# Gravity Tides Extracted from Relative Gravimeter Data by Combining Empirical Mode Decomposition and Independent Component Analysis

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## Abstract

The static observation data from a relative gravimeter contain noise and signals such as gravity tides. This paper focuses on the extraction of the gravity tides from the static relative gravimeter data for the first time applying the combined method of empirical mode decomposition (EMD) and independent component analysis (ICA), called the EMD-ICA method. The experimental results from the CG-5 gravimeter (SCINTREX Limited Ontario Canada) data show that the gravity tides time series derived by EMD-ICA are consistent with the theoretical reference (Longman formula) and the RMS of their differences only reaches 4.4 μGal. The time series of the gravity tides derived by EMD-ICA have a strong correlation with the theoretical time series and the correlation coefficient is greater than 0.997. The accuracy of the gravity tides estimated by EMD-ICA is comparable to the theoretical model and is slightly higher than that of independent component analysis (ICA). EMD-ICA could overcome the limitation of ICA having to process multiple observations and slightly improve the extraction accuracy and reliability of gravity tides from relative gravimeter data compared to that estimated with ICA.

## Keywords

Gravity tides relative gravimeter data CG-5 gravimeter empirical mode decomposition independent component analysis EMD-ICA## 1 Introduction

It is of great significance to study the static gravity observation, structure of the Earth’s interior, and geodynamics. The change of gravity tides can also be used in research of seismic precursor monitoring. In addition, the analysis of continuous gravity recordings and the other deformation components is one of the most important factors to understand physical processes of earthquakes, slow deformation of the Earth, and the determination of geodynamic parameters (Crescentini et al. 1999; Kasahara 2002; Métivier et al. 2009).

The acquisition and research of gravity tides has been an important content of classical geodesy for decades. In 1997, a variety of space observation techniques were applied in the research project of “gravity tides in space geodesy technology” (Haas 2001), such as Global Navigation Satellite System (GNSS), satellite altimetry, Satellite Laser Ranging (SLR), Doppler Orbitography and Radio-positioning Integrated by Satellite (DORIS), Lunar Laser Ranging (LLR), and Very Long Baseline Interferometry (VLBI). The relevant information about the tidal effect was obtained, and the inconsistency between the results of the different spatial observations was analyzed. With the continuous efforts of many scholars, softwares and methods for harmonic analysis and calculation of gravity tides have been continuously improved. The program packages ETERNA developed by Wenzel (1997) and BAYTAP by Tamura et al. (1991) are considered two of high standard harmonic analysis softwares of gravity tides in the world. BAYTAP is based on the Bayesian principle for harmonic analysis and ETERNA 3.4 is currently the only earth tide data processing package with a model accuracy better than 1 nGal (Wenzel 1997). A lot of theoretical studies on the gravity tides in gravity measurements have also been done by many scientists and the gravity tides model has achieved high accuracy (Sun et al. 1999; Venedikov et al. 2003; Zhou et al. 2009), which corresponds to the tidal potential developments achieved using precise analysis methods (Cartwright and Tayler 1971; Cartwright and Edden 1973; Tamura 1987; Xi 1989; Hartmann and Wenzel 1995). With the development of ground observation technique and the improvement of the accuracy of gravimeters, the relative gravity measurement has been one of the main observation approaches of the gravity tides (Ducarme and Sun 2001; Sun et al. 2001; Timofeev et al. 2017). The superconducting gravimeter produced by GWR company is currently recognized by the international counterparts as the relative gravimeter with the highest accuracy, continuity, and sensitivity, whose sensitivity can achieve up to \(0.001\; \times \; 10^{ - 8} {\text{m}}/{\text{s}}^{ 2}\) in the frequency domain. The CG5 relative gravimeter is a new type of improved automatic electronic reading gravimeter designed and produced by Scintrex Company in Canada. This gravimeter adopts a microprocessor device to realize automatic measurement. The sensor is designed with a static fused quartz spring, so the gravimeter’s accuracy can reach \(5\; \times \; 10^{ - 8} {\text{m}}/{\text{s}}^{2}\), and the reading resolution can be up to \(1\; \times \; 10^{ - 8} {\text{m}}/{\text{s}}^{2}\), both in the time and frequency domain. However, the recordings of gravimeters are usually influenced by many factors, especially non-tidal changes such as hydrological effects, air mass, and loading changes, which consequently are superimposed by the real change of the gravity tides. Based on the gravity observation data, how to obtain the accurate time series of the gravity tides has become the main emphasis of this study.

