Coordinate Determination of Domestic SLR Stations Based on Long Arc Orbit Determination Method Using Laser Ranging Data from Lageos1/2 Satellites

doi:10.21203/rs.3.rs-3140938/v1

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Coordinate Determination of Domestic SLR Stations Based on Long Arc Orbit Determination Method Using Laser Ranging Data from Lageos1/2 Satellites

https://doi.org/10.21203/rs.3.rs-3140938/v1

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Satellite Laser Ranging (SLR) technology is one of the main technologies in the field of space geodesy, it has played an extremely important role in laser ranging data application and research. As a commonly used geodynamic satellites, the main contribution of Lageos1/2 is the solution of orbit determination and station coordinates. At present, there are 7 analysis centers of the International Laser Ranging Service (ILRS) to release the precise orbit products for Lageos1/2 satellites. However, there is currently no relevant research on the analysis of the consistency of these orbits. So, the authors would evaluate the accuracies of them, and the results show that the orbital accuracies of Lageos1/2 satellites are 3 ~ 5 cm. SLR is regarded as one of the important input data of the International Terrestrial Reference Frame (ITRF), the analysis center of ILRS provides weekly solutions of station coordinates for ITRF. But, due to the limitation of SLR observation conditions, for example observations can only be made almost exclusively at night and cannot be observed in rainy days, the orbit solution of the seven-day arc may have the problem of less data. Therefore, this paper proposed a method for calculating the station coordinates based on long arc orbit determination, and we adopted the method to solve the station coordinates of five SLR stations in China. The results show that the obtained orbital precision is basically superior to 3.70 cm, and the differences between station coordinates solved by us and the coordinates published by ITRF2014 are basically smaller than 0.013 m, which is consistent with research results domestic and overseas.

Satellite Laser Ranging (SLR)

Lageos1/2 satellites

Long arc orbit determination

Orbital precision

Satellite Laser Ranging (SLR) technology was initiated by the National Aeronautics and Space Administration (NASA) in the early 1960s, and after decades of development, a single measurement can now be accurate in millimeters. With the continuous development of science and technology, SLR has become one of the main technologies in the field of space geodesy, it has played an extremely important role in determining and checking satellite orbit, Time Transfer by Laser Link (T2L2), monitoring continental plate movement and crustal deformation, solving coordinate frame, earth rotation parameters and low-order coefficient of Earth gravity field, observing space debris and so on (Guillemot et al., 2006; Sośnica et al., 2019; Steindorfer et al., 2020; Bury et al., 2021). At the same time, it participates in building and maintaining the International Terrestrial Reference Frame (ITRF) together with Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), Global Navigation Satellite Systems (GNSS), and Very Long Baseline Interferometry (VLBI). Compared with the other three geodetic techniques, SLR has sufficient sensitivity to the geocentric position and scale factor, and the ITRF scale factor determined by this technique is less than 0.7 ppb (Appleby et al., 2016).

The main contribution of SLR in ITRF is the solution of station coordinates using laser data from geodynamic satellites. Lageos satellites are commonly used as geodynamic satellites, 426 corner reflectors (CCRs) are arranged in 20 rings and distributed on the surface of the satellite, this configuration is easy to observe. Due to the spherical structure, Lageos satellites have a high surface-to-mass ratio, so it is less affected by non-conservative forces such as atmospheric drag (Kucharski et al., 2013; Hattori and Otsubo, 2019). Since the launch of them, there have been many studies about the orbit modeling of Lageos satellites, including the earth's radiation pressure, solar pressure, center of mass corrections (CoM), etc (Scharroo et al., 1991; Chreston et al., 1996; Rodríguez et al., 2019). Based on these researches, the orbit accuracy of Lageos satellites can reach the centimeter level. Through the high-precision orbit determination of Lageos satellite, we can accurately determine the low-order coefficient of the earth's gravity field, the coordinates of the station, and the observation quality of the station (Rutkowska et al., 2010; Kang et al., 2019; Otsubo et al., 2019). At present, there are 7 analysis centers of the International Laser Ranging Service (ILRS, https://ilrs.gsfc.nasa.gov/) to release the precise orbit products for Lageos satellites. Besides, according to the combination of the orbits of these analysis centers, ILRS generated the LAGEOS comprehensive orbit products ILRSA and ILRSB through weighted least squares and variance component estimation methods, respectively (Koch, 1999; Davies and Blewitt, 2000). However, there is currently no relevant research on the analysis of the consistency of these orbits. In this paper, the authors evaluate the consistency between nine groups of orbits, and the results show that the orbital accuracies of Lageos satellites are at the centimeter level.

