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\DOI{10.5802/crgeos.270}
\datereceived{2024-03-05}
\daterevised{2024-06-12}
\dateaccepted{2024-06-25}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Reception at very large distances of monochromatic Rayleigh wave
of the 2018--2019 Mayotte crisis: attenuation factor of \textit{Ms} magnitude}

\alttitle{R\'{e}ception \`{a} tr\`{e}s grande distance des trains
d'ondes de Rayleigh monochromatiques de la crise de Mayotte 2018--2019 :
facteur d'att\'{e}nuation de l'\'{e}chelle de magnitude \textit{Ms}}

\author{\firstname{Jacques} \lastname{Talandier}}
\address{55 rue des 2 villages, 33220 Saint Avit Saint Nazaire, France}
\email{Jac.talandier@wanadoo.fr}

\keywords{\kwd{Mayotte crisis}\kwd{Monochromatic Rayleigh wave trains 
(VLPs)}\kwd{Magnitude \textit{Ms}}}

\altkeywords{\kwd{Crise de Mayotte}\kwd{Trains d'ondes de Rayleigh 
monochromatiques (VLP)}\kwd{Magnitude \textit{Ms}}}

\begin{abstract}
{During the strong 2018 and 2019 submarine volcano-seismic events on
the} Fani Maor\'{e} {volcano on the eastern flank of the island of
Mayotte (Indian Ocean), numerous unusual long-period monochromatic
Rayleigh wave trains (VLPs) were recorded. Similar perfectly
monochromatic signals but with much lower intensities were previously
observed in 2011, 2013 and 2017 from the Rocard submarine volcano
(Society hot spot, French Polynesia). The VLPs of Mayotte and Rocard
are the only signals of this type observed so far.
Perfectly monochromatic and recorded up to the antipodes, the Rayleigh
wave trains of Mayotte allow verifying experimentally the attenuation
factor of the \textit{Ms} magnitude scale, thus justifying the reception of
these signals from sources of modest energy at very large distances.}
{\vspace*{-2pt}}
\end{abstract}

\begin{altabstract}
Lors des forts \'{e}v\'{e}nements volcanosismiques sous-marins de 2018
et 2019 du volcan Fani Maor\'{e}, sur le flanc Est de l'\^{i}le de
Mayotte (Oc\'{e}an Indien), de nombreux trains d'ondes de Rayleigh
monochromatiques inhabituels \`{a} longue p\'{e}riode (VLP) ont
\'{e}t\'{e} enregistr\'{e}s. Issus du volcan sous-marin Rocard (point
chaud de la Soci\'{e}t\'{e}, Polyn\'{e}sie fran\c{c}aise), des signaux
similaires parfaitement monochromatiques mais d'intensit\'{e}s beaucoup
plus faibles ont \'{e}t\'{e} pr\'{e}c\'{e}demment observ\'{e}s en 2011,
2013 et 2017. Les VLP de Mayotte et du Rocard sont les seuls signaux de
ce type observ\'{e}s \`{a} ce jour. Parfaitement monochromatiques et
enregistr\'{e}s jusqu'aux antipodes, les trains d'ondes de Rayleigh de
Mayotte permettent de v\'{e}rifier exp\'{e}rimentalement le facteur
d'att\'{e}nuation de l'\'{e}chelle de magnitude \textit{Ms}, justifiant ainsi la
r\'{e}ception \`{a} tr\`{e}s grande distance de ces signaux issus de
sources d'\'{e}nergie modeste.
\end{altabstract}

\maketitle

\twocolumngrid

\end{noXML}

\section{Introduction}\label{sec1}

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 \textit{Ms}, obtained at a
period close to 20 s has become the most widely used measure of
teleseismic events. 
{\begin{equation}\label{eq1}
\mathit{Ms} = \log_{10} (A/T) + 1.66 \,\log_{10} ({\rmDelta}) + 3.3
\end{equation}}\unskip
where $\rmDelta$ is the epicentral distance, $A$ is the amplitude of the
zero-to-peak vertical displacement in $\rmmu $m, and $T$ is the
signal period.

