, I., 1986. The Atmospheric Delay of Radio Waves: Modelling the Elevation Dependence on a Global Scale. Licentiate Thesis, Technical Report, p. 38, 1986. Jin, S., Luo, O.F., Gleason, S., 2009. Characterization of diurnal cycles in ZTD from a decade of global GPS observations. J. Geodes. 83 (6), 537–545. Jin, S.G., Luo, O., Ren, C., 2010. Effects of physical correlations on longdistance GPS positing and zenith tropospheric delay estimates. Adv. Space Res. 46, 190–195. Kim, J., Song, J., No, H., Han, D., Kim, D., Park, B., Kee, C., 2017. Accuracy improvement of DGPS for low-cost single-frequency receiver using modified fl€ achen korrektur parameter correction. ISPRS Int. J. Geo-Inf. 6 (7), 222. Krueger, E., Schueler, T., Hein, G.W., Martellucci, A., Blarzino, G., 2004. Galileo tropospheric correction approaches developed within GSTB V1. Proc. ENC GNSS. 1619. Lagler, K., Schindelegger, M., B€ ohm, J., Kr� asn� a, H., Nilsson, T., 2013. GPT2: empirical slant delay model for radio space geodetic techniques. Geophys. Res. Lett. 40 (6), 1069–1073. Li, W., Yuan, Y., Ou, J., Chai, Y., Li, Z., Liou, Y.A., Wang, N., 2015. New versions of the BDS/GNSS zenith tropospheric delay model IGGtrop. J. Geodes. 89 (1), 73–80. Li, W., Yuan, Y., Ou, J., Li, H., Li, Z., 2012. A new global zenith tropospheric delay model IGGtrop for GNSS applications. Chin. Sci. Bull. 57 (17), 2132–2139. Oropeza, V., Sacchi, M., 2011. Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis. Geophysics 76 (3), V25–V32. Saastamoinen, J., 1972. Atmospheric Correction for the Troposphere and Stratosphere in Radio Ranging Satellites. The Use of Artificial Satellites for Geodesy, pp. 247–251. Schüler, T., 2014. The TropGrid2 standard tropospheric correction model. GPS Solut. 18 (1), 123–131. Song, J., Park, B., Kee, C., 2016. Comparative analysis of height-related multiple correction interpolation methods with constraints for Network RTK in mountainous areas. J. Navig. 69, 991–1010. Sun, Z., Zhang, B., Yao, Y., 2019. A global model for estimating tropospheric delay and weighted mean temperature developed with atmospheric reanalysis data from 1979 to 2017. Rem. Sens. 11 (16), 1893. Wong, M.S., Jin, X., Liu, Z., 2015. Multi-sensors study of precipitable water vapour over mainland China. Int. J. Climatol. 35 (10), 3146–3159. Yao, Y., He, C., Zhang, B., Xu, C., 2013. A new global zenith tropospheric delay model GZTD. Chinese Journal of Geophysics-Chinese Edition 56 (7), 2218–2227. Yao, Y., Hu, Y., Yu, C., Zhang, B., Guo, J., 2016. An improved global zenith tropospheric delay model GZTD2 considering diurnal variations. Nonlinear Process Geophys. 23, 127–136. Yao, Y., Xu, C., Shi, J., Cao, N., Zhang, B., Yang, J., 2015. ITG: a new global GNSS tropospheric correction model. Sci. Rep. 5, 10273. Yao, Y., Xu, X., Xu, C., Peng, W., Wan, Y., 2019. Establishment of a real-time local tropospheric fusion model. Rem. Sens. 11 (11), 1321. Ye, S.R., Zhang, S.C., Liu, J.N., 2008. J. Wuhan Univ. (Nat. Sci. Ed.) vol. 33, 788–791. Precision Single Point Positioning Method Is Used to Estimate the Accuracy of Tropospheric Delay. Zhang, B., Liu, L., Khan, S.A., van Dam, T., Zhang, E., Yao, Y., 2017. Transient variations in glacial mass near Upernavik Isstrøm (west Greenland) detected by the combined use of GPS and GRACE data. J. Geophys. Res.: Solid Earth 122 (12), 10–626. Zhao, Q., Yao, Y., Yao, W., 2019a. Studies of precipitable water vapour characteristics on a global scale. Int. J. Rem. Sens. 40 (1), 72–88. Zhao, Q., Yao, Y., Yao, W., Zhang, S., 2019b. GNSS-derived PWV and comparison with radiosonde and ECMWF ERA-Interim data over mainland China. J. Atmos. Sol. Terr. Phys. 182, 85–92.