A total of five fixed SLR stations in China (Changchun, Beijing, Wuhan, Kunming, and Shanghai) have joined the ILRS station network, the site information of them is given in Table 1, and Changchun and Kunming stations are affiliated to National Astronomical Observatories of the Chinese Academy of Sciences (NAOC, CAS) (Fumin, 2001). The Chinese station network began to have a laser ranging capability of kHz in 2009. Nowadays, all stations are operating with capability of kHz, equipped with 1 mJ pulsed lasers, C-SPAD detectors and EventTech event timers, they have achieved mm-level observation accuracy, and that Kunming, Shanghai, and Changchun stations also have the capability to observe space debris (Hao et al., 2016; Li et al., 2016; Wilkinson et al., 2019; Liang et al., 2019). The addition of Chinese station network has greatly improved the observation situation of ILRS. In 2021, Chinese SLR stations contributed 21% of the observed passes and 15% of normal point data to ILRS, respectively.

Table 1

Chinese SLR network
site ID	site name	location name	Agency
7237	CHAL	Changchun	Changchun Observatory, NAOC, CAS
7249	BEIL	Beijing	Chinese Academy of Surveying and mapping
7396	JFNL	Wuhan	Innovation Academy for Precision Measurement Science and Technology, CAS
7819	KUN2	Kunming	Yunnan Observatory, NAOC, CAS
7821	SHA2	Shanghai	Shanghai Astronomical Observatory, CAS

High-precision station coordinates are the basis of ITRF implementation. In February, 2015, the United Nations General Assembly adopted a resolution on the "Global Geodetic Reference Framework for Sustainable Development" (GGRF United Nations General Assembly, 2015). For most applications that require knowledge of the kinematics of the Earth's surface, it is generally accepted that an ITRF (Global Geodetic Observing System, GGOS) supporting a ground station coordinate accuracy of 1 mm and a stability of 0.1 mm per year is required (Plag and Pearlman, 2009). However, the current accuracy and stability of ITRF's long-term origin that can be realized by SLR data is at a position of 5 mm and a time change level of 0.5 mm per year, which is still a little far from the accuracy required by GGOS (Altamimi et al., 2023). As one of the four input data of ITRF (DORIS, SLR, GNSS, VLBI), the analysis center of ILRS provides weekly solutions of station coordinates and Earth Orientation Parameters (EOP) for ITRF. However, due to the limitation of SLR observation conditions, (such as observations can only be made almost exclusively at night, a small part stations can achieve daytime laser ranging and cannot be observed in rainy days), the orbit solution of the seven-day arc may have the problem of less data. Figure 1 shows the observation data of the Lageos1 satellite at Wuhan station in 2021. From the figure, we can see that this site has the less observation data in the second quarter, with an average of only 63 normal points per month. The reason for this phenomenon is that the second quarter is the rainy season in southern China, with more cloudy and rainy weather, which is not suitable for observation. Therefore, we proposed a method for calculating the station coordinates based on long arc orbit determination, and we adopted the method to solve the station coordinates of five SLR stations in China.

In our case, this paper proposed a strategy and method for orbit calculation. Based on that, the long arc orbit determination of Lageos1/2 satellites using the normal point data is preliminarily realized, and the LAODGEO software (Long Arc Orbit Determination Software for Geodynamic Satellite) is also developed successfully. Besides, for knowing what level of our orbit precision, this paper also analyzed the consistency of Lageos1/2 satellite precision orbits provided by the nine analysis centers. Finally, we have solved the coordinates of five domestic SLR sites based on long arc orbit determination, and the results are basically at the same precision level as the coordinate data published by ITRF2014.

This paper is structured as follows. In section 2, the method and data are introduced, mainly including observation and dynamic models, solution of satellite motion equation and data collection. All results and discussion obtained through the above methods and theories are shown in Section 3. Conclusions are given in Section 4.

In this section, the observation equations and dynamical model are presented, and some measurement and dynamical models for the satellite are introduced. On this basis, the equations of motion of the satellite are derived in detail. In order to achieve the goal of long arc orbit determination for Lageos1/2 satellites, this section also analyzes the characteristics of distribution for normal point data used in the orbit determination.

