# Earthquake Source Spectra and Site Attenuation in Northwestern France

Marion Alloncle, Mickaël Bonnin, Antoine Mocquet

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Marion Alloncle, Mickaël Bonnin, Antoine Mocquet. Earthquake Source Spectra and Site Attenuation in Northwestern France. Bulletin of the Seismological Society of America, 2025, 10.1785/0120240122. hal-05034252

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# Earthquake Source Spectra and site
attenuation in Northwestern France

Marion Alloncle¹, Mickaël Bonnin¹, and Antoine Mocquet¹

## ABSTRACT

Northwestern France is a stable continental region (SCR) submitted to a high rate of small to moderate magnitude earthquakes, which source characteristics are not well known. The deployment of permanent seismological networks during the last decade allows, for the first time in this region, to study their source spectra in order to determine their seismic moment $M_0$, associated magnitude $M_W$, corner frequency $f_c$, and stress drop $\Delta\sigma$. In France, two national agencies, RéNaSS and CEA-LDG, provide local magnitudes. The spectra of 108 events of $2.0 \le M_{LRéNaSS} \le 5.3$ that occurred between 2015 and 2024 are analyzed in the [0.01 - 50] Hz frequency range under the assumption of an omega-square source model. The effect of attenuation on $f_c$ is tested by correcting the observations with two different absorption-band models, and an empirical parameter $\bar{\kappa}_0$ is assessed at each station to account for the near-site loss of energy above ~5 Hz. Linear relationships $M_{LRéNaSS} \propto 1.13 M_W - 0.21$ and $M_{LLDG} \propto 1.20 M_W - 0.19$ are obtained for $1.7 \le M_W \le 5.0$. A Brune $\Delta\sigma$ in the range [5.4 - 9.4] MPa is derived for the largest magnitude ($M_W$ 5) of the data set, which is similar to results obtained in other intraplate regions. When combined with the $M_0 - f_c$ relations, the regressions between local magnitudes and $M_W$ indicate that the slope of the $M_{LLDG} - M_W$ relation is governed by attenuation effects, while the $M_{LRéNaSS} - M_W$ relation may be explained either by attenuation effects or by an increase of $\Delta\sigma$ with $M_0$. However, a more precise description of the attenuation is required before concluding about a possible deviation from self-similarity between small and large magnitude earthquakes.
## KEY POINTS

* Magnitude and source scaling relations are investigated for intraplate earthquakes in northwestern France.

* Site attenuation, and the relations $M_0 - f_c$, and $M_L - M_W$, are assessed for this region.

* A Brune stress drop in the range [5.4 – 9.4] MPa is obtained for the largest magnitude earthquake ($M_W$ 5).

## Supplemental Material

## INTRODUCTION

The Armorican Massif, in northwestern France, is a stable continental region (SCR) subject to a high rate of low-to-moderate magnitude earthquakes whose source characteristics are not well known. The densification of permanent seismic networks during the last decade (Figure 1) now provides a better visibility of this seismicity (Beucler et al., 2021; Hourcade et al., 2022) and allows, for the first time in this region, to study the earthquake sources in terms of seismic moment $M_0$, associated magnitude $M_W$, corner frequency $f_c$, and stress drop $\Delta\sigma$.

The seismic moment $M_0$ and the associated moment magnitude $M_W$ are essential parameters for earthquake studies and seismic hazard assessment. While the empirical parameters controlling the local magnitude, $M_L$, initially defined by Richter (1935), and commonly used by seismic monitoring agencies — such as, e.g. for France, BCSF-RéNaSS (Bureau Central Sismologique Français – Réseau National de Surveillance Sismique) and CEA-LDG (Alternative Energies and Atomic Energy Commission – Geophysical and Detection Laboratory) — may vary from one country to another, the moment magnitude, $M_W$, defined by the seismic moment, $M_0$, uniquely relies on physical basis (Hanks & Kanamori, 1979):

