Multivariate analysis of GPS position time series of JPL second reprocessing campaign

Abstract

The second reprocessing of all GPS data gathered by the Analysis Centers of IGS was conducted in late 2013 using the latest models and methodologies. Improved models of antenna phase center variations and solar radiation pressure in JPL’s reanalysis are expected to significantly reduce errors. In an earlier work, JPL estimates of position time series, termed first reprocessing campaign, were examined in terms of their spatial and temporal correlation, power spectra, and draconitic signal. Similar analyses are applied to GPS time series at 89 and 66 sites of the second reanalysis with the time span of 7 and 21 years, respectively, to study possible improvements. Our results indicate that the spatial correlations are reduced on average by a factor of 1.25. While the white and flicker noise amplitudes for all components are reduced by 29–56 %, the random walk amplitude is enlarged. The white, flicker, and random walk noise amount to rate errors of, respectively, 0.01, 0.12, and 0.09 mm/yr in the horizontal and 0.04, 0.41 and 0.3 mm/yr in the vertical. Signals reported previously, such as those with periods of 13.63, 14.76, 5.5, and 351.4 / n for $n=1,2,{\ldots },8$ days, are identified in multivariate spectra of both data sets. The oscillation of the draconitic signal is reduced by factors of 1.87, 1.87, and 1.68 in the east, north and up components, respectively. Two other signals with Chandlerian period and a period of 380 days can also be detected.

Appendices

Appendix 1: models employed within the second IGS reanalysis campaign

1.1 Yaw attitude variations

Inconsistent yaw attitude models affects the precision of the IGS combined clock solutions (Hesselbarth and Wanninger 2008). Therefore, the reliability of the IGS combined clocks is impaired. To diminish the effect of the eclipsing satellites on the IGS clock solutions, consistent modeling of attitude changes is needed (Ray 2009). Distortions in the orientation of the eclipsing satellites follow a simplified yaw attitude model for Block II/IIA and Block IIR satellites (see Kouba 2009a). Attitude behavior of the Block IIF-1 (launched on May 27, 2010) spacecraft during the eclipse has been studied by Dilssner (2010). In addition, with complete modernization of the GLONASS satellites, ACs should include GLONASS observations as well. An appropriate yaw attitude modeling of these satellites may follow the model proposed by Dilssner et al. (2011).

1.2 Modeling of orbit dynamics

Urschl et al. (2007) observed anomalous pattern in the plot of GPS-SLR residuals which they attributed to the GPS orbit mismodeling. This anomalous pattern (particularly, the GPS draconitic year signal) was also identified in the geocenter Z-component (Hugentobler et al. 2006) and GPS position time series (Ray et al. 2008).

One of the potential sources for GNSS orbit mismodeling is the deficiencies in the Earth radiation pressure (ERP) model. Not all IGS ACs are yet modeling ERP. Utilizing a model for Earth radiation, proposed by Rodriguez-Solano et al. (2012), results in the reduction in root mean square (RMS) of orbit’s height component by about 1–2 cm and smaller perturbations of other components related to the orbit. Rodriguez-Solano et al. (2012) showed that the model can compensate the SLR residual bias observed.

The GPS orbit perturbations due to ERP depend on the relative position of Sun, Earth, and satellite. Parts of the observed periodic patterns in GPS time series may stem from failure to correctly model ERP (Rodriguez-Solano et al. 2012). They found that the inclusion of the ERP model results in the reduction in the sixth draconitic signal for the north component at a submillimeter level (equal to reduction of around 38%). Ray (2009) also suggested taking ERP into consideration. Hence, the model proposed by Rodriguez-Solano et al. (2011) has been used within the IGS in the operational reprocessing.

