**3. NA-MEMD Method**

To further reduce the mode-mixing effect in an MEMD, a noise-assisted MEMD method, named NA-MEMD, was introduced by adding white noise. The method is different from the ensemble EMD method in that multiple realizations of white noise are added to the input signal before being decomposed via EMD [13]. The NA-MEMD method adds extra multivariate channels, such as *l* dimensional independent white noise to the original *n* dimensional multivariate signal, and then processes such an *n+l* dimensional composite signal via MEMD. The NA-MEMD algorithm is described as follows:


For adding Gaussian white noise, the NA-MEMD method has better binary filtering characteristics than the MEMD method. Both the mode mixing and misalignment problems were significantly reduced. Figure 3 illustrates the benefits of noise-assisted MEMD algorithms for reducing mode mixing via a bivariate signal example. The Z component signal contains a 20 Hz sinusoid and two intermittent signals with frequencies 8 Hz and 80 Hz. The X component was generated by rotating the Z component 90 degrees on the vertical direction. For convenience, we plotted only the IMFs of the Z component. We can see that mode mixing in a single channel is obvious for the standard EMD in Figure 3a. A single-frequency scale mode exists in both IMFs. In addition, different frequency modes can also be seen in the same IMF. With two extra channels of white noise, the decomposition of the same bivariate signal by the proposed NA-MEMD method is shown in Figure 3b. This demonstrates that mode mixing is significantly reduced, and each IMF only contains a single frequency mode.

**Figure 3.** The EMD decompositions of a 20Hz sinusoid contaminated with two 8 Hz and 80 Hz intermittent signals: (**a**) IMFs of EMD method; (**b**) IMFs of the NA-MEMD method.

### **4. Ground Roll Attenuation via NA-MEMD**

This three-component geophones record two horizontal components and one vertical component of the incident seismic data, which are an oscillation stack of different frequency sub-bands. The ground roll, which is located in the low-frequency range, is different from the reflection wave. The alignment of frequency sub-bands can be obtained from different channels of the multivariate signal by inserting them into noisy channels via the NA-MEMD method. As a result, the IMFs were aligned according to their frequency content. Then, we can utilize the low oscillation property to separate the ground roll from the reflection wave.

Attenuating the ground roll based on the NA-MEMD method includes two steps: The first step is to use NA-MEMD to decompose multicomponent seismic data into a set of IMFs. The IMFs of ground roll are present in high-index IMFs, and their amplitude spectra are computed. A complete estimate of IMFs is reduced to identify these high-index IMFs in that the ground roll is dominant. In the second step, we remove these high-index IMFs and reconstruct the remaining low-index IMFs to obtain the seismic data without ground roll. In this workflow, the key parameter is the number of IMFs of the high-index ground roll. The number choice is a trade-off between less damage to the reflection waves and more ground roll suppression.