Declaration of competing interest The authors declare no conflict of interest. Acknowledgments The author would like to thank IGRA for providing access to the webbased IGRA data. The Crustal Movement Observations Network of China (CMONOC) is also acknowledge for providing GNSS data and ZTD products. In addition, the GPT2_1w model also can be downloaded from https://github.com/dazhipeter/gpt2_1w.m. This research was sup ported by the National Natural Science Foundation of China (41904036). References Ahn, Y.W., Kim, D., Dare, P., 2006. Local tropospheric anomaly effects on GPS RTK performance. In: Proceedings of ION GNSS 2006, pp. 1925–1935. Askne, J., Nordius, H., 1987. Estimation of tropospheric delay for microwaves from surface weather data. Radio Sci. 22 (3), 379–386. Bałdysz, Z., Nykiel, G., Figurski, M., 2015. Investigation of the 16-year and 18-year ZTD time series derived from GPS data processing. Acta Geophys. 63 (4), 1103–1125. B€ ohm, J., Heinkelmann, R., Schuh, H., 2007. Short note: a global model of pressure and temperature for geodetic applications. J. Geodes. 81 (10), 679–683. B€ ohm, J., M€ oller, G., Schindelegger, M., Pain, G., Weber, R., 2015. Development of an improved empirical model for slant delays in the troposphere (GPT2w). GPS Solut. 19 (3), 433–441. Chen, Q.M., Song, S.L., Zhu, W.Y., 2012. An analysis of the accuracy of zenith tropospheric zenith delay calculated from ECMWF/NCEP data over Asia. Chin. J. Geophys. 55 (3), 275–283. Collins, J.P., Langley, R., 1998. The residual tropospheric propagation delay: how bad can it get?. In: Proceedings of ION GPS, vol. 11. Institute of Navigation, pp. 729–738. Collins, J.P., Langley, R.B., 1997. A Tropospheric Delay Model for the User of the Wide Area Augmentation System. Department of Geodesy and Geomatics Engineering, University of New Brunswick. Davis, J.L., Herring, T.A., Shapiro, I.I., Rogers, A.E.E., Elgered, G., 1985. Geodesy by radio interferometry: effects of atmospheric modeling errors on estimates of baseline length. Radio Sci. 20 (6), 1593–1607.
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Research Paper
Improved GPT2w (IGPT2w) model for site specific zenith tropospheric delay estimation in China Zheng Du a, Qingzhi Zhao a, *, Wanqiang Yao a, Yibin Yao b a b
College of Geomatics, Xi’an University of Science and Technology, Xi’an, 710054, China School of Geodesy and Geomatics, Wuhan University, Wuhan, 430079, China
A R T I C L E I N F O
A B S T R A C T
Keywords: GNSS ZTD residual GPT2w IGPT2w Periodic signals
Despite being a systematic error source in global navigation satellite system (GNSS) positioning and navigation, tropospheric delay is a key parameter in GNSS meteorology. Therefore, deriving a zenith tropospheric delay (ZTD) value as accurately as possible from an empirical model without using auxiliary data is a prerequisite for the high-precision application of GNSS positioning and navigation. To reach the goal above, this paper proposes an improved model to estimate the ZTD based on the Global Pressure and Temperature 2 wet (GPT2w) model, which is called the improved GPT2w (IGPT2w) model. The GPT2w-derived ZTD is first calculated as the initial value of the IGPT2w model, and the time series of ZTD residual can thus be obtained between the GNSS- and GPT2w-derived ZTDs over GNSS stations. Analysis of the long time series variation of ZTD residuals using the multichannel singular spectrum analysis method reveals evident periodic signals. The Lomb–Scargle method is then used to determine the specific values of these periodic signals, and different periods are identified at various GNSS stations. Therefore, a ZTD residual model that considers annual, semi-annual, and seasonal periods is established. The IGPT2w model, in which the ZTD value is obtained by combining the estimated ZTD residual and the GPT2w-derived ZTD, can be acquired. A total of 188 GNSS stations in China throughout 2015 to 2017 are selected to validate the IGPT2w model. In the model, the GNSS-derived ZTD is obtained using the GAMIT/ GLOBK software, and the accuracy is validated using radiosonde data with root mean square and bias of 1.9 and 0.1 cm, respectively, at 33 collocated stations in China. Statistical results reveal that the accuracy of the IGPT2wderived ZTD is improved by 13.7% compared with that of the GPT2w-derived ZTD when the GNSS-derived ZTD is regarded as the reference. Such result indicates that the proposed IGPT2w model outperforms the GPT2w model in China.