2.1 Observation Model

The principle of SLR is to use the laser emission system of the ground station to send laser pulses to the satellite, after the reflection of the reflector on the satellite, the laser pulses are returned to the receiving system of the station, thus the timing system on the ground can determine the time interval between the laser round trip, and the distance between the station and the satellite is calculated. Here, the SLR observations can be written as:

$${\rho _{SLR}}=\left| {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }_{groud}} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }_{sat}}} \right|+\varepsilon$$

where ${\rho _{SLR}}$ represents the measurement of laser ranging, ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} _{groud}}$ is position coordinates of the ground station, ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} _{sat}}$ is position coordinates of the satellite. $\varepsilon$ stands for other systematic errors and random errors during signal propagation, which mainly includes tropospheric error, satellite phase center offset, station eccentricity error, station displacement caused by plate movement and earth tide, and the influence of relativistic effect on distance observation. These errors can be basically corrected by the model, Table 2 would present the revised models adopted in this paper.

2.2 Dynamic Model

For a satellite in orbit, the equation of motion in the inertial frame can be expressed as follows (Montenbruck and Gill, 2000):

$$\ddot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }= - \frac{{GM}}{{{{\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} } \right|}^3}}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} +{a_r}+{a_{rtn}}$$

Here, GM is geocentric gravitational constant, $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}$ and $\ddot {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }$ stand for the satellite position and acceleration in the inertial coordinate system, respectively; ${a_r}$ is main perturbation acceleration excluded Earth center gravity, which includes the Sun, Moon and other planet perturbations, atmosphere drag, solar radiation pressure, Earth radiation pressure, solid Earth tide, ocean tide. ${a_{rtn}}$ refers to the RTN (Radial Tangential Normal) perturbation, which can make up for residual error in the force models (Rim et al., 2002a, 2002b; Zhao, 2004). And, the tangential and normal RTN perturbation parameters would be estimated in our orbit determination.

2.3 Solution of Satellite Motion Equation

The equation of motion in the above is a two-order differential equation. In order to facilitate linearization, it can be transformed into a first order differential equation as follows:

$$\left\{ \begin{gathered} \dot {r}=v \hfill \\ \dot {v}= - \frac{{GM}}{{{{\left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} } \right|}^3}}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} +{a_r}+{a_{rtn}} \hfill \\ \dot {p}=0 \hfill \\ \end{gathered} \right.$$

The initial state of the satellite can be expressed:

$$\left\{ \begin{gathered} {\left. r \right|_{t0}}={r_0} \hfill \\ {\left. v \right|_{t0}}={{\dot {r}}_0} \hfill \\ {\left. p \right|_{t0}}={p_0} \hfill \\ \end{gathered} \right.$$

where stands for the satellite velocity, is dynamic parameters to be estimated in satellite dynamics equation, which includes atmospheric drag coefficients, solar radiation pressure coefficients and so on. If we set $x={\left( {r,\dot {r},p} \right)^T}$, then the equation of motion can be written in the following:

$$\left\{ \begin{gathered} \dot {x}=F\left( {x,t} \right) \hfill \\ {\left. x \right|_{t0}}={x_0} \hfill \\ \end{gathered} \right.$$

When an initial state ${x_0}$ is set, the reference orbit ${x^ * }$ of the satellite can be obtained by integrating the equation of motion. At this point, the Eq. 5 is expanded by first order Taylor at ${x^ * }$, the detailed expression is shown as follows:

$$\dot {x}={\dot {x}^ * }+{\left. {\frac{{\partial F\left( {x,t} \right)}}{{\partial x}}} \right|_ * }\left( {x - {x^ * }} \right)$$