Many scientists have applied different methods to process different signals via their essential characteristics. Independent component analysis (ICA) (Forootan and Kusche 2012, 2013), wavelet analysis in combination with independent component analysis (WICA) (Lin and Zhang 2005), and empirical mode decomposition (EMD) (Mijović et al. 2010a, b; Cai and Chen 2016) are widely used for signal processing. ICA is a kind of blind source separation method, which can be used to extract dominating signals from the observations even if no apriori knowledge of the component is existing (Hyvärinen and Oja 1997; Cheung and Lei 2001; Stone 2002; Davies and James 2007; Guo et al. 2014). ICA is based on the assumption that the components belong to non-Gaussian distribution or not more than one meets Gaussian distribution. Besides, ICA assumes that the components are statistically independent from each other (Bell and Sejnowski 1995; Amari et al. 1997; Hyvärinen and Oja 2000; Zarzoso and Comon 2010). ICA can separate the independent signals, here the tides, from the superimposed signals, but it needs to deal with the multiple series of observations (Mijović et al. 2010a, b). WICA first uses the wavelet to decompose the signal into its sub-bands to expand a one-dimensional signal to two dimensions, and then ICA is applied to extract the source signals. However, some limitations remain in analysis for processing a practical signal (Lin and Zhang 2005; Shah et al. 2010): (1) there is no uniform criterion for the selection of an appropriate mother wavelet to extract signals, and we make decisions only by the intuition and experience of the analysts; (2) determining appropriate wavelet parameters without a prior knowledge of the signal is very difficult. EMD is a decomposition method for a non-stationary signal (Mijović et al. 2010a, b; Cai and Chen 2016), which can decompose one time series into a set of spectrally independent oscillatory modes called intrinsic mode functions (IMFs). However, EMD usually leads to the problem of model aliasing.

Since the static relative gravimeter data contain noise and signals, besides other signals, such as gravity tides and ocean tides, we can effectively combine the advantage of the empirical mode decomposition (EMD) and the independent component analysis (ICA) to determine the tides in the gravity observations recorded at a station. Therefore, the main focus of this paper is to use EMD-ICA to extract the gravity tides from the gravimeter time series to study the reliability and accuracy of this method, which could provide a new idea to obtain the gravity tides for the relative gravimetry correction.

The rest of this paper is organized as follows: Sect. 2, respectively, gives an introduction to the theory of EMD-ICA and ICA, and especially elaborates the boundary effect processing method. In Sect. 3, the case study and analysis of the results are made to verify the reliability and applicability of EMD-ICA method by comparing with ICA and the theoretical model. Section 4 then presents the conclusions of the study.

## 2 Methodology

### 2.1 Empirical Mode Decomposition

Empirical mode decomposition (EMD) is a signal decomposition method proposed by Huang et al. (1998), which mainly aims at analyzing non-stationary and non-linear signals (Wu and Huang 2004; Mijović et al. 2010a, b; Jiang et al. 2015; Mariyappa et al. 2015; Cai and Chen 2016). Since the decomposition is based on the local characteristic time scale of the data, complicated signals can be adaptively decomposed into a finite set of intrinsic mode functions (IMFs) whose instantaneous frequency is generated from the high frequency to the low frequency.

- 1.
Identify all local minimum points and maximum points of the gravity data, and select all local minimums and maximums to one data set.

- 2.
Using the cubic spline interpolation method, the local maxima and minima are connected by two special lines which are called the upper envelopes \(e_{\hbox{max} } (t)\) and the lower envelopes \(e_{\hbox{min} } (t)\). The original gravity record \(x(t)\) lies between the upper and lower envelopes.

- 3.
The mean of these two envelopes at any time is designated as \(m_{ 1} (t) = [e_{\hbox{max} } (t) + e_{\hbox{min} } (t)]/\text{2}\), i.e., the instantaneous mean of both envelopes. The local mean function of the original gravity signal \(x(t)\) is denoted by \(m_{\text{1}} (t)\), and the difference between \(x(t)\) and \(m_{\text{1}} (t)\) is \(h_{\text{1}} (t)\), that is \(h_{\text{1}} (t) = x(t) - m_{\text{1}} (t)\).

