Towards the tropospheric ties in the GPS, DORIS, and VLBI combination analysis during CONT14

Abstract

In space geodetic data analysis, improving tropospheric delay modelling is motivated by the correlation between tropospheric zenith total delay (ZTD) and the station height. The gradients are correlated with the horizontal displacements. In the microwave techniques such as GPS, DORIS, and VLBI, the tropospheric delay effects are correlated over the collocation sites. This correlation allows for the estimation of common tropospheric parameters, often referred to as tropospheric ties. These tropospheric ties provide valuable complementary information for the computation of the terrestrial reference frame (TRF) and have been the subject of investigation in many studies. In this study, we investigate the effects of tropospheric ties on the daily TRF combination at the observation level using a batch least-squares estimation. The observations of GPS, DORIS, and VLBI were collected from 06 May 2014 to 20 May 2014 during the CONT14 campaign of VLBI. The tropospheric delay and gradient ties are computed using different numeric weather prediction (NWP) data sets provided by ECMWF and NCEP. We examined different levels of tropospheric ties 0.01, 5, and 10 mm for ZTD and 0.001, 0.5, and 1.0 mm for gradients in the combination of techniques. The results show that the combined solution with tropospheric ties derived from the four NWP data sets does not exhibit significant differences. For VLBI, the repeatability of station coordinates and network scale were found to be improved by around 20% and 30%, respectively. The stronger tropospheric ties show a higher improvement in VLBI baseline repeatability. However, applying tropospheric ties at GPS-DORIS collocation sites does not significantly affect the repeatability of station coordinates and network scale. Both ZTD and gradient ties enhance the repeatability of polar motion components in EOPs, while no observable contribution is observed for dUT1 and celestial pole offsets.

Appendices

Appendix A: Computation of local ties from ITRF2014

A weekly local tie constraint $\Delta {\varvec{L}}_{ij}$ between stations i and j can be calculated by:

$$\begin{aligned} \Delta {\varvec{L}}_{ij} = {\varvec{L}}_{ij} - (\hat{{\varvec{X}}}_i - \hat{{\varvec{X}}}_j), \end{aligned}$$

(A1)

where ${\varvec{L}}_{ij}$ is the local tie between stations i and j and the $\hat{{\varvec{X}}}$ vectors are the estimated station positions.

Local ties can be observed from local surveys. In this study, both the local ties and the a priori datum are generated from ITRF2014 following two rules: 1) the height difference less than ${100}\,\hbox {m}$ and 2) the horizontal distance less than $10\,\hbox {km}$ between collocation sites. This leads to $\Delta {\varvec{L}}_{ij}={\varvec{0}}$ implicitly introducing additional datum constraints with these local ties in the combination. This strategy has already been applied by Glaser et al. (2019) for the investigation of the impact of local ties.

The local tie ${\varvec{L}}_{ij}$ between stations i and j at the epoch t reads as:

$$\begin{aligned} {\varvec{L}}_{ij} = {\varvec{X}}_i(t) - {\varvec{X}}_j(t), \end{aligned}$$

(A2)

where ${\varvec{X}}$ vectors are the positions of stations i and j expressed in ITRF2014. The station positions ${\varvec{X}}(t)$ at the epoch t are propagated from the station coordinates at the reference epoch $t_0$

$$\begin{aligned} {\varvec{X}}(t) = \begin{bmatrix} {\varvec{I}}_{3}&(t-t_0){\varvec{I}}_{3} \end{bmatrix} \cdot \begin{bmatrix} {\varvec{X}} \\ \dot{{\varvec{X}}} \end{bmatrix} = {\varvec{K}} \begin{bmatrix} {\varvec{X}} \\ \dot{{\varvec{X}}} \end{bmatrix}, \end{aligned}$$

(A3)

where ${\varvec{I}}_{3}$ is the identity matrix with the dimension of $3 \times 3$.