1. Introduction Tropospheric delay is a major error source in satellite navigation and positioning but an important parameter in global navigation satellite system (GNSS) meteorology. Therefore, this error should be carefully processed in such applications as GNSS positioning and navigation (Jin et al., 2010). Tropospheric delay is generated when radio signals penetrate the troposphere with values of approximately 25 m at low elevation angles and 2.5 m in the vertical direction (Chen et al., 2012). The zenith tropospheric delay (ZTD) is an average value projected by the value of slant path delay with different azimuth and elevation angles into the vertical direction using a mapping function. ZTD comprises zenith hydrostatic delay (ZHD) and zenith wet delay (ZWD). ZHD can be precisely modeled using the observed surface pressure. Meanwhile,
describing ZWD is difficult because it is highly affected by tempora l–spatial variations in atmospheric water vapor. Therefore, the ZTD model cannot be indirectly described by combining ZHD and ZWD models. Nevertheless, certain empirical models directly estimate the ZTD, thereby improving the accuracy of GNSS navigation and positioning. In general, three types of ZTD models, including very-long-baseline interferometry, real-time kinematic, and differential global positioning system, are commonly used for tropospheric corrections in GNSS (Ahn et al., 2006; Song et al., 2016; Kim et al., 2017; Yao et al., 2019). The first type involves established models based on real-time GNSS-derived ZTDs in regional networks. Tao (2008) estimated ZWD parameters using real-time orbit and clock products provided by the Jet Propulsion Lab oratory (JPL) based on precise point positioning (PPP) technique. Ye
* Corresponding author. E-mail address: [email protected] (Q. Zhao). https://doi.org/10.1016/j.jastp.2020.105202 Received 30 November 2019; Received in revised form 31 December 2019; Accepted 2 January 2020 Available online 28 January 2020 1364-6826/© 2020 Elsevier Ltd. All rights reserved.
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et al. (2008) highlighted the feasibility of this method, and the accurate ZTD was estimated using the PPP technique. Generation strategies for real-time and near-real-time tropospheric products were respectively developed using Kalman filtering and backward smoothing based on the PPP technique by Dou�sa et al. (2018). The second type comprises ZTD models based on single/multi-meteorological parameters obtained from the ground, and the commonly used models include Saastamoinen (1972), Hopfield (1969), and Ifadis (1986) models. The ZTD accuracy estimated by these models ranges from the decimeter-to centi meter-level under different conditions (Yao et al., 2019). For example, the ZTD accuracy estimated by the Hopfield model is low at stations with high elevation because this model is subject to elevation. By contrast, the Saastamoinen model is slightly affected by station height. The third type includes ZTD empirical models by which the ZTD parameter can be estimated without any auxiliary information. Several examples of series models include the University of New Brunswick (Collins and Langley, 1997, 1998), global pressure and temperature (Bohm et al., 2007; Lagler et al., 2013; Bohm et al., 2015), IGGtrop (Li et al., 2012, 2015), TropGrid (Krueger et al., 2004; Schüler, 2014), global ZTD (GZTD) (Yao et al., 2013, 2016) and Global Tropospheric Model (GTrop) (Sun et al., 2019) series models. Currently, the ZTD accuracy estimated by empirical models is approximately 4 cm on a global scale (Yao et al., 2015). Although the accuracy of ZTD derived from empirical models has significantly improved, further increasing the accuracy of obtaining the ZTD parameter remains a research point in GNSS positioning and nav igation and GNSS meteorology, especially for regional areas. To achieve the aforementioned goal, this study proposes an improved GPT2w (IGPT2w) model for estimating ZTD parameters to further improve the quality of the GPT2w model in China. In the IGPT2w model, the GPT2wderived ZTD is considered the initial value, and the time series of ZTD residual is modeled considering the periodic signals. Therefore, the final ZTD parameter of IGPT2w is obtained by combining the GPT2w-derived ZTD and the ZTD residual estimated from the established residual model. This paper is organized as follows. Section 2 introduces the GPT2w model and the retrieval of the ZTD parameter. Section 3 de scribes the experimental description and establishment of the IGPT2w model. Section 4 presents the analysis and validation of the IGPT2w model. Finally, Section 5 provides the conclusion.