$$M_W = \frac{2}{3} (\log_{10} M_0 - 9.1). \qquad (1)$$

In the context of tectonic plate boundaries, characterized by large earthquakes ($M_W \ge 6.0$) and high deformation rates, the high level of seismic energy release and the density of the seismic networks usually enable a precise determination of source parameters, such as $M_W$, and the establishment of a seismic energy budget, which can be compared to other measurements of the deformation, such as GNSS data. Conversely, stable continental regions, remote from active plate boundaries, although characterized by very low deformation rates close to the precision threshold of GNSS observations (e.g., Masson et al., 2019, for a study of France), can display a high occurrence rate of low-to-moderate earthquakes ($M_W \le 4.0$). In addition to far-field tectonic stresses, several phenomena involving different time scales have been associated with such background seismicity (Levandowski et al., 2018), as summarized in Beucler et al. (2021): spatial variations of gravitational energy over Myr periods (Camelbeeck et al., 2013), paleo-stress release (Calais et al., 2016), short-term (Steer et al., 2014; Ribes et al., 2020) and long-term (Champagnac et al., 2007; Calais et al., 2010; Vernant et al., 2013; Mazzotti et al., 2023) adjustments to erosion and
sedimentation, glacial cycles (Steffen et al., 2014; Olivieri & Spada, 2015; Damon et al., 2023), oceanic loading (Luttrell & Sandwell, 2010), and hydrological (Leclère & Calais, 2019) or meteorological (Bollinger et al., 2007) processes. The study of this SCR seismicity and its link with these processes is made difficult by the low magnitude of the earthquakes and the low density of seismological networks.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Major shear zones (after Chantraine et al., 2003, see infoterre.brgm.fr) and seismic networks in northwestern France. The limits of the Proterozoic and Paleozoic metasedimentary and crystalline basement outcrops (Armorican Massif and Massif Central) are drawn in red. Sensors acronyms are BB: Broad-Band, ACC: Accelerometer. GB: Great Britain Seismograph Network (British Geological Survey, 1970), operated by the British Geological Survey. Networks of the European Plate Observing System France (Epos-France) include FR (Réseau Large Bande Permanent, RESIF, 1995a), RA (Réseau Accélérométrique Permanent, RESIF, 1995b), and RD (CEA/DASE Seismic Network, RESIF, 2018). FR stations which started their operations before and after 2019 are represented by open triangles (13 stations) and solid triangles (23 stations), respectively. The stations mentioned in this article are named on the map. Information on stations is provided in the Supplemental Materials (see Data and Resources). It is also available at seismology.resif.fr and earthquakes.bgs.ac.uk for the French and British networks, respectively.</figcaption></figure>
Although improvements in the seismic network in France, and particularly in NW France, now enable the detection of a greater number of events, the local magnitudes determined by BSCF-RéNaSS, $M_{LRéNaSS}$, remain significantly higher for northwestern France with respect to the rest of the country (Beucler et al., 2021). Similarly, while the computation method for the local magnitude determined by CEA-LDG, $M_{LLDG}$, remained the same over time (Duverger et al., 2021), the use of stations with a minimum epicentral distance of 90 km induces a noticeable bias towards high magnitudes. Recent studies investigating conversion laws between local magnitudes and $M_W$, or inferring $M_W$ through coda wave amplitude analysis, highlighted the necessity to consider both local, such as site effects, and regional properties (Cara et al., 2015; Denieul et al., 2015). The maritime context of the northwest of France (Atlantic Ocean and English Channel, Figure 1), inducing a strong microseismic signal of oceanic origin (Beucler et al., 2015), and the low attenuation of its paleozoic basement with respect to western Europe (Arroucau et al., 2006; Mayor et al., 2018b), are critical parameters. Failing to account for these regional properties can lead to an overestimation of local magnitudes (Beucler et al., 2021; Cara et al., 2015). In order to limit the uncertainties on the estimation of magnitudes and to enhance the seismic hazard assessment in northwestern France, a robust determination of $M_W$ is therefore needed (Laurendeau et al., 2022). Here, we attempt to determine $M_W$ through the assessment of the seismic moment, $M_0$, and the corner frequency, $f_c$, of low magnitude earthquakes in this region using spectral methods. From the subsequent $M_0 - f_c$ relation, we could derive a range for the Brune stress drop $\Delta\sigma$, a crucial parameter for estimating strong ground motions in hazard assessments. We pay a specific attention to the signal-to-noise (S/N) ratio, to the attenuating properties of the crust, and to local site effects, which are known to affect significantly the spectra of low-to-moderate magnitude earthquakes (e.g., Anderson & Hough, 1984; Abercrombie, 1997; Stork et al., 2014). The studied catalog consists of 108 earthquakes detected by the BCSF-RéNaSS (https://renass.unistra.fr/en/zones/france/) between 2015 and 2024, of $M_{LRéNaSS}$ between 2.0 and 5.3.

