Preamble

Astronomical Techniques and Instruments, Vol. 2, September 2025, 327–337 Article Open Access Instant determination of polar motion with tri-static common view lunar laser ranging

Zhipeng Liang 1,2 , Chengzhi Liu 1,3* , Yanning Zheng 1,2 , Xue Dong 1

1 Changchun Observatory of National Astronomical Observatories Chinese Academy of Sciences Changchun 130117, China Sciences Nanjing 210008, China *Correspondence:

INTRODUCTION

Determining Earth’s orientation in inertial space, also known as the Earth Rotation Parameters (ERPs), is impor- tant in astrometric studies that relate terrestrial and celes- tial coordinates. The ERP data consists of three numbers: the PM components , and the UT1. Currently, ERPs are usually provided using the Very Long Baseline Interferometry (VLBI) technique. The international VLBI service can determine PM to an accuracy of 50 μas and UT1 to an accuracy of 3 μs , and delivery of the ERP product with temporal resolution of 1 day normally requires 8–10 days Determination of ERPs can also be conducted using the LLR technique, which involves measurement of the round-trip time between a ground-based laser ranging sta- tion (hereafter, station) and a lunar retro-reflector array (hereafter, reflector). The use of LLR data to determine ERPs was first explored soon after the data initially became available . Over the past decade, a number of LLR stations have produced precise data, with range preci- sion enhanced to the centimeter level . Recent LLR analy- sis studies typically selected observation nights with 10–15 NPs, with the minimum requirement of 5 NPs per night . The most up-to-date studies have achieved an accuracy of several milliarcseconds for PM coordinates, with temporal resolution of 1 day The concept of organizing LLR observations in a com- mon view manner, or LLRCV, was proposed by Leick It was claimed that the LLRCV method could provide ERP components with measurement accuracy on the Earth’s surface of approximately 30 cm for LLR in the early 1980s, equivalent to 10 mas of PM or 0.6 ms of . Another form of LLRCV, proposed by Müller et

2 University of Chinese Academy of Sciences , Beijing 101408, China

3 Key Laboratory of Space Object and Debris Observation Purple Mountain Observatory Chinese Academy of © 2025 Editorial Office of Astronomical Techniques and Instruments, Yunnan Observatories, Chinese Academy of Sciences. This is an open access article under the CC BY 4.0 license ( Citation: Liang, Z. P., Liu, C. Z., Zheng, Y. N., et al. 2025. Instant determination of polar motion with tri-static common view lunar laser ranging.

Astronomical Techniques and Instruments (5): 327−337. ati2025027

Abstract

A method is presented for determining instant values of Earth’s polar motion (PM) using a set of lunar laser ranging (LLR) measurements acquired simultaneously by tri-static common view (TCV) at three LLR stations in Europe. We developed a model of the LLR TCV measurements, then formulated the linear equation for solving PM.

Although there was no actual TCV event in the data, we conducted a two-phase study to test our method using actual LLR normal points (NPs) acquired by the European stations during 2012–2022. In the first phase, we simulated TCV events and PM solutions. The robustness of our method was assessed by introducing Universal Time (UT1) errors and per-station range errors in this phase. In the second phase, we augmented the actual LLR NPs with simulated data to generate realistic TCV events and solutions, using the ‘1+2’ and ‘2+1’ strategies, which differed in terms of data composition. Results indicated that a UT1 error of 0.1 ms caused PM errors of <18 mas, while a uniform range error of 50 mm resulted in PM errors of <180 mas. In the augmentation phase, the maximum solution errors were 752 and 899 mas, and 88.5% and 91.2% of the solutions were better than the predictions for the ‘1+2’ and ‘2+1’ strategies, respectively. The presented approach relies on precise geodetic data, and therefore, it is not intended to replace the traditional method. However, this study demonstrated that instant determination of PM is feasible and robust, although the accuracy requires further enhancement.

Keywords

Equatorial coordinate system; LLR; PM; Satellite laser ranging; Space geodesy

, involves the placement of an optical transponder on the lunar surface to illuminate multiple laser ranging sta- tions simultaneously. However, follow-up studies on the LLRCV concept have been limited.