- 4.
Replace \(x(t)\) with \(h_{\text{1}} (t)\) and repeat the above steps until \(h_{{\text{1}k}}\) becomes a function as IMF, i.e., \(imf_{\text{1}} (t) = h_{{\text{1}k}}\). Then, subtract the first \(imf_{\text{1}} (t)\) from \(x(t)\) to obtain a new time series \(r_{\text{1}} (t) = x(t) - imf_{\text{1}} (t)\) and make \(x(t) = r_{\text{1}} (t)\). To determine whether \(h_{\text{1}} (t)\) is IMF, Huang et al. (1998) proposed the standard stopping iterative criterion \({\text{SD}} = \sum\nolimits_{t = 0}^{T} {\left| {h_{1(k - 1)} (t) - h_{1k} (t)} \right|}^{2} /\sum\nolimits_{t = 0}^{T} {h_{1(k - 1)}^{2} (t)}\), where SD is generally between 0.2 and 0.3.

- 5.
Repeat the process as described in step (4) until \(r_{n} (t)\) or \({\text{imf}}_{n} (t)\) is smaller than the predetermined value, or the residue \(r_{n} (t)\) becomes a monotonic function or a constant. \(r_{n} (t)\) is a residual component after the complete decomposition and it is a trend term which represents the average trend of the signal. Then, the decomposition process of the gravity record stops.

### 2.2 Independent Component Analysis

### 2.3 Emd-Ica

The original gravity observation can be adaptively decomposed to several IMFs and a residue, but there usually exists a mode aliasing phenomenon. ICA can extract the dominating signals such as gravity tides signal from other non-tidal signals. However, ICA needs to process multiple observations. Otherwise, the underdetermined problem of ICA will not be solved. Therefore, taking good use of the advantages of the two signal processing methods, a new method of combining EMD and ICA is gradually developed for signal denoising and signal extraction (Hyvärinen and Oja 2000; Mijović et al. 2010b; Forootan and Kusche 2012).

- 1.
The gravity record \(x(t)\) is decomposed with EMD and the matrix \(\varvec{imf}_{\varvec{i}}\) is obtained.

- 2.
The correlation coefficients between each component of

and the original gravity observation signal are calculated and analyzed.**IMF** - 3.Several
**IMF***s*are selected which have a relatively strong independence from \(x(t)\) to construct a virtual noise series, which is expressed as$$\varvec{ref\_EMD} = \left[ {\varvec{imf}_{ 1} ,\varvec{imf}_{ 2} , \ldots ,\varvec{imf}_{m} } \right],$$(8)

where \(1, 2, \cdots ,m\) are the subscripts of the * IMF*, which correspond to the

**IMF***s*with a relatively weak correlation with \(x(t)\). Then, \(\varvec{ref\_EMD}\) and \(x(t)\) are used together as the input matrix of ICA, which can solve the underdetermined problem of ICA. Then, use ICA to realize effectively the separation of signals and noise to extract the tidal signal.

*m*

*components are obtained as follows:*

**IMF***with a relatively weak correlation with \(\varvec{G}\). For this relative gravimeter data as listed in Table 1, if the correlation coefficient is smaller than 0.01, we define that the*

**IMF***or*

**IMF***has a relatively weak correlation with \(\varvec{G}\). The matrix (10) can be expressed as follows:*

**r**Correlation coefficients between each IMF and the original signal **G**

IMF | IMF | IMF | IMF | IMF | IMF | IMF | |
---|---|---|---|---|---|---|---|

Correlation coefficient | 0.0021 | 0.0048 | 0.0047 | 0.0041 | 0.4161 | 0.7877 | 0.4288 |

IMF | IMF | IMF | IMF | IMF | IMF | r | |

Correlation coefficient | 0.0210 | 0.0076 | 0.0174 | 0.0027 | 0.0049 | 0.0004 | 0.0083 |

ICA is used to realize the extraction of the gravity tides from the matrix \(\varvec{G\_ICA}\) and to achieve the separation of the gravity tides and noise, other signals like hydrological, air pressure effects, or other mass changes.

