## 1. Introduction

The rapid measurement, if possible in real time and at a single seismic observatory, of the “size” of an earthquake remains an important aspect of observational Seismology. Among the various magnitude scales previously developed, the surface-wave magnitude Ms, obtained at a period close to 20 s has become the most widely used measure of teleseismic events.

$$\mathit{Ms}={log}_{10}\left(A/T\right)+1.66\phantom{\rule{2.77695pt}{0ex}}{log}_{10}\left(\Delta \right)+3.3$$ | (1) |

In a comprehensive study of this magnitude scale Ms, Okal [1989] concludes that despite several problems at shorter distances, the use of the correction 1.66 Log_{10}(Δ) in the 20–160° range can be justified. The purpose of the present approach is to experimentally study this attenuation factor.

It has long been known, however that any magnitude scale measured at a constant period T saturates when the duration of rupture along the fault becomes comparable to T. This phenomenon was explained in detail by Aki [1967, 1972] and later by Geller [1976], Geller and Kanamori [1977], as being due to destructive interference occurring when the source breakup time exceeds the period used for measurement, in this case 20 s. To avoid this saturation effect, Okal and Talandier [1989] and Okal [1989] developed the Mm (Mantle Magnitude) magnitude scale using the longest period of the frequency spectrum, defined by the source time and therefore the fault rupture length. Mm is directly related to the seismic moment by the relation:

$$\mathit{Mm}={\text{Log}}_{10}\left(\mathit{Mo}\right)+13$$ | (2) |

In a similar approach, to avoid this Ms saturation effect, Hanks and Kanamori [1979] developed the Mw mantle magnitude scale:

$$\mathit{Mw}=2/3\phantom{\rule{2.77695pt}{0ex}}{\text{Log}}_{10}\left(\mathit{Mo}\right)-6.07.$$ | (3) |

But we will note that here Mw is obtained from Mo while on the contrary Mo is deduced from Mm which makes it possible to use this last scale of magnitude for the rapid measurement of earthquakes and to elaborate alerts in particular to Tsunamis [Talandier et al., 1987, Talandier and Okal, 1989].

## 2. Monochromatic Rayleigh wavetrains (VLPs) of the 2018–2019 Fani Maoré (Mayotte) and 2011, 2013, and 2017 Rocard volcano/seismic crises

Figure 1 shows the 11/11/2018 Rayleigh wave trains from Mayotte and 29/04/2013 from Rocard with perfectly monochromatic signals of 15.6 and 17.1 s and duration of ca. 2000 and 3000 s. respectively. Figure 2 illustrates the time series and frequency spectra of the Rayleigh waves of the VLP of 11/11/2018 from Mayotte in 5 stations located from 2800 to 16,000 km away from the source. Consistency of the times and spectra confirms the total absence of frequency dispersion characterizing the propagation of this type of phase: a textbook case, never encountered in practice, these wave trains are perfectly monochromatic. Figure 3 shows that the monochromatic Rayleigh wave train corresponds to the resonance of a chamber excited by an earthquake.

## 3. Experimental verification of the attenuation factor 1.66 log (Δ) of the Ms magnitude relation

Figure 4 shows the amplitude versus distance of the Mayotte Rayleigh wave trains of 11/11/2018. Close of 1 the slope of the regression line, defined with 41 records for a wide range of distances Δ (700 to 17,000 km), corresponds to the geometric expansion in 1/Δ.

The Ms magnitude measured over a period obviously cannot characterize the source of these Rayleigh wave trains comprising some 160 oscillations. However, it is interesting to document the evolution of Ms over a wide range of distances since the most intense of these signals (31/07/2018 and 11/11/2018) were recorded as far as Tahiti and Hawaii located 16,300 km and 17,300 km away from the source respectively.

