Introduction

The November 11, 2018, Mayotte event was first introduced in the media by Wei-Haas (2018) in National Geographic as a strange earthquake in which seismic waves were recorded by instruments around the world, but, unusually, nobody felt shaking. The event occurred offshore around 9:28 UTC near Mayotte Island in the Mozambique Channel between the northern tip of Madagascar and the eastern coast of Africa. Mayotte Island is one of the four main islands in the volcanic Comoros archipelago. The November 11 Mayotte event, in the absence of body waves, caused large, long-lasting, monotonic long-period surface waves that traveled around the globe. Before May 2018, the seismic activity in this archipelago was dispersed and moderate, with only a few earthquakes having a magnitude greater than 4. But in the year following the Mayotte volcano-seismic sequence on May 10, 2018, 32 earthquakes with a magnitude greater than 5 were recorded (Lemoine et al. 2020). The biggest earthquake ever recorded in the area was a Mw 5.9 event that happened on May 15, 2018 (USGS). Figure 1 shows the recorded waveforms for the May 15, 2018, earthquake, and the Mayotte event at a regional station, ABPO, Ambohimpanompo, Madagascar (part of the global IRIS/IDA seismic network), and at a teleseismically distant station, ASAO, Ashtian, Iran (part of the Iranian National Broadband Seismic Network). The epicenters of these two events were located very close to each other, and the epicentral distances were 713 and 5261 km from the ABPO and ASAO stations, respectively. The waveforms show clear differences in the nature of the propagated waves between the regular earthquake and the Mayotte event. The most obvious feature of the Mayotte event is that while large long-period monotonic surface waves are visible, body waves are absent. The long-period waves appeared for more than 30 min at both the regional and teleseismic distances. Moreover, several smaller tremors with similar characteristics were recorded during the 2018–2019 Mayotte seismo-volcanic crisis (Cesca et al. 2020). Thus, there is still much to learn about these very-long-period (VLP) signals occurring during the Mayotte event. We call the November 11 event the “monotonic very-long-period” signal and abbreviate it as MVLP. The event has been described in the framework of underwater volcanic activity, as the result of the resonance of a deep magma reservoir, in response to the sagging of the reservoir roof (Cesca et al. 2020). In this study, we investigate this event using the waveforms recorded by the Iranian National Broadband Seismic Network. The objective of our study is to characterize the seismic signal and its source by analyzing seismic data recorded by the Iranian National Broadband Seismic Network: in particular, we aim at (1) evaluating its spectral properties, (2) estimating the magnitude, and (3) assessing the occurrence of an earthquake just before the event for possible interaction that may have triggered the event.

Fig. 1
figure 1

(Left) Map of area of the November 11 Mayotte event (called MVLP), and locations of Iranian broadband seismic stations (triangles). The red triangles show the stations that were selected for array analysis. The inset enlarges the view of the locations of the biggest regular earthquake (black star behind red star), Mw 5.9 event of May 15, 2018 (USGS), and the MVLP (red star) (Lemoine et al. 2020). The blue lines indicate great circles connecting the epicenter and ABPO station, which is located SSE about 6.4° from the events, and ASAO station, which is located NNE about 47.5° from the events. (Right) Seismic waves for the two events recorded by ABPO (IRIS/IDA seismic network) on the left and by ASAO (Iranian National Broadband Seismic Network) on the right. The duration of the seismograms is 4000 s, and their start time, which is given at the lower left of each plot, corresponds to the origin time of the May 15 event at 15:48:09 UTC and for the MVLP at 09:28:00 UTC. The waveforms are vertical components and have been filtered in the range of 0.01 to 0.1 Hz

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This work extends previous results (Cesca et al. 2020; Lemoine et al. 2020), which partially addressed these objectives, by investigating the largest VLP signal in the light of teleseismic recordings at a dense regional seismic network.

