In this study, we use a nonlinear and non-stationary time series analysis method, the ensemble empirical mode decomposition method (EEMD), to analyze the polar motion (PM) time series (EOP C04 series from 1962 to 2013) to find a 531-day-period wobble (531 dW) signal. The 531 dW signal has been found in the early PM series (1962–1977), but cannot be found in the recent PM series (1978–2013) using conventional analysis approaches. By virtue of the demodulation feature of EEMD, the 531 dW can be confirmed to be present in PM based on the differences of the amplitudes and phases between different intrinsic mode functions. Results from three sub-series divided from the EOP C04 series show that the period of the 531 dW is subject to variations, in the range of 530.9–524 days, and its amplitude is also time-dependent (about 2–11 mas). Synthetic tests are carried out to explain why the 531 dW can only be observed in recent 30-year PM time series after using EEMD. The 531 dW is also detected in the two longest available superconducting gravimeter (SG) records, which further confirms the presence of the 531 dW. The confirmation of the 531 dW existence could be significant in establishing a more reasonable Earth rotation model and may effectively contribute to the prediction of the PM and its mechanism interpretation.

It is recognized that the polar motion (PM) contains two dominant components:
the annual wobble (AW) with a 12-month period and the Chandler wobble (CW)
with a 14-month period. Some researchers have suggested that the CW is highly
variable with respect to its amplitude (e.g., Carter, 1981, 1982;
Höpfner, 2003; Chen et al., 2009, 2013a, b), some have considered it to
have double or multiple frequencies (e.g., Chao, 1983; Pan, 2012), and some
have considered its frequency to be invariant (e.g., Okubo, 1982; Vicente and
Wilson, 1997; Gross et al., 2003; Seitz and Schmidt, 2005). If the CW is
frequency modulated as Carter (1981, 1982) suggested, namely, the frequency
is governed by the magnitude, it will create an infinite number of sidebands,
arranged symmetrically about the carrier and spaced at integer multiples of
the modulating frequency (Carter, 1981). The first upper and lower sidebands
could be at 1 cpy (cycles per year) and 0.69 cpy, respectively, when the
beat frequency is 0.157 cpy (CW's frequency
0.843 cpy

Generally, the traditional Fourier analysis method cannot observe this 531 dW signal (see Fig. 2d–f); hence, in this study, a nonlinear and non-stationary time series analysis method, the ensemble empirical mode decomposition (EEMD) (Huang and Wu, 2008; Wu and Huang, 2009), is applied to a PM time series as a filter bank to detect the 531 dW signal. (To know the differences and relations between different spectral analysis methods, we suggest the readers to refer some other literatures, such as Tary et al., 2014).

Here, the PM time series, the EOP C04 series spanning from 1962 to 2013 with
a 1-day sampling interval from the International Earth Rotation and Reference
System Service (IERS)
(

EEMD was proposed to overcome the disadvantages existing in the empirical mode decomposition (EMD) (Huang et al., 1998; Huang and Wu, 2008), such as the mode-mixing problem and the end effect (see Fig. 2 in Shen and Ding (2014)), although EMD has demonstrated its applicability in a wide range of geoscience studies over the last 15 years (e.g., Pee and McMahon, 2006; Thomas et al., 2009; Franzke, 2009; Jackson and Mound, 2010; Jeng and Chen, 2011; Lee et al., 2011; Shen and Ding, 2014; Chambers, 2015). The details of EMD and EEMD can be found in many relevant literatures (e.g., Huang et al., 1998; Huang and Wu, 2008; Wu and Huang, 2009; Shen and Ding, 2014).

A given time series

Identify all local maxima and local minima of

Use

Generally,

Then, letting

Repeat the above steps, and then the

The above process is the EMD. In these steps, two different iterative loops
exist. The first loop is employed to obtain

Given that EMD has the mode-mixing problem and end effects, EEMD is developed
with the following procedures (Huang and Wu, 2008; Shen and Ding, 2014).

Add a white noise series to the targeted time series

Decompose the series with added white noise into IMFs.

