Keywords

1 Introduction

The Beidou-3 project is a global satellite navigation system independently built by China, including three geosynchronous orbit GEO satellites. Due to the influence of the orbit motion, the geosynchronous satellites have long-term, long-period, medium-long and short terms. In order to maintain the fixed position, the satellite needs to overcome long-term and long-period perturbation. This raises a high demand for the identification and separation of periodic motion. When the geosynchronous orbit satellites perform position maintenance (especially for colocation control), in order to eliminate the coupling effect of the positioning target, the short term perturbation needs to be separated from the satellite ephemeris [1]. In order to improve the accuracy of navigation calculations and reduce the computational load of on-board computers, it is also necessary to accurately compress the satellite oscular ephemeris. The mean orbit elements and quasi-mean elements using analytical methods are important method for separating the short terms of the periodic perturbation motion of geosynchronous satellites. In 1959, Kozai used the idea of averaging in nonlinear analytical mechanics to propose the mean orbit element for the perturbation motion of satellites in the Earth’s non-spherical gravitational field [2]. The mean elements method can be used to construct the small deviation perturbation form solution of the satellite perturbation equation, which plays an important role in the satellite orbit determination and quantitative analysis of satellite motion [3]. The analysis method has several problems. The form of the power series will be very complicated when the solution order is high. The coherence of the solution also makes it difficult to identify and separate the periodicity of the perturbation motion. It is also difficult to accurately separate the satellite short term perturbation caused by Three-body gravity and solar radiation pressure. Only the lunar gravity causes the eccentricity to be half-month short-term term of about 0.00005. With the development of dynamic model and computer technology, the orbit accuracy of geosynchronous satellites has reached centimetres [4]. At the same time, the theory and method of time-frequency signal analysis such as wavelet analysis have also made great progress. From the perspective of engineering application, this paper uses the numerical calculation method to identify and separate the periodic perturbation information of the satellite precise ephemeris, and obtain the high-precision mean orbit elements.

2 Mean Orbit Elements Calculation

2.1 Analytical Method

The mean orbit elements method was firstly applied by Kozai [1] to the analytical solution of the earth satellite under the influence of the perturbation of the earth’s non-spherical gravitational force (mainly with harmonics j2, j3 and j4). The mean elements can represent the sum of the zero-order term, the first-order long-term change term, the second-order long-term change term, and even the higher-order long-term change term. The mean elements including only the second-order long-term change term is constructed as follows:

$$ \left\{ {\begin{array}{*{20}l} {\bar{\sigma }\left( t \right) = \bar{\sigma }^{\left( 0 \right)} \left( t \right) + \sigma_{1} \left( {t - t_{0} } \right) + \sigma_{2} \left( {t - t_{0} } \right) + \cdots } \hfill \\ {{\bar{\varvec{\upsigma}}}^{\left( 0 \right)} \left( t \right) = \left[ {\begin{array}{*{20}c} {a_{0} } & {e_{0} } & {i_{0} } & {\Omega _{0} } & {\omega_{0} } & {M_{0} + n_{0} \left( {t - t_{0} } \right)} \\ \end{array} } \right]^{T} } \hfill \\ {{\varvec{\upsigma}}_{1} \left( {t - t_{0} } \right) = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & { - \frac{{A_{2} }}{{\bar{p}^{2} }}\bar{n}\cos \bar{i}\left( {t - t_{0} } \right)} & {\frac{{A_{2} }}{{\bar{p}^{2} }}\bar{n}\left( {2 - \frac{5}{2}\sin^{2} \bar{i}} \right)\left( {t - t_{0} } \right)} & {\frac{{A_{2} }}{{\bar{p}^{2} }}\bar{n}\left( {1 - \frac{3}{2}\sin^{2} \bar{i}} \right)\sqrt {1 - \bar{e}^{2} } \left( {t - t_{0} } \right)} \\ \end{array} } \right]^{T} } \hfill \\ {\Omega _{2} \left( {t - t_{0} } \right) = - \frac{{\bar{n}\cos \bar{i}}}{{p^{4} }}\left[ {A_{4} \left( {\left( {\frac{6}{7} + \frac{9}{7}\bar{e}^{2} } \right) - \sin^{2} \bar{i}\left( {\frac{3}{2} + \frac{9}{4}\bar{e}^{2} } \right)} \right) + A_{2}^{2} \left( {\left( {\frac{3}{2} + \frac{1}{6}\bar{e}^{2} + \sqrt {1 - \bar{e}^{2} } } \right) - \sin^{2} \bar{i}\left( {\frac{5}{3} - \frac{5}{24}\bar{e}^{2} + \frac{3}{2}\sqrt {1 - \bar{e}^{2} } } \right)} \right)} \right]} \hfill \\ {\omega_{2} \left( {t - t_{0} } \right) = \frac{{A_{4} }}{{\bar{p}^{4} }}\bar{n}\left[ {\left( {\frac{12}{7} + \frac{27}{14}\bar{e}^{2} } \right) - \sin^{2} \bar{i}\left( {\frac{94}{14} + \frac{27}{4}\bar{e}^{2} } \right) + \sin^{4} \bar{i}\left( {\frac{21}{4} + \frac{81}{16}\bar{e}^{2} } \right)} \right] + } \hfill \\ {\quad \quad \quad \quad \frac{{A_{2}^{2} }}{{\bar{p}^{4} }}\bar{n}\left[ {\left( {4 + \frac{7}{12}\bar{e}^{2} + 2\sqrt {1 - \bar{e}^{2} } } \right) - \sin^{2} \bar{i}\left( {\frac{103}{12}{ + }\frac{3}{8}e^{2} + \frac{11}{2}\sqrt {1 - \bar{e}^{2} } } \right) + \sin^{4} \bar{i}\left( {\frac{215}{48} - \frac{15}{32}\bar{e}^{2} + \frac{15}{4}\sqrt {1 - \bar{e}^{2} } } \right)} \right]} \hfill \\ {M_{2} \left( {t - t_{0} } \right) = \frac{{A_{2}^{2} }}{{\bar{p}^{4} }}\bar{n}\sqrt {1 - \bar{e}^{2} } \left\{ {\left[ {\frac{1}{2}\left( {1 - \frac{3}{2}\sin^{2} \bar{i}} \right)^{2} \sqrt {1 - \bar{e}^{2} } + \left( {\frac{5}{2} + \frac{10}{3}\bar{e}^{2} } \right) - \left( {\frac{19}{3} + \frac{26}{3}e^{2} } \right)\sin^{2} \bar{i} + \left( {\frac{233}{48} + \frac{103}{12}\bar{e}^{2} } \right)\sin^{4} \bar{i} + } \right.} \right.} \hfill \\ {\quad \quad \quad \quad \left. {\left. {\frac{{\bar{e}^{4} }}{{1 - \bar{e}^{2} }}\left( {\frac{35}{12} - \frac{35}{4}\sin^{2} \bar{i}{ + }\frac{315}{32}\sin^{4} \bar{i}} \right)} \right] + \left( {\frac{{A_{4} }}{{A_{2}^{2} }}} \right)\bar{e}^{2} \left( {\frac{9}{14} - \frac{35}{14}\sin^{2} i + \frac{45}{16}\sin^{4} \bar{i}} \right)} \right\}} \hfill \\ {\bar{n} = \bar{a}^{{ - {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} ,\,\bar{p} = \bar{a}\left( {1 - \bar{e}^{2} } \right),\,A_{2} = \frac{3}{2}J_{2},\,A_{3} = - J_{3},\,A_{4} = - \frac{35}{8}J_{4} } \hfill \\ \end{array} } \right. $$
(2.1)