Major geodynamic events, such as earthquakes, can cause rapid changes in ERPs [ 13 , 14 ] . While conventional LLR analysis extracts a daily solution using single-night observations, the common view method provides an instant solution at higher temporal resolution, which is defined by the accumulation time of LLR NPs, i.e., typi- cally 15 min or less. This represents a substantial improve- ment in monitoring capability for transient geodynamic phe- nomena by LLR.

In this paper, we explore the instant determination of Earth’s PM using LLR data in the form of TCV measure- ments. Linear equations can be formulated upon three simul- taneous range residuals from three cooperating stations to 753.1 km METHOD AND DATA This paper explores instant determination of Earth’s PM using LLR data in the form of TCV measurements.

We first established the method for determining Earth’s PM from the LLR model, the range derivatives to ERPs, the PM prediction model, and the least squares solu- tion. Then, we conducted a two-phase study to evaluate the method. The first or simulation phase involved generat- ing LLR TCV simulation data to test the robustness of the PM determination method by introducing UT1 and range errors. In the second or augmentation phase, we com- bined simulated data with actual LLR NPs to create realis- tic TCV events and solve for PM. These results were com- pared with reference values to generate solution error data. Statistical analysis was conducted to assess the accu- racy of the solutions.

LLR Model and Data The lunar ranging measurement can be formulated as solve for PM in near real time.

Three European stations were selected from the Interna- tional Laser Ranging Service network to form a station group that we named GMW. The stations are located in Grasse France (GRSM/7845), Matera Italy (MATM/7941), and Wettzell in Germany (WETL/8834).

The baselines between the stations are 753 km (GW), 877 km (GM), and 990 km (MW), see Our approach is grounded in contemporary geodetic infrastructures, which include the International Terrestrial Reference Frame (ITRF) , International Earth Rotation Service (IERS) , high-precision planetary ephemerides fundamental astronomical software , and the IERS con- ventions 2010, which provide key algorithms . The ref- erence ERP data were extracted from the IERS C04 20 series, which was accessed via the IERS official website 990.1 km follows: tropo where represents the time of flight measured at the rang- ing station. The distances represent the light path from the station to the reflector and from the reflec- tor to the station, respectively, accounted in barycentric coordinates. The relativistic delay represents the gravita- tional delay effect on the flying pulse. The tropospheric delay represents atmospheric refraction delay dur- ing the propagation of light. Tags 12 and 23 represent the outbound flight and the inbound flight, respectively. An empirical term was introduced and fitted to compen- sate for the remaining periodic effect. tropo Time-of-flight measurements performed ground stations in the clock rate of International Atomic Time (TAI), while planetary motions are calculated using barycentric dynamical time (TDB). The difference in the

clock rate between TAI and TDB is considered negligi- ble in this paper.

Light paths are the Euclidean distances in barycentric inertial vacuum space, where represents the path from the station at transmit time to the reflec- tor at hit time , and represents the path from the reflector at to the station at receive time . Thus, we have the following decompositions:

where , , , and are vectors in barycentric inertial space with B/E/M/S/A meaning barycenter, Earth center, moon center, Earth observing station, and lunar reflector array, respectively. The barycentric vectors and and the selenocentric vector are computed with position vectors and lunar libration angles provided by DE430 planetary ephemeris, using TDB. The geocen- tric vector is computed with station coordinates from ITRF2020 and IERS Earth Orientation Parameter data.

Minor adjustments were made to the terrestrial and lunar coordinates to minimize the range residuals.

We selected 9 747 LLR NP data records, which were retrieved from the EUROLAS Data Center . The data were acquired by European LLR stations GRSM, MATM, and WETL, with the observation time spanning from May 2, 2012 to December 7, 2022.

The data were recorded in the Consolidated Laser Ranging Data (CRD) format. For each NP data record, the CRD format includes the following details: the observ- ing station, the observed reflector, the mean epoch of the data, the NP duration (or NP bin size), the number of pho- toelectrons accumulated within the NP bin, and the root mean square (RMS) value. The NP RMS value is equiva- lent to the 1-σ standard deviation of the range measure- ments in the NP.

We discarded 0.56% (55 of 9 747) outlier NPs for which the range residuals were >0.4 m. In the remaining 99.44% (9692 of 9747) of the NP data, the overall range measurement residual against our LLR model was 35.9 mm in RMS. The RMS values were 36, 48, and 22 mm for stations GRSM, MATM, and WETL, respectively 2 .