Repeat procedures 1 and 2 iteratively, but with different white noise series each time.

Obtain the (ensemble) means of the corresponding IMFs of the decompositions as the final IMF.

In the decomposition using EEMD, the added white noise series cancel each other out in the final mean of the corresponding IMF. Note that in the EEMD process, procedure 2 is just the whole EMD process. The means of the IMFs remain within the natural filter windows, significantly reducing the possibility of mode mixing and preserving the filter property (Huang and Wu, 2008; Shen and Ding, 2014).

By using EEMD, the chosen time series can be decomposed into a finite number
of simple IMFs, and different-scale signals in the series will be re-combined
by proper IMFs based on the fact that different IMFs have different frequency
bands. Hence, for the chosen EOP C04 series,

A frequency modulation signal can be expressed as follows (Carter, 1981,
1982):

The

The first 10 IMFs of the

The waveforms of the two components, the

The observed frequencies (cpy) and amplitudes (mas) of the CW, AW and the target wobble.

The vector differences of the amplitudes of IMF 5 and IMF6 for the three sub-series (frequency domain).

Based on the conventional Fourier analysis approach, the results as shown in
Fig. 3 clearly indicate that only the target peak in the spectra of the
1962–1977 series (see Fig. 3b) is over their corresponding background noise
level (see Fig. 3d; note that a 0.57 cpy peak in Fig. 3f is also over the
noise level, but it is not the interesting signal of this study), which is
consistent with previous studies (Carter, 1981, 1982; Morgan et al., 1982).
Without using EEMD, our estimates for the target 531 dW, CW, and AW (the
annual wobble) are tabulated in Table 1 (note that the estimations for the
amplitudes, frequencies and their corresponding error bars are obtained by
using the AR (Autoregressive) method of Chao and Gilbert, 1980; more details can be found in
Ding and Shen, 2013, and Ding and Chao, 2015). For the

The three sub-series divided from the EOP C04 series (1962–2013)
and their corresponding amplitude spectra. The vertical dashed line is
located at 0.687 cpy, and the horizontal lines denote the RMS noise
amplitudes of the frequency bands 0.2–0.75 cpy and 1.05–2.0 cpy, which
are used as the background noise levels of their corresponding spectra. For
the

The amplitudes and phase spectra of the IMF 5 and IMF 6 and the sum
of the other IMFs of the

After applying EEMD to the three sub-series, only IMF5 and IMF6 contain the
531 dW signal, the corresponding spectra being shown in Fig. 4 (for the

Same as Fig. 4 but for the

The chosen amplitudes of the CW (0.8437 cpy) and AW (1 cpy) for the synthetic series, and the corresponding estimated values from the synthetic series. (Unit: mas.)

As shown in Figs. 4 and 5, the phases for CW (and AW) in IMF5 and IMF6 are the same (except for IMF 6 in Figs 4c and 5c, where there is no peak for AW), whereas the phases for the 531 dW in IMF5 and IMF6 are quite different. The corresponding amplitude and frequency estimates are listed in Table 1. Taking into account the same phases as shown in Figs. 4 and 5, one can find that the amplitudes from the IMFs for CW and AW are consistent with the results without using EEMD (see Table 1). As for the amplitudes of the 531 dW in the three sub-series, we use the vector difference to estimate the relevant values. The corresponding results are shown in Table 2. Considering the estimation errors, Table 2 also indicates that the 531 dW cannot be found in the spectra by using the Fourier analysis (as shown in Fig. 3e and f).

Wahr (1983) implied that there exists an 18-month (

The amplitude spectra of the synthetic records (noise-free) based on
a conventional approach

By carefully examining Table 1, we can find that the amplitudes of the
531 dW in IMF6 perhaps have some proportional relation to the amplitudes of
CW in their corresponding

The input parameters for the six synthetic series. (FM: frequency modulation.)

The expression of the frequency modulation signal can be found in Eq. (2) in
Sect. 2. From Table 1, we can get the amplitudes of the

Now we set

The amplitude spectra of the synthetic series without using EEMD.

