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focal depth may strongly bias an estimation of magnitude made from macroseismic
observations as we will see for the 1967 Arette earthquake in the French Pyrenees.
Furthermore, I
0
may differ from the maximum intensity I
max
when the epicentre
is outside a zone of observation, making I
0
estimate difficult. The situation is not
much better when using the whole set of macroseismic areas A(I) due to the strong
dependence of the macroseismic attenuation law on the focal depth.
To avoid the uncertainties due to the attenuation law, site effects, or shift in
frequency with epicentral distance, Cara et al. (2005) have proposed to com-
pare directly the intensities of a recent instrumentally-known earthquake with the
historical-earthquake intensities at large distances from the epicentre. Looking at
(2), it is clear that for two earthquakes located at the same hypocentral distance
R, the difference of intensity
I is proportional to the difference of their magni-
tude
M:
=
,
I
b
M
(3)
where the constant factor “b” is determined experimentally in the region of interest.
For two earthquakes located at the same epicentral distance, R may be confounded
with D far from the observation point. As a rule of thumb, we propose to work at
distances D larger than three times the standard 10-15 km focal depths of crustal
earthquakes in continents, a difference between D and R of a few kilometres being
negligible for a macroseismic investigation.
Using the isoseismal areas A(I) to estimate D(I), as in Cara et al. (2005), further-
more acts as a smoothing filter on the azimuthal radiation pattern at the source and
on the possible site effects. The investigated zone may then be broad enough to cover
densely-populated regions, making the average intensity observations more robust
and reliable than the epicentral intensity I
0
for estimating an earthquake magnitude.
The main source of uncertainty comes from the parameter “b” of relationship (3).
As the linearity of this relationship probably fails when it is applied to a too broad
magnitude range, it is safe to estimate “b” from a set of events with magnitudes
not too far from those under study. In France, Levret et al. (1994) found b
27
for a large set of data based on a homogeneous set of local magnitudes (4-5.8 M
L
)
issued by the Laboratoire de Detection Geophysique (LDG) of the French commis-
sion of atomic energy, while Souriau (2006) found b
=
2
.
17 from a smaller set of
recent earthquakes and magnitudes issued by the Reseau National de Surveillance
Sismique (ReNaSS) (3.0-5.4 M
L
). Accordingly, a value b
=
2
.
2 will be used in the
present paper for application to France in the moderate-size magnitude range 4.5-
6M
w
. The fact that we use a factor “b” determined from M
L
catalogues to com-
pute M
w
should not be a problem if we refer to Braunmiller et al. (2005). These
authors have shown that the slope of the M
w
versus M
L
relationship is close to 1
for the different European catalogues they have investigated in the neighbouring
countries of Switzerland. Only the intercept differs, M
w
being smaller than M
L
.
The difference reaches 0.2 for both the Swiss Seismological Service (SED) and the
Karlsruhe catalogues and 0.6 for the LDG catalogue. Note also that in the differ-
ential macroseismic method proposed here, an error on “b” will only affect the
=
2
.
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