The number of NPs used was 8 191, after applying the lunar declination limit, as mentioned in Section 3.1.

The ERPs represent the rotation axis of Earth in the ter- restrial reference frame and the rotation angle in the celes- tial reference frame. The three ERP components, (Earth’s rotation angle), are related to frame rota- tions about the , and axes, respectively. is quantified within the celestial reference frame and represents the angle between Earth’s prime meridian and the equinox. It is conventionally represented by UT1.

The formula for conversion from the Earth rotation angle (modulo (modulo , with units of seconds and arcseconds for UT1 and the Earth rota- tion angle, respectively.

x p ( t ) = 23 . 513 + 7 . 614 1( t − t 0 ) , (4)

y p ( t ) = 358 . 891 − 0 . 628 7( t − t 0 ) , (5)

During 2012–2022, the PM prediction errors relative to the reference data were both within Determination of PM by LLR To investigate the variation of lunar range measure- ment with respect to variations in ERP components , and , or the partial derivatives, we consider the fol- lowing geometrical relation:

δρ = H ( θ, ϕ ) δ r GCRS , (6)

H ( θ, ϕ ) = ( − cos θ cos ϕ, − sin θ cos ϕ, − sin ϕ )

where is the unit vector along the line of sight from the reflector to the station, transposed; angles are the celestial right ascension and declination of the reflector, respec- tively, as seen from the station; and the variation vector is the variation of the station’s geocentric inertial vector, caused by ERP variation.

According to the IERS Conventions 2010 , the for- mula for transformation from an international terrestrial ref- erence system (ITRS) to a geocentric celestial reference sys- tem (GCRS) is as follows:

2. The supplementary material provides further details regard-

ing the annual RMS data.

3. The supplementary material provides a visualization of the

polar motion.

where the bracketed term repre- sents the effects of the ERPs. The term is the transfor- mation matrix arising from the motion of the celestial pole in the celestial reference system, i.e., precession and nutation, while , and are matrices for the coordi- nate frame rotation about axes , and , respectively.

We adapted the equation to its current form using commuta- tivity between the terrestrial intermediate origin locator and the Earth rotation The variations in astronomical angles and due to ERP changes are negligible. If we apply partial deriva- tive rules on Equation (7), and let the ERP components be predicted values , then we have the follow- ing partial derivatives:

∂ρ ∂ y p = M HQR R 3 ( − ψ 0 ERA ) R 2 ( x 0 p ) R ′ 1 ( y 0 p ) · r ITRS (8)

∂ρ ∂ x p = M HQR R 3 ( − ψ 0 ERA ) R ′ 2 ( x 0 p ) R 1 ( y 0 p ) · r ITRS (9)

∂ρ ∂ψ ERA = − M HQR R ′ 3 ( − ψ 0 ERA ) R 2 ( x 0 p ) R 1 ( y 0 p ) · r ITRS (10)

where , and the matrices denote the derivatives of each rotation matrix with respect to the rotation angle.

We consider that the variation in ERPs represented by the vector is the exclusive source of the variation in the laser ranging measurements,

p = ( δ y p , δ x p , δψ ERA ) T

δρ δρ = ∂ρ ∂ y p δ y p + ∂ρ ∂ x p δ x p + ∂ρ ∂ψ ERA δψ ERA δρ

, i.e., , where the change in the laser ranging measurement, and the par- tial derivatives represent the sensitivity of the ranging mea- surement to changes in each ERP component. To deter- mine the PM components, we assume UT1 is known; hence, . At the epoch of the common view mea- surement, with the group of stations , and , we can form a linear equation , where the following holds:

δψ ERA = 0

Mp = q

   

, p = [ δ y p δ x p

] , q =

where is called the design matrix, while is the range residual vector, i.e., the difference between the measure- ments acquired and the measurements predicted by the theo- retical model. It is an overdetermined problem, with three equations to solve for two unknowns. We apply the least squares technique to the problem, and the normal equa-

M T Mp = M T q

tion is formed as , giving the least squares

p = ( M T M ) − 1 M T q δ q

estimate . If any perturbation added to the measurement residual, then the solution would perturbed correspondingly, i.e.,

δ p = ( M T M ) − 1 M T δ q

The PM predictions were calculated with a linear model as described in subsection 2.2. After solving , we adjust the prediction with to form estimates , i.e., . The esti- mated ERP components were then compared with the refer- ence data to form the solution error.