Observations

The broadband seismic network of the Iran International Institute of Earthquake Engineering and Seismology (IIEES) started operating with 4 stations in 1998 and currently has 31 stations (IIEES 2020). The stations are equipped with Güralp CMG-3 T sensors with a flat velocity response from 120 s (0.0083 Hz) to 50 Hz. Twenty-six of the stations (triangles in Fig. 1) were in operation on November 11, 2018, and recorded the MVLP. The stations are located from 4542 to 5772 km north-northeast of the MVLP’s epicenter, which was assumed from the study of Lemoine et al. (2020). A 1.5-h trace view of the vertical component of the broadband records filtered between 0.01 and 0.1 Hz, and the frequency spectra of their unfiltered waveforms, are displayed in Fig. 2. All the stations show clear long-period seismic waves arriving around 10:00 UTC on November 11, 2018, and a frequency analysis of the seismograms shows a clear peak in the spectral amplitudes. The peak is in a narrow frequency band of 0.05–0.07 Hz. Among the stations, no significant changes are observed in the monochromatic wave trains and the narrow-band spectral peak.

Fig. 2
figure 2

One-and-a-half hour trace view of the vertical components of MVLP observed by Iran’s 26 broadband stations. Columns (a) and (c) show the normalized waveforms, and columns (b) and (d) show the corresponding normalized Fourier amplitude spectra. The traces are ordered from top to bottom by increasing epicentral distance (station name and epicentral distance are displayed). The waveforms are bandpass filtered between 0.01 and 0.1 Hz. Long-lasting, long-period seismic waves are observed to arrive at around 10:00 UTC on November 11, 2018. The Fourier spectra are for unfiltered seismograms. The peak of the spectra (indicated by the gray arrow in the two bottom spectra) is at 0.064 Hz (15.6 s/cycle)

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Seismic waves and focal mechanisms

We compared the seismic waves for the MVLP and the largest regular earthquake (called “May 15” in Fig. 1 and hereafter). Figure 3 shows the seismic waves for the three components observed at the ASAO station. The ground noise level for the ASAO station is low. It is clear that the observed waveforms for the two earthquakes differ greatly even though the epicenters are close. The MVLP waveforms are dominated by long-lasting monochromatic surface waves, and body waves are lacking altogether. The maximum amplitude of the transverse component is less than a half of the radial component. Clear P- and S-phase arrivals are observed on the May 15 seismograms, and dispersion of the surface waves clearly appears in all three components on the seismograms. The maximum amplitude of the transverse component is 1.2 times that of the radial component. Figure 4a displays the focal mechanisms for the MVLP and May 15 events as beach-ball representations of their moment tensors (MTs). The MT components of the MVLP are proposed by averaging the MTs of 21 events with monochromatic long-duration and VLP signals that are provided in the seismic catalogs of the 2018–2019 crises that occurred east of Mayotte Island (Cesca et al. 2019). To eliminate the seismic moment (event size) effect, the components of each moment tensor were first normalized to 1 by dividing by the maximum absolute value of its elements, and then the normalized components were averaged (Eq. 1).

$$\overline{\boldsymbol{M}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {\frac{{\boldsymbol{M}_{i} }}{{\left| {\boldsymbol{M}_{i} } \right|_{\max } }}} \right){,}}$$
(1)
Fig. 3
figure 3

Comparison of the vertical, radial, and transverse components for the MVLP (top) and May 15 (bottom) events observed at station ASAO. The waveforms show a length of 45 min and were filtered between 0.01 and 0.1 Hz. Vertical short red lines in the waveforms in the bottom show the arrivals of P and S phases

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Fig. 4
figure 4

a Focal mechanisms for the MVLP (top) and May 15 (bottom) events. The MVLP MT, which is defined by averaging the MTs of 21 VLPs proposed by Cesca et al. (2019), shows a vertical CLVD, while the focal mechanism for May 15 indicates dominant strike-slip faulting with a NW–SE-directed P-axis and NE–SW-directed T-axis. The relative components of the MVLP MT are defined as Mrr = − 1.0, Mtt = 0.7, Mpp = − 0.1, Mrt = − 0.3, Mrp = 0.8, and Mtp = − 0.2, and the May 15 MT is obtained from the GCMT catalogue. b Azimuthal radiation pattern for surface waves at a frequency of 0.06 Hz considering the moment tensors for the MVLP (top) and May 15 (bottom) events with source depths at 28 and 17 km, respectively. The black circles show the relative spectral displacement amplitude on a linear scale for each pair of Rayleigh and Love waves. The red arrow points to station ASAO