horizontal direction. P ¼ P0 ⋅e
gm ⋅dM Rg ⋅Tv ðh
h0 Þ
Tv ¼ T0 ð1 þ 0:6077⋅QÞ T ¼ T0 þ dT*dh
(1)
e0 ¼ Q*P0 =ð0:622 þ 0:378*QÞ=100 e ¼ e0 ðP*100=P0 Þλþ1 where P0 , T0 , and e0 are the pressure, temperature, and water vapor pressure, respectively, at the grid point; P, T, and e are the pressure, temperature, and water vapor pressure, respectively, at the target location; gm is gravity and has a value of 9.80665 m/s2; dM is the molar mass of dry air with a value of 28.965 � 10 3 kg/mol; Rg is the universal gas constant with a value of 8.3143 J/K/mol; Tv is the virtual temper ature (Kelvin); Q is the specific humidity; dT is the temperature lapse rate (degree/km);λ is the water vapor decrease factor; h and h0 represent station height and grid height, respectively. 2.2. Retrieval of ZTD Only the nine meteorological parameters can be obtained from the GPT2w model. Therefore, the ZTD derived from GPT2w must be calculated. ZTD comprises ZHD and ZWD. In this study, ZHD and ZWD are calculated separately based on corresponding formulas and then added together to obtain the ZTD. The ZHD parameter is calculated as follows (Davis et al., 1985): ZHD ¼
1
0:0022768⋅P 0:00266⋅cos 2 ϕ 0:00028⋅h
(2)
where P is the surface pressure (hPa), h is the geoid height of the station (km), and ϕ is the latitude of station (radian). ZWD can be calculated based on the formula proposed by Askne and Nordius (1987): � � k3 Rd 0 ZWD ¼ 10 6 k 2 þ es Tm ðλ þ 1Þgm � (3) k’2 ¼ k2 k1 Mw Md
2. GPT2w model and retrieval of ZTD parameter
Rd ¼ R=Md
2.1. Description of the GPT2w model
where R is the molar gas constant with a value of 8.314 J/(mol K); Mw and Md are the molar masses of water and dry air with values of 18.0152 and 28.9644 g/mol, respectively; k1 , k2 , and k3 are refractivity formula constants with values of 77.604 � 0.014 K/mbar, 64.79 � 0.08 K/mbar, and (3.776 � 0.004) � 105 K2 /mbar, respectively; Rd denotes the spe cific gas constant for the dry constituents; gm is the gravity acceleration at the mass center of the vertical column of the atmosphere; Tm is the weighted mean temperature; and λ is the water vapor reduction factor. According to the above formulas, only four of the nine meteorological parameters derived from GPT2w (P, e, Tm , and λ) are required to calculate the ZTD in this work.
The GPT2w model is an empirical model that does not use auxiliary information and considers the annual and semi-annual amplitudes of certain meteorological parameters, such as pressure, temperature, and water vapor pressure. This model was established based on the monthly mean pressure level data derived from the fourth product of European Center for Medium-Range Weather Forecasts (ECMWF ERA-Interim) reanalysis data with a horizontal resolution of 1� � 1� throughout 2001 to 2010, which can be downloaded from (http://ggosatm.hg. tuwien.ac.at/DELAY/SOURCE/GPT2w/MATLAB/gpt2_1w.m). Only the ellipsoidal coordinates (latitude, longitude, and height) of GNSS stations and the corresponding modified Julian data are required as the input information because the GPT2w model is a blinded model. The nine outputs are as follows: hydrostatic and wet mapping function co efficients, temperature in degrees Celsius and its lapse rate in degrees per km, mean temperature of the water vapor in Kelvin, pressure in hPa, water vapor pressure in hPa, water vapor decrease factor, and geoid €hm et al., 2015). The specific procedure of GPT2w undulation in m (Bo for estimating the corresponding parameters is performed as follows. (a) The four grid points surrounding the input location are first determined. (b) The corresponding parameters at the desired height are calculated using the data of the four grid points at the given grid height based on Eq. (1). (c) The parameters from the four grid points are interpolated to the target location using the bilinear interpolation method in the
3. Improved GPT2w (IGPT2w) model for ZTD estimation 3.1. Experiment description A total of 188 GNSS stations in China are selected from 2014 to 2017 for the experiment. The GNSS observations are processed using the GAMIT/GLOBK (Ver. 10.4) software, and the elevation cutoff mask of 10� is selected in this research. The tropospheric parameters of ZTD and gradients in the east/west and north/south directions are both estimated with an interval of 1 h. The specific configurations of GAMIT software have been presented in Table 1. Additionally, the accuracy of the GNSSderived ZTD is compared with that obtained from the radiosonde data at 2
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height less than 100 m between the GNSS and radiosonde stations are regarded as a criterion for selecting the collocated stations. Therefore, 33 collocated stations in China are chosen, and Fig. 2 provides the root mean square (RMS) and bias of the ZTD differences derived from GNSS and radiosonde. The long-term ZTD residuals between GNSS-derived and radiosonde-derived ZTD are first obtained at collocated stations and then calculated the RMS value of each station. It can be observed from Fig. 2 (a) that the RMS value of the ZTD difference derived from the GNSS observation ranges from 0 cm to 3 cm when compared with radiosonde data. The average RMS and bias are approximately 1.9 and 0.1 cm, respectively, for the 33 selected stations from 2015 to 2017.