### 2.4 Boundary effect processing for EMD

When dealing with a discrete gravity record, i.e., \(t \in [t(\text{1}),t(\text{2}), \ldots ,t(n)] = [t_{\text{1}} ,t_{\text{2}} , \ldots ,t_{n} ]\), \(X(t) \in [x(t_{\text{1}} ),x(t_{\text{2}} ), \ldots ,x(t_{n} )] = [x_{\text{1}} ,x_{\text{2}} , \ldots ,x_{n} ]\), with the number of samples \(T = t_{n} - t_{\text{1}} + \text{1}\), the traditional EMD performs the cubic spline interpolation at all the upper and lower extreme points of the time series. Then, the mean of the envelope \(m(t)\) and the difference \(h_{\text{1}} (t)\) can be obtained according to step (3) in Sect. 2.1. However, in the EMD algorithm of Huang et al. (1998), there is no signal on both ends of gravity observation and the cubic spline interpolation can create a boundary effect, which results in distortion near two endpoints and contamination for the gravity observation. To suppress the boundary effect, many researchers (Wang et al. 2007; Ye 2013; Jaber et al. 2014; An et al. 2015) take the whole period in the form of symmetry to extend the signal on the boundary in electronics and we take the half of the series, which achieved the desired results. The specific steps are detailed as follows.

- 1.Carry out the periodic symmetric extension processing on the boundaries of the gravity record, in other words, make \(x(t)\) to mirror half of the series at both boundaries. Connect with \(x(t)\) to create a new signal \(x_{\text{1}} (t)\) with the series length of 2T, which is:$$\begin{aligned} t \in [t(0.5n), \ldots ,t(1),t(1), \ldots ,t(n),\;t(n), \ldots ,t( 0.5n)] = [t_{0.5n} ,\; \ldots ,t_{1} ,\;t_{1} , \ldots ,t_{n} ,\;t_{n} , \ldots ,t_{ 0.5n} ] \hfill \\ x_{ 1} \text{(}t\text{)} \in \text{[}x\text{(}t_{0.5n} \text{),} \ldots ,\;x\text{(}t_{1} \text{),}x\text{(}t_{1} \text{)} \ldots \text{,}x\text{(}t_{n} \text{),}x\text{(}t_{n} \text{)} \ldots \text{,}x\text{(}t_{0.5n} \text{)]} = [x_{0.5n} , \ldots x_{1} ,x_{1} , \ldots ,\;x_{n} ,x_{n} , \ldots ,\;x_{0.5n} ]. \hfill \\ \end{aligned}$$(12)
- 2.
Perform EMD sifting process (Ref. Sect. 2.1) for \(x_{ 1} (t)\), and then, \(m(t)\) and \(h_{\text{1}} (t)\) will be obtained. Intercept the corresponding section of the original gravity record and discard the parts with the length 0.5T on the left and right boundaries, respectively. Then, the corresponding section of the original gravity record after EMD processing is obtained.

## 3 Case Study and Analysis

### 3.1 Data

The static relative gravimeter data used in this experiment were observed inside the laboratory of Shandong University of Science and Technology, about 10 km far from the coast. The 30-day observations were obtained from April 5 to May 5 2016 with the CG5 gravimeter (serial number 140541221). In this period, the observations are used as a statistical variable. Because CG5 relative gravimeter itself has the function of seismic filtering, the low-frequency noise can be filtered and the high-frequency noise that is six times higher than the standard deviation can be discarded. In addition, adopting a 6 Hz sample frequency (a time resolution of 1 s), the readings of 1 min are averaged to get a final reading to which has been applied the tilt correction and temperature compensation (SCINTREX LIMITED 2009). Therefore, 1440 × 30 gravimeter readings are obtained in 30 days with the sample rate of 1 min.

Except the influence of the gravity tides, it is necessary to apply other corrections. The influence of temperature and ground vibration can be compensated by the gravimeter’s own function. The drift characteristic of the CG-5 gravimeter is nearly linear, and hence, its effects can be corrected by the least square fit method. Since the ocean loading has the extremely similar frequencies with the gravity tides, we cannot separate them from each other. Therefore, the ocean tides (Lei et al. 2017) must be subtracted from the observational data. We compute the ocean tide loading values on the website http://holt.oso.chalmers.se/loading/ provided by Bos and Scherneck (2011). The website provides our first 11 tidal component parameters (M2, S2, N2, K2, K1, O1, P1, Q1, Mf, Mm, and Ssa from the ocean tide model NAO.99b) just when we select an ocean tide model, tick the required type of loading, and fill in the coordinates of the stations. Then, these parameters are used to generate the ocean loading with the software Tsoft (Van Camp and Vauterin 2005). The preprocessing is carried out to weaken the influence of gross errors, and a time series mainly containing the gravity tidal signal has been obtained. In this paper, the main focus is to use EMD-ICA for the extraction of the gravity tides from the time series to study the reliability and accuracy of this method.

