Separation of geophysical signals in the LAGEOS geocentre motion based on singular spectrum analysis

SUMMARY

We recompute the 26-yr weekly Geocentre Motion (GCM) time-series from 1994 to 2020 through the network shift approach using Satellite Laser Ranging (SLR) observations to LAGEOS1/2. Then the Singular Spectrum Analysis (SSA) is applied for the first time to separate and investigate the geophysical signals from the GCM time-series. The Principal Components (PCs) of the embedded covariance matrix of SSA from the GCM time-series are determined based on the w-correlation criterion and two PCs with large w-correlation are regarded as one periodic signal pair. The results indicate that the annual signal in all three coordinate components and semi-annual signal in both X and Z components are detected. The annual signal from this study agrees well in both amplitude and phase with those derived by the Astronomical Institute of the University of Bern and the Center for Space Research, especially for the Y and Z components. Besides, the other periodic signals with the periods of (1043.6, 85, 28), (570, 280, 222.7) and (14.1, 15.3) days are also quantitatively explored for the first time from the GCM time-series by using SSA, interpreting the corresponding geophysical and astrodynamic sources of aliasing effects of K₁/O₁, T₂ and M_m tides, draconitic effects, and overlapping effects of the ground-track repeatability of LAGEOS1/2.

1. INTRODUCTION

The origin of the International Terrestrial Reference Frame (ITRF) is determined by using the geodetic techniques such as Doppler Orbitography and Radiopositioning Integrated on Satellite (DORIS, Crétaux et al. 2002; Feissel-Vernier et al. 2006; Willis et al. 2006; Gobinddass et al. 2009), Global Navigation Satellite System (GNSS, Rebischung et al. 2014; Zajdel et al. 2019a; Kosek et al. 2020), and Satellite Laser Ranging (SLR, Cheng et al. 2013; Kosek et al. 2020). Since the stations of the global observation network are located at the crust of the Earth, their coordinates are related to the centre of network (CN) rather than the centre of figure (CF), whereas Earth satellites orbit around the centre of mass (CM) of the entire Earth system including atmosphere, solid Earth, continental water and oceans (Chen et al. 1999; Dong et al. 2003; Cheng et al. 2013; Altamimi et al. 2016). The definition of geocentre motion (GCM) is the offset of CM relative to CF (Chen et al. 1999; Metivier et al. 2010; Zannat & Tregoning 2017a) or, oppositely, the offset of CF relative to CM (Klemann & Martinec 2011). Due to a finite number of inhomogeneous distribution of global SLR stations which cause especially the lack of observation coverage over the Southern Hemisphere and vast ocean area (Sośnica et al. 2013; Wu et al. 2006, 2012; Zajdel et al. 2019b), the discrepancy between CN and CF leads to the so-called network effect (Dong et al. 2014; Riddell et al. 2017; Feissel-Vernier et al. 2006; Ries, 2013, 2016; Zannat & Tregoning 2017b).

The main methods for determining GCM include the network shift approach, the dynamic approach (i.e. derivation of the gravity field spherical harmonics of degree-1), kinematic approach, and inverse surface load-displacement method (Kosek et al. 2020; Kang et al. 2019; Wu et al. 2002; Greff-Lefftz et al. 2010). Different methods are sensitive to different error sources and the option of tracking data type (Kosek et al. 2020; Gobinddass et al. 2009). The network shift approach performs the 7-Helmert transformations between the realized CM frame (e.g. the estimated network station coordinates) and a priori frame to estimate the three translation parameters which represent the GCM (Kang et al. 2019; Collilieux et al. 2009; Dong et al. 2003; Ries 2016). For the dynamic approach, GCM is expressed as the linear relationship with the non-zero degree-1 spherical harmonic coefficients under the a priori ITRF rather than the CM frame (Sośnica et al.2014,2015; Cheng et al. 2013). It is important to note that the combined use of multiple satellites with different inclination angles is needed to realize the reliable solution due to the strong correlation between the harmonic coefficients (Sośnica et al.2015). The kinematic approach directly estimates the three coordinate offsets between CF and CM at each epoch, then GCM is computed by averaging the offsets during a specific period (Couhert et al. 2018; Kang et al. 2019), thereby it needs a dense and uniform distribution of observation stations. The Earth is an elastic body, and the loading change will lead to surface deformation, causing a movement of the CF relative to the CM. Hence, the inverse surface load–displacement method uses the surface deformation observed through the globally distributed geodetic network for the recovery of GCM (Blewitt et al. 2001; Abbondanza et al. 2020).