We generate six synthetic noise-free time series. The length of the synthetic
series I and II is equal to that of the 1962–1977 series, the length of the
synthetic series III and IV is equal to that of the 1978–1994 series, and
the length of the synthetic series V and VI is equal to that of the
1995–2013 series, with a 1-day sampling interval. The synthetic series I
(III/V) contains three sinusoidal components without frequency modulation,
and the frequencies and amplitudes of CW and AW (see Table 4) are set to make
the results for CW and AW after tuning the frequency modulation of CW to be
the same as with the observations from the

The frequency modulation mechanism of CW is an open question. If we accept the modulation mechanism of CW addressed above, the results obtained from the PM series after using EEMD can be appropriately explained. However, the excitation sources of the 531 dW in IMF5 still cannot be confirmed. We do not intend to answer this question in this study, but we will try to confirm the 531 dW in the independent gravity records in the following section to further confirm its presence.

The amplitudes of the significant peaks (from left to right:
531 dW, CW, AW) are marked by the arrows. The spectra of synthetic series I

The Fourier spectra of the two residual SG records.

If the 531 dW can be excited by the atmospheric/oceanic angular momentum, or
if it is the normal mode of the Earth, it may be found in the gravity
records. Hence, we choose the superconducting gravimeter (SG) records from
the Global Geodynamic Project (GGP) network to further confirm the existence
of this signal. The GGP network has been operating about 25 years (from
1997), but the longest available continuous records (without very large gaps)
are only about 16 years long, namely, the records from SG stations

No matter how, our results clearly show that the demodulation feature of EEMD is helpful for detecting the 531 dW signal in the PM series, and we confirm that the amplitude and frequency of the 531 dW vary with time.

After applying EEMD to the 1962–2013 PM time series (EOP C04), a 531 dW is
clearly found with a mean amplitude of about 7 mas (with much larger
amplitudes in the IMF5 and IMF6 series, respectively), but in the spectra
without using EEMD, this signal cannot be found. The 531 dW has been found
by previous studies in the 1962–1977 PM series with a lower SNR. To confirm
previous observations (Carter, 1981, 1982) and to further study this signal,
we divide the whole PM series into three sub-series, the 1962–1977,
1978–1994 and 1995–2013 PM time series. Without using EEMD, the results for
the 1962–1977 PM series are consistent with previous studies, while the
531 dW signal disappears from the Fourier spectra of the 1978–1994 and
1995–2013 PM time series. However, after applying EEMD to those three
sub-series, the 531 dW signals can be found in each sub-series with
different outstanding amplitudes (based on the spectra of the decomposed IMF5
and IMF6 series), which (taking

Although the frequency modulation mechanism of CW is an open question, we
find that if the modulation index

Though various previous studies have confirmed that the incomplete orthogonality of EEMD hardly affects the analysis of many geophysical processes, it might still affect our PM time series analysis process. Owing to the frequency and amplitude of the 531 dW signal being time-variant (generally, they might be caused by the excited process and the background noise in different time spanning), it becomes quite difficult to explore its excitations. It might be caused by the core dynamics, or even by some random processes (e.g., Chao et al., 2014), but from the phase results of the 531 dW of King (2015), we may prefer to consider that the 531 dW seems to be excited by some kind of unknown process, which needs further investigations. Anyway, as a final conclusion, we suggest that the 531 dW signal exists in nature, but left its mechanism open. With the accumulation of high-precision geodetic observation data, the long-period terms, such as 531 dW, are to be further studied, and this may help us better understand the Earth system.

We thank Jim Ray, Wei Chen and Benjamin Fong Chao for fruitful discussions
that improved the manuscript, and thank Christian Bizouard for helpful
comments on an early draft of this paper. We also thank M. King and two
anonymous reviewers for their valuable comments and suggestions, which
significantly improved the manuscript. The atmospheric/hydrological/non-tidal
ocean loadings are provided by J. P. Boy (which can be downloaded from