After introducing the data set of 1033 three-component seismograms recorded by the French and British permanent arrays, we describe their analysis in the frequency domain under the hypothesis of an omega-square source model. This analysis incorporates the determination of site attenuation and geometrical factor in order to correct the spectra for propagation effects. Additionally, two different absorption-band models are tested to evaluate the influence of regional attenuation on the source models. This approach provides a first estimation of an attenuation coefficient $\bar{\kappa}_0$ for each recording site. We then propose new $M_{LRéNaSS} - M_W$ and $M_{LLDG} - M_W$ scaling relationships as well as $M_0 - f_c$ scaling laws for the region. We further discuss our results in terms of Brune stress drops and the link between $M_0 - f_c$ and $M_L - M_W$ relations. Finally, we highlight the need for a more precise characterization of regional attenuation properties before inferring from these results a possible deviation from self-similarity between low and large magnitude earthquakes.

## DATA SET

The 108 earthquakes studied in this paper, with magnitudes $M_{LRéNaSS}$ ranging from 2.0 to 5.3, were detected and cataloged between 2015 and 2024 by the BCSF-RéNaSS (Figure 2). The initial data set (1033 seismograms) is composed of three-
component waveforms continuously recorded by 49 stations of the French (FR: Réseau Large Bande Permanent (RESIF, 1995a), RA: Réseau Accélérométrique Permanent (RESIF, 1995b) and RD: CEA/DASE Seismic Network (RESIF, 2018)) and British (GB: Great Britain Seismograph Network (British Geological Survey, 1970)) permanent networks displayed in Figure 1. The characteristics of the stations used are provided in the Supplemental Materials. The seismic array is mostly composed of broadband velocimeters, with a sampling rate of 100 samples per second (sps, 50 sps for RD stations) and four accelerometers (100 sps).

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Location of the 108 earthquakes studied in this paper with respect to the geographical distribution of the permanent seismic stations (as it is) in 2024. The size and color of the symbols represent the local magnitude determined by BCSF-RéNaSS and the hypocentral depth for each event, respectively. The stars correspond to the 2019-09-04 ($M_{L_{RéNaSS}}$ 2.0) Dinan (D) and the 2023-06-16 ($M_{L_{RéNaSS}}$ 5.3) La Laigne (L) earthquakes used as examples in this study. The stations are represented by green triangles.</figcaption></figure>
All 108 earthquakes are located at shallow depth (84% ≤ 20 km) in the crystalline basement of the Armorican Massif (either inland or in the continental shelf), in the northwestern part of the Massif Central, and under the sedimentary cover of the adjacent Paris and Aquitaine basins (Figure 2). The source parameters are provided in the Supplemental Materials, together with the distance range and number of stations used for each event. The data set combines 92 earthquakes of magnitude $M_{L_{RéNaSS}} \ge 3.0$ that occurred since January 2015, and 16 events of magnitude $2.0 \le M_{L_{RéNaSS}} < 3.0$ that occurred between 2017 and 2024. These later earthquakes were selected in the BCSF-RéNaSS bulletin from 2017 onwards in order to be able to work with at least one station at an epicentral distance lower than 100 km. Before 2017, the low density of the network did not allow us to include $M_{L_{RéNaSS}} < 3.0$ events in the database. The densification of the seismic network now allows us to accurately observe, for the first time in this SCR, the details of the source spectra, especially since 2015, when the FR stations started to be equipped, whenever possible, with boreholes seismometers buried at ~ 10 m depth (see the supplemental material). These sub-surface installations improve the S/N ratio of the signals, but seem sensitive to local site effects, which tend to minimize the energy at high frequencies. Despite this densification, the network remains constrained by the coastline, which truncates the azimuthal coverage of the events. Though located at more than 100 km, BGS (British Geological Survey) stations circumvent to some extent this difficulty for the earthquakes occurring in the northern part of the Armorican Massif.