Liu et al. proposed the concept of quasi-mean elements [5]. The quasi-mean elements only separated the short term of the perturbation solution. The difference is that the quasi-mean elements also includes a first-order long-period term, a second-order long-period term, and even a higher-order long-period term, that is,

$$ \begin{array}{*{20}l} {\bar{\sigma }\left( t \right) = \bar{\sigma }^{\left( 0 \right)} \left( t \right) + \left( {\sigma_{1} \left( {t - t_{0} } \right) + \sigma_{2} \left( {t - t_{0} } \right) + \cdots } \right)} \hfill \\ { + \left( {\sigma_{L}^{(1)} \left( {t - t_{0} } \right) + \sigma_{L}^{(2)} \left( {t - t_{0} } \right) + \cdots } \right)} \hfill \\ \end{array} . $$
(2.2)

Since the formula of the mean elements containing only the second-order long-term term is already quite complicated, the analytical expression is not given here. It should be pointed out that the long-period term of the eccentricity \( e \) and the orbital inclination \( i \) is not 0, although. The first-order long-period term of a \( a_{L}^{(1)} \left( {t - t_{0} } \right) = 0 \), but its second-order long-period term \( a_{L}^{(2)} \left( {t - t_{0} } \right) \) is not 0. Therefore, the six mean orbit elements contain periodic variation terms, and these periodic variation terms cannot be eliminated by polynomial fitting.

The advantage of the analytic method is that the physical meaning is clear, and it can be converted between the oscular elements and mean elements. However, the high-order term structure is complex, the rounding error is large, and the analytical solution for the dissipative perturbation is low, the conversion accuracy too low to meet the needs of high-precision numerical calculation.

2.2 Harmonic Decomposition

Determining the coefficients in the expansion is a classic time domain fitting problem when the long-term term order and the periodic term frequency are known. For the orbit elements of any epoch time, there is:

$$ f\left( t \right) \approx \sum\limits_{k = 0}^{m} {a_{k}\Phi _{k} \left( t \right)} =\Phi ^{T} \left( t \right)P,\;t \in \left[ {t_{0} ,t_{f} } \right]. $$
(2.3)

\( \Phi \left( t \right) = \left( {\begin{array}{*{20}c} {\Phi _{0} \left( t \right)} \\ \vdots \\ {\Phi _{m} \left( t \right)} \\ \end{array} } \right) \) is basis function column vector. \( P = \left( {\begin{array}{*{20}c} {a_{0} } \\ \vdots \\ {a_{m} } \\ \end{array} } \right) \) is coefficient vector corresponding to the basis function.

The orbit elements at time series \( t_{i} \in \left[ {t_{0} ,t_{f} } \right],\;i = 1,2, \cdots ,n \) Can be described as:

$$ \left( {\begin{array}{*{20}c} {\Phi ^{T} \left( {t_{1} } \right)} \\ \vdots \\ {\Phi ^{T} \left( {t_{n} } \right)} \\ \end{array} } \right)P = \left( {\begin{array}{*{20}c} {f\left( {t_{1} } \right)} \\ \vdots \\ {f\left( {t_{n} } \right)} \\ \end{array} } \right) $$
(2.4)

The above equations are generally over determined equations, and the least squares solution is:

$$ \hat{P} = \left( {\Phi ^{T}\Phi } \right)^{ - 1}\Phi ^{T} f, $$
(2.5)

where \( f = \left( {\begin{array}{*{20}c} {f\left( {t_{1} } \right)} \\ \vdots \\ {f\left( {t_{n} } \right)} \\ \end{array} } \right) \), \( \Phi = \left( {\begin{array}{*{20}c} {\Phi ^{T} \left( {t_{1} } \right)} \\ \vdots \\ {\Phi ^{T} \left( {t_{n} } \right)} \\ \end{array} } \right) \).

When the long-period term is decomposed by the ephemeris signal with a short time, the conditional number of the regular matrix will be very large, close to the ill-conditioned matrix, and the more stable coefficient matrix singular value decomposition algorithm is often used in practical.