### 3.2 Results and Analysis

Tidal wave groups of the frequency spectrum

Wave | Frequency/(1/h) | Amplitude/mGal | Period/h | |
---|---|---|---|---|

E | K2 | 0.086975 | 0.007368 | 11.49754386 |

B | S2 | 0.083313 | 0.029442 | 12.00293040 |

A | M2 | 0.080566 | 0.060012 | 12.41212121 |

C | N2 | 0.078735 | 0.023474 | 12.70077519 |

D | 2N2 | 0.076904 | 0.011270 | 13.00317460 |

G | MNS2 | 0.074158 | 0.005721 | 13.48477366 |

J | K1 | 0.043945 | 0.006758 | 22.75555556 |

H | S1 | 0.042114 | 0.030144 | 23.74492754 |

I | M1 | 0.039368 | 0.025889 | 25.40155039 |

K | Q1 | 0.036621 | 0.006470 | 27.30666667 |

L | 2Q1 | 0.034790 | 0.002801 | 28.74385965 |

Statistical results of the gravity tides derived from ICA, EMD-ICA, and GT (mGal)

Statistics | ICA | EMD-ICA | GT | I-G | E-G |
---|---|---|---|---|---|

MAX | 0.1092 | 0.1101 | 0.1070 | 0.0173 | 0.0127 |

MIN | − 0.1540 | − 0.1512 | − 0.1490 | − 0.0337 | − 0.0157 |

MEAN | 0.0000 | 0.0000 | 0.0015 | − 0.0015 | − 0.0015 |

RMS | 0.0572 | 0.0573 | 0.0577 | 0.0054 | 0.0044 |

As can be seen from Fig. 6, the results estimated by ICA and EMD-ICA are basically consistent with those of GT. In Table 3, the results of the ICA and GT are virtually identical in terms of the root mean squares (RMS). In addition, RMS of I-G is 5.4 μGal, which results the slightly systematic deviation of the ICA results from GT as the ‘residuals’ distributed in Fig. 7. The maximum value reaches 33 μGal, but the number of larger differences is quite low and most of the values focus between -10 μGal and +10 μGal by analyzing the histogram (Fig. 7).

Correlation coefficients among ICA, EMD-ICA, and GT

Method | ICA/GT | EMD-ICA/GT | EMD-ICA/ICA |
---|---|---|---|

Correlation Coefficient | 0.9964 | 0.9974 | 0.9968 |

## 4 Conclusions

Since CG5 relative gravimeter observations are obtained from a fixed station, the gravimeter data set is mainly affected by the gravity tides. This paper primarily focuses on the estimation of gravity tides with EMD-ICA from the relative gravimeter data. The original gravity observations can be adaptively decomposed to several IMFs and a residue with EMD, but there usually exists mode aliasing phenomenon. ICA can separate the components from other non-tidal signals, but ICA needs to process multiple signals. Therefore, taking the advantage of the two signal processing methods into consideration, the method of combining EMD and ICA is proposed for the first application in the static relative gravimetric data processing, which can obtain the gravity tides for the relative gravimetry correction. Based on the characteristics of CG5 relative gravity data, EMD-ICA is used to decompose the gravity record into different components and then reconstruct them to obtain the gravity tides.

The experimental results indicate that the gravity tides derived from ICA, EMD-ICA, and GT, respectively, are of rather high agreement. The results from ICA, EMD-ICA, and GT display strong correlations with correlation coefficients greater than 0.99, but the correlation between EMD-ICA and GT is slightly closer to 1.0 than that between ICA and GT. Obviously, EMD-ICA is more suitable for the extraction of the gravity tides from the relative gravimeter data with a accuracy comparable to that of the theoretical model and slightly higher than that of ICA. Based on the above calculated results, the relative gravimeter observations after the gravity tides correction still present some small non-tidal changes, and the changes remain within ± 2 μGal in the mean, but within 4 or 5 microGal in fact.

## Notes

### Acknowledgements

The authors are grateful to the editors and anonymous reviewers for their helpful comments, which led a significant improvement in this paper. This study is partially supported by the National Natural Science Foundation of China (Grant No. 41774001, 41374009 & 41574072), the Special Project of Basic Science and Technology of China (Grant No. 2015FY310200), the Shandong Natural Science Foundation of China (Grant No. ZR2013DM009), and the SDUST Research Fund (Grant No. 2014TDJH101).

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