## SPECTRAL METHOD

Considering the $\omega^{-2}$ source model defined by Aki (1967) and generalized by Brune (1970), Sato & Hirasawa (1973), Madariaga (1976), and Boatwright (1980), the values of $M_0$ and $f_c$ are determined for each event using a shear wave spectral fitting method derived from the overall approach of Satriano (2022)'s SourceSpec software. The algorithm describing the successive steps of the process is sketched in Figure 3. The following subsections briefly recall its theoretical background and describe its application to our data set, with some adaptations to the SourceSpec algorithm that we performed in the course of the study.

## Theoretical background

The S-wave displacement spectrum $S(f)$ recorded at the hypocentral distance $r$ is modeled using (e.g., Abercrombie, 1995; Ide et al., 2003)

$$S(f) = \frac{1}{G(r)} \times \frac{2 R_{\theta\phi}}{4\pi \rho V_S^3} \times \frac{M_0}{[1 + (f/f_c)^{nm}]^{1/m}} \times \exp[-\pi\kappa(r, f)f], \quad (2)$$

with $f$ the frequency, $1/G(r)$ a geometrical spreading factor depending on the hypocentral distance $r$, $\rho$ the density, $V_S$ the S-wave propagation velocity, $M_0$ the seismic moment, $f_c$ the corner frequency, $\kappa(r, f)$ the frequency- and path-dependent attenuation model, and $R_{\theta\phi}$ the radiation pattern coefficient. In (2) the elastic structures are considered identical at the source and at the stations. We use the average value $R_{\theta\phi} = 0.62$ (Boore & Boatwright, 1984) because the distribution of the stations
still allows for only a few focal mechanism determinations. The shape of the source spectrum is governed by the exponents $n$ and $m$ [for a review see, e.g., Abercrombie (2021)]. Here, we consider Brune (1970) model, with $n = 2$ and $m = 1$.

**Geometrical spreading** While the recent densification of the seismic network now allows us to investigate in more detail the seismicity of the Armorican Massif, the spacing between stations remains greater than 50 km. Such spacing requires the inclusion of distant stations (up to around 500 km) in the analysis, consequently leading to the use of a window length ($t = 60$ s) large enough to consider large hypocentral distances. Furthermore, when studying small earthquakes, the microseismic noise, combined with the uncertainties on the attenuation properties of the region, reduces the accuracy required for body wave inversion alone. The selected window therefore includes the entire signal beyond the S-wave arrival. When correcting the spectra with a simple geometrical factor $G(r) = r$, we observed, in agreement with previous studies, a significant increase of the $M_0$ assessment beyond an hypocentral distance $r_0$, also known as the crossover distance, that varied among the studied earthquakes. This increase occurs at hypocentral distances where $L_g$ waves have a stronger impact than the S-wave on the spectrum, leading to an overestimation of $M_0$. Following e.g., Street et al. (1975), Herrmann & Kijko (1983), Ordaz & Singh (1992), and Boatwright et al. (2002), we thus use a geometrical factor of the form

$$G(r) = \begin{cases} r & ; \quad r \le r_0 \\ r_0 (r/r_0)^{\gamma(f)} & ; \quad r > r_0 \end{cases} \quad (3a)$$

$$G(r) = \begin{cases} r & ; \quad r \le r_0 \\ r_0 (r/r_0)^{\gamma(f)} & ; \quad r > r_0 \end{cases} \quad (3b)$$

where $\gamma(f)$ is a frequency-dependent factor comprised between 0.5 and 0.7 (see eq. (6) in Boatwright et al., 2002), which accounts for the predominance of body-waves, $L_g$ and surface waves, at short ($r \le r_0$) and large ($r \ge r_0$) hypocentral distances, respectively. The value of the crossover distance $r_0$ is assessed for each event during a preliminary inversion step described in the subsection below.