The limitation of the harmonic decomposition method is that the basis function frequency is fixed and cannot adapt to the processing of time-varying signals. However, there is a frequency coupling phenomenon in the fine perturbation modelling, and the actual perturbation period is not fixed. This method is also subject to the incompleteness of rounding error and perturbation modelling, and the adaptability is not strong.

2.3 Hilbert-Huang Transformation

In 1996, American Chinese Norden. Huang et al. created a time-frequency analysis method for Hilbert-Huang Transformation (HHT) for the concept of instantaneous frequency [6, 7]. The method proposes the concept of Intrinsic Mode Functions (IMF) and the method of decomposing arbitrary signals into intrinsic mode functions—Empirical Mode Decomposition (EMD), which establishes the instantaneous frequency as a basic variable that characterizes the alternating signal, and a time-frequency analysis method in which IMF signal is the basic time domain signal.

Since the orbital perturbation cannot be fully modelled and the perturbation frequency is coupled, the unmodelled periodic components cannot be completely separated using a finite fixed basis function. Regardless of the physical constraints, the Fourier decomposition will bring false periodic signals and false frequency phenomena. Limited by the uncertainty of Heisenberg, the time-frequency analysis method based on Fourier analysis theory cannot describe the transformation of frequency with time. The perturbation orbit is a typical nonlinear non-stationary time series. The HHT theory adaptively decomposes the orbital signal into a finite number of IMFs and residual signals that characterize the trend of the signal through EMD. It has strong adaptive performance and sensitive frequency detection. By performing a HHT on the sequence of oscular orbit element, the mean orbit elements that characterize the motion can be obtained.

The characteristic of HHT is that it does not directly perform Hilbert transform on the signal, but first studies the reasonable definition of the IMF signal, and uses EMD method to decompose the signal into IMFs, and calculates each component separately by the definition of instantaneous frequency. HHT not only utilizes the reasonable definition of instantaneous frequency, but also consistent with the multi-component result recognition of the signal, which can more accurately reflect the physical characteristics of the system.

3 Analysis of Algorithms

3.1 Dynamic Perturbation

Due to the \( J_{22} \) term of the ellipsoid of the Earth’s equatorial plane, that is, the triaxiality of the Earth, the stationary orbiting satellite at the nominal longitude \( \lambda \) receives an additional tangential gravitational acceleration. The tangential acceleration will cause the semi-major axis of the geostationary orbit to change, which will cause the orbital flat motion to be inconsistent with the angular velocity of the Earth’s rotation, causing the satellite to deviate from the nominal fixed-point longitude. The perturbation equation for the semi-major axis of the stationary orbit caused by the non-spherical perturbation of the Earth is:

$$ \begin{aligned} & \frac{da}{dt} = - \frac{2}{3}\frac{{a_{c} }}{{n_{c} }} \cdot\Gamma _{\lambda } \\ &\Gamma _{\lambda } = - 3n_{c}^{2} \left\{ {\begin{array}{*{20}l} { - 6J_{22} \left( {\frac{{R_{e} }}{{a_{c} }}} \right)^{2} \sin 2\left( {\lambda - \lambda_{22} } \right)} \hfill \\ {\quad + \frac{3}{2}J_{31} \left( {\frac{{R_{e} }}{{a_{c} }}} \right)^{3} \sin \left( {\lambda - \lambda_{31} } \right)} \hfill \\ {\quad - 45J_{33} \left( {\frac{{R_{e} }}{{a_{c} }}} \right)^{3} \sin 3\left( {\lambda - \lambda_{33} } \right)} \hfill \\ \end{array} } \right\} \\ \end{aligned} $$
(3.1)

Where \( a_{c} \) is the semi-major axis of the geostationary orbit. \( n_{c} \) is the angular velocity of the earth’s rotation. \( J_{nm} \) and \( \lambda_{nm} \) are the Earth’s gravitational field association Legendre polynomial coefficient and geographic longitude, respectively. E is the satellite longitude of the satellite.