Earth albedo radiation (EAR) is another source for orbit modeling deficiencies. This radiation consists of both visible reflected light and infrared emitted radiation. Most AC contributors have not taken into account the effect of EAR. The albedo acceleration may have a significant effect on the orbit of GPS satellites (a mean reduction in the orbit radial component by 1–2 cm) (Hugentobler et al. 2009). They concluded that for the high-precision GPS orbit determination, EAR and antenna thrust should be taken into consideration. However, regarding the spectra of geocenter and position time series, no significant impact has been observed when the model for EAR was used (Hugentobler et al. 2009). This indicates that there could be still unmodeled effects on the GPS orbit which can be larger than the albedo radiation.

1.3 Geopotential field

In terms of the geopotential model, a new model referred to as EGM2008 has been defined (see Ray 2009). EGM2008 exhibits significant improvements compared to its previous counterpart EGM96, thanks to the availability of CHAMP and most importantly GRACE data in the 2000s. Compared to EGM96, used for the 1st processing campaign, EGM2008 has been modified in the following aspects:

• 1. Its degree and order have been increased by a factor of 6.

• 2. Updated value for secular rate of low-degree coefficients.

• 3. A new model for the mean pole trajectory was proposed.

• 4. Model for geopotential ocean tide has been updated for FES2004.

• 5. A new ocean pole tide model has been introduced.

For more information, the reader is referred to IERS 2010 conventions (Petit and Luzum 2010).

1.4 Tidal effects

Tidal effects are categorized to the following two contributions. (1) Tidal displacement of station positions; (2) Tidal EOP variations. For the former, within the new processing campaign, a new model which is introduced for the mean pole trajectory IERS 2010 (Petit and Luzum 2010) has been used for the pole tide correction. Moreover, model for ocean pole tide loading presented by Desai (2002) should be used. For the latter, the Earth rotation axial component in terms of UT1 contains small diurnal and subdiurnal signals. Thus, the tidal gravitation effect on those features of Earth’s mass distribution results in the astronomical precession-nutation of Earth rotation (Brzeziński 2008). A minor part of the astronomical variations, called libration, is a result of the tidal gravitation effect on the non-zonal terms of geopotential (Brzeziński 2008). In case of UT1, the perturbation is semidiurnal with total amplitude up to $75\,\upmu \hbox {as}$. Brzeziński and Capitaine (2009) studied the subdiurnal libration in UT1. They derived a solution for the structural model of the Earth composing of an elastic mantle and a liquid core not coupling to each other.

A key expectation in tidal EOP variations modeling compared to the 1st reprocessing campaign is the addition of the UT1 libration effect introduced by Brzeziński and Capitaine (2009). It is noted that the maximum effect of UT1 libration is about $105\,\upmu \hbox {as}$, or 13 mm at GPS altitude. It probably severely aliases into the orbit parameters.

1.5 Tropospheric propagation delay

In the second reprocessing, a new slant delay model (GPT2) was suggested. It improves its older models GPT/GMF with refined horizontal resolution, enhanced temporal coverage, and increased vertical resolution (37 isobaric levels compared to 23 ones utilized for GPT/GMF) (Lagler et al. 2013). In addition to mean value, $a_0$, and annual amplitude, A, estimated using the least-squares method in GPT/GMF, semiannual harmonics are incorporated within GPT2. This better accounts for regions with very rainy or dry periods. As for the temperature reduction, in contrast to the GPT/GMF in which a constant $-6.5\,^{\circ }\hbox {C}/\hbox {km}$ was assumed, mean, annual, and semiannual variations of temperature lapse rate are determined each grid point in GPT2. Regarding the pressure reduction, unlike the GPT/GMF which utilizes an exponential formula based on the standard atmosphere, GPT2 deploys an exponential formula based on virtual temperature (Lagler et al. 2013). The improved performance of GPT2 compared to the previous model GPT/GMF has been examined by Lagler et al. (2013). They have recommended to replace GPT/GMF with GPT2 as an empirical model.

Because of the partial compensation of the atmospheric loading by mismodeling the zenith hydrostatic delays (ZHDs) (Kouba 2009b), GPT-derived ZHDs give rise to a better station height repeatability compared to ECMWF ZHDs if atmospheric loading is not corrected for (Steigenberger et al. 2009). On the other hand, if one needs to examine the coordinates time series to reveal atmospheric loading signals, application of ZHDs derived from numerical weather models is a key element.