**Regional attenuation and site effects** The frequency-dependent attenuation term $\kappa(r, f)$ greatly affects the results on source parameters (e.g., Anderson, 1986; Boore, 1986), in particular the corner frequency determination of low-to-moderate magnitude earthquakes. It is usually composed of two parts: a term that describes the frequency-dependent absorption along ray-path, which depends on the quality factor $Q^{-1}(f)$, and a station-dependent parameter which accounts for near site effects (e.g., Anderson & Hough, 1984; Abercrombie, 1997). This latter parameter mostly affects the high-frequency range of the spectrum, which is critical for small earthquakes. In the original version of SourceSpec, both terms are merged into a single frequency-independent parameter $t^*$, which is inverted for each station and event, together with $M_0$ and $f_c$. A preliminary inversion test of our data set showed that this procedure resulted in a large scatter of attenuation and $f_c$ values for a single event, induced by the strong covariance between both parameters. Indeed, the SourceSpec algorithm is mostly efficient for moderate and large magnitude earthquakes, for which the observed spectrum is mainly dependent on the source term, but, for smaller earthquakes, corner frequencies may belong to the high frequency range where attenuation decreases with
increasing frequency. In order to stabilize the inversion of $f_c$, we thus consider an absorption band model of the form

$$ \kappa(r, f) = \frac{r}{V_s} Q^{-1}(f) + \bar{\kappa}_0, \qquad (4) $$

with $Q^{-1}(f)$ provided by a priori regional models, and where its frequency-dependence is described by a frequency $f_0$, beyond

which a power law decrease of the attenuation at high frequencies is controlled by a coefficient $\alpha$

$$ Q^{-1}(f) = \begin{cases} Q_0^{-1} & ; \quad f \le f_0 \\ Q_0^{-1} (f/f_0)^{-\alpha} & ; \quad f > f_0. \end{cases} \qquad (5a) $$

$$ Q^{-1}(f) = \begin{cases} Q_0^{-1} & ; \quad f \le f_0 \\ Q_0^{-1} (f/f_0)^{-\alpha} & ; \quad f > f_0. \end{cases} \qquad (5b) $$

The near site parameter $\bar{\kappa}_0$ is assessed during a preliminary inversion test for each event and each station following the

procedure described in the following subsection. At the end of the analysis, the mean $\bar{\kappa}_0$ value of a given station corresponds

to the average of the values obtained for each event.

## Source spectrum assessment

The following paragraphs describe the different steps of the signal processing and source parameters assessment sketched in

Figure 3.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Overview of the algorithm used in this study, illustrating the sequential stages: data acquisition, relocation, and pre-processing. The algorithm then applies corrections for $G(r)$ and $\kappa(r, f)$ (initially without considering $\bar{\kappa}_0$), followed by a preliminary inversion to determine $\bar{\kappa}_0$. The final inversion incorporates $\bar{\kappa}_0$ into the correction stage, yielding the main parameters: $M_0$, $f_c$, $M_W$, and $\Delta\sigma$.</figcaption></figure>
**Relocation and crustal seismic velocities** Initially, all P- and S-wave phases are picked manually, and the events are relocated using the method of Haugmard (2016), which corresponds to step (1) in Figure 3. The details of this method and the results are presented in the Supplemental Materials. When the signal to noise ratio (S/N) is not sufficiently high to allow for an accurate manual picking of the S-wave arrival time ($t_S$), we assess this arrival time theoretically, with an estimate of $V_S$ obtained from a Chatelain diagram (Chatelain, 1978). This diagram, derived from the method of Wadati & Oki (1933), allows the determination of a $V_P/V_S$ ratio regardless of the uncertainties on the location and origin time of the events. For a given region and set of events, denoting by $\Delta t_{P_{ij}}$ and $\Delta t_{S_{ij}}$ the differences of the observed arrival times between two stations i and j for P- and S-waves, respectively, the value of the $V_P/V_S$ ratio is provided by the linear fit to the distribution $\Delta t_{S_{ij}} = f(\Delta t_{P_{ij}})$. Indeed, Figure 4 shows a linear distribution for all considered events and stations. A value $V_P/V_S = 1.70$ is obtained, which is consistent with the expected value for a felsic continental crust (Sui et al., 2022). The linearity of the distribution also confirms the relatively homogeneous structure of the Armorican Massif, with average crustal velocities $V_P = 6.05$ km s$^{-1}$, obtained from the location process presented in the Supplemental Materials, and, derived from this $V_P$ average and the previously established $V_P/V_S$ ratio, $V_S = 3.56$ km s$^{-1}$. These values are in a good agreement with previous estimates by Perrot et al. (2005), Arroucau (2006), Haugmard (2016) and Gaudot et al. (2020) in the region.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Chatelain diagram (Chatelain, 1978) used to determine the $V_P/V_S$ ratio (see text for explanation). TP and TS are the arrival times of P- and S-waves. The events which occurred before and after 2019 are represented by triangles and stars, respectively. The linear fit of the distribution provides $V_P/V_S = 1.70$, which corresponds to a Poisson ratio $\nu = 0.23$.</figcaption></figure>
**Pre-processing** For each earthquake, we select two 60 s time windows for each component of each station, starting 1 s before $t_S$ for the signal window, and 61 s before the origin time $t_0$ for the noise window (Figure 5a). After removing the instrumental response, a high-pass filter is applied to both windows in order to adjust the studied frequency range to the local magnitude of the events. The lowest bound of the filter is comprised between 0.05 Hz and 0.5 Hz for $M_{L_{RéNaSS}} < 4.5$ events, and between 0.01 Hz and 0.02 Hz beyond this magnitude.