Sun and moon perturbation does not produce long-period perturbations on the semi-major axis of the stationary orbit. The half-long axis change rate of the orbit in one circle is zero, but a short period is generated for the semi-major axis of the orbit, and the period is half a solar day. The amplitude of the short term change rate is related to the elevation angle of the sun and the moon to the celestial equator and the phase of the sun and the moon. The short term oscillation of the semi-major axis of the sun and moon perturbation orbit is expressed as:

$$ \frac{da}{dt} = - 3a_{s} \frac{{n_{k}^{2} }}{{n_{e} }}\cos^{2} \delta_{k} \sin 2\lambda_{k} $$
(3.2)

Where \( a_{s} = 42164.2\;{\text{km}} \) is the semi-major axis of the stationary orbit under the two-body problem. \( n_{k}^{{}} \) is the angular velocity of the third body (day, month) around the center of the earth, \( n_{e} \) is the angular velocity of the earth’s rotation, \( \delta_{k} \) is the vertical angle of the third body, and \( \lambda_{k} \) is the difference between the right ascension of the satellite and the third body.

The short term perturbation of the semi-major axis of the stationary orbit caused by the sun’s light pressure, is a period of one day. The amplitude of the semi-major axis caused by light pressure is greatest when the sun is at the vernal equinox or the equinox. The specific expression is:

$$ \frac{da}{dt} = \frac{2}{n}C_{R} \left( {\frac{S}{m}} \right)P_{0} \cos \delta_{s} \sin \left( {l - \alpha_{s} } \right) $$
(3.3)

Where \( n \) is the orbital angular velocity, \( C_{R} \) is the optical pressure coefficient, \( \frac{S}{m} \) is the area to mass ratio, \( P_{0} \) is the solar radiation pressure per unit area, \( (\alpha_{s} ,\delta_{s} ) \) is the solar right ascension declination, and \( l \) is the satellite mean right ascension.

Similarly, it modeled by the perturbation potential, to establish and eccentricity perturbed inclination motion equation is as follows:

$$ \begin{aligned} & \frac{de}{dt} = \frac{{1 - e^{2} }}{{na^{2} e}}\frac{\partial R}{\partial M} - \frac{{\sqrt {1 - e^{2} } }}{{na^{2} e}}\frac{\partial R}{\partial \omega } \\ & \frac{di}{dt} = \frac{\cot i}{{na^{2} \sqrt {1 - e^{2} } }}\frac{\partial R}{\partial \omega } - \frac{\csc i}{{na^{2} \sqrt {1 - e^{2} } }}\frac{\partial R}{{\partial\Omega }} \\ \end{aligned} $$
(3.4)

The effects of the perturbation on the long-term, long-period, medium and short period and short-term caused by the number of orbits are organized as follows (Table 1):

Table 1. The perturbation effects on orbit elements
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Where \( n_{s} = 0.0172 \), \( n_{m} = 0.23 \), \( n_{e} = 6.283 \) (radian/day), the eccentricity vector is defined as \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{e} = e\,\cos (\Omega + \omega ) + \text{je}\,\sin (\Omega + \omega ) \), and the inclination vector is defined as \( i = i\,\cos (\Omega ) + \text{j}\,i\,\text{sin}(\Omega ) \).

3.2 Empirical Mode Decomposition

The intrinsic mode function IMF defined in the HHT transform is a type of signal that satisfies the physical interpretation of a single component signal [8]. At each moment, there is only a single frequency component, which is the instantaneous frequency in the physical sense. The IMF component must satisfy the following two conditions: one is that the number of extreme points and the number of zero crossings are the same or at most one, and the other is that the upper and lower envelopes are locally symmetric about the time axis.