The study of GCM is critically important for understanding the geodynamic processes of global mass redistributions within the Earth system especially those induced by surface water, atmosphere, Mean Sea Level Changes (MSLCs), Earth tides, glacial isostatic adjustment, mantle convection and liquid core oscillations (Wu et al. 2012; Collilieux et al. 2009; Feissel-Vernier et al. 2006; Bergmann-Wolf et al. 2014; Yu et al. 2018). Plag & Rothacher (2006) pointed out that maintaining the accuracy for ITRF at a level of 1 mm and the stability at a level of 0.1 mm yr^–1 was a long-term scientific goal of the global geodetic observing system, which was to satisfy some more precise requirements such as MSLCs at a rate of a few millimetres per year (Rebischung & Garayt 2013). Wu et al. (2012) and Metivier et al. (2010) indicated that more accurate and consistent ITRF realizations and improved GCM determination were expected to estimate and interpret MSLCs at an accuracy of 0.1 mm yr^–1. The origin of ITRF is the strict average of the long-term CM by considering the tidal impact (Watkins & Eanes 1997), while the seasonal variations are not corrected (Kang et al. 2019; Wu et al. 2012). Melachroinos et al. (2013) indicated that ignoring the annual GCM correction can result in an obvious 0.14 mm yr^–1 systematic error in the MSLCs estimates at 2-yr intervals. Sun et al. (2016) pointed out that the ocean mass change displayed an obvious discrepancy of 173, 95 and 190 Gt on the global scale, caused by 1-mm GCM error in all the three geocentre components. Therefore, accurate quantification and analysis of the annual component of GCM are of great importance to the establishment and maintenance of ITRF and its geodetic and geophysical applications. At present, the GCM time-series solved by different research institutions, such as the Centre for Space Research (CSR) and the Astronomical Institute, University of Bern (AIUB), is different to a certain extent. Besides, these GCM time-series by using SLR data are relatively short and the longest time-series is from CSR just until 2017 (http://download.csr.utexas.edu/pub/slr/geocenter/, Ries 2016). Therefore, we recompute the geocentre motion time-series from 1994 to 2020, extending the GCM time-series by 3 yr.

Besides, only an evident annual signal and a very weak semi-annual signal have been studied extensively in the GCM time-series with different approaches and techniques (Feissel-Vernier et al. 2006; Collilieux et al. 2009; Wu et al. 2012). However, the GCM time-series may also consist of other periodic signals which are seldom studied. Kosek et al. (2020) showed that a periodic oscillation of shorter than 120 d was detectable from GNSS and DORIS amplitude spectra and the annual signals were detected in all residuals by combinedly using Hilbert Transform and Fourier Transform Band Pass Filter (HT + FTBPF), attributing these oscillations to the technical systematic errors caused by the mismodelling of the satellite orbit. Kuzin et al. (2010) used the adaptive dynamic regression model to analyse the GCM estimates, finding out several other harmonic signals with different periods, but without giving their geophysical interpretation. Thus, this motivates us to explore the periodic signals in the LAGEOS GCM and their geophysical and astrodynamic interpretations.

As for signal analysis, the singular spectrum analysis (SSA), as a powerful non-parametric spectrum estimation method, can effectively realize the separation of the periodic sources from a time-series (Vautard & Ghil 1989; Vautard et al. 1992). SSA can decompose a time-series into various components, where those dominating the spectrums can be interpreted as trends or periodic signals, and the remaining components are usually regarded as noise (Vautard et al. 1992; Hassani 2007). SSA allows us effectively to follow the evolution in amplitude and phase of a signal, maintaining the original characteristics of the signal to the greatest extent. When missing data exist in a time-series, SSA is improved by Shen et al. (2015) and Wang et al. (2020) for directly processing the incomplete time-series with the single- and multichannel SSA, respectively. As a special application of the empirical orthogonal function, SSA conducts the construction of the principal components of a time-series and its prediction by using the w-correlation criterion (Hassani 2007; Shen et al. 2018). Besides, no one has applied SSA to study GCM time-series so far. Therefore, SSA is used for the separation and construction to analyse the derived GCM time-series in this study.

The paper is organized as follows. Section ‘Geocentre Motion Solution from LAGEOS 1/2’ summarizes the strategy of GCM determination and comparison. Section ‘Annual Geocentre Motion Analysis’ presents the theory of SSA and the analysis of annual GCM. Section ‘Other Periodic Signal Analysis’ provides the analysis of other periodic signals and their geophysical interpretation. The last section concludes the paper.

2. GEOCENTRE MOTION FROM LAGEOS 1/2

2.1. Processing scheme and models

We determine the GCM time-series from SLR observations to LAGEOS satellites (Pearlman et al. 2019a) using the Bernese GNSS Software 5.2. The force models, the list of parameters and the data processing scheme used in our solution refer to Zajdel et al. (2019b). Table 1 shows the scheme differences among this study, CSR and AIUB. The weekly solution scheme of LAGEOS data is adopted by choosing the length of arc for 7 d. The satellite orbits of LAGEOS1/2, the Earth rotation parameters, station coordinates, range biases and GCM parameters are simultaneously estimated as listed in Table 1. Due to the high correlations between the range biases and the vertical component of station coordinates, the range biases are estimated as one set per 7-d arc only for selected stations. Estimating range biases only for the non-core stations (as shown in Fig. 1) rather than all stations is expected to reduce the scatter and improve the deviation along the X- and Z-axes as well as mitigate the aliasing of draconitic errors (Spatar 2016). Ries (2016) from CSR introduced the estimation of a bias for each 30- or 60-d solution, but with a somewhat tight a priori constraint. This allows enough freedom for the local site loading (especially impact on the height of station) to go into the range bias, remaining the GCM time-series relatively uncorrupted by the local loading. SLR data is provided by a global network of stations coordinated by the International Laser Ranging Service (ILRS), whereas the highest-quality SLR data modelling standards are being developed by the ILRS Analysis Standing Committee (ASC, Luceri et al. 2019).

Table 1.