Here, the expression of $\frac{{\partial F\left( {x,t} \right)}}{{\partial x}}$ is given as follows:

$$\frac{{\partial F\left( {x,t} \right)}}{{\partial x}}=\left( \begin{gathered} \frac{{\partial \dot {r}}}{{\partial r}}\quad \frac{{\partial \dot {r}}}{{\partial v}}\quad \frac{{\partial \dot {r}}}{{\partial p}} \hfill \\ \frac{{\partial \dot {v}}}{{\partial r}}\quad \frac{{\partial \dot {v}}}{{\partial v}}\quad \frac{{\partial \dot {v}}}{{\partial p}} \hfill \\ \frac{{\partial \dot {p}}}{{\partial r}}\quad \frac{{\partial \dot {p}}}{{\partial v}}\quad \frac{{\partial \dot {p}}}{{\partial p}} \hfill \\ \end{gathered} \right)=\left( \begin{gathered} \;\,0\quad \;\;\,I\quad \;\;0 \hfill \\ \frac{{\partial \ddot {r}}}{{\partial r}}\quad \frac{{\partial \ddot {r}}}{{\partial \dot {r}}}\quad \frac{{\partial \ddot {r}}}{{\partial p}} \hfill \\ \;\,0\quad \;\;0\quad \;\;0 \hfill \\ \end{gathered} \right)$$

If $X=x - {x^ * }$, then the following expression is obtained:

$$\left\{ \begin{gathered} \dot {X}=AX \hfill \\ X\left( {{t_0}} \right)={X_0} \hfill \\ \end{gathered} \right.$$

where $A={\left. {\frac{{\partial F\left( {x,t} \right)}}{{\partial x}}} \right|_ * }$. Besides, the observation Eq. 1 is also expanded by first order Taylor at ${x^ * }$, the detailed expression is shown as follows:

$${\rho _i}={H_i}{X_i}+{V_i},\quad \quad i=1,2, \cdot \cdot \cdot l$$

here $H={\left. {\frac{{\partial \rho \left( {x,t} \right)}}{{\partial x}}} \right|_ * }$, it stands for the partial derivative of the observations with respect to each estimated parameter, represents the number of observations involved in orbit calculation.

By solving the differential Eq. 8, the solution can be written as follows:

$$X=\Phi \left( {t,{t_0}} \right){X_0}$$

where $\Phi \left( {t,{t_0}} \right)$ is state transition matrix, it satisfies the following expression:

$$\left\{ \begin{gathered} \dot {\Phi }\left( {t,{t_0}} \right)=A\Phi \left( {t,{t_0}} \right) \hfill \\ \Phi \left( {t,{t_0}} \right)=I \hfill \\ \end{gathered} \right.$$

This transfer matrix is solved by numerical integration, this paper would use the KSG (Shampine and Gordon, 1975) integral method to calculate satellite orbit.

Based on the numerical solution of state transition matrix, the observation equation can be written as follows:

$${\rho _i}={H_i}\Phi \left( {t,{t_0}} \right){X_0}+{V_i}$$

So far, satellite orbit determination becomes a state estimation problem, this paper would adopt the least square method to conduct parameter estimation.

2.4 Data Collection

In order to analyze the reliability of method, this paper selects three years of normal point data from the EUROLAS Data Center (EDC, Torrence et al., 1985) for Lageos1/2 satellites to conduct orbit determination. Figure 2 and Table 2 show the statistics result of normal point data for Lageos1/2 satellites in the period from 2020 to 2022.

As can be seen from the figure and table, the number of normal points for Lageos1 satellite is more than that of Lageos2 satellite each year, which is likely to related to that Lageos1 satellite was launched more than a decade before Lageos2 satellite. In addition to, the data volume of Lageos1/2 satellites is decreasing with the increase of year, but their range are basically 60000 ~ 70000 and 50000 ~ 60000, respectively.

Table 2

The statistics of normal point data for Lageos1/2 satellites from 2020 to 2022
	2020	2021	2022
Lageos1	68639	67641	62054
Lageos2	60372	57833	49866
Total	129011	125474	111920

In this section, this paper mainly analyzes and discusses experiment results. At first, we show the processing strategies in the orbit determination, and introduce self-developed software and its computational flow. Then, the residuals and precision of orbit determination have been analyzed in detail. On other hand, for multi-directional comparison of our orbit determination results, this paper also compares and analyzes the one-year orbital determination results of Lageos1/2 from nine ILRS analysis centers. Finally, we show coordinate determination of domestic SLR tracking stations and analyze their accuracies.

3.1 Processing Strategies and Computational Flow of Software

In order to realize the orbit and SLR station coordinates determination based on the long arc orbit determination method, this paper independently developed a long arc orbit determination software for geodynamic satellite combined with the relevant theoretical method derivation, which is named LAODGEO (Long Arc Orbit Determination software for GEOdynamic satellite). This software has processed the laser ranging data of Lageos1/2 for the period from 1 January, 2020 to 31 December, 2022. Table 3 gives the models and strategies used for Lageos1/2 satellites in the orbit determination.