The empirical mode decomposition EMD method assumes that any signal is composed of different IMFs [9]. The method steps are: first finding the maximum value and the minimum value of the signal \( x(t) \), and obtaining the upper envelope \( U_{x} (t) \) and the lower envelope \( L_{x} (t) \) of the signal by cubic spline fitting, and calculating the mean value of the upper and lower envelopes:

$$ m_{1} (t) = \frac{{U_{x} (t) + L_{x} (t)}}{2} $$
(3.5)

Subtracting this mean \( m_{1} (t) \) from the original signal x(t) signal yields the first component \( h_{1} (t) \), i.e.: \( h_{1} (t) = h_{1} (t) - m_{1} (t) \) Check whether \( h_{1} (t) \) is the two conditions that satisfy the IMF component. If it is not satisfied, perform the re-screening, that is, calculate the mean line \( m_{1} (t) \) through the upper and lower envelopes of \( h_{1} (t) \), and obtain a new component. Repeat the above process k times:

$$ h_{1k} (t) = h_{1(k - 1)} (t) - m_{1k} (t) $$
(3.6)

Until \( h_{1k} (t) \) satisfies the condition of the IMF component, the \( h_{1k} (t) \) obtained at this time is taken as the first IMF component \( C_{1} = h_{1k} (t) \), which contains the local high frequency portion of the original signal \( x(t) \).

Subtracting \( C_{1} \) from the original signal \( x(t) \) yields the residual signal \( r_{1} (t) \):

$$ r_{1} (t) = x(t) - C_{1} $$
(3.7)

The residual signal is subjected to screening as described above to obtain more IMF until the final IMF is separated. The final residual signal \( r_{n} (t) \) is a constant or constant trend change, which is the long-term trend of the number of mean orbital element in this scenario.

At this point, the signal \( x(t) \) is decomposed into the sum of n IMFs and residual signal \( r_{n} (t) \): \( x(t) = \sum\limits_{1}^{n} {C_{i} (t)} + r_{n} (t) \).

3.3 HHT Perform

The mathematical basis of the Hilbert Transform (HT), proposed by the German mathematician Hilbert, is an important tool in signal analysis [10, 11]. Perform a Hilbert transform on each IMF with the expression:

$$ \overline{{C_{i} }} (t) = C_{i} (t)\,*\,\frac{1}{\pi t} = \frac{1}{\pi }P\int_{ - \infty }^{ + \infty } {\frac{{C_{i} (\tau )}}{t - \tau }d\tau } , $$
(3.8)

where P is the Cauchy principal value. The HT can be understood as the IMF passing a linear time-invariant system with a unit impulse response of \( \frac{1}{\pi t} \).

Constructing an analytical signal:

$$ z_{i} (t) = C_{i} (t) + j\overline{{C_{i} }} (t) = a_{i} (t)e^{{j\phi_{i} (t)}} $$
(3.9)

where \( a_{i} (t) = \sqrt {C_{i}^{2} (t) + \overline{C}_{i}^{2} (t)} \) is the amplitude and \( \phi_{i} (t) = \arctan (\overline{{C_{i} }} (t)/C_{i} (t)) \) is the phase. The instantaneous frequency of each IMF is defined as \( \omega_{i} (t) = \frac{{d\phi_{i} (t)}}{dt} \), then the signal can be expressed as:

$$ x(\text{t}) = \text{Re}\sum\limits_{i = 1}^{n} {a_{i} (\text{t})\text{e}^{{j\int {\omega_{i} (\text{t})\text{dt}} }} } + r_{n} (\text{t}) $$
(3.10)

The signal amplitude can be expressed as a function of time t and instantaneous frequency \( \omega_{i} (t) \), so called the Hilbert spectrum. Due to the introduction of instantaneous frequency, the limitation of Fourier variation is broken, so that HHT transform can be applied to the processing of nonlinear systems and non-stationary signals.