TCV Simulation Regarding the simulation phase, we simulated TCV events for the GMW station group, according to the exist- ing NP data at the European stations. First, we identified the available NPs at the European stations in the LLR data, taking each NP epoch as the TCV epoch. Second, ranging measurement data were generated at the NP epochs from the GMW stations to the observed reflector.

Finally, the same number of PM solutions were gener- ated from the simulated TCV events by solving the underly- ing equation

Mp = q

To assess the robustness of the determination method, we introduced perturbations to the range residual vector , and then computed the perturbed solution . The per- turbed solution was compared with the original solution to determine the effects of the perturbation.

We started with the control case denoted as S0. Regard- ing the perturbation on UT1, an error of μs was intro- duced to the UT1 prediction to observe its effect, referred to as SU1 and SU2 in . Regarding the perturba- tion on the ranging measurement, we have six cases rang- ing from SG1 to SW2, for which uniform range errors of were introduced to each of the three stations.

These perturbations were applied independently, although the linear nature of the model allows them to accumulate.

In the augmentation phase, we combined the existing NP data with the simulated data. The aim was to assess the accuracy of the solutions with common view events occurring in real-world observation epochs. In this phase, two augmentation strategies were employed in what were termed the ‘2+1’ stage and the ‘1+2’ stage; their settings are listed in In the ‘1+2’ stage, we identified the available NPs at the European stations in the LLR data, which were regarded as monostatic events. Taking each NP epoch as the TCV epoch, we simulated the missing data with range residuals of 48, and 22 mm for the stations GRSM, MATM, and WETL, respectively. The ‘1+2’ stage yielded 12 cases, from AG1 to AW4, which are detailed in In the ‘2+1’ stage, we first identified the bi-static com-

Phases/Stages Case ID Simulation Augmentation stage ‘1+2’ Augmentation stage ‘2+1’ mon view events at the European stations in the LLR data. The search for bi-static common view events was based on the criterion that the temporal separation between two NPs at different stations should be <1 h. We then calculated the mid-point of the pair of NP epochs to be the TCV event epoch. Finally, we simulated the range residual at the missing station in the same way as in the ‘1+2’ stage. The ‘2+1’ stage yielded four cases, as listed The outputs of both stages were evaluated using statisti- cal methods to estimate how the TCV solution would per- form under real-world conditions. If a TCV event occurs in reality, the solution should behave similarly to that of our results.

RESULTS AND ANALYSIS Preliminary Analysis Prior to solution of the determination problem, two important attributes can be assessed: the sensitivity and the stiffness. The sensitivity is represented by the partial derivatives, and the stiffness is represented by the condi- tion number.

The sensitivity of a station’s ranging measurement to variations in the ERPs can be quantified by the mean abso- lute values of the partial derivatives, which are measured Range residuals

in units of meters per arcsecond (m arcsec −1 ). In our

dataset, the mean absolute value of varies from 14 to 17 m arcsec among the stations, larger than that of which varies from 8 to 10 m arcsec . The WETL sta- tion exhibited the largest mean absolute values for both par- tial derivatives. The difference can be attributed to the geo- graphic location of the stations, specifically their distance from the , and axes of the ITRF. The lower sensitiv- ity of the range measurement to compared with those is due to the stations being located closer to the axis than to the Another key attribute is the stiffness of the least squares estimate problem , which can be quantified by the condition number of the normal matrix , i.e., . Here, the matrix norm is the 2-norm. The condition num- is inversely related to , which is the lunar declination. When approaches zero, the condition number soars to infinity. As an example, the rela- tion between is shown in , where the condition number is plotted on a logarithmic scale. In

M T Mp = M T q c Normal M T M c Normal = ∥ M T M ∥∥ ( M T M ) − 1 ∥ ∥·∥ c Normal δ moon δ moon c Normal

Normal

4. The supplementary material provides a table displaying the

per-station partial derivatives.

| δ moon | > 5 ◦

applying the criterion , 8 191 remained out of the original 9 692 TCV events. The condition numbers of the remaining TCV events could be confined to an upper limit of . Among the remaining 8 191 events, the GRSM station accounted for 93% (7 646 events), the MATM station accounted for 4% (331 events), and the WETL station accounted for 3% (214 events).