The scheme difference between this study, CSR and AIUB (S - same as this study).

	Description
Scheme and Models	This study	AIUB	CSR
Approach	Network shift	S	Dynamic approach
Software	Bernese 5.2	S	UTOPIA
Reference frame	SLRF2014	SLRF2008	S
Arc length	7 d	S	30 or 60 d
Estimated parameters
Satellite orbits	One set per 7-d arc, 6 Keplerian elements; 5 empirical parameters; A constant along-track acceleration; once-per-revolution parameters in along-track and cross-track	S	Orbit elements, one set per 30- or 60-d arc; Once-per revolution parameters in along-track and cross-track every 3.5 d; A constant along-track acceleration every 1.75 d
Station coordinates	One set per 7-d arc, X, Y, Z components for every station	S	Station coordinates fixed
Range biases	One set per 7-d arc, only for selected SLR stations	S	One set per 30- or 60-d arc for every station with a relatively tight a priori (∼2.5–3.0 times the a posteriori sigma or ∼5 mm for the 60-d solutions)
			(alternatively, heights can be estimated with the same a priori constraint)
Earth rotation parameters	8 parameters per 7-d arc using piece-wise-linear parametrization; Pole X and Y coordinates and UT1-UTC with one parameter fixed to the a priori IERS-14-C04 series	S	Not estimated; IERS-14-C04 used
Geocentre coordinates	One set per 7-d arc	S	One set per 30- or 60-d arc

Figure 1.

Ground network distribution of SLR stations from 1994 to 2020. Blue dots denote all stations and red pentagrams are the core stations.

The network shift approach is used for computing the GCM time-series. For the realization of the reference frame, the minimum constraint conditions are imposed on the core station network (Fig. 1). The 7-parameter Helmert transformation is adopted for the transformation of the realized frame and the a priori frame as a verification of the core stations selected for the minimum constraints. The estimated translation parameters represent GCM. Because Earth rotation parameters, station coordinates, and orbits are simultaneously estimated, the no-net-rotation (NNR) is mandatorily applied to invert a normal equation matrix and remove singularities. Besides, the datum definition of global networks typically requires the no-net-translation when estimating GCM as an additional parameter. Zajdel et al. (2019a,b) discussed the details of minimum constraint conditions and their implications in SLR and GNSS networks, respectively.

2.2. Geocentre motion solution and comparison

The time-series of GCM solutions are derived using the methodology from Section 2.1 and presented with a blue line in Fig. 2 for the time-series of estimated GCM and in Fig. 3 after applying smoothing with a 1-month window (Zajdel et al. 2019b). To validate the solutions, the GCM weekly solutions from AIUB (http://ftp.aiub.unibe.ch/GRAVITY/GEOCENTER/, Sośnica et al. 2014) and monthly solution from CSR are also shown in Fig. 2 with the pink line and Fig. 3 with the red line, respectively. Moreover, the results from AIUB and CSR are demonstrated in Fig. 4 with the same colour as in Figs 2 and 3. In Figs 2 –4, ‘SLR-weekly’, ‘SLR-AIUB-weekly’ and ‘SLR-CSR-monthly’ denote the GCM estimates from this study, AIUB, and CSR. ‘SLR-AIUB-weekly’ is equivalent to the internal validation and ‘SLR-CSR-monthly’ to the external validation.

Figure 2.

Weekly GCM time-series estimated in this study and from AIUB.

Figure 3.

GCM time-series estimated in this study and CSR.

Figure 4.

GCM time-series estimated by AIUB and CSR.

Fig. 2 shows that this solution agrees well with that of AIUB. Table 2 gives the statistical values of the differences between this solution and AIUB. The RMS of the Z-component difference reaches 9 mm, whereas it is at the 5 mm level in the two equatorial components. Moreover, the mean difference is about 1 mm in the Y component, indicating an offset between the two solutions. In 1997, a deviation occurs, which is caused by different modelling standards employed in two solutions, that is different a priori reference frames, different a priori range biases, and independent selection of the stations used as core stations for the no-net-translation network constraint. The GCM time-series from this study is consistent with those from CSR (see Fig. 3), but associating with high-frequency variations. This mainly depends on the arc length used. CSR employs 30- or 60-d arc (see Table 1). The longer arcs including more SLR observations can make the solution more stable and less noisy, which was elaborated in details based on the analyses conducted by Lutz et al. (2016) using GPS and GLONASS 1-d and 3-d orbit arcs. In addition, each arc length has its own aliasing periods with other signals (including geophysical signals and the satellite revolution period) and longer arcs are less sensitive to aliasing issues with draconitic and other spurious periods with the cost of a poorer temporal resolution (Stewart et al. 2005). In this study, the 7-d LAGEOS arcs are used, which is similar to the standard products of the ILRS Analysis Standing Committee. After 2011, an obvious discrepancy occurs between CSR and AIUB in the two equatorial components with a slight shift of the amplitude and phase in the Y component due to different a priori reference frames used (see Fig. 4). AIUB solutions used SLRF2008 as the newest reference frame available in 2014 (Sośnica 2015a,b), whereas CSR and in this study SLRF2014 was used. SLRF2008 was based on solutions until 2008, therefore a degradation is observed in 2010–2015 for AIUB solutions. The GCM time-series from this study better agrees with those from CSR than those from AIUB in the two equatorial components. However, the GCM time-series from AIUB has a better consistency with those from CSR than those from our GCM for the Zcomponent.