Table 3

Force models, measurement models and parameter strategies for Lageos1/2 satellites employed in the LAODGEO software
Project	Selection/Description
Force model
Earth gravity filed	EIGEN-GRGS.RL04.MEAN-FIELD 50×50 (Lemoine et al., 2019)
Atmosphere drag	MSIS00 density model (Picone et al., 2002)
Solar radiation pressure	Cr = 1.13
Sun, moon and planet ephemeris	JPL DE405 (Standish, 1998)
Earth radiation pressure	Knocke-Ries-Tapley model (Knocke et al., 1998)
Empirical force model	RTN perturbation
Ocean tide (Petit and Luzum, 2010)	FES2004 (Lyard et al., 2006)
Solid Earth tide	TIDE2000 (Petit and Luzum, 2010)
Relativity	Post-Newtonian correction
Reference frame
Inertial reference frame	J2000.0
Station coordinate	SLRF2014 for epoch 2010 (Altamimi et al., 2016)
Polar motion	C04 IERS
Earth orientation parameters	IERS EOP 14 C04 (Bizouard et al., 2019), IAU2000A precession-nutation model
Estimated parameters
Satellite state vector	Three initial positions and velocities
Atmospheric drag coefficient	Cd/3 days
RTN perturbation parameters	Tangential and normal/3 days
Station information	Range bias of part of tracking stations are estimated
Measurement model
Data type	Normal point data from EUROLAS Data Center
Laser pulse wavelength	532 nm
Tropospheric model	Mendes-Pavlis model (Mendes, 2002, 2004)
Center of mass correction	0.248 m
Residual elimination criteria	3D sigma
Cut-off elevation	15°
Numerical integration
Integration method	KSG
Orbit integration step size	120 s
Arc length	A year arc dynamic solution

As can be shown from Table 3, the force models used by the software consist of earth center gravity, atmosphere drag, solar radiation pressure, third-body gravity, earth radiation pressure, empirical RTN perturbation, solid earth tide, ocean tide and relativity perturbation. For the dynamic parameters, the software respectively estimates tangential, normal RTN perturbation parameters and the atmospheric drag coefficient Cd per 3 days, but the solar radiation pressure coefficient Cr is set 1.13 (Appleby et al., 2016). On the measurement parameters, only the range bias value for part of SLR tracking stations has been estimated.

Besides, this section also gives the calculation flow chart for the LAODGEO software processing, the detailed contents are shown in the following Fig. 3. As can be seen from Fig. 3, this software mainly consists of six functional modules, which are respectively data preparation, data pre-processing, generating observation equation, numerical integration, parameter estimation, residuals analysis and generating orbital products.

3.2 Residuals Analysis

We select three years of normal point data to conduct orbit determination, but this section only shows analysis results from 2020, other two years results are similar to it. Figure 4 shows the variation of residuals in the orbit determination for Lageos1/2 satellites using SLR data from 2020. As can be seen in the picture, about 98.41 percent of residual values are located within ± 4.00 cm for Lageos1 satellite, and about 92.11 percent for Lageos2 satellite, these residuals have no significant fluctuations. Combined with the statistics of residuals in the Table 4, we can find that the average value and standard deviation of residuals are respectively 0.191 cm and 0.987 cm for Lageos1 satellite, while 0.092 cm and 1.087 cm for Lageos2 satellite, which is consistent with research results domestic and overseas (Schillak et al., 2021; Zhang et al., 2022).

Table 4

The statistics of residuals in the orbit determination for Lageos1/2 satellites (unit: cm)
	Mean	Std
Lageos1	0.191	0.987
Lageos2	0.092	1.087

3.3 Precision Analysis

In addition, this article also compares the one year of calculated orbits for Lageos1/2 satellites with precise orbits provided by ILRS. In order to obtain the orbit precision using the LAODGEO software, this section analyzes the orbit differences of Lageos1/2 satellites in radial, tangential and normal directions. Figure 5 and Fig. 6 show the variation of orbit differences in three directions for Lageos1/2 satellites in 2020 between the calculated orbits and the ILRSA orbits (UTC time), respectively.