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where Mi is the MT of the ith event and \(\left| {{\boldsymbol{M}}_{i} } \right|_{\max }\) is the maximum absolute value of its elements, and n is the number of MTs. The focal mechanism for May 15, from the (GCMT) catalog, indicates a dominant strike-slip with NW–SE-directed compression. Cesca et al. (2020) suggested a vertical compensated linear vector dipole (CLVD) for the Mayotte VLP events, which are not always exactly vertical. In a vertically symmetric CLVD, no Love waves or azimuthal variations in Rayleigh waves are expected, but the proposed focal mechanism for the MVLP shows no vertical symmetry axes. Previous studies discussed the relation between active volcanism and the occurrence of rarely observed non-double-couple earthquakes (e.g., Shuler et al. 2013a, b). Their study used a standard source time function as in a GCMT solution, while Cesca et al. (2020) modeled the VLP events with a damped resonator source time function. Although the present study cannot draw definitive conclusions, to compare the MVLP with a regular earthquake, we have to make a model for the radiation pattern of surface waves considering the MTs. Figure 4b shows the theoretical seismic energy radiation patterns for Rayleigh and Love waves at a frequency of 0.06 Hz. The spectral amplitudes based on the considered source mechanism were calculated with an azimuthal spacing of 1° using the data product (Rösler and van der Lee 2020) of the IRIS DMC’s Surface-Wave Radiation Pattern. Seismic radiation for the MVLP is dominated by Rayleigh waves displaying a nearly oval form with larger elongation in the northeast-southwest direction. Love waves are clearly less energetic than Rayleigh waves and show more azimuthal variation in the maximum amplitude. Seismic radiation for May 15 shows larger spectral amplitudes for Love waves than for Rayleigh waves. The spatial variation of the maximum amplitude shows a clear four-lobe pattern for Love waves; while for Rayleigh waves, the radiation pattern has a two-lobe shape. The broadband station in Iran is located in the azimuth range of 359°–21° from the MVLP and lies close to the maximum radiation point for the surface waves.

Figure 5 presents spectrograms of the power spectral density (PSD) in (m/s)2/Hz, Fourier spectra, and dominant periods for the MVLP and May 15 vertical and transverse components at the ASAO station (the seismograms are shown in Fig. 3). As we expected from the model of the radiation pattern for the MVLP, the surface waves in the vertical components (dominated by Rayleigh waves) have larger amplitudes than the transverse components (Love waves). In addition, the MVLP shows four clear differences from regular earthquakes. The first obvious difference is the very long duration of its surface waves. The second difference can be seen through the spectrograms, where a dominant monochromatic surface wave (around 0.06 Hz) extends throughout the signal duration. The Fourier spectra show a clear large peak at 0.064 Hz. The third and fourth differences can be observed in the graph of the dominant period over time; the dominant period for the transverse component is lower than that for the vertical component, and the signals in both components propagate with nearly the same velocity. A convergence of surface wave velocities can also be seen in the graph of the dominant period for May 15 around a period of 16.5 s (0.06 Hz), which corresponds to the Airy phase with a minimum in the group velocity dispersion curves (Bullen and Bolt 1985). The Airy phases in the transverse component can be observed by large amplitudes after an abrupt termination in the wave trains (see Fig. 3, bottom). The gap between the red lines around 1600 s in the dominant period graph (right-bottom in Fig. 5b) for the transverse component are due to this termination, and also our criterion to select power values greater than the mean value of the PSD.