Table 2.

The statistical differences between this study and AIUB (mm).

	MEAN	RMS	STD
X	–0.18	5.15	5.14
Y	–1.30	4.82	4.64
Z	0.34	8.95	8.94

3. ANNUAL GEOCENTRE MOTION ANALYSIS

3.1. Singular spectrum analysis

By using SSA, the original time-series can be decomposed into a series of components, where each component can be determined as either noise component, quasi-periodic or periodic component, or trend component (Hassani 2007). For the GCM time-series {g_i, i = 1, 2, …, L} with the length L, the trajectory matrix G can be formed by a window size N (N < L/2) as follows:

(1)

Then the Toeplitz lagged correlation matrix can be expressed as

(2)

where the unbiased autocovariance function can be computed by

(3)

The correlation matrix is decomposed as , where is a diagonal matrix with the diagonal values sorted in descending order and is an orthogonal matrix with row eigenvectors . The temporal matrix of principal components (PCs) can be generated as after projecting the series onto . The PCs at row and column of A is expressed as

(4)

where . The reconstructed component (RC) can be given by

(5)

where is the ith component from the PCs. Since is sorted in descending order, the first several numbers of RCs can approximate the original GCM time-series, while the remaining RCs can be regarded as noise. Therefore, if one wants to reconstruct the GCM time-series , it can be obtained by the sum of the first p RCs as follows:

(6)

The window size is a key issue of SSA and is chosen empirically. The optimal window size should maximize the separation of the contained signals (Kondrashov et al. 2010) and is usually proportional to the periods of main signals. The window length N is selected to best reflect the separability of SSA. The properties ‘separability’ of SSA is also called the weighted correlation or w-correlation (Hassani 2007), which is a natural measure of dependence between two decomposed series and in eq. (7),

(7)

where The smaller absolute w-correlation value means the more w-orthogonal of two series, and the zero value indicates the two series are completely separable. On the contrary, the two components will be far from w-orthogonal and strongly dependent. Therefore, we possibly regard these two PCs with large w-correlation as one periodic signal pair.

3.2. Singular spectrum analysis of geocentre motion series

Since the annual variation is the most significant signal, we select the widow lengths with different integer years to determine the best separable window length based on the w-correlations. For the 26-yr GCM time-series of weekly solution (1359 weeks in total), the window length N with 157 points (3 yr) can best reflect the separability of SSA and is most suitable to separate the various short-periodic signals from the GCM time-series. The eigenvalues of the first 30 RCs of the three components are illustrated in Fig. 5. The first two eigenvalues of the two equatorial components together with the second and third eigenvalues of the Z component are much larger than other eigenvalues, which indicates that the signals of the RCs (1,2) for the two equatorial components and RCs (2,3) for the Z component are much more significant than other components. By applying fast Fourier transform (FFT) analysis, these periodic pairs are the annual terms of GCM. The annual GCM and its spectral analysis in the three components are illustrated in Fig.   6. The amplitudes are 2.8, 3.0 and 3.9 mm in the three components, respectively. The most significant RC (1) of the Z component will be analysed in Section 4.

Figure 5.

Eigenvalues of the first 30 RCs of the X, Y and Z component.

Figure 6.

Annual periodic signals in the three components.

3.3. Annual geocentre motion analysis

The annual variation is the most significant signal in the GCM time-series, which is mainly induced by the large-scale mass redistribution within continental hydrology (Klemann & Martinec 2011), the ocean and atmosphere (Dong et al. 2014; Chen et al. 1999; De Viron et al. 2005). In the first three solutions of Table 3, both the phases and amplitudes of the monthly smooth LAGEOS SLR solutions from this study show good agreement with the CSR solution for the Y and Z components, whereas an apparent disagreement is displayed in the X component. Compared to the AIUB solution, the phases of our solution are a bit smaller in the Z and Y components. The predicted annual GCM from the geophysical models tends to agree well with the SLR solutions of this study. As implied in Kang et al. (2019), Couhert et al. (2018), Cheng et al. (2013) and Collilieux et al. (2009), the geocentre motions of X and Z are more sensitive to the network effects because of the uneven and sparse distribution of SLR stations (see Fig. 1) with most high-performance stations in Europe thus close to the X Earth axis in the Northern Hemisphere. This is especially obvious for the GCM time-series in the X component, which also displays more scatter than the Y component as shown in Fig. 3. A changeable network configuration, where the number of measurements and SLR stations varies with arcs, may be another cause of higher scatter in the X component. Moreover, some studies (Collilieux et al. 2009; Ries 2016 ; Couhert et al. 2018) indicated that estimating range biases and station heights for all stations can mitigate the effect of the vertical errors on the SLR-based GCM in the Z and X components, whereas biases for only selected stations are estimated in this study as recommended by the ILRS ASC. These two aspects may more or less lead to the overestimating annual amplitude of the GCM time-series in the X component. However, for the Y component, the GCM time-series does not seem to be apparently sensitive to the network effects and the correlation with the station heights (Couhert et al. 2018; Kang et al. 2019), suggesting a better distribution of SLR station for the geocentre Y-component recovery.

Table 3.

