Magnitude M _w in metropolitan France

Abstract

The recent seismicity catalogue of metropolitan France Sismicité Instrumentale de l’Hexagone (SI-Hex) covers the period 1962–2009. It is the outcome of a multipartner project conducted between 2010 and 2013. In this catalogue, moment magnitudes (M _w) are mainly determined from short-period velocimetric records, the same records as those used by the Laboratoire de Détection Géophysique (LDG) for issuing local magnitudes (M _L) since 1962. Two distinct procedures are used, whether M _L-LDG is larger or smaller than 4. For M _L-LDG >4, M _w is computed by fitting the coda-wave amplitude on the raw records. Station corrections and regional properties of coda-wave attenuation are taken into account in the computations. For M _L-LDG ≤4, M _w is converted from M _L-LDG through linear regression rules. In the smallest magnitude range M _L-LDG <3.1, special attention is paid to the non-unity slope of the relation between the local magnitudes and M _w. All M _w determined during the SI-Hex project is calibrated according to reference M _w of recent events. As for some small events, no M _L-LDG has been determined; local magnitudes issued by other French networks or LDG duration magnitude (M _D) are first converted into M _L-LDG before applying the conversion rules. This paper shows how the different sources of information and the different magnitude ranges are combined in order to determine an unbiased set of M _w for the whole 38,027 events of the catalogue.

Appendix

The standard errors of the slope α and constant β in (9) can be written in different ways. The least square fit of a normal distribution of K data points (x _k , y _k ) by a straight line y = a(x − <x>) + b is associated with the following standard errors σ _a and σ _b (Draper and Smith 1981):

$$ {\sigma}_a=\frac{S}{\sqrt{Sxx}}\ \mathrm{and}\ {\sigma}_b=\kern0.6em S\sqrt{\frac{1}{K}+\frac{{\left(x-<x>\right)}^2}{Sxx}}, $$

where

$$ \begin{array}{l}S=\frac{1}{K-2}{\displaystyle \sum_{k=1}^K{\left({y}_k-{\widehat{y}}_k\right)}^2}\kern0.5em \mathrm{with}\kern0.5em {\widehat{y}}_k=\kern0.5em a\kern0.1em \left({x}_k-<x>\right)+b,\\ {}\mathrm{and}\\ {}Sxx={\displaystyle \sum_{k=1}^K{\left({x}_k-<x>\right)}^2}\kern0.5em \mathrm{with}<x>=\frac{1}{K}{\displaystyle \sum_{k=1}^K{x}_k}.\end{array} $$

When x = < x >, σ _b becomes

$$ {\sigma}_b=\sqrt{\frac{S}{K}}={\left\{\frac{1}{K\left(K-2\right)}{\displaystyle \sum_{k=1}^K{\left({y}_k-{\widehat{y}}_k\right)}^2}\right\}}^{1/2}. $$

Introducing the correlation coefficient

$$ r=\frac{Sxy}{\sqrt{Sxx\kern0.5em Syy}} $$

and recalling that

$$ a=\frac{Sxy}{Sxx}\ \mathrm{with}\ Sxy={\displaystyle \sum_{k=1}^K\left({x}_k-<x>\right)\left({y}_k-{\widehat{y}}_k\right)} $$

And

$$ S=\frac{1}{K-2}\left(Syy-\frac{Sx{y}^2}{Sxx}\right)\ \mathrm{with}\ Syy={\displaystyle \sum_{k=1}^K{\left({y}_k-\widehat{y}\right)}^2}, $$

simple computation yields

$$ {\sigma}_a=\frac{a}{\sqrt{K-2}}\sqrt{\frac{1}{r^2}-1}. $$

Expressions (11) of standard errors of the slope α and constant parameter β are obtained by setting

$$ \alpha =a,{M}_k={y}_k\ \mathrm{and}\ {M}_{\mathrm{w}k}={\widehat{y}}_k. $$