1.6 Higher-order ionospheric terms

A linear combination of multi-frequency observations allows for taking into consideration the first-order ${\sim }\frac{1}{f^{2}}$ ionospheric term (Hofmann-Wellenhof et al. 2008). The first-order ionospheric delay is in the order of 1 to 50 meters, which depends on the satellite elevation, ionospheric activities, local time, season and solar cycle (Kedar et al. 2003). The higher-order ionospheric terms, which are in the order of submillimeters to several centimeters, are usually neglected. Kedar et al. (2003) stated that the effect of second-order ionospheric term introduced by Bassiri and Hajj (1993) can likely improve the position repeatability and reduce the small biases in geocenter estimates. Fritsche et al. (2005) and Hernández-Pajares et al. (2007) showed that the second-order ionospheric term affects the geocenter Z-component estimates. Fritsche et al. (2005) processed the double difference phase observation of a global network and compared solutions with and without the higher-order ionospheric terms. They concluded that applying these higher terms will became a standard part of precise GPS applications. IERS 2010 conventions (Petit and Luzum 2010) suggested that while the first- and second-order ionospheric terms are to be considered for GNSS applications, the third order is at the limited significance and the fourth order can be neglected.

1.7 Analysis constraints

Ferland (2010) found that high-frequency smoothing may be due to unremovable continuity constraints for some ACs. Ray (2009) suggested that, for the 2nd reprocessing campaign, ACs constraints and procedures should be reconsidered from the following aspects: (1) Reviewing the necessity of applying constraints, (2) Paying particular attention to the constraint on the orbit and UT1/LOD, (3) Elimination and minimization of the constraints as many as possible, and (4) Better understanding of the impacts of constrains retained is necessary. Accordingly, in the IGS2008 recommendations (http://igs.org/overview/pubs/IGSWorkshop2008/), all ACs should report their a-priori constraints. Although removable constraints are acceptable, unconstrained solutions are preferred. Inner constraints (origin, orientation, scale) are acceptable.

Appendix 2: rate errors in multivariate model

Having r time series available, a multivariate linear model is of the form (Koch 1999)

$$\begin{aligned} E\left( {\hbox {vec}\left( Y \right) } \right) =\left( {I_r \otimes A} \right) \hbox {vec}\left( X \right) ,D\left( {\hbox {vec}\left( Y \right) } \right) =Q_{\mathrm {vec}\left( Y \right) }\nonumber \\ \end{aligned}$$

(1)

where vec is the vector operator and $\otimes$ is the Kronecker product. $I_r$ is the identity matrix of size r. X and Y are the matrices of the sizes $n\times r$ and $m\times r$ collecting unknown parameters and observations from r number of series, respectively. A and $Q_{\mathrm {vec}\left( Y \right) }$ are, respectively, the functional and stochastic models describing all deterministic effects and statistical characteristics of the observables. E indicates the expectation operator, and D is the dispersion operator.

The following structure for the stochastic model, referred to as the more practical model, is used (Amiri-Simkooei 2009)

$$\begin{aligned} D\left( {\hbox {vec}\left( Y \right) } \right) ={\Sigma }\otimes Q={\Sigma }\otimes \mathop \sum \nolimits _{k=1}^p s_\mathrm{k} Q_\mathrm{k} \end{aligned}$$

(2)

where $Q_\mathrm{k}$’s are the known cofactor matrices of size $m\times m$. The matrix ${\Sigma }$ and the unknown factors $s_\mathrm{k}$ are to be estimated using LS-VCE.