Fig. 5
figure 5

Spectrograms of the power spectral density (PSD) in the vertical component (left-top) and transverse component (left-bottom), Fourier spectra (right-top), and the dominant periods over time (right-bottom) for the vertical (Z) and transverse (T) components at ASAO station for MVLP (a) and May 15 (b). The dominant periods are plotted by dots, and the lines are piecewise linear interpolations through the data points. Note that periods with a power less than the mean values for the spectrogram are removed from the graph. The gray circle on the Fourier spectra for May 15 indicates a peak of around 0.06 Hz related to the Airy phase of surface waves. This peak is also clearly displayed by the maximum values of PSD on the spectrograms of transverse components

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Phase and group velocities of the surface waves

We used two analysis methods, frequency–wavenumber (f–k) analysis and regression analysis, to estimate the group velocity of the MVLP surface waves. We applied f–k analysis to a group of nine relatively closely spaced stations as an array using the python library Obspy (Beyreuther et al. 2010). The selected stations are marked by red triangles in the map in Fig. 1. The array response function for the directional sensitivity and its resolution power are shown in Fig. 6. This function was computed for a frequency band of 0.05–0.07 Hz. There are no distinct side lobes in the array response.

Fig. 6
figure 6

Array response function at 0.05 to 0.07 Hz. The color scheme corresponds to the normalized amplitude of the relative power, as shown by color bar on the right

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The semblance technique (Neidell and Taner 1971) was used to measure the slowness and back-azimuths of the coherent wave phases that crossed the array. The analysis was applied to 1-h sections from 2018-11-11, 09:45:00, to 2018-11-11, 10:45:00. After the waveform data were filtered using a bandpass filter with a bandwidth of 0.05–0.07 Hz, the semblance coefficient was calculated for a sliding window length of 20 s in steps of 1 s. The range of slowness was searched from 0.1 to 1.5 s/km in steps of 0.05 s/km. The quality of the stacked signal was measured by the semblance coefficient, which is a dimensionless quantity between 0 and 1, representing no coherency and perfect coherency, respectively. The semblance values show a coherency peak at about 75% for the radial (Rayleigh waves) components (Fig. 7a). According to Lemoine et al. (2020), the latitude and longitude of the MVLP were 12.777° S and 45.590° E, respectively, corresponding to a back-azimuth of ~ 185° from the center of the array. From Fig. 7a, the back-azimuth was determined to be ~ 200°, but its error was large. In contrast, for the transverse (Love waves) components, the semblance values did not show a unique peak and the coherency was higher than 55% (Fig. 7b). For the regression analysis, we manually picked the wave group arrivals for bandpass-filtered radial and transverse components for all 26 stations. One station was taken as a reference, and the relative delays, positive or negative, in arrival times for the other stations were calculated. Assuming a planar wavefront propagating horizontally with a constant apparent velocity across the array, the tau-p method (Havskov and Ottemöller 2010) was used. The apparent velocity and back-azimuth were estimated by fitting a plane wave, that is, a least-squares regression line, to the time delays. The procedure was repeated using another station as a reference. The averages of the apparent velocities in the transverse and radial directions were calculated as 3.31 and 2.97 km/s, respectively. These correspond to the phase velocities of Love and Rayleigh waves, respectively, in the frequency band of 0.05–0.07 Hz. Figure 8 shows the average back-azimuth (black arrows). They are generally in good agreement with the back-azimuth to the MVLP epicenter (red arrows) according to Lemoine et al. (2020). In this figure, a clear pattern of back-azimuth errors from positive to negative moving from west to east in Iran is found. Such systematic errors can be due to velocity variations in the crust and upper mantle, while we assumed a homogenous structure.

Fig. 7
figure 7

Semblance values, back-azimuth in degrees, and apparent slowness in s/km for radial components (a) and transverse components (b) of MVLP. Data from nine array stations (red triangles in Fig. 1) were used. The pair of vertical gray lines in a indicates the solution with the maximum semblance value of 0.75%

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Fig. 8
figure 8

Estimated back-azimuth (black arrow) at each station using both Rayleigh and Love waves of MVLP (see text). Red arrows show the direction to the epicenter of MVLP (Lemoine et al. 2020) from each station