In a comprehensive study of this magnitude scale \textit{Ms}, \citet{Okal1989}
concludes that despite several problems at shorter distances, the use
of the correction $1.66\, \mathrm{Log}_{10} (\rmDelta)$ in the
20--160\textdegree\  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 $T$ \mbox{saturates} when the duration of rupture along the
fault becomes comparable to $T$. This phenomenon was explained in detail
by \citet{Aki1967,Aki1972} and later by \citet{Geller1976,GellerKanamori1977},
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, \citet{OkalTalandier1989} and
\citet{Okal1989} developed the \textit{Mm} (Mantle Magnitude) magnitude scale
using the longest period of the frequency spectrum, defined by the
source time and therefore the fault rupture length. \textit{Mm} is directly
related to the seismic moment by the relation:
{\vspace*{-2pt}}
{\begin{equation}\label{eq2}
\mathit{Mm} = \mathrm{Log}_{10 }(\mathit{Mo}) + 13
\end{equation}}\unskip
where \textit{Mo} is in N-m.

\begin{figure*}
\includegraphics{fig01}
\caption{\label{fig1}Long period Rayleigh wave trains generated during
the Mayotte (right panel) and Rocard (left panel) volcano seismic
events. Perfectly monochromatic long duration signals of 15.6 and 17.1
sec period.}
\end{figure*}

\begin{figure*}
\includegraphics{fig02}
\caption{\label{fig2}Time series and frequency spectra of monochromatic
Rayleigh wave trains from Mayotte on 11/11/2018 in 5 stations located
from 2800 to 16,000 km away from the source. Peak-to-peak amplitudes
in $\rmmu $m. Consistency of times and spectra confirms the total
absence of frequency dispersion characterizing the propagation of this
type of phase.}
\end{figure*}


In a similar approach, to avoid this \textit{Ms} saturation effect,
\citet{HanksKanamori1979} developed the \textit{Mw} mantle magnitude scale:
{\vspace*{-2pt}}
{\begin{equation}\label{eq3}
\mathit{Mw} = 2/3\, \mathrm{Log}_{10 }(\mathit{Mo}) - 6.07.
\end{equation}}\unskip

But we will note that here \textit{Mw} is obtained from \textit{Mo} while on the contrary
\textit{Mo} is deduced from \textit{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\break
\citep{Talandieretal1987,TalandierOkal1989}.


\section{Monochromatic Rayleigh wavetrains (VLPs) of the 2018--2019
Fani Maor\'{e} (Mayotte) and 2011, 2013, and 2017 Rocard
volcano/{\ubreak}seismic crises}\label{sec2}

Figure~\ref{fig1} 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~\ref{fig2} 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~\ref{fig3} shows that the monochromatic
Rayleigh wave train corresponds to the resonance of a chamber excited
by an earthquake. 


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}(a) VLP and its peak to peak amplitude. The
amplitude of the Short Periods signal is normalized. (b) CP signal with
evidence of the increasing amplitude of the VLP. We see that the
resonance is excited by an earthquake.}
\end{figure}



\section{Experimental verification of the attenuation factor 1.66 log
($\rmDelta$) of the \textit{Ms} magnitude relation} 

Figure~\ref{fig4} 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
$\rmDelta$ (700 to 17,000~km), corresponds to the geometric expansion in
$1/\rmDelta$.


\begin{figure*}
{\vspace*{-2pt}}
\includegraphics{fig04}
{\vspace*{-2pt}}
\caption{\label{fig4}Amplitude versus distance of the Mayotte Rayleigh
wave trains of 11/11/2018. In green the slope line ${-}$1.66 of the
attenuation coefficient of \textit{Ms}. In red the regression line with the
slope obtained of ${-}$1.06. In blue the regression line by imposing a
slope of ${-}$1.0. In dashed the high and low limits encompassing 90\% of
the data. Despite a rather high dispersion due to the impact of the
structure of the earth's crust on the amplitude of these Rayleigh waves
of relatively short wavelengths, close of 1 the regression line defined
with 41 records implies a geometric expansion in $1/\rmDelta$.}
\end{figure*}


The \textit{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
\textit{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~\ref{fig5} shows \textit{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 ${[}1.66\, \log(\rmDelta){]}$, minus the geometric expansion in
$1/\rmDelta$. 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 $1.66
\,\log (\rmDelta)$ of Richter's \textit{Ms} formula we also justify the reception of
Rayleigh wave trains from Mayotte at very large distances. 