Condition number Normal −10 −5 Declination of moon Solutions Overview A total of 106 845 TCV solutions were obtained for simulation and augmentation cases, as listed in In each solution, a set of PM components a TCV event was evaluated.

In the simulation part, 8 191 simulated TCV events were solved one time in the unperturbed control case S0, and 8 more times in the perturbed cases from SU1 to SW2.

In the augmentation part, two stages were investi- gated. In the ‘1+2’ stage, 8 191 TCV events were aug- mented in four combinations. Augmentation solutions were categorized by station and residual signs into 12 cases, from AG1 to AW4. In the ‘2+1’ approach, there were 181 TCV events, each solved twice.

Solutions and Assessment of TCV Simulations In the simulation phase, we simulated TCV events at 191 epochs selected from the actual LLR data. Then, we compared the solved PM to the reference values to gen- erate the solution error data. In the control case S0, the solution errors were trivially zero. The positively per- turbed cases (SU1, SG1, SM1, and SW1) are depicted in . The negatively perturbed cases are symmetrical with the positive cases about the origin. It can be seen from the plots that each perturbation type displays a dis- tinct directional pattern in the error distribution.

The focus in the simulation phase was to examine the effects of perturbations in UT1 and range on the solution.

The effects were quantified by using the mean absolute error (MAE), RMS error (RMSE), and absolute maxi- mum (MaxAbs) metrics for the PM components and the pole position error . With the introduc- tion of the UT1 error of 100 μs, the resulting solution errors were only several milliarcseconds, as indi- cated by both the RMSE and the MAE. The maximum angular separation between the perturbed pole and the refer- ence pole reached 18 mas, with an average of 6 mas. For comparison, range perturbations of 50 mm resulted in solution errors that were 4–10 times greater. Detailed statis- tics are provided in The results imply that the UT1 prediction error of 100 μs could, on average, introduce an error of several milliarcseconds into the PM solutions. Furthermore, the range error of 50 mm could result in RMSEs of around 20 mas in the solution and nearly 50 mas in the solu- tion. The range error at the WETL station contributed less to the pole position error because of its higher latitude and therefore the greater distance from the of the ITRF, which is related to the fact that the partial derivatives were larger at the WETL sta- tion. The perturbation results are useful for assessing solu- tion errors under a certain amount of ranging error.

Solutions and Assessment of TCV Augmentations The augmentation phase focuses on evaluating solu- tion errors under real-world settings. displays all the augmentation results by case ID, the settings of which are listed in In the ‘1+2’ stage of the augmentation phase, we simu- lated two range residuals for each selected range residual data. This stage yielded 12 cases from AG1 to AW4.

Among the cases, the maximum pole position error was 752 mas. When comparing the pole position data to the lin- ear pole prediction, 88.5% of the events (29 000 out of 32 yielded more accurate results than the prediction. The component errors were markedly larger than those of the component owing to the geographical locations of the GRSM and MATM stations, i.e., the distance to the axis of the ITRF is less than that to the -axis. The WETL cases, labeled AW1–AW4, yielded the lowest errors, while the MATM cases ranked in the middle and the GRSM cases exhibited the highest errors. This is partly attributable to the low range residuals of the WETL station, and also to the limited size of the WETL dataset.

In the ‘2+1’ stage of the augmentation phase, we identi- fied 181 bi-static common view events from the actual NP data without applying the lunar declination filter. In the selected bi-static events, the range residuals varied from −0.097 m to +0.141 m and the RMS was 0.034 m.