Table 1 Some configurations of GAMIT software for GNSS processing. Term
Value
Reference frame
Troposphere
Satellite quality center correction Satellite altitude angle Tide model
Inertial coordinate system Axial precession Nutation Prior zenith tropospheric delay Interval Mapping function Bock I Bock II/IIA Bock IIR
J2000.0 Ifadis, 1986 Ifadis, 1986 0.5m hourly HGMF、DGMF x,y,z: 0.2100, 0.0000, 0.8540m x,y,z: 0.2790, 0.0000, 1.2300m x,y,z: 0.0000, 0.0000, 0.0000m 10� IERS2003 model; FES2004 model and Polar tide model
3.2. Time series analysis of ZTD residual After the ZTD values derived from GPT2w and GNSS are calculated over the 188 GNSS stations, the time series of the ZTD residual between GPT2w and GNSS can be further obtained. Examples of time series of ZTD residuals between GPT2w and GNSS at XJML, XJSS, and XJKE stations throughout 2015–2017 are presented in Fig. 3. In this figure, the existence of periodic signals is possibly observed in the time series of
the collocated stations. The distribution of GNSS and radiosonde stations is presented in Fig. 1. Herein, the horizontal distances in latitudinal and longitudinal directions less than 30 km, and the vertical difference in
Fig. 1. Geographical distribution of GNSS and radiosonde stations selected in China over throughout 2015–2017.
Fig. 2. Comparison of RMS and bias of ZTD difference between GNSS and radiosonde throughout 2015–2017 at 33 collocated stations in China. 3
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Fig. 3. Time series of ZTD difference between GNSS and GPT2w at stations XJML, XJSS, and XJKE throughout 2015–2017.
Fig. 4. First RC (mm) corresponding to the 10 largest eigenvalues when MSSA method is applied to the time series of ZTD residual at XJKE throughout 2015–2017.
ZTD residuals for the three selected GNSS stations. At present, spectrum analysis is widely used to discuss the periodic characteristics of ZTD (Jin et al., 2009; Li et al., 2012; Bałdysz et al., 2015). The multi-channel singular spectrum analysis (MSSA) method, which is an extension of singular spectrum analysis (SSA), is first introduced to verify the exis tence of periodic signals in the time series of ZTD residuals (Oropeza and Sacchi, 2011). The SSA method is designed to extract useful information from a noisy time series, whereas its multivariate form MSSA is effective in determining the periodic variations of time series (Zhang et al., 2017). The theory of MSSA can be summarized as follows: if the time series of ZTD residual includes n complex exponentials, then the corresponding Hankel matrix of the time series is a matrix of rank n. The missing data and noise error will increase the rank of the Hankel matrix, and this issue can be resolved only by rank reduction (Hua, 1992). Fig. 4 shows the first reconstructed components (RC) (mm) corresponding to the 10 largest eigenvalues when the MSSA method is applied to the time series of the ZTD residual at the XJKE station. The figure depicts the evident existence of periodic signals in the time series of the ZTD residual; semi-annual, seasonal, and annual periods appeared in RC1-2, RC4-5,
and RC6, respectively. However, the specific values of periodic signals cannot be determined by the MSSA method, and this incapability is the defect of this method. Therefore, the Lomb–Scargle periodogram method is used to deter mine the specific values of periodic signals of ZTD residuals between GNSS and GPT2w at the selected GNSS stations. The periodogram method introduced in this paper, which can handle time series with gaps or uneven sampling intervals (Zhao et al., 2019a), is similar to Fourier spectrum analysis. Fig. 5 presents the periodic variations for XJML, XJSS, and XJKE stations using the Lomb–Scargle periodogram method throughout 2015–2017. Findings show the following: only the annual period exists at XIML station; the annual and semi-annual periods exist at XJSS station; and the annual, semi-annual, and seasonal periods exist at XJKE station. Such results further verify the existence of some periods in the time series of the ZTD residuals. Additionally, the time series of the ZTD residuals at different GNSS stations in China have different periodic signals. Additionally, the Lomb–Scargle periodogram method is used to analyze the time series of ZTD residuals at the 188 selected GNSS 4
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Fig. 5. Periodic variations for XJML, XJSS, and XJKE stations obtained using the Lomb–Scargle periodogram method, in which the first to third columns represent the annual, semi-annual, and seasonal periods.