As can be seen from Figs. 5 and 6, the variations in the radial direction for Lageos1/2 satellites have small fluctuation, which are basically located within ± 1.50 cm. And, the fluctuation in the tangential and normal directions are relatively bigger, but most of differences are located within ± 7.00 cm, and it has no significant fluctuations, which shows that the models and strategies used for Lageos1/2 satellites in the orbit determination is reasonable. Besides, to analyze the orbit differences of 2020 visually, Table 5 gives the mean and RMS (root mean square) values of orbit differences in three directions for Lageos1/2 satellites. We can find that the mean values are at millimeter level, which illustrates that these orbit differences are basically unbiased and have no obvious systematic errors. For Lageos1 satellite, the RMS values of radial, tangential, normal and three-dimensional (3D) directions can achieve 0.605 cm, 2.615 cm, 2.522 cm, and 3.683 cm respectively, and these values are respectively 0.613 cm, 2.667 cm, 2.512 cm, and 3.714 cm for Lageos2 satellite. These statistical results show that the obtained orbit precision based on the current self-developed LAODGEO software is basically superior to 3.70 cm, and the radial accuracy is better than 6 mm, that is consistent with the single ranging accuracy of Lageos1/2 satellites, and these study results are also consistent with research results domestic and overseas (Schillak et al., 2021; Zhang et al., 2022).

Table 5

Accuracy statistics of Lageos1/2 satellites using SLR data from 2020 (unit: cm)
		Mean	RMS
Lageos1	Radial	0.289	0.605
	Tangential	-0.063	2.615
	Normal	-0.067	2.522
	3DRMS	3.683
Lageos2	Radial	-0.039	0.613
	Tangential	-0.249	2.667
	Normal	0.011	2.512
	3DRMS	3.714

3.4 Orbit Comparison from Different Analysis Centers of ILRS

In order to obtain more comparison results, this paper also analyzes the orbital precision of different analysis centers of ILRS. These analysis centers consist of ASI, BKG, DGFI, ESA, GFZ, ILRSA, ILRSB, JCET and NSGF, Table 6 shows the full name of these centers. It should be pointed that the product results of ILRSA and ILRSB are combination solution of the other 7 analysis centers, the products of which are basically agreement, so this section only gives the orbit differences of Lageos1/2 satellites between the ILRSA and other 8 analysis centers.

Table 6

Overview for analysis centers of ILRS
Analysis Centers	Code
Agenzia Spaziale Italiana	ASI
Bundesamt für Kartographie und Geodäesie	BKG
Deutsches Geodaetisches Forschungsinstitut	DGFI
European Space Agency, European Space Operations Centre	ESA
German Research Centre for Geosciences	GFZ
ILRS primary combination solution	ILRSA
ILRS backup combination solution	ILRSB
Joint Center for Earth Systems Technology	JCET
NERC (Natural Environment Research Council) Space Geodesy Facility	NSGF

Figure 7 to Fig. 12 show the variation of orbit differences in three directions for Lageos1/2 satellites in 2020 between the orbits of each analysis centers and the ILRSA orbits (UTC time), respectively. As can be seen from these comparison results, the variations in the radial direction for Lageos1/2 satellites have small fluctuation, which are basically located within ± 1.40 cm, especially it is located within ± 1.00 cm for ILRSB. In the tangential and normal directions, except that the variations of difference for DGFI and JCET orbits are slightly bigger, most of differences for other analysis centers are located within ± 6.00 cm, and it has no significant fluctuations, which shows that the orbit products of Lageos1/2 satellites provided by each analysis centers are reliable. Besides, in order to show the orbit differences between each analysis centers and the ILRSA orbits visually, Table 7 gives the accuracy statistics of Lageos1/2 satellites for each analysis centers. For Lageos1 satellite, the RMS values in R, T, N and 3D directions are 0.555 cm, 2.315 cm, 2.322 cm, and 3.325 cm respectively, and these values are respectively 0.563 cm, 2.467 cm, 2.312 cm, and 3.427 cm for Lageos2 satellite. These statistical results are agreement with the orbit precision calculated by the current self-developed LAODGEO software, which illustrates that our orbit determination results are reliable.