Of all the bi-static events, 69.6% (126 of 181) were between GRSM and MATM and 30.4% (55 of 181) were between GRSM and WETL. This stage yielded four cases from AGM1 to AGW2, as listed in . The statisti- cal values of the component are obviously larger than

error/mas error/mas error/mas error/mas those of the component owing to the same geographi- cal reasons. Comparison of the linear PM predictions with the two perturbed cases combined reveals that 91.2% of the solutions provided values that were more accurate than the predictions. Among all cases, the maximum pole position error was 899 mas, which was attributable primar- ily to the high value of the maximum . The AGW1&2 cases yielded lower errors owing to the low range of the residual values of the WETL station. error/mas error/mas error/mas error/mas the percentage of the solutions that outperformed the predic- tions, were comparable. This indicates that the two augmen- tation strategies did not introduce substantial bias in simulat- ing reality. of 32 764) of the solutions in the ‘1+2’ augmentation, and SU1, SG1, SM1, and SW1, respectively.

Solution variation/mas Perturbations MaxAbs MaxAbs MaxAbs Control Case (S0) (trivial zero) UT1 ±100 μs (SU1, SU2) GRSM ±50 mm (SG1, SG2) MATM ±50 mm (SM1, SM2) WETL ±50 mm (SW1, SW2)

Case ID This paper presents a method for determining Earth’s PM using tri-static range measurements in real time. We formulated the linear equations for this purpose and applied the least squares method to solve them. The solu- tions were initiated using linear PM predictions. Prelimi- nary analysis showed that the partial derivatives were asso- ciated with the geographical locations of the stations, and that the condition numbers were strongly correlated with the lunar declination. The solutions were compared with the IERS Earth Orientation Parameter 20 C04 time series to generate solution error data. Our method was exam- ined through a two-phase study, including a simulation phase and an augmentation phase. error/mas error/mas

During the simulation phase, we simulated TCV events to the actual range data epochs. Predefined errors were introduced in the UT1 and range to observe the conse- quent variations in the solution. The results showed that the UT1 prediction error of 100 μs led to minor varia- tions of several milliarcseconds in the solutions, and that the uniform range errors of 50 mm could lead to errors of 18–23 mas in the solution and 24–45 mas in the solution (RMSE). Perturbation patterns were revealed in the solution variations.

During the augmentation phase, two strategies were used to simulate realistic TCV scenarios. In the first strat- egy named ‘1+2’, we simulated two additional measure- ments for each NP entry. Realistic range errors were intro- duced to yield solution errors of 26 and 53 mas in error/mas error/mas Solution error/mas MaxAbs MaxAbs MaxAbs ‘1+2’ summary ‘2+1’ summary

(RMSE), respectively. In the second strategy named as ‘2+1’, we identified the actual bi-static common view events and augmented them into tri-static events using simu- lated data, to yield solution errors of 34 and 99 mas in (RMSE), respectively. Both strategies revealed that the solution error of the component was markedly larger than that of the component. The largest pole posi- tion error was 899 mas. The realistic scenarios within the ‘1+2’ and ‘2+1’ augmentation strategies yielded solutions that were more accurate than the linear pole prediction for 88.5% and 91.2% of the cases, respectively. Cases involv- ing WETL data had notably smaller error statistics owing to the low range of the residuals. The results presented in this paper are compared with those of other reported meth- Parameter However, our method cannot replace traditional long- term determination methods, partly because of its depen- dence on precise geodetic data and partly because it lacks the capability to estimate additional parameters.

ACKNOWLEDGEMENTS This research was supported by the National Natural Science Foundation of China (NSFC) (11673082 and 11903059). Current LLR data are collected, archived, and distributed under the auspices of the International Laser Ranging Service (ILRS). The LLR data used in this paper were retrieved from the EUROLAS Data Center. We acknowledge with thanks that the processed LLR data, acquired since 1969, have been obtained through the efforts of the personnel at the Observatoire de la Côte d’Azur in France, the LURE Observatory in Hawaii (USA), the McDonald Observatory in Texas (USA), the Apache Point Observatory in New Mexico (USA), the Mat- era Laser Ranging Observatory in Italy, and the Wettzell Laser Ranging System in Germany. We greatly acknowl- edge all the developers of the software packages used in this paper, i.e., SOFA, PyMsOfa, IERS Conventions 2010 software collection, and Astropy.

AI DISCLOSURE STATEMENT Deepseek was employed for language and grammar checks within the article. The authors carefully reviewed, edited, and revised the Deepseek-generated texts to their own preferences, assuming ultimate responsibility for the content of the publication. ods in . Our method could yield errors that are 20–60 times larger than LLR long-term solutions, or 300– 000 times larger than VLBI solutions. However, the tem- poral resolution of our method is the LLR NP bin size, i.e., approximately 15 min, which is much shorter than that of the other methods considered.

