The previous section showed that short-period OBSs deployed near known VLFEs epicenters recorded tremors associated with each VLFE. Conversely, this section investigates the tremors that are not associated with the reported VLFEs since tremors are detected more frequently than VLFEs (Ghosh et al. 2015; Ohta et al. 2019; Nishikawa et al. 2019). Generally, the envelope correlation method (Obara 2002) is effective in detecting tremors. However, the method did not work well for detecting and locating tremors, including those identified by visual inspection of the seismograms around VLFE timing. This might be because most of the tremor sources were located outside the OBS network used in the study, which consisted of only five stations, making tremor detection very difficult.
Instead, the amplitude patterns (Fig. 4) and power spectral densities (Fig. 5) can be used for identifying additional tremors because these characteristics should be similar among tremors #1–7 and new tremors that are not associated with the reported VLFEs.
We quantified the following two characteristics: First, the deviation of the peak amplitude pattern (hereafter amplitude deviation) from the averaged peak amplitude pattern of tremors #1–7. The amplitude deviation was defined by the difference between the peak amplitude distribution of samples and the average distribution for the seven tremors, and is calculated as follows:
$${\text{Amplitude }}\;{\text{deviation}} = \max \left( {{\text{AMF}}_{i} } \right)$$
and
$${\text{AMF}}_{i} = \left| {\hat{A}_{i}^{{{\text{sample}}}} - \overline{{\hat{A}_{i}^{{{\text{ref}}}} }} } \right|/2\sigma_{i}$$
\({\widehat{A}}_{i}^{\mathrm{sample}}\) is the peak horizontal amplitude at station i, normalized to the peak amplitude at station AO.S05. The peak values of the root-mean-squared horizontal amplitude of the sample waveforms of the records of the two horizontal components were used. \(\overline{{\widehat{A} }_{i}^{\mathrm{ref}}}\) is the average of the peak amplitudes of the reference seismograms at station i, including tremors #1–7. The deviation from the reference at station i was scaled by \({\sigma }_{i}\), the standard deviation of the peak amplitudes of the seven tremors at the corresponding station i. The amplitude deviation decreased with the increase in the similarities, in terms of peak amplitude distribution, between the sample and the reference. After applying a band-pass (2–4 Hz) filter, a frequency band in which the tremor signal was most pronounced, sample seismograms were obtained from continuous seismograms using a sliding time window of 30 s with a time interval of 15 s.
Second, the residual sum of squares between the power spectral density of a sample waveform and that of tremor #5, which was observed with the highest signal-to-noise ratio (S/N), was calculated as a measure of similarity between power spectral densities. The residual sum termed “spectral deviation” is calculated as follows:
$${\text{Spectral}}\;{\text{deviation}} = \mathop \sum \limits_{j = 1}^{N} \left( {{\text{log}}_{10} {\text{PSD}}_{j}^{{{\text{sample}}}} - {\text{log}}_{10} {\text{PSD}}_{j}^{{{\text{ref}}}} } \right)^{2}$$
where \({\mathrm{PSD}}_{j}^{\mathrm{sample}}\) represents the power spectral density of a sample seismogram and \({\mathrm{PSD}}_{j}^{\mathrm{ref}}\) is the power spectral density of the reference seismogram at a frequency band j. N = 90 is the number of power spectral densities within a frequency range of 1–4 Hz, where high S/N is expected. We used the records at station AO.S05 for tremor #5 for the reference seismogram, since the seismogram had the highest S/N. A higher similarity yielded a smaller spectral deviation. The sample power spectral densities were calculated from the seismograms extracted using sliding time windows identical to the amplitude deviation calculation. Spectral deviation was calculated in the frequency band of 1–4 Hz. Husker et al. (2019) developed a tremor detection method based on the single-station tremor spectrum template. Our spectral deviation is essentially the same as that used in Husker et al. (2019), but different cost functions. Both amplitude deviation and spectral deviation take smaller values when a sampled seismogram resembles that of the tremors. We determined the length of the sliding window as 30 s after a test using different lengths (Additional file 1: Fig. S2).
A 15 s window is too short to characterize the given event in terms of maximum amplitude due to the short-term irregular fluctuation of seismograms. By contrast, a 60 s window is too long to discriminate successive occurrences of tremors.
Figure 6 shows the two-hour spectrograms with temporal variations of amplitude and spectral deviations for the period in which tremor #5 occurred. Both amplitude and spectral deviations had the smallest values at the time of tremor #5. As expected, the amplitude and spectral deviations for the seismograms obtained in the seven windows including the occurrence times of tremors #1–7 were considerably smaller than the rest of the seismograms in the six days when the seven tremors with the VLFEs were detected (Fig. 7).
Notably, spectral deviation of tremor #1 was relatively large due to low signal levels (Figs. 3 and 5). Tremor #6, which was depleted in the low (1 Hz) frequency components more than other tremors (Fig. 5) and slightly deviated from other tremors in terms of peak amplitude distribution (Fig. 4), had a larger amplitude and spectral deviations. For more conservative tremor detection, these two events (tremors #1 and #6) were excluded while defining the thresholds of amplitude and spectral deviations. Therefore, based on the amplitude and spectral deviations for tremors #2–5 and #7, the thresholds were set at 0.675 and 0.17 for amplitude and spectral deviations, respectively (Fig. 7).
Based on the proximity of the epicenters of repeating earthquakes to those of the VLFEs, the amplitude deviations of repeating earthquakes are expected to be as small as those of tremors. The amplitude deviation of repeating earthquake-A was not smaller than the criteria (Fig. 7), whereas that of repeating earthquake-B was smaller than the threshold for tremor detection. This suggests that repeating earthquake-B, a regular earthquake, occurred very close to the sources of the tremors. By contrast, spectral deviations of repeating earthquakes are different from those of tremors.
In the seismogram of the remote earthquake shown in Fig. 1, spectral deviation is smaller, but amplitude deviation is larger than the thresholds (Fig. 7). However, spectral deviation of the remote earthquake is similar to those of tremors.
Therefore, tremors can be distinguished from local and remote regular earthquakes by amplitude and spectral deviations. As shown by the triangle in Fig. 7, a seismogram satisfied the criteria of tremor detection during the six days. Figure 6 shows the corresponding record as an event 10 min before tremor #5. The newly detected event was similar to tremor #5 in the seismograms, and it can be regarded as another tremor event. There were a few events that satisfied the tremor detection threshold of spectral deviation but not that of amplitude deviation (e.g., events at 87 min and 100 min in Fig. 6). The slightly larger amplitude deviation but acceptably small spectral deviation of these events indicate that these events can be tremors with different source locations from the group of tremors identified in the study.