Table 7

Accuracy statistics of Lageos1/2 satellites for each analysis centers comparing with ILRSA orbits (units: cm)
Analysis centers	Lageos1				Lageos2
Analysis centers	R	T	N	3DRMS	R	T	N	3DRMS
ASI	0.431	2.203	2.162	3.12	0.39	2.45	2.31	3.39
BKG	0.522	1.882	1.450	2.43	0.47	1.87	1.76	2.61
DGFI	0.632	2.610	3.041	4.06	0.69	2.66	3.18	4.21
ESA	0.572	2.173	2.063	3.05	0.50	2.36	2.11	3.20
GFZ	0.533	1.621	1.513	2.28	0.45	1.28	1.73	2.20
ILRSB	0.361	0.823	0.864	1.24	0.28	0.96	0.94	1.37
JCET	0.592	3.292	2.421	4.12	0.85	4.99	2.12	5.49
NSGF	0.451	2.403	2.143	3.25	0.43	2.04	2.29	3.10

3.5 Coordinate Determination of Domestic SLR Stations

In order to validate the performance of the LAODGEO software, this section also utilizes that software to conduct coordinate determination of five domestic SLR sites based on the laser ranging data from 2020 to 2022. Figure 13 shows the geographic distribution for five sites, we can see that the five stations are evenly distributed in the northern and southern latitudes of China.

Besides, we also counted the amount of normal point data for each station in different years. The ID number of five domestic SLR sites are respectively 7237, 7249, 7396, 7819 and 7821, and their corresponding site name are CHAL, BEIL, JFNL, KUN2 and SHA2. Figure 14 gives the statistics result for each station in the period from 2020 to 2022.

As can be shown from Fig. 14, the CHAL has more data volume than other four sites, its data amount is 4000 ~ 7000, and it is basically 2 to 4 times than other sites, which is likely to related to that the CHAL achieved daytime ranging and had more sunny days during the year. Although the data volume for other four sites are less compared with the CHAL, their amount is increasing with the increase of year, and the largest amount of data is also over 4000.

Table 8

Analysis of Coordinate Determination for Domestic SLR Stations
Site	Year	X	Y	Z	ILRS (XYZ)
7237	2020	-2674387.0915	3757189.1931	4391508.2818	-2674387.0805
	2021	-2674387.0902	3757189.1892	4391508.2806	3757189.1940
	2022	-2674387.0908	3757189.1864	4391508.2827	4391508.2868
	RMS	0.0103	0.0052	0.0052	3D: 0.0127
7249	2020	-2148760.7587	4426759.5462	4044509.6146	-2148760.7599
	2021	-2148760.7685	4426759.5437	4044509.6048	4426759.5478
	2022	-2148760.7695	4426759.5378	4044509.5976	4044509.6064
	RMS	0.0075	0.0063	0.0070	3D: 0.0120
7396	2020	-2279755.6878	5004737.4376	3219791.7448	-2279755.6807
	2021	-2279755.6823	5004737.4280	3219791.7387	5004737.4341
	2022	-2279755.6843	5004737.4264	3219791.7417	3219791.7385
	RMS	0.0047	0.0060	0.0041	3D: 0.0087
7819	2020	-1281301.3480	5640724.6142	2682905.7147	-1281301.3232
	2021	-1281301.3492	5640724.6189	2682905.7144	5640724.5935
	2022	-1281301.3511	5640724.6214	2682905.7184	2682905.6869
	RMS	0.0263	0.0248	0.0290	3D: 0.0463
7821	2020	-2830744.5928	4676580.2235	3275072.7815	-2830744.5970
	2021	-2830744.5972	4676580.2232	3275072.7807	4676580.2287
	2022	-2830744.6031	4676580.2212	3275072.7823	3275072.7841
	RMS	0.0043	0.0062	0.0027	3D: 0.0080

Here, this section gives the results of coordinate solutions for each station in different years. As illustrated in Table 8, it shows the X, Y and Z coordinate values of each station calculated by the LAODGEO software, and that also gives the coordinate results of ILRS solution. According to using our results and the ILRS solution to make a difference comparison, we obtain the RMS values in different coordinate directions and 3DRMS values for each station. Based on the analysis results in Table 8, we can find that the 3DRMS values are basically smaller than 0.013 m for 7237, 7249, 7396 and 7821, even can achieve 0.008 m for 7396 and 7821, which basically illustrates that the coordinates calculated by the LAODGEO software is reliable. However, the 3DRMS value for 7819 is slightly bigger than other four sites, it has achieved 0.046 m, which may have something to do with many small earthquakes occurring in Kunming from 2020 to 2022.