\begin{figure*}
\includegraphics{fig05}
{\vspace*{-2pt}}
\caption{\label{fig5}Magnitude \textit{Ms} as a function of distance. Same data
set as Figure~\ref{fig4}. In green materialization of the concept of
magnitude independent of the distance at which the measurement is made.
From 0.60, the slope of the regression line (red) is close to the 0.66
slope (blue) corresponding to the attenuation factor of Richter's
formula ${[}1.66\, \log(\rmDelta){]}$ minus the geometric expansion in
$1/\rmDelta$. Defined for the measurement of earthquakes with dispersed
Rayleigh waves of broad frequency spectrum, this relationship does not
apply to the perfectly monochromatic VLPs of Rocard and Mayotte. This
means that the factor 0.66 characterizes the frequency dispersion of
earthquake Rayleigh waves.}
\end{figure*}


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 \textit{Ms} magnitude cannot be applied to
monochromatic signals. The values obtained by \citet{Cescaetal2020},
\citet{SadeghiSuzuki2021} and \citet{Bhattacharya2022} cannot therefore
be significant.

\section{Comments on the articles concerning the application of the
\textit{Ms} magnitude to Rayleigh waves in Mayotte}

It has been shown that for monochromatic waves, therefore not
dispersed, the application of the magnitude scale \textit{Ms} leads to values
dependent on the distance at the measuring point, the negation of the
notion of magnitude. \citet{Cescaetal2020} use the magnitude \textit{Ms} to
characterize the November 11, 2018 VLP. Referring to Figure~\ref{fig5}
showing the evolution of \textit{Ms} as a function of distance, it is therefore
surprising\break that these authors obtain a coherent value of 
\textit{Ms} ($Ms=5.1$).
Indeed, they use data from 4 stations at unspecified distances after
eliminating those for which a deviation from a magnitude ratio of
${\sim}$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~\ref{fig6} 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.


\begin{figure*}
{\vspace*{2pt}}
\includegraphics{fig06}
{\vspace*{2pt}}
\caption{\label{fig6}This extension of Figure~\ref{fig5} shows the
azimuths station seen from the source, in blue. We can see that the
radiation from these source of these Rayleigh waves has no directional
\mbox{effect.}}
{\vspace*{2pt}}
\end{figure*}


In a similar way \citet{SadeghiSuzuki2021} use seismic data from the 26
Iranian seismic stations, estimate the \textit{Ms} of 5.07 to characterize this
VLP of 11 November 2018. Likewise, \citet{Bhattacharya2022} obtained \textit{Ms}
magnitudes between 3.15 and 4.94 for 251 VLPs of the Mayotte crisis
{using the following formula:}  
{\begin{eqnarray*}
\mathit{Ms}&=& \mathit{Ms}\text{\_BB (Broad Band)}\\
\mathit{Ms} &=& \log_{10 }(V_{\max}/2\rmpi) + 1.66\, 
\log_{10}(\rmDelta) + 0.3
\end{eqnarray*}}\unskip
where $V_{\max}$ is maximum ground velocity (nm/s), zero to peak, of
surface waves in vertical component and $\rmDelta$ is epicentral
distance $2\text{\textdegree}\leq \rmDelta \leq 160\text{\textdegree}$
\citep{BormannDewey2012}.

The \textit{Mo} is deduced from \textit{Mw}, which in turn is deduced from \textit{Ms}. This
amounts to estimating the \textit{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, \citet{Cescaetal2020}, \citet{SadeghiSuzuki2021}
and \citet{Bhattacharya2022} treat these monochromatic VLPs as dispersed
Rayleigh waves of earthquakes. 

\section{Conclusions}

We have shown that:

\begin{enumerate}[(i)]
\item The Rayleigh wave trains (VLPs) generated during the Rocard and
Mayotte volcano seismic events are perfectly monochromatic.

\item These Rayleigh wave trains allow us to demonstrate that the
attenuation factor, 1.66, of the \textit{Ms} magnitude scale takes into account
the geometric attenuation in $1/\rmDelta$ and the frequency dispersion
effect corresponding to the coefficient 0.66.

\item 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
$1/\rmDelta$.

\item The radiation from the source of these monochromatic Rayleigh
wave trains has no directional effect.
\end{enumerate}

\section*{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.

\section*{Acknowledgment}
Thank you to the anonymous reviewer for these pertinent and
constructive comments.

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