Annual GCM time-series estimated by various approaches and techniques as well as predicted by geophysical models (Wu et al. 2012; Sun et al. 2016; Kang et al. 2019).

	X		Y		Z
mm	amp	pha	amp	pha	amp	pha	Comments and reference
SLR NS (This study)	2.8	31	3.0	292	3.9	52	1994.01-2020.01, L1/L2, 7-d estimates
SLR NS (This study)	2.8	28	3.0	304	3.9	61	Monthly-smooth
SLR NS (AIUB)	2.2	32	2.2	328	3.7	71	1994.01-2015.02, L1/L2, 7-d estimates (Sośnica 2015a,b;2014)
SLR DA (CSR)	1.7	32	2.8	298	4.3	57	1994.01-2017.01, L1/L2, 30-d estimates (Ries 2016)
SLR NS	2.8	35	2.1	319	3.8	65	2008.01-2020.01, L1/L2, 7-d estimates (Drozdzewski et al. 2019)
SLR NS	1.9	50	2.8	321	3.9	28	1993.01-2016.12, L1/L2, 30-d estimates, mean of 4 SLR solutions using ITRF2005 and ITRF2014 (Ries 2016)
SLR DA	0.9	61	2.6	324	2.2	26	1993.01-2015.01, L1/L2, (Couhert et al. 2020)
SLR NS	2.1	48	2.0	327	3.5	43	1993.01–1996.08 (Bouillé et al. 2000)
SLR NS	2.6	32	2.5	309	3.3	36	1993.01–2000.01 (Crétaux et al. 2002)
SLR NS	3.5	26	4.3	303	4.6	33	1993.01–2001.07 (Moore & Wang 2003)
SLR NS	2.7	50	3.8	309	5.4	5	1993.01–2007.06 (Angermann & Müller 2009)
SLR NS	2.7	45	3.8	327	3.6	4	1993.01–2006.01 (Collilieux et al. 2009)
SLR NS	2.6	42	3.1	315	5.5	22	1983.0–2009.0 (Altamimi et al. 2011)
SLR NS	3.0	55	2.7	328	5.4	22	2002.03–2009.01 (Wu & Heflin 2015)
SLR KA/NS	1.5	21	3.1	302	5.9	21	2008.07-2015.06, Jason-2, 10-d estimates (Couhert et al. 2018)
DORIS KA/NS	1.6	113	3.2	322	6.4	18	2008.07-2015.06, Jason-2, 10-d estimates (Couhert et al. 2018)
GPS NS	2.9	363	3.2	319	3.0	168	1997.01–2009.01 (Rebischung et al. 2014)
SLR KA	2.2	60	3.2	303	2.8	46	1992.07–1997.01 (Eanes et al. 1997)
SLR KA	2.1	48	2.0	327	3.5	43	1993.01–1996.08 (Bouille et al. 2000)
SLR KA	2.6	32	2.5	309	3.3	36	1993.01–2000.01 (Cretaux et al. 2002)
SLR KA	3.5	26	4.3	303	4.6	33	1993.01–2001.07 (Moore & Wang 2003)
SLR KA	3.2	31	2.6	305	4.3	31	2002.01–2010.06 (Cheng et al. 2013)
SLR KA	3.5	59	2.9	321	4.6	28	1995.01–2015.01 (Spatar 2016)
SLR KA	2.8	45	2.5	322	5.7	32	1995.01–2015.01 CSR monthly
SLR KA	2.9	49	2.7	324	4.2	33	2002.06–2014.05 (Cheng et al. 2013)
SLR KA	1.1	65	3.4	325	2.9	22	2003.01-2016.06 (Kang et al. 2019)
GPS/LEO KA	3.0	32	2.4	353	4.0	288	2003.01–2007.05 (Kang et al. 2009)
INV	2.3	52	2.8	327	2.9	69	2002.06-2014.05 (Sun et al. 2016)
INV	1.9	41	2.9	329	3.7	25	1993.01-2004.01 (Wu et al. 2006)
INV	2.0	21	2.6	334	3.6	24	2002.01-2007.01 (Jansen et al. 2009
INV	2.0	62	3.5	322	3.1	19	2003.01-2009.01 (Rietbroeck et al. 2012)
INV	1.8	49	2.7	329	4.2	31	2002.01-2009.01 (Wu et al. 2010)
INV	1.9	25	3.3	330	3.7	21	2002.01-2014.01 (Wu & Heflin 2014)
INV	3.9	22	2.7	25	7.6	57	1999.05–2004.05 (Kusche & Schrama 2005)
INV	1.8	46	2.5	329	3.9	28	2002.03–2004.02 (Wu et al. 2006)
INV	1.1	52	2.7	325	1.2	55	2003.01–2007.01 (Swenson et al. 2008)
CLM	4.2	47	3.2	295	3.5	36	1992.01–1995.01 (Dong et al. 1997)
CLM	2.4	26	2.0	360	4.1	43	1992.07–1997.01 (Chen et al. 1999)
CLM	2.1	28	2.1	342	2.7	49	1993.01–2006.01 (Collilieux et al. 2009)
CLM	1.6	34	1.8	326	3.1	16	1993.01–1996.08 (Bouillé et al. 2000)
CLM	2.3	16	2.0	352	3.4	30	1993.01–2001.07 (Moore & Wang 2003)
GPS UA	2.1	39	3.2	346	3.9	74	1997.02–2004.02 (Lavallée et al. 2006)
GPS UA	0.1	40	1.8	342	4.0	22	1994.01–2008.01 (Fritsche et al. 2010)
SLR KFS	2.1	45	2.7	321	3.9	21	2002.03–2009.03 (Wu et al. 2015)
GPS RC	0.1	39	1.8	342	4.0	22	1994.01–2008.01 (Fritsche et al. 2010)
GNSS RC	1.7	73	2.6	295	6.6	59	1994.01–2011.01 (Glaser et al. 2015)
Mean	2.3	49.3	2.8	315.4	4.0	44.0	Mean values of amplitudes and phases from all these models