This paper proposed a strategy and method to determinate orbits of Lageos1/2 satellites, on the basis of it, a long arc orbit determination software for geodynamic satellite is independently developed, which is named LAODGEO (Long Arc Orbit Determination software for GEOdynamic satellite). This software has processed the laser ranging data of Lageos1/2 for the period from 1 January, 2020 to 31 December, 2022, and also conducted coordinate determination of five domestic SLR sites. The specific conclusions are as follows:

Firstly, the variation of residuals in the orbit determination for Lageos1/2 satellites are basically located within ± 4.00 cm, and the average value and standard deviation of residuals are respectively 0.191 cm and 0.987 cm for Lageos1 satellite, while 0.092 cm and 1.087 cm for Lageos2 satellite. Besides, according to the comparison results with precise orbits provided by ILRS, the 3DRMS values based on the current self-developed LAODGEO software are basically superior to 3.70 cm, which is consistent with research results domestic and overseas.

Secondly, through analyzing the orbital precision of different analysis centers of ILRS, this paper shows a conclusion that the orbital accuracies of Lageos1/2 satellites provide by these analysis centers are 3 ~ 5 cm, which is agreement with the orbit precision calculated by the current self-developed LAODGEO software.

Finally, we utilize that software to conduct coordinate determination of five domestic SLR sites based on the laser ranging data from 2020 to 2022. The analysis results show that the differences compared with the coordinates published by ITRF2014 are basically smaller than 0.013 m, which illustrates that the coordinates calculated by the LAODGEO software is also reliable.

Funding This study is supported by Knowledge Innovation Program of Wuhan-Shuguang Project (No. 2022010801020138), National Natural Science Foundations of China (Nos. 42174222, 41904165), the Open Fund of Hubei Luojia Laboratory (No. 220100062) and Scientific Instrument and Equipment Development Project of Chinese Academy of Sciences (2019) (No. YJKYYQ20190062).

Competing interests All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interests or nonfinancial interests in the subject matter or materials discussed in this manuscript.

Authors’ contributions Chongchong Zhou and Junao Wang provided the initial idea for this study; Chongchong Zhou and Junao Wang collected the experimental data and conducted the experiment; Chongchong Zhou and Junao Wang wrote the paper; Chongchong Zhou, Junao Wang, Shiming Zhong, Bibo Peng analyzed the results of the experiment; Shiming Zhong, BiBo Peng, Jie Zhang, Zhao Guo reviewed and improved the manuscript.

Ethics approval and consent to participate This study does not involve ethical issues, and affiliates agree with the author's submission. Informed consent was obtained from all individual participants included in the study.

Consent for publication The participant has consented to the submission of the study to the journal.

Acknowledgments The authors would like to thank the satellite information of Lageos1/2 provide by ILRS. The authors are also grateful to EDC and ILRS for providing the normal point data and precise orbit products of Lageos1/2, respectively. We also appreciate the reviewers for their valuable suggestions on improving the manuscript. This work is funded by Knowledge Innovation Program of Wuhan-Shuguang Project (No. 2022010801020138), National Natural Science Foundations of China (Nos. 42174222, 41904165), the Open Fund of Hubei Luojia Laboratory (No. 220100062) and Scientific Instrument and Equipment Development Project of Chinese Academy of Sciences (2019) (No. YJKYYQ20190062).

Data Availability Our sincere thanks to the EDC for providing the normal point data of Lageos1/2; ILRS for providing the coordinates of all SLR tracking stations, satellite information and precise orbit products of Lageos1/2; IERS for providing Earth orientation products.

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No competing interests reported.

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Coordinate Determination of Domestic SLR Stations Based on Long Arc Orbit Determination Method Using Laser Ranging Data from Lageos1/2 Satellites

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Abstract

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1 Introduction

2 Method and Data

2.1 Observation Model

2.2 Dynamic Model

2.3 Solution of Satellite Motion Equation

2.4 Data Collection

3 Results and Discussion

3.1 Processing Strategies and Computational Flow of Software

3.2 Residuals Analysis

3.3 Precision Analysis

3.4 Orbit Comparison from Different Analysis Centers of ILRS

3.5 Coordinate Determination of Domestic SLR Stations

4 Conclusions

Declarations

References

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