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Estimation of magnitudes

Using seismic data from the 26 Iranian seismic stations, we estimated the surface wave magnitudes (Ms) of the MVLP and the regular earthquakes occurring around Mayotte Island. We used Eq. (2), named the Prague formula (Karnik et al. 1962; Vanek et al. 1962):

$$Ms = {\text{ log }}\left( {A/T} \right) \, + {1}.{\text{66 log}}\Delta + { 3}.{3}0,$$
(2)

where A is the peak amplitude of the surface waves in micrometers, T is the period in seconds, and Δ is the epicentral distance in degrees. First, the surface wave magnitude for the MVLP was calculated to be Ms = 5.07 ± 0.22. Cesca et al. (2020) estimated a similar value of Ms = 5.1. Second, we calculated the surface wave magnitudes for the 24 regular earthquakes with body wave magnitudes (mb presented by USGS) > 4.5 occurring from May 13 to June 1, 2018. The surface wave magnitudes for the selected events were calculated using averages of at least 10 amplitude readings. These Ms values and body wave magnitudes (mb) are listed in the Table in the Appendix. It is important to note that the biggest earthquake is mb = 5.8 and a lower limit of mb = 4.5 was defined based on the signal strength and quality for reliable amplitude measurements. Then, we compared the relationship between Ms and mb, as shown in Fig. 9. Assuming that the best regression equation between Ms and mb has the form mb = a + bMs, we obtained Eq. (3):

$$mb = {2}.{54 }\left( { \pm \, 0.{29}} \right) + 0.{56}\left( { \pm \, 0.0{7}} \right)Ms, {\text{std}} = 0.{15}, R^{{2}} = 0.{76}.$$
(3)
Fig. 9
figure 9

Relation between the calculated surface wave magnitude and the body wave magnitude estimated by USGS

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Equation 3 shows that the average Ms in the range of 4.5–5.8 is lower by 0.5–0 than mb on average. Scordilis (2006) studied empirical global relations for magnitude scale conversions and found nearly the same result for Ms < 6.2. We defined the converted body wave magnitude, mb(conv), using the following Eq. (4):

$$mb\left( {conv} \right) \equiv 2.54 \, + \, 0.56Ms.$$
(4)

The obtained mb(conv) values are listed in Table 1 for the three largest earthquakes and the MVLP. The hypocenters of the three regular earthquakes were obtained from USGS, while the location of the MVLP was assumed from the source location that best fit the ground deformation after the early July 2018 event reviewed by Lemoine et al. (2020).

Table 1 Hypocenter location and magnitudes [mb, Ms, and mb(conv)] for the three largest earthquakes occurring during May 2018 and the MVLP and JBE occurring on November 11, 2018
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Trigger earthquake just before MVLP

Lemoine et al. (2020) stated that no strong earthquakes preceded the MVLP event, but some earthquakes are embedded in its seismic record, especially at the beginning of the monochromatic MVLP signal. Here, we want to take a closer look at the beginning of the signal to investigate the possibility of an earthquake occurring before and triggering the MVLP. Figure 10 shows the waveform and time variation of the dominant period at the ASAO station for the vertical component of the MVLP. It is clear from the figure that immediately before the arrival of the long-lasting monochromatic signal of the MVLP, the surface waves (Rayleigh waves) are dispersed. The period decreases from 17.5 to 16 s during the time period from 09:56:56 to 09:58:38 UTC, for about 100 s, as shown in Fig. 10b. This could be the result of the occurrence of a regular earthquake just before the MVLP event. We call this event the “just before the monochromatic event” (JBE) in this paper. Compared with the arrival time of the equivalent dominant periods of the regular earthquakes (listed in Table 1) that occurred in the vicinity of the MVLP, the origin time of the JBE was estimated to be 09:27:56 UTC ± 13 s. The surface wave magnitude of Ms = 4.3 ± 0.11 was calculated for JBE. Cesca et al. (2019) detected a volcano-tectonic earthquake at 09:27:59.575 on November 11, 2018, at a latitude of 12.8058° S and a longitude of 45.4736° E. They determined its magnitude based on a regional moment tensor inversion following the record of Mw 3.8 at a single strong-motion station on Mayotte Island. This earthquake must be the JBE. Now, the question is, how long before the MVLP did it occur? An answer can be obtained by comparing the frequency of these two events. Figure 10b shows that the dominant period curve for the JBE is bent at around the 16-s period at 09:58:38 UTC (red arrow). This may suggest that the surface wave of the JBE lasted until at least 09:58:38 UTC. For a more accurate determination, we applied a notch (a band-stop with narrow stopband) filter to separate the two merged events. In the spectrum of the MVLP, the largest peak is at 15.6 s. Therefore, we filtered out the 15.6-s signal from the waveform using the notch filter to remove the main component of the MVLP signal. The results are shown in Fig. 10c. This figure shows that the filter removed the main content of the waveform corresponding to the long-duration monochromatic signal of the MVLP after 09:59:09 UTC (black arrow). This suggests that the MVLP surface wave arrived at 09:59:09 UTC at the latest. When we assume that the difference in surface wave velocities between the periods of 15.6 and 16 s are nearly the same, the time difference between 09:58:38 UTC and 09:59:09 UTC may suggest that the JBE occurred less than 31 s before the start of the MVLP. Because the interpretation of the MVLP triggering mechanism may differ depending on the accuracy of the delay time between the JBE and MVLP, we examined the record of the ABPO broadband station in Madagascar (Fig. 1). Figure 11 shows the recorded waveforms filtered at different frequency bands. The presence of several volcano-tectonic earthquakes accompanying the MVLP is seen in Fig. 11a. The first is the JBE, which makes the arrival of the P and S waves more clearly visible when enlarged (Fig. 11b). The first arrival time for the JBE Rayleigh wave and the time interval for the MVLP arrival are marked in Fig. 11c. Regardless of the difference in group velocity due to the Rayleigh wave period, the MVLP surface wave appears to start ~ 7 s after the JBE arrival, as shown in Fig. 11c. However, it is difficult to mark the MVLP onset accurately.