4 Calculation Results

The Beidou-3 satellite project is a global satellite navigation system independently built by China, including five GEO satellites pointing at 60°, 80°, 110°, 140° and 160° east longitude. Beidou 3G01 is the first geostationary orbit satellite of the Beidou-3 project. It was launched at the Xichang Satellite Launch Center on November 1, 2018, Beijing time and was successfully positioned on November 9th, Beijing time. It can provide large-capacity short message service, and satellite-based enhancement services for civil aviation users. In this paper, 3 months numerical predicted oscular orbit of Beidou3G01 is analyzed.

4.1 EMD Decomposition

HHT is a repetitive cycle decomposition process. In theory, the decomposition ends when the signal margin cannot be decomposed into more IMFs. However, the last few IMF amplitudes are often too small to lose physical meaning, but if the conditions are too loose, the useful signal components may be lost. Since the evolution of the orbit affected by perturbation is a deterministic nonlinear process, the frequency components of main perturbation can be estimated. Therefore, the number of iterations of the decomposition termination is consistent with the main perturbation period of Sect. 3.1, which has a clear physical meaning. The EMD decomposition of the semi-major axis and the eccentricity vector is shown in figure, where the eccentricity vector is processed as a complex signal (Figs. 1, 2 and 3):

Fig. 1.
figure 1

EMD decomposition of semi-major axis

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Fig. 2.
figure 2

EMD decomposition for real part of eccentricity vector

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Fig. 3.
figure 3

EMD decomposition for imaginary part of eccentricity vector

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It can be seen from the figure that the semi-major axis EMD decomposes 2 IMFs, corresponding to the daily and half-day periods of the semi-major axis perturbation evolution. The eccentricity vector decomposes 4 IMFs, corresponding to the year, month, day, and half-day periods of the eccentricity vector perturbation evolution.

4.2 Frequency Analysis

Decomposition semi-major axis as an example, perform HT to 2 IMF signals, the instantaneous frequency of the information changes over time, as shown (Fig. 4):

Fig. 4.
figure 4

Instantaneous frequency of IMFs of semi-major axis

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As it can be seen from the figure, since the coupling perturbation period and other factors, the two frequencies obtained from IMF component are not constant but varies with time. HHF visually shows this change. The blue component IMF1 frequency is approximately twice the red component IMF2, corresponding to the semi-major axis perturbation day period term and half-day period term, respectively, and the physical meaning is clear.

4.3 Mean Orbit Elements Evolution

Apply the analytical method and the HHT methods respectively to the semi-major axis (topical), eccentricity and inclination vector time series. The comparison results of the mean orbit elements shown below (Figs. 5, 6 and 7).

Fig. 5.
figure 5

Semi-major axis evolution

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Fig. 6.
figure 6

Eccentricity vector evolution

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Fig. 7.
figure 7

Inclination vector evolution

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It can be seen from the figure that the mean semi-major axis and the mean eccentricity vector solved by the HHT method have significantly reduced jitter, and the stability and accuracy are better than the analytical method. Due to the inclination of the trend vector is much greater than perturbation fluctuations, it shows rarely difference between the two methods.

5 Conclusions

For the deficiencies of the analytical method and the Fourier spectrum analysis method, the Hilbert-Huang transform is introduced. Under the premise that no preset basis function needed, the HHT transform adaptively decomposes the oscular orbit elements ephemeris data into multiple IMF components and residual trend information. The instantaneous frequency information with time is obtained after perform HT to separated IMF component, which is consistent with the perturbation analysis result. The mean semi-major axis and the mean eccentricity vector solved by the HHT method have significantly reduced jitter, and the stability and accuracy are better than the analytical method. However, the inclination vector is less obvious because its perturbation fluctuation is far weak than trend.

The instantaneous frequency fluctuations of the separated IMF signals reflect the error portion of the perturbation period coupling and modelling and real perturbation, including the influence of the lunar high-order gravitational field and the lunar orbital eccentricity. The HHT decomposition termination condition problem, the influence of the fitting method used in searching the IMF, and the endpoint problem causing the inaccurate frequency estimation [12], etc., can be further explored.