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Comptes Rendus
Géoscience
Sciences de la Planète
Jacques Talandier
Reception at very large distances of monochromatic Rayleigh wave of the
2018–2019 Mayotte crisis: attenuation factor of Ms magnitude
Volume 356 (2024), p. 117-122
Online since: 1 August 2024
https://doi.org/10.5802/crgeos.270
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Géoscience — Sciences de la Planète are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-7025
Comptes Rendus. Géoscience — Sciences de la Planète
2024, Vol. 356, p. 117-122
https://doi.org/10.5802/crgeos.270
Research article
Reception at very large distances of monochromatic
Rayleigh wave of the 2018–2019 Mayotte crisis:
attenuation factor of Ms magnitude
Jacques Talandier a
a 55 rue des 2 villages, 33220 Saint Avit Saint Nazaire, France
E-mail: Jac.talandier@wanadoo.fr
Abstract. During the strong 2018 and 2019 submarine volcano-seismic events on the Fani Maoré
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 Ms
magnitude scale, thus justifying the reception of these signals from sources of modest energy at very
large distances.
Keywords. Mayotte crisis, Monochromatic Rayleigh wave trains (VLPs), Magnitude Ms.
Manuscript received 5 March 2024, revised 12 June 2024, accepted 25 June 2024.
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 obser-
vational Seismology. Among the various magnitude
scales previously developed, the surface-wave mag-
nitude Ms, obtained at a period close to 20 s has be-
come the most widely used measure of teleseismic
events.
Ms = log10(A/T )+1.66 log10(∆)+3.3 (1)
where∆ is the epicentral distance, A is the amplitude
of the zero-to-peak vertical displacement in µm, and
T is the signal period.
In a comprehensive study of this magnitude scale
Ms, Okal [1989] concludes that despite several prob-
lems at shorter distances, the use of the correction
1.66Log10(∆) in the 20–160° range can be justified.
The purpose of the present approach is to experi-
mentally study this attenuation factor.
It has long been known, however that any magni-
tude 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 destruc-
tive 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:
Mm = Log10(Mo)+13 (2)
where Mo is in N-m.
In a similar approach, to avoid this Ms saturation
effect, Hanks and Kanamori [1979] developed the Mw
mantle magnitude scale:
Mw = 2/3Log10(Mo)−6.07. (3)
ISSN (electronic): 1778-7025 https://comptes-rendus.academie-sciences.fr/geoscience/
118 Jacques Talandier
Figure 1. 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.
Figure 2. 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 µm. Consistency of times and spectra confirms the total absence of frequency dispersion
characterizing the propagation of this type of phase.
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 mag-
nitude 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 per-
fectly monochromatic signals of 15.6 and 17.1 s and
duration of ca. 2000 and 3000 s. respectively. Fig-
ure 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 dis-
persion characterizing the propagation of this type of
phase: a textbook case, never encountered in prac-
tice, these wave trains are perfectly monochromatic.
Figure 3 shows that the monochromatic Rayleigh
wave train corresponds to the resonance of a cham-
ber excited by an earthquake.
3. Experimental verification of the attenuation
factor 1.66 log (∆) of the Ms magnitude rela-
tion
Figure 4 shows the amplitude versus distance of the
Mayotte Rayleigh wave trains of 11/11/2018. Close
Jacques Talandier 119
Figure 3. (a) VLP and its peak to peak ampli-
tude. The amplitude of the Short Periods sig-
nal is normalized. (b) CP signal with evidence
of the increasing amplitude of the VLP. We see
that the resonance is excited by an earthquake.
of 1 the slope of the regression line, defined with
41 records for a wide range of distances ∆ (700 to
17,000 km), corresponds to the geometric expansion
in 1/∆.
The Ms magnitude measured over a period ob-
viously cannot characterize the source of these
Rayleigh wave trains comprising some 160 os-
cillations. 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(∆)],
minus the geometric expansion in 1/∆. Defined
for the measurement of earthquakes with dispersed
Rayleigh waves of broad frequency spectrum, this re-
lation 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 atten-
uation factor 1.66 log(∆) 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 sin-
gle 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 mag-
nitude scale Ms leads to values dependent on the dis-
tance at the measuring point, the negation of the no-
tion of magnitude. Cesca et al. [2020] use the magni-
tude 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 cor-
rect.
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:
Ms = Ms_BB (Broad Band)
Ms = log10(Vmax/2π)+1.66 log10(∆)+0.3
where Vmax is maximum ground velocity (nm/s), zero
to peak, of surface waves in vertical component and
∆ is epicentral distance 2° ≤ ∆ ≤ 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,
120 Jacques Talandier
Figure 4. 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 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/∆.
Figure 5. Magnitude Ms as a function of distance. Same data set as Figure 4. 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(∆)] minus the geometric expansion in 1/∆. 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.
Jacques Talandier 121
Figure 6. This extension of Figure 5 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 effect.
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:
(i) The Rayleigh wave trains (VLPs) generated
during the Rocard and Mayotte volcano seis-
mic events are perfectly monochromatic.
(ii) 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/∆
and the frequency dispersion effect corre-
sponding to the coefficient 0.66.
(iii) 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 prop-
agation damping limited to geometric atten-
uation in 1/∆.
(iv) 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 bene-
fit from this article, and have declared no affiliations
other than their research organizations.
Acknowledgment
Thank you to the anonymous reviewer for these per-
tinent and constructive comments.
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