Fig. 10
figure 10

a Enlargement of the right side of the top plot of Fig. 3. JBE and MVLP waveforms can be seen. b Dominant periods of the waveforms in a. The dispersion of the surface waves is clear, and in the range of periods from 17.5 to 16.5 s is visible in the waveform (marked by red line). The red arrow indicates 09:58:38 UTC. c Notch filter output of the waveform with the notch in the period of 15.6 s and a bandwidth of 1 s. The black arrow indicates 09:59:09 UTC

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Fig. 11
figure 11

Filtered seismic records of the vertical component at the ABPO broadband station in Madagascar. a Multiple volcano-tectonic signals accompanying the waveform of MVLP. b Part of panel a enlarged to show arrivals of the P and S phases of JBE. c Part of panel b enlarged to compare the arrival of the MVLP surface wave and JBE Rayleigh wave. The blue line shows the record through a 0.07- to 2-Hz filter. The blue arrow indicates that the JBE Rayleigh wave arrived at 09:31:08. The MVLP surface wave seems to arrive between 09:31:15 (first red arrow) and 09:31:39 (second red arrow)

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Globally observed long-lasting monochromatic surface waves with dominant periods of about 230 and 270 s, generated by the 1991 Mt. Pinatubo eruption (Philippines), were reported by Watada and Kanamori (2010). They demonstrated that the mechanism of these harmonic ground motions can be explained by the resonant oscillation between the atmosphere acoustic waves and the Rayleigh waves in the solid Earth. Both MVLP and Pinatubo Rayleigh waves are monochromatic but their wave periods are quite different. The mechanism of what controls the 16 s wave period of MVLP is not clear yet. We hope the existence of JBE before MVLP confirmed by this study serves as a stepping stone to unveil the generating mechanism of MVLP.

Conclusions

We studied the Mayotte MVLP, which occurred on November 11, 2018, using the records of the Iranian National Broadband Seismic Network. Long-lasting, monotonic long-period surface waves are visible and dominate the waveforms. The frequency of the monochromatic signals is 0.064 Hz. The radiation pattern of the surface waves supports a vertical CLVD source mechanism. Apparent phase velocities of 3.31 and 2.97 km/s, respectively, for the Love and Rayleigh waves of the event were obtained from the results of our array analysis. The surface wave magnitude of the MVLP was estimated to be Ms = 5.1. Evidence of dispersion of the surface waves just before the arrival of the monochromatic event suggests that the MVLP was triggered by a regular earthquake with a magnitude of Ms = 4.3, which occurred around 09:28 UTC.