stations have three types of periods. Additionally, it can be observed from Fig. 6 that the distribution of GNSS stations with different types of periods for ZTD residuals exhibits certain regional features. Such phe nomenon is mainly related to the distribution of atmospheric water vapor (Zhao et al., 2019b).
Table 2 Number of GNSS stations and the corresponding periods, where A, B, C, and D represent annual, semi-annual, and seasonal periods with 120 and 90 days, respectively. Type
Number
AþBþCþD BþCþD AþCþD CþD AþC AþBþD AþBþC BþC BþD AþB AþD A B C
27 25 25 23 21 16 11 8 2 8 5 6 8 3
3.3. Establishment of ZTD residual model Owing to the existence of some periodic signals in the time series of ZTD residuals between GNSS and GPT2w at GNSS stations, fitting those values is possible by establishing the corresponding periodic model. Herein, a periodic model of the time series of ZTD residuals of 188 GNSS stations is established considering the annual, semi-annual, and seasonal periods and the average of ZTD residuals. Finally, the periodic model of ZTD residuals can be determined as a combination of four periodic items and the average term, which is expressed as follows: � � � � doy doy 2π ϕ1 þ A2 cos 4π ϕ2 þ … ΔZTD ¼ A0 þ A1 cos 365:25 365:25 � � � � doy doy A3 cos 6π ϕ3 þ A4 cos 8π ϕ4 365:25 365:25 (4)
stations in China, and the statistical results further reveal that different stations show various periodic signals. Table 2 shows the number of GNSS stations and the corresponding periods. Fig. 6 provides the first six distributions of GNSS stations with the maximum number of period. Table 1 reveals that the time series of ZTD residuals at most GNSS sta tions has at least two types of periods, and more than 50% of GNSS
where ΔZTD refers to the ZTD residual; A0 is the average of ZTD re sidual; A1 A4 represent the annual, semi-annual, and seasonal periods of 120 and 90 days, respectively, while ϕ1 ϕ4 refer to their 5
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Fig. 6. First six distributions of GNSS stations with the maximum number of period.
corresponding phases. These unknown parameters are estimated by the least square method in this study. Therefore, the final IGPT2w model can be obtained by combining the initial value of ZTD derived from the GPT2w model and the ZTD residual estimated by the fitted model in Eq. (4).
derived/GTrop-derived ZTD. Such result indicates the good perfor mance of the established IGPT2w model, and a good consistency existed between IGPT2w and Gtrop models. In addition, Table 3 gives the sta tistical result of ZTD accuracy based on different models in China. It can be concluded that IGPT2w model has the highest accuracy in China, Gtrop model is the second, and GPT2w model is the worst. The accuracy of the IGPT2w model proposed has been improved by approximately 10% when compared with GTrop-ZTD model in China.