Overall, these results indicate that tri-static determina- tion of PM coordinates is possible. Our method requires one set of data from three stations tracking the moon simul- taneously, instead of a long period of data accumulation.

This makes it a potentially viable tool for detecting tran- sient changes in Earth’s rotation. It could also prove useful for validating ERPs to accuracy of several milliarcseconds.

AUTHOR CONTRIBUTIONS

Zhipeng Liang was responsible for the methodology, software development, validation processes, and the prepara- tion of the original draft. Chengzhi Liu and Xue Dong undertook the supervision of the project and contributed to its conceptualization. Zhipeng Liang and Yanning Zheng were involved in data curation, visualization cre- ation, and investigative tasks. Chengzhi Liu, Xue Dong, and Yanning Zheng participated in the reviewing and edit- ing of the manuscript. All authors have read and approved the final manuscript.

DECLARATION OF INTERESTS

Chengzhi Liu is an editorial board member for Astro- nomical Techniques and Instruments and he was not in- volved in the editorial review or the decision to publish this article. The authors declare no competing interests.

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SUPPLEMENTARY MATERIAL For the sake of completeness, additional data tables and figures that are not included in the main text of the manuscript are provided here.

S.1 Preprocessing of LLR data In the case of traditional LLR analysis studies, it is usual to estimate numerous physical and empirical model parameters in addition to the ERP components. Our approach, however, cannot solve for extra parameters while solving for instant ERP components. For the long- term physical effects, we performed a global preprocess- ing on the complete 11-year dataset of LLR residuals. In the preprocessing of the LLR residuals, we fitted for the fol- lowing parameters: (1) ITRF coordinate adjustments of the GRSM, MATM, and WETL stations: 9 parameters; (2) Lunar coordinate adjustments of the Apollo- 11/14/15 and Luna-17/21 reflectors, X component only: 5 parameters; (3) Periodical terms, including triannual, biannual, annual, synodic, semi-synodic, lunar-day, and solar-day terms: 14 parameters; (4) Other empirical terms, including constant and lunar solid tide factor: 2 parameters.

The range residuals plotted per station are shown in Fig. S1

0.20 GRSM

Range residual/m −0.05 −0.10 −0.15 −0.20 Modified Julian day Fig. S1. LLR residuals at the European stations.

Table S1 presents the root mean square (RMS) tempera- ture values by station and year from 2012 to 2022, along with the overall RMS across all years.

Fig. S2 displays the annual range residual statistics for each station.

S.2 Polar motion on the ground For interested readers, Fig. S3 provides a clearer and more intuitive visualization of the polar motion.

Fig. S3 a map near Earth’s north pole, where we provide the direc- tion to 0° longitude (Greenwich) and 90°W longitude (Chicago). The ITRF north pole is in the upper-left direc- tion outside this map, where pole offsets both zero.

Fig. S4 shows the prediction errors obtained by compar- ing actual polar motion data with predictions from the mean pole model defined in the IERS Conventions (2010).

S.3 Partial Derivatives Table S2 displays the mean absolute values of the par-

Table S1. RMS of range residuals by station and year RMS of range residual/m Fig. S2. RMS of range residuals of the stations in the selected time span.

Reference pole Predicted pole To Greenwich To Chicago

Fig. S3. Predicted and reference pole offset and from 2010 to 2022. Note that the axis is inverted according to geographical convention. The reference pole (black solid curve) drifts around within a certain area, while the predicted pole (dashed line) moves linearly. Deviations between the reference and predicted poles are <300 mas within the selected time span. In the lower-right corner, a 1-m rule is shown to scale (1 m 32.3 mas).

Prediction error error/mas error/mas Table S2. Mean of the absolute values of the ranging partial derivatives to all ERP components ������ ������ ������ ������ ����� ����� ����� ����� Stations /(m s /(m arcsec /(m arcsec /(m arcsec tial derivatives of the ranging measurements with respect to the ERP components, where the UT1 component is listed in both the and the UT1 columns, with units of meters per arcsecond and meters per second, respec- tively.

Stations RMS of range residuals/mm