*2.2. Waveform Estimation by Stacking*

The time window, whose length is *<sup>T</sup>*, is chosen and is represented as *si*. The chosen time window *si* is approximated as the noise waveform; however, it is affected by white Gaussian noise. A stacking method is widely used to improve the SNR of the seismic profile [19], because it weakens white Gaussian noise and emphasizes seismic waves. We stack the approximate waveforms for an accurate noise waveform:

$$w\_{\vec{j}} = \sum\_{i=1}^{n} \overline{s}\_{i} \tag{5}$$

where *wj* is the noise waveform on the trace *hj* and its length *<sup>T</sup>* is equal to that of *si*. The waveforms of different traces are similar to the near traces *h*1, *h*2, ... , *hm* when the traces are interfered with by the same noise source. Therefore, we can stack the similar waveforms *w*1, *w*2,..., *wm* along different traces:

$$w = \sum\_{j=1}^{m} w\_j \tag{6}$$

where *w* is the noise waveform estimated from the traces *h*1, *h*2, ... , *hm*. We note that waveforms *w*1, *w*2, ... , *wm* obtained by Equation (5) are not in phase. Before stacking, waveforms *wj*(*j* = 1) need to be cyclically shifted to the same phase as *w*<sup>1</sup> by scanning the shift length in an interval 0, 1, . . . , *T* − 1 . The suitable shift length corresponds to the maximum correlation coefficient between *wj*(*j* = 1) and *w*1.
