As the global climate and environmental issues become more serious with the passage of time, more accurate monitoring of land water storage, sea level change, glacial melt, and redistribution of surface mass is important to this problem. The successful GRACE (Gravity Recovery and Climate Experiment) satellite, jointly developed by NASA and DLR, has made it possible to provide highly accurate global gravity field observations, paving the way for the observation of global climate change (Feng, 2013; Lu et al., 2015; Ning et al., 2016). The GRACE plays an important role in the study of regional hydrology (Landerer et al., 2010), ice sheet balance (Velicogna & Wahr, 2013) and ocean mass redistribution (Chambers & Bonin, 2012). And through GRACE, an Earth gravity field map with a spatial resolution of several hundred kilometers and a time resolution of 1 month is provided (Bettadpur et al., 2012; Watkins & Yuan, 2012; Dahle et al., 2013).

The GRACE satellite is subject to a number of factors in its orbit that can cause a certain amount of error in the spherical harmonic (SH) coefficients of the gravity field model it provides, such as satellite orbit error, instrument error, and satellite attitude measurements (Tapley et al., 2004). These errors can have a combined effect on the time-varying Earth’s gravity field model. The mass density of the earth’s surface change seriously affected the time-varying gravity field model inversion. The serious north-south (NS) striping errors in the gravity field inversion is mainly due to the configuration of satellite orbits for the GRACE mission, which can mask the true geophysical signal and be detrimental to the subsequent work, so some filtering methods are needed. Currently commonly used filtering methods can be divided into two categories (Guo et al., 2018). The first type of filtering algorithm is the introduction of filtering factors to reduce the weight of higher order terms in the data processing, which is called spatial filtering and so as to achieve the purpose of removing the striping error, mainly including Gaussian filtering (Wahr et al., 1998), Wiener filtering (Sasgen et al., 2007), Fan filtering (Zhang et al., 2009), and DDK filtering (Kusche et al., 2007; Kusche et al., 2009), etc. However, there are certain limitations of this type of filtering, with the increase of the filtering radius, although the noise is effectively removed, the real signal is also gradually weakened, that is, at the expense of the spatial resolution is sacrificed to achieve the removal of stripe noise (Zhan et al., 2011). The second type of filtering is decorrelation filtering, which uses polynomial fitting to achieve the purpose of decorrelation, including polynomial fitting (PnMm) and sliding window polynomial fitting, such as the commonly used P4M6 (Chen et al., 2007), P4M15 (Chambers et al., 2012), Duan (Duan et al., 2009) and so on. Some scholars also introduced the temporal and spatial filtering, mainly includes empirical orthogonal functions (Schrama et al., 2007; Wouters & Schrama, 2007), the stochastic filter (Wang et al., 2016), multichannel singular spectrum analysis (Guo et al., 2018; Prevost et al., 2019; Wang et al., 2020), and the least square filter (Crowley & Huang, 2020).

Temporal filtering treats the NS striping noise as white/random noise, and it is suggested that this random signal can be identified with the growth of time information (Yi et al., 2022). Nowadays, a combined filtering approach is the preferred choice. The adaptive time-frequency localization analysis approach empirical mode decomposition (EMD) (Huang et al., 1998), was used to post-process the time-varying GRACE gravity field models, demonstrated that EMD can better remove the strong noise with less signal leakage (Huan et al., 2022; Ai et al., 2022). Considering the mode mixing problem existed in EMD approach, an ensemble EMD (EEMD) approach was developed by Wu et al. (2009). In view of the advantage of EEMD approach with respect to EMD, and the good performance of EMD for filtering GRACE gravity field models, here in this study we will try to apply EEMD to extract the geophysical signals from the SH coefficients of GRACE time-varying gravity field models, together with EMD approach for comparisons. The rest of this paper is organized as follows: Section 2 introduces the theories of EEMD and EMD. In Section 3, we describe and analyze the results in the spectrum domain and spatial domain and Section 4 is the simulation experiment, Section 5 is the summary of the results.