Note: CLM, climate model; INV, inverse method; UA, unified approach; NS, network shift; KA, kinematic approach; DA, dynamic approach; KFS, Kalman filter and Rauch–Tung–Striebel smoother; RC, rigorous parameter combination.

The annual GCM estimates with other approaches and other techniques, as well as the geophysical prediction models, are listed in Table 3 and illustrated in Fig. 7. The GCM estimates derived by the network shift (NS) and the kinematic approach (KA) from SLR data has always been proven to be relatively reliable. In contrast, even though the GNSS and DORIS techniques have the advantage of dense and evenly distributed tracking station networks, the GCM from these techniques is usually affected by the orbit mismodelling and the correlations between parameters (Couhert et al. 2018; Gobinddass et al. 2009). In recent years, the inverse estimates of GCM from geophysical predictions have been increasingly applied by combining the ocean bottom pressure (OBP), Gravity Recovery and Climate Experiment satellite mission (GRACE) and loading deformation data (Sun et al. 2016). As displayed in Fig. 7, the inverse estimates which are hardly affected by the network effects show small X and Z annual amplitudes, whereas the SLR-based estimates with NS and KA methods represent relatively large X annual amplitudes, which agrees with Spatar (2016). Besides, the X annual amplitude from the dynamic approach is smaller than NS and KA methods. The agreement is shown between our results and other SLR estimates for Y and Z components, while the phase is less consistent for the Y component. Between the two equatorial geocentre coordinates, the Y component exhibits greater annual variability for the SLR-based solutions with the NS method than the KA and inverse methods.

Figure 7.

Phasor diagrams of the annual GCM estimates listed in Table 3.

Moreover, the mean amplitudes and phases of different techniques and approaches listed in Table 3 are illustrated in Fig. 8. There exist slightly systematic errors for the amplitudes and phases among different techniques in the X and Z components, whereas relatively better consistency is shown in the Y component. The limitation of the inhomogeneous distribution of SLR stations results in an overestimating amplitude and a small phase than the other two satellite tracking techniques. The effect of the orbit mismodelling and the correlations between parameters (Couhert et al. 2018) on the DORIS Z component may be the reason that leads to the obvious deviation of the amplitude from that of SLR and GNSS. The phases in the Z component display disagreement among the three techniques. Besides, the amplitudes in both Y and Z components and the phases in the Y component agree very well by using different approaches. Slightly different amplitudes in the X component and phases in both X and Z components by different approaches are also demonstrated in Fig. 8. To sum up, regardless of the amplitude and phase, there is no obvious systematic error in the Y component among different techniques and approaches, but a slight systematic deviation in the X and Z components.

Figure 8.

Histogram of mean amplitudes and phases of the geocentre annual signal from different techniques and different approaches.

The last row of Table 3 lists the mean values of amplitudes and phases from all GCM estimates. The amplitudes of our solutions are very consistent with the mean values with differences of 0.5, 0.2 and 0.1 mm, respectively, in the three components, while the deviations of the phases exist between this study and the mean values in the three components due to the large variation range of the phase values estimated by various approaches and techniques.

4. OTHER PERIODIC SIGNAL ANALYSIS

In addition to annual signal (and semi-annual signal, Wu et al. 2012), the GCM time-series contains various periodic signals due to modelling errors in orbit determination (e.g. infrared radiation, Earth's albedo, solar radiation pressure, thermal-induced effects, such as Yarkovsky-Schach, Sośnica et al. 2014), and tidal effects (e.g. solid tides, ocean tides, atmospheric tides and pole tides). To identify these periodic signals, we compute the absolute w-correlation values of the first 30 RCs from Fig. 5 and show them in Fig. 9 for all three components. For the X component, the larger absolute w-correlation values of the RCs (5,6), (11,12), (13,14), (15,16) and (17,18) indicate that these pairs are strongly correlated. Therefore, the RCs (5,6), (11,12), (13,14), (15,16), (17,18) are apparent five pairs of periodic components, which are shown in Fig. 10 in detail. The pairs of RCs derived from the Y and Z components are illustrated in Figs 11 and 12, respectively. Table 4 lists and quantifies the periodic signals for all three coordinate components.

Figure 9.

The first 30 w-correlations of RCs of the X, Y and Z components.

Figure 10.

Main merged periodic signals in the X component.

Figure 11.

Main merged periodic signals in the Y component.

Figure 12.

Main merged periodic signals in the Z component.

Table 4.

The periodic signals detected by SSA with a window length of 3 yr.