The least-squares estimate of X reads then (Koch 1999)

$$\begin{aligned} {{\hat{X}}} =\left( {A^{T}Q^{-1}A} \right) ^{-1}A^{T}Q^{-1}Y \end{aligned}$$

(3)

The covariance matrix of the nr-vector$\hbox { vec}\!\left( {{{\hat{X}}}} \right)$ is

$$\begin{aligned} Q_{\mathrm {vec}\left( {{{\hat{X}}} } \right) } ={\Sigma }\otimes \left( {A^{T}Q^{-1}A} \right) ^{-1}={\Sigma }\otimes N^{-1} \end{aligned}$$

(4)

where $N=A^{T}Q^{-1}A$ is the normal matrix. Here, we assume that the functional model contains two columns for the linear regression terms plus two columns for each of the annual, semiannual, and tri-annual signal. A is thus of size $m\times 8$. Its ith row at the time instant $t_i$ is

$$\begin{aligned} \left[ {1 \quad \! {t_i }\quad {\cos 2\pi t_i }\quad \! {\sin 2\pi t_i }\quad \! {\cos 4\pi t_i }\quad \! {\sin 4\pi t_i }\quad \! {\cos 6\pi t_i } \quad \! {\sin 6\pi t_i } } \right] \nonumber \\ \end{aligned}$$

(5)

Therefore, the unknown parameters are the intercept, rate, and amplitudes of the annual, semiannual, and tri-annual signals. The covariance matrix of the slopes (for all series) is given as $Q_r ={\Sigma }\times (N^{-1})_{22}$, where $(N^{-1})_{22}$ is the second diagonal element of $N^{-1}$. It is further assumed that Q matrix has the form

$$\begin{aligned} Q=s_\mathrm{w} I+s_\mathrm{f} Q_\mathrm{f} +s_{\mathrm{rw}} Q_{\mathrm{rw}} \end{aligned}$$

(6)

where $s_\mathrm{w}$, $s_\mathrm{f}$, $s_{\mathrm{rw}}$ are the white, flicker, and random walk noise amplitudes, respectively. $Q_\mathrm{f}$ and $Q_{\mathrm{rw}}$ are the flicker and random walk noise cofactor matrices, respectively. LS-VCE has been employed to estimate $s_\mathrm{w}$, $s_\mathrm{f}$, $s_{\mathrm{rw}}$, and ${\Sigma }$. As the three coordinate components of all stations have been processed simultaneously, ${\Sigma }$ is of the size $r\times r$. Its corresponding, north, east, and up components are referred to as ${\Sigma }_N$, ${\Sigma }_E$, and ${\Sigma }_U$, respectively (block diagonals). To compute the white, flicker, and random walk noise rate errors for the east components, matrix Q in Eq. (4) is substituted with $Q_\mathrm{w} =s_\mathrm{w}I$, $Q_\mathrm{f} =s_\mathrm{f} Q_\mathrm{f}$ or $Q_{\mathrm{rw}} =s_{\mathrm{rw}} Q_{\mathrm{rw}}$, respectively. Matrices $N_\mathrm{w}, N_\mathrm{f}$, $N_{\mathrm{rw}}$ are then obtained. The rate errors of the east component read

$$\begin{aligned}&\sigma _\mathrm{r}^\mathrm{w} =\sqrt{\hbox {diag}\left( {{\Sigma }_\mathrm{E} N_\mathrm{w}^{-1} \left( {2,2} \right) } \right) } \end{aligned}$$

(7)

$$\begin{aligned}&\sigma _\mathrm{r}^\mathrm{f} =\sqrt{\hbox {diag}\left( {{\Sigma }_\mathrm{E} N_\mathrm{f}^{-1} \left( {2,2} \right) } \right) } \end{aligned}$$

(8)

$$\begin{aligned}&\sigma _\mathrm{r}^{\mathrm{rw}} =\sqrt{\hbox {diag}\left( {{\Sigma }_E N_{\mathrm{rw}}^{-1} \left( {2,2} \right) } \right) } \end{aligned}$$

(9)

where $\sigma _\mathrm{r}^\mathrm{w}$, $\sigma _\mathrm{r}^\mathrm{f}$ and $\sigma _\mathrm{r}^{\mathrm{rw}}$ are the vector of rate errors for the east component of all stations. Their mean indicate the average error rates over all stations. The corresponding values for the north and up components can accordingly be obtained.