The variance–covariance matrix of $\Delta {\varvec{L}}_{ij}$ is propagated from the variance–covariance matrices of positions and velocities for stations i and j. Let $\varvec{\Sigma }$ be the variance–covariance matrix of $6 \times 6$ size for ${\varvec{X}}$ and $\dot{{\varvec{X}}}$, the variance–covariance of $\Delta {\varvec{L}}_{ij}$ can be computed by:

$$\begin{aligned} {\varvec{\Sigma }}_L = \begin{bmatrix} {\varvec{K}}&-{\varvec{K}} \end{bmatrix} \begin{bmatrix} \varvec{\Sigma }_{ii} &{} \varvec{\Sigma }_{ij} \\ \varvec{\Sigma }^{\top }_{ij} &{} \varvec{\Sigma }_{jj} \end{bmatrix} \begin{bmatrix} {\varvec{K}}^{\top } \\ -{\varvec{K}}^{\top } \end{bmatrix}. \end{aligned}$$

(A4)

where $\varvec{\Sigma }_{ij}$ is the variance–covariance of positions and velocities between stations i and j. Note that all variance–covariance information is obtained from ITRF2014 and the post-seismic deformation model of ITRF2014 has been considered.

Appendix B: Computation of ZTD and tropospheric ties

The ZTD is the sum of ZHD and ZWD, which are integrated through the hydrostatic refractivity ($N_h$) and wet refractivity ($N_w$), respectively, from the station height (h) along the vertical direction using the NWP data:

$$\begin{aligned} \text {ZHD} = 10^{-6} \int ^{\infty }_{h}{N_h(z) \text {d} z} \approx 10^{-6} \sum _{i}{N_{h,i} \Delta z_i}, \end{aligned}$$

(B1)

$$\begin{aligned} \text {ZWD} = 10^{-6} \int ^{\infty }_{h}{N_w(z) \text {d} z} \approx 10^{-6} \sum _{i}{N_{w,i} \Delta z_i}, \end{aligned}$$

(B2)

where $N_h$ and $N_w$ are functions of air pressure, temperature, and water vapour pressure following Thayer (1974). The geopotential height in the NWP data is first converted to geometric height. The profiles of $N_h$ and $N_w$ are exponentially interpolated in the vertical direction. Another method to compute the ZTD is based on Saastamoinen’s model (Saastamoinen 1972) and the surface meteorological data interpolated from NWP data sets, e.g. ERA and ERA5. The interpolation follows the procedure described in Bock et al. (2010) and Pollet (2011).

The tropospheric ties in this study include the ZWD ties and gradient ties. In the GINS software, the tropospheric parameters $\text {ZWD}=\text {ZTD}-\text {ZHD}_{\text {GPT2}}$ are estimated; therefore, the tropospheric ties in Eq. 3 between stations i and j are calculated by:

$$\begin{aligned} \Delta \text {ZWD}= & {} \left( \text {ZTD}^{NWP}_{i}-\text {ZTD}^{GPT2}_{i}\right) \nonumber \\{} & {} - \left( \text {ZTD}^{NWP}_{j}-\text {ZTD}^{GPT2}_{j}\right) \nonumber \\= & {} \text {ZWD}_{i} - \text {ZWD}_{j}. \end{aligned}$$

(B3)

The NWP data provide the atmospheric profile, which can be interpolated to calculate the ZTD considering the height difference. Hence, the ZWD ties applied in our study include the topography-related tropospheric delay difference.

For gradients, the difference between collocation sites calculated by the GPT3/VMF3 and GRAD grid-based product from VMF Data Server is very close to 0, suggesting that the gradients are not sensitive to the height difference. As a result, the gradients ties in Eq. 4 are set to 0 with given standard deviations.

Appendix C: Calculation of repeatability

The repeatability in this study is calculated as the weighted standard deviation (WSTD):

$$\begin{aligned} \text {WSTD} = \sqrt{\frac{ \sum { (X_t - {\bar{X}})^2 / \sigma _t^2}}{\sum {1/\sigma _t^2}} }, \end{aligned}$$

(C1)

where $X_t$ is the value at epoch t and $\sigma _t$ is the corresponding standard deviation, ${\bar{X}}$ is the weighted mean value:

$$\begin{aligned} {\bar{X}} = \frac{ \sum { X_t / \sigma _t^2}}{\sum {1/\sigma _t^2}}. \end{aligned}$$

(C2)