X (days, mm)				Y (days, mm)				Z (days, mm)
RC	Period	Frequency	Amplitude	RC	Period	Frequency	Amplitude	RC	Period	Frequency	Amplitude
RC5+6	570	0.64	0.67	RC5+6	570	0.64	0.65	RC4+5	1043.6	0.35	2.04
RC11+12	52.6	6.94	0.58	RC9+10	107.8	3.39	0.24	RC6+7	570	0.64	1.80
RC13+14	255.4	1.43	0.44	RC15+16	122.2	2.99	0.35	RC8+9	222.7	1.64	1.43
RC15+16	42.7	8.56	0.40	RC17+18	48.9	7.47	0.22	RC10+11	280.0	1.30	1.03
RC17+18	86.3	4.23	0.32	RC19+20	14.1	25.89	0.20	RC12+13	140	2.62	1.01
一	一	一	一	RC21+22	61.3	5.96	0.26	RC15+16	20.7	17.63	0.68
一	一	一	一	RC23+24	36.9	9.89	0.25	RC20+21	125.9	2.90	0.59
一	一	一	一	RC25+26	20.5	17.78	0.26	RC22+23	15.3	23.85	0.41

As shown in Figs 10 –12 and Table 4, the submillimetre oscillations at a period of 570 d in both X and Y components and an obvious periodic signal of 570 d with a magnitude of 1.8 mm in the Z component are identified. Moreover, a significant periodic signal of about 1043.6 d, the millimetre periodic signals of 280 and 140 d exist in the Z component, and weak periodic signals of 14.1 and 15.3 d exist in the Z and Y components, respectively. Moreover, a weak signal with an about 21-d period exists in the Y and Z components.

The periodic signals with 570 d (close to 560 d) and 280 d in Table 4 can be interpreted as the draconitic astrodynamic effect caused by the repeatability of the Sun elevation above the LAGEOS-1 orbital plane which has the same period as the resonance between the semi-diurnal tide S₂ and diurnal tide S₁ and the LAGEOS-1 satellite orbit. The period of draconitic effect can be estimated with the following expression (Sośnica 2015a,b),

(8)

where denotes the prograde drift, for LAGEOS-1 () d, then d. Therefore, the periodic signals of 280 and 140 d in the Z component correspond to the semi- and quarter-draconitic effects of LAGEOS-1. Similarly, for LAGEOS-2 (), its retrograde drift d, then d when derived from eq. (8). Therefore, the detected periodic signal of 222 d in Table 4 corresponds to the draconitic effect by S₁ to the LAGEOS-2. The draconitic errors are typically caused by modelling errors of direct solar radiation pressure and derivative effect modellings, such as thermal Yarkovsky and Yarkovsky–Schach effects that are caused by inhomogeneous heating of satellite components depending on the relative Sun-satellite-Earth configuration. Despite that, draconitic signals have the same periods as the resonance periods of LAGEOS-1 orbit and S₁/S₂/S₄ tides, they should be assigned, to the greatest extent, to the orbit modelling issues and not to the errors caused by the a priori tidal models.

The periods of 1043.6 d in the Z component is equal to the alias period of K₁/O₁ tide for LAGEOS-1. In other words, errors in K₁/O₁ tide modelling with the resonance period of 1043.6 d for the LAGEOS-1 imposes perturbation on the LAGEOS-1 satellite to affect the GCM. These errors may be attributed to the deficiency of the ocean tide model or the subdaily polar motion model. The period of 14.1 d detected in the Y component and 15.3 d in the Z component can be interpreted as the overlapping effects of the annual tidal signal S_a, the satellite's ground-track repeatability and generated 7-d arcs, which is computed by (Sośnica et al. 2015)

(9)

where represents the ground repeat period for LAGEOS-1 and for LAGEOS-2, then the d for LAGEOS-1 and d for LAGEOS-2 are obtained. Therefore, the periodic signals of 14.1 and 15.3 d in Table 4 are the overlapping effects of the ground-track repeatability of LAGEOS-1 and LAGEOS-2, respectively. However, when the monthly GCM solutions from CSR is analysed with SSA, the periods of 14.1-d and 15.3-d, the overlapping effects of the orbit repeatability of LAGEOS1/2 with respect to observing stations, cannot be detected in the three components from the CSR monthly estimates, due to the lower temporal resolution of the monthly solution. Moreover, a weak semi-annual signal is detectable in the Y component, whereas the weekly estimates cannot find this signal in the Y component.

After removing the annual signal and all the periodic signals in Table 4 from the time-series of the original GCM time-series, the residual series is processed again using SSA to analyse other geophysical signals. The most significant RC (1) of the Z component mentioned in Section 3 is also left in the residual time-series. The larger window length N as 485 points, which corresponds to the half of 18.6-yr, is employed to extract the long-periodic signals. The derived the first 30 absolute w-correlations of RCs in the X, Y and Z components are illustrated in Fig. 13. By applying FFT analysis to the residual time-series, the spectrum analysis results are displayed in Fig. 14 and Table 5, where the signals including 19.6-, 13- and 39.2-yr terms are detected in all three geocentre components. The long term 39.2-yr can be recognized as a secular drift from the GCM time-series. However, the GCM time-series is too short to give the exact interpretation of the nature of these signals of 39.2-, 19.6- and 13-yr. These periods are artifacts related to the length of the GCM time-series and the limitations of the method used.

Figure 13.