In today’s signal processing field, there are many signal processing approaches, such as empirical mode decomposition, variational mode decomposition, local mean decomposition, etc. However, most of the signals we face are non-linear and non-stationary, in order to achieve our desired decomposition effect, we need to use more efficient and convenient analysis approach. EEMD is an adaptive spectrum analysis approach which improved the mode mixing problem on the basis of EMD, has been widely applied in many research fields. For example, global navigation satellite system (GNSS) data processing (Niu et al., 2018), mechanical vibration analysis (Lei et al., 2009), diagnosis of winding faults in a transformer (Mejia-Barron et al., 2017), and so on.

Though we have validated the advantage of EEMD for filtering the noise and extracting the geophysical signals from GRACE SH coefficients, the simulation experiments are also performed in this study. The CSR RL06 Mascon gridded data are converted to SH coefficients and SH coefficients are up to degree and order (d/o) 60, which is used as the true SH signals. To simulate the real GRACE type noise, we generated the noise from the time-varying gravity field model by DDK7 & EMD filtering, similar to Vishwakarma et al. (2016). Then the noise was added to the true signals to generate the noisy GRACE SH coefficients. Here we take March 2008 as an example to show the global mass changes extracted by EEMD, EMD and Gaussian smoothing 300 km combined with DDK7 filtering approaches in Figure 10. It is obviously to find that the EEMD and EMD approaches can both better filtering the noise with less signal leakage than Gaussian smoothing 300 km.

To evaluate the quality of extracted signals, we use the latitude weighted root mean squared errors (RMSE) of global mass change differences between true (Mascon) signal and extracted signal by EEMD, EMD and Gaussian smoothing 300 km (Wang et al., 2020), which are calculated as follows, R M S E Ω = ∑ ∀ i ∈ Ω Δ v i 2 . cos ⁡ φ i ∑ ∀ i ∈ Ω cos ⁡ φ i , (8) where Δ v i represents the differences between the true (Mascon) signals and the reconstructed signals, φ i is the corresponding latitude, Ω represents a spatial range and can be global or regional. And all grids within Ω are summed based on their area weights reflected by latitudes.

The relative improvement percentage (IMP) of spatial RMSE of EEMD with respect to EMD and Gaussian methods is calculated by, I M P E = R M S E E M D − R M S E E E M D R M S E E M D × 100 % (9) I M P G = R M S E G a u s s i a n − R M S E E E M D R M S E G a u s s i a n × 100 % (10)

Here we compute the spatial RMSEs of five selected global and regional areas including global, ocean, land, Amazon and Yangtze basins. Figure 11 shows the spatial mean RMSEs of all available months for five selected regions over April 2002 to August 2016 and the corresponding IMPs. It is clearly to find that all the spatial RMSEs of EEMD are smaller than EMD and 300 km Gaussian smoothing approaches, and the relative improvements of EEMD with respect to 300 km Gaussian smoothing are more significant than those of EMD approach for five selected regions, which indicate that EEMD can extract the geophysical signals more accurately than EMD and 300 km Gaussian smoothing approaches. Besides, we further compute the mean mass changes over four regions (Land, Ocean, Amazon and Yangtze) and shown in Figure 12. The corresponding RMSEs of three filtering approaches in terms of the mean mass changes are presented in Table 2. Through the simulation experiments, we can draw the conclusion that EEMD can extract closer geophysical signals than EMD and 300 km Gaussian smoothing approaches.

In this paper, EEMD approach is first applied to filter the time-varying gravity field models, together with the EMD approach. We evaluate the filtering efficiency of EEMD in both spectral and spatial scale, respectively. For the real GRACE SH coefficients analysis, the fitting errors of all SH coefficients by EEMD approach are smaller than those of EMD approach. The mean RMS_ratios of all available months for EEMD is 3.45, higher than 3.40 of EMD approach. The results show that EEMD can better filter the noise and extract more geophysical signals. Besides, the simulation results show that all the mean RMSEs of EEMD are smaller than EMD for global, ocean, land, Amazon and Yangtze, indicating that EEMD can extract the closer geophysical signals than EMD with respect to the true signals from CSR mascon data. In summary, we can believe that EEMD is a good choice for extracting the geophysical signals and filtering the noise from GRACE time-varying gravity field models.