4. Validation of IGPT2w model for ZTD estimation The quality of the established IGPT2w model is the key to evaluating its further usage. Therefore, the fitted ZTD residual model and the ZTD time series estimated by the IGPT2w model are first compared at three GNSS stations. Afterward, the accuracy of the established IGPT2w model and its improvement rate are further analyzed. In this work, two indexes are introduced to evaluate the established IGPT2w model, namely, RMS and mean absolute error (MAE), which are computed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m �2 1 X RMS ¼ ZTDmodel ZTDGNSS i i m i¼1 (5) m � � 1 X model GNSS �ZTD MAE ¼ ZTDi � i m i¼1
4.2. Validation of internal and external accuracies of IGPT2w model The above analysis presents only the result of the IGPT2w model at the three selected GNSS stations and thus cannot reflect the total accu racy of the IGPT2w model in China. Therefore, the validation of internal and external accuracies of the IGPT2w model is further analyzed to evaluate its performance in entire China. In this section, only the ZTD data derived from GNSS and the GPT2w model throughout 2015–2016 are used to establish the IGPT2w model, while the ZTD data for the year 2017 are used to perform the external validation at 188 GNSS stations. Fig. 8 shows the internal/external validation of the IGPT2w model (first and second rows, respectively); the first and second columns are the RMS and MAE, respectively. The values of RMS and MAE decrease from the southeast to the northwest of China, and this pattern corresponds to the distribution of atmospheric water vapor in the country (Zhao et al., 2019b). The maximum values of RMS and MAE appear in the southeast of China for the internal and external validations. In addition, Table 4 provides the statistical results regarding the internal/external accuracies of GPT2w and IGPT2w models. The accuracy of the ZTD derived from the IGPT2w model (approximately 2.2 cm) is improvement to that from the GPT2w model (approximately 2.5 cm) when the GNSS-derived ZTD is considered to be the reference.
where ZTDmodel refers to the ZTD value derived from the GPT2w or IGPT2w model; ZTDGNSS is the ZTD value derived from GNSS observa tions, which is estimated using GAMIT software and considered as the reference in the comparison; and m is the total number of ZTD pairs. 4.1. Analysis of fitted ZTD residual model To analyzed the fitted ZTD residual model at different stations, an example of ZTD residuals is given in three selected stations randomly in Fig. 7. The ZTD residual model is first presented at stations XJML, XJSS, and XJKE in this section (first column of Fig. 7). Findings show that the fitted model can effectively describe time series variation of ZTD re siduals. In addition, the ZTD time series derived from GNSS, GPT2w, GTrop and IGPT2w are presented at these stations (second column of Fig. 7). It can be observed from Fig. 7 that The IGPT2w-derived ZTD agrees well with that from GNSS when compared with the GPT2w-
4.3. Improvement rate of IGPT2w for ZTD estimation The improvement rate, which refers to the proportion between the absolute value of RMS difference between IGPT2w and GNSS and the RMS value of GPT2w, is introduced in this section to further assess the performance of the proposed IGPT2w model. Herein, the coefficients of 6
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Fig. 7. ZTD residual model and the ZTD time series derived from GNSS, GPT2w, GTrop and IGPT2w at XJML, XJSS, and XJKE stations throughout 2015–2017; the first column represents the ZTD residual model, and the second column refers to the ZTD time series.
difference between GNSS-GPT2w and GNSS-IGPT2w models are calcu lated. Finally, the improvement rate of the IGPT2w model for ZTD estimation compared with GPT2w can be obtained (Fig. 9); the com parison reveals that the accuracy of estimated ZTD values at 188 GNSS stations is improved to different degrees. Statistical results reveal that the average improvement rate of the 188 stations in China is 12.0%, which indicates the good performance of the proposed IGPT2w model. In addition, the frequency distributions of the RMS values derived from GPT2w, GTrop and IGPT2w at 188 GNSS stations are analyzed (Fig. 10). These distributions show that most GNSS stations derived from the IGPT2w model with RMS values less than 3 cm account for 87.2% of all GNSS stations. Meanwhile, stations derived from the GPT2w model with RMS values less than 3 cm account for only 70.7% of all stations. Findings further reveal that the average RMS/MAE values of the ZTD difference between GNSS-GPT2w and GNSS-IGPT2w at the 188 selected GNSS stations over the entire 2017 (2015–2017) are 2.58/2.03 cm and
Table 3 Statistical accuracy of ZTD estimated by three models in China (Unit: cm). Model
RMS
MAE
GPT2w GTrop IGPT2w
2.58 [1.13,5.80] 2.52 [1.16, 5.67] 2.23 [0.95,5.44]
2.03 [0.88, 4.87] 1.98 [0.92, 4.74] 1.67 [0.69, 4.42]
the ZTD residual model in Eq. (4) are first estimated using the least square method with the data throughout 2015–2016 and adopted to estimate the ZTD residual in the year 2017. The initial ZTD value derived from the GPT2w model for 2017 is then calculated. Finally, the ZTD value derived from the IGPT2w model can be obtained by combining the estimated ZTD residual and the GPT2w-derived ZTD. The GPT2w- and IGPT2w-derived ZTD values are compared with those from 188 GNSS stations over the entire 2017, and the RMS values of the ZTD 7
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Fig. 8. Validations of the IGPT2w model in China at 188 stations, where the first and second rows represent the internal/external validation, respectively while the first and second columns are the RMS and MAE, respectively. Table 4 Statistical result of internal/external accuracies of GPT2w and IGPT2w models in China when compared with the GNSS-derived ZTD (unit: cm). Model
Type Internal
GPT2w IGPT2w
External
RMS
MAE
RMS
MAE
2.61 2.25
2.06 1.72
2.53 2.23
1.98 1.67
Fig. 10. Frequency distributions of RMS derived from GPT2w, GTrop and IGPT2w at 188 GNSS stations in China.