The first 30 w-correlations of the RCs of the X, Y and Z components for the residual time-series.

Figure 14.

Main merged periodic signals of all three components for the residual time-series.

Table 5.

The periodic signals detected by SSA with a window length of 9.3 yr.

X (days, mm)				Y (days, mm)				Z (days, mm)
RC	Period	Frequency	Amplitude	RC	Period	Frequency	Amplitude	RC	Period	Frequency	Amplitude
RCr1+2	39.2 yr	0.026	0.87	RCr1+2	39.2 yr	0.051	1.263	RCr1+2	39.2 yr	0.026	3.00
	19.6 yr	0.051	0.76		19.6 yr	0.026	0.92		19.6 yr	0.051	1.43
	13 yr	0.076	0.88		13 yr	0.076	0.68		13 yr	0.076	0.43
RCr6+7	30.3	12.07	0.50	RCr7+8	18.7	19.57	0.35	RCr3+4	14.4	25.45	0.83
RCr8+9	Semi-annual	2.00	0.43	RCr14+15	84.7	4.31	0.31	RCr5+6	14.9	24.49	0.67
RCr10+11	76.7	4.76	0.39	RCr16+17	28.0	13.05	0.26	RCr7+8	44.4	8.22	0.56
RCr12+13	103.8	3.52	0.25	RCr18+19	45.1	8.10	0.29	RCr9+10	59.3	6.16	0.58
RCr17+18	37.2	9.83	0.31	一	一	一	一	RCr12+13	28.4	12.87	0.74
一	一	一	一	一	一	一	一	RCr14+15	73.5	4.97	0.46
一	一	一	一	一	一	一	一	RCr16+17	Semi-annual	1.99	0.60
一	一	一	一	一	一	一	一	RCr18+19	42.3	8.64	0.37
一	一	一	一	一	一	一	一	RCr20+21	21.3	17.15	0.55
一	一	一	一	一	一	一	一	RCr22+23	83.4	4.38	0.42

In addition to long periodic signals in Table 5, the semi-annual periodic terms with the magnitude of submillimetre can be only detected in the X and Z components. The observability of the semi-annual geocentre variations is very consistent with the solutions from Couhert et al. (2018). The periodic signals of close to 85 d in the X (Table 4) and Y components and 28 d in the Y and Z components are related to the tidal aliasing perturbations of the T₂ tide on LAGEOS-2 and Mm tide on LAGEOS1/2, respectively, similar to those affecting GNSS orbits (Zajdel et al. 2020).

Through the above analysis, the Z component contains more other periodic signals than the equatorial components, suggesting that the Z component is more interrupted by the modelling errors or other systematic errors. After reducing draconitic effects in GCM by removing the PCs with the draconitic periods, via the eq. (6) for the construction of the GCM as shown in Fig. 15, the SSA series (in black dotted-line) are more consistent with CSR than that of the original GCM time-series (in blue solid-line). The consistency is high especially for the Z component and mitigates the relatively large difference between the CSR and this study in the Z component illustrated in Fig. 3.

Figure 15.

The constructed GCM time-series by SSA and the comparison with that of CSR. ‘SLR-weekly-SSA’ means the results from the SSA.

5. CONCLUSIONS

Using SLR observations to LAGEOS1/2 satellites, we determine the 26-yr GCM time-series from 1994 to 2020 through the network shift approach. The results show that SSA is effective to realize the separation of the periodic terms by decomposing the estimated GCM time-series. Moreover, the spurious draconitic effects, which have no geophysical interpretation, can be detected, separated and excluded from the time-series of GCM by applying SSA. Each coordinate component of the 26-yr GCM time-series contains various periodic signals. Most of the detected periodic signals are generated by the orbit mismodelling issues such as draconitic effects (periods of 570-, 280-, 222-d, etc.) and aliasing perturbation effects of K₁/O₁, T₂ and Mm tide (periods of 1043.6-, 85- and 28-d), and overlapping effects of the orbit repeatability of LAGEOS1/2 with respect to observing stations (periods of 14.1-d and 15.3-d), consequently quantifying and interpreting the corresponding astrodynamic and geophysical mechanism of these periodic signals. The explored dominant periodic sources related to the modelling issues and correlations should be mitigated when determining the GCM time-series.

The GCM time-series determined in this study is consistent with the estimates from CSR, AIUB and a large number of other techniques and methods employed for the geocentre recovery. The annual GCM estimates derived from our solutions generally agree with other annual GCM time-series estimated by various approaches and techniques as well as geophysical models. The phases and the amplitudes of the annual GCM estimated by this study are 31°, 292°, 52° and 2.8, 3.0 and 3.9 mm, respectively in the three components. Compared to the annual GCM derived by CSR, both amplitude and phase agree well, except that the amplitude is 1.1 mm larger than that of CSR in the X component, due to the stronger impact of the network effects on the X component than the other two components. The amplitudes of our solutions are very consistent with the mean values with differences of 0.5, 0.2 and 0.1 mm, respectively, in the three components, while the deviations of the phases exist because of the large variation range of the phase values estimated by various approaches and techniques. Regardless of the amplitude and phase, there is no obvious systematic error in the Y component among different techniques and approaches, but a systematic deviation in the X and Z components, also suggesting a better distribution of SLR station for the Y-component geocentre recovery.