Figure 5 shows Ms as a function of distance. The slope of 0.60 of the regression line is close to the attenuation factor of Richter’s formula [1.66 log(Δ)], minus the geometric expansion in 1/Δ. Defined for the measurement of earthquakes with dispersed Rayleigh waves of broad frequency spectrum, this relation does not apply to the perfectly monochromatic ones of Rocard and Mayotte. This means that the factor 0.66 characterizes the frequency dispersion of Rayleigh waves of earthquakes. Validating the attenuation factor 1.66 log(Δ) of Richter’s Ms formula we also justify the reception of Rayleigh wave trains from Mayotte at very large distances.

The maximum amplitude of the Rayleigh wave train, shown in these figures, is used to calculate the magnitude. In addition to the fact that the source cannot be characterized by a measurement on a single period, it is clear that the Ms magnitude cannot be applied to monochromatic signals. The values obtained by Cesca et al. [2020], Sadeghi and Suzuki [2021] and Bhattacharya [2022] cannot therefore be significant.

## 4. Comments on the articles concerning the application of the Ms magnitude to Rayleigh waves in Mayotte

It has been shown that for monochromatic waves, therefore not dispersed, the application of the magnitude scale Ms leads to values dependent on the distance at the measuring point, the negation of the notion of magnitude. Cesca et al. [2020] use the magnitude Ms to characterize the November 11, 2018 VLP. Referring to Figure 5 showing the evolution of Ms as a function of distance, it is therefore surprising that these authors obtain a coherent value of Ms (M s = 5.1). Indeed, they use data from 4 stations at unspecified distances after eliminating those for which a deviation from a magnitude ratio of ∼1, which they attribute to the radiation pattern of the source, and LSZ and FURI stations due to changes in instrumental responses. And this whereas Figure 6 indicates that the radiation of these Rayleigh waves does not present a directional effect. Finally, it was verified that in 2018 the instrumental responses of FURI and LSZ from the USGS IU network were correct.

In a similar way Sadeghi and Suzuki [2021] use seismic data from the 26 Iranian seismic stations, estimate the Ms of 5.07 to characterize this VLP of 11 November 2018. Likewise, Bhattacharya [2022] obtained Ms magnitudes between 3.15 and 4.94 for 251 VLPs of the Mayotte crisis using the following formula:

$$\begin{array}{ccc}{\displaystyle}\mathit{Ms}& =& {\displaystyle}\mathit{Ms}\text{\_BB(BroadBand)}\\ {\displaystyle}\mathit{Ms}& =& {\displaystyle}{log}_{10}\left({V}_{max}/2\pi \right)+1.66\phantom{\rule{2.77695pt}{0ex}}{log}_{10}\left(\Delta \right)+0.3\end{array}$$ |

_{max}is maximum ground velocity (nm/s), zero to peak, of surface waves in vertical component and Δ is epicentral distance 2°⩽Δ⩽160° [Bormann and Dewey, 2012].

The Mo is deduced from Mw, which in turn is deduced from Ms. This amounts to estimating the Mo on the basis of the amplitude of one of the signal periods, which comprise 150 to 190 of them!

Not taking into account the resonant nature of the sources and their durations, from 30 to 60 min, Cesca et al. [2020], Sadeghi and Suzuki [2021] and Bhattacharya [2022] treat these monochromatic VLPs as dispersed Rayleigh waves of earthquakes.

## 5. Conclusions

We have shown that:

- The Rayleigh wave trains (VLPs) generated during the Rocard and Mayotte volcano seismic events are perfectly monochromatic.
- These Rayleigh wave trains allow us to demonstrate that the attenuation factor, 1.66, of the Ms magnitude scale takes into account the geometric attenuation in 1/Δ and the frequency dispersion effect corresponding to the coefficient 0.66.
- Originating from moderate energy events, the largest of these signals (31/07/2018, 11/11/2018) were correctly recorded up to very large distances (PPT-Tahiti: 16,300 km; HAW-Hawaii: 17,300 km). The cause is propagation damping limited to geometric attenuation in 1/Δ.
- The radiation from the source of these monochromatic Rayleigh wave trains has no directional effect.

## Declaration of interests

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

## Acknowledgment

Thank you to the anonymous reviewer for these pertinent and constructive comments.