2.23/1.67 cm, respectively. Such a result further indicates the capability of the IGPT2w model to improve the accuracy of the GPT2w model by 13.7% for ZTD estimation in China. 4.4. Accuracy of planar distribution for IGPT2w model under different seasons The two-dimensional (2D) distributions of ZTD residuals between GNSS-GPT2w and GNSS-IGPT2w and the improvement rate of the IGPT2w model compared with the GPT2w model under different sea sons are presented in Fig. 11 to analyze the accuracy of IGPT2w model in entire China under different seasons. The four seasons are divided based on the standard meteorological practice: spring is from March to May, summer is from June to August, autumn is from September to
Fig. 9. Improvement rate of IGPT2w model for ZTD estimation at 188 GNSS stations compared with GPT2w model in China over the entire 2017.
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Fig. 11. 2D distributions of ZTD residuals; the first and second columns refer to the ZTD residuals between GNSS-GPT2w and GNSS-IGPT2w, respectively, and the third column shows the improvement rate of the IGPT2w model compared with the GPT2w model under the four seasons.
November, and winter is from December to February. Fig. 11 shows that the 2D distribution of the ZTD residual between GNSS and IGPT2w models is smaller than that between GNSS and GPT2w models, espe cially in summer and autumn. Additionally, the third column of Fig. 11 shows that the improvement rate of the IGPT2w model is concentrated in summer and autumn. Such result indicates that the established IGPT2w model performs better than the GPT2w model in these seasons, but the capabilities of the GPT2w and IGPT2w models for estimating ZTD in China are similar in spring and winter. The reason for such phenomenon may be related to the distribution of atmospheric water vapor in China. Water vapor fluctuates greatly in summer and autumn, but less in winter and spring (Wong et al., 2015), and the established IGPT2w model can be used to correct those large fluctuations in summer and autumn. Therefore, the IGPT2w model is found better than GPT2w model in summer and autumn. In spring and winter, the water vapor fluctuation is inactive and the ZTD difference is small, therefore, the improvement of IGPT2w model is not clear but it also has some improvement.
5. Conclusion The IGPT2w model is proposed in this study for ZTD estimation in China. The GPT2w-derived ZTD is regarded as the initial value of the IGPT2w model. The MSSA and Lomb–Scargle periodogram methods are introduced to analyze the ZTD residuals between GNSS and GPT2w model, and determine the existence of some periodic items. Afterward, a corresponding ZTD residual model is established by considering the influence of annual, semi-annual, and seasonal periods. Finally, the ZTD of the IGPT2w model can be obtained by combining the initial ZTD value derived from the GPT2w model and the ZTD residual estimated from the established residual model. A total of 188 GNSS stations and 33 collocated radiosonde stations in China throughout 2015–2017 are selected for the experiment. The GNSS observations are processed by the GAMIT/GLOBK software, and the RMS and bias of the GNSS-derived ZTD are 1.9 cm and 0.1 cm, respec tively, when compared with the 33 collocated radiosonde stations. GNSS data of 2015–2016 are used to establish the IGPT2w model and applied for the ZTD estimation for the entire 2017. External validation shows 9
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that the IGPT2w model outperforms the GPT2w model. When the GNSSderived ZTD is regarded as the reference, the average RMS and MAE of the ZTD difference of GNSS-IGPT2w (2.23/1.67 cm) are lower than those of GNSS-GPT2w (2.58/2.03 cm). Furthermore, 2D comparison of the IGPT2w model over entire China reveals that the improvement rate of ZTD estimation mainly appears in summer and autumn when compared with that of the GPT2w model, but the capacities of these models are similar in spring and winter. This finding is due to the relatively large ZTD value in summer and autumn and the failure of only the blinded model (GPT2w) to describe the ZTD variation well. There fore, the ZTD residual model plays a supplementary role in the IGPT2w model, especially for cases with relatively large atmospheric water vapor variations in summer and autumn.
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