<figure><img src="image_6.png" /></figure>

**Figure 5.** Examples of data processing for the 2019-09-04 ($M_{L_{RéNaSS}}$ 2.0) Dinan earthquake (left panels) and 2023-06-16 ($M_{L_{RéNaSS}}$ 5.3) La Laigne earthquake (right panels), recorded at stations GUEF and SDOF, respectively. The hypocentral distance equals 57 km in both examples. (a) Displacement waveforms of the three components. The sampled noise and signal windows are labelled N and S, respectively. (b) S/N ratio as a function of frequency. The vertical lines delimit the frequency band used in the source analysis, for which the S/N ratio is greater than the $S/N = 3.0$ threshold (horizontal line). (c) and (d) : corresponding displacement and acceleration source spectra, respectively. These source spectra were corrected for the geometrical factor and regional attenuation (see Source spectrum assessment).
Next, we determine the S-wave and noise displacement spectra using the Fourier transform of the seismograms in the frequency band [0.01–50] Hz, constrained by the 100 sps sampling rate of most of our stations. The S/N ratio is computed at each frequency for each component to select the optimal frequency range for each station. The S-wave spectra are then limited by two frequency bounds, $f_{min}$ and $f_{max}$, that exclude low and high frequencies where the S/N ratio is lower than 3.0 (Figure 5b). We also ensure that $f_{min}$ is not higher than 6 Hz and $f_{max}$ is not lower than 15 Hz. In order to maintain a continuous frequency range that ensures an optimal determination of $M_0$ and $f_c$, we only select stations where $f_{max}$ is at least 10 times higher than $f_{min}$ and where the average S/N ratio between these two bounds is greater than 4.0. Following these pre-processing stages, represented as step (2) in Figure 3, we finally calculate at each remaining station the root sum of squares $S(f)$ of the 3 components.

**Preliminary inversion** After this data pre-processing stage, an iterative procedure is implemented to determine the source parameters, $M_0$ and $f_c$, for the considered earthquakes. First, we need to correct the spectra for the geometrical factor $G(r)$ and attenuation parameter $\kappa(r, f)$ (see step (3) in Figure 3). Initially, only the regional attenuation term, which depends on the quality factor $Q^{-1}(f)$, is considered. Both parameters, introduced in the section, are described below. While the attenuating properties of the Armorican Massif are still poorly known, both macroseismic (Arroucau et al., 2006) and coda-wave (Mayor et al., 2018b) studies suggest that a low attenuation induces an overestimation of magnitudes in this region (Cara et al., 2015, 2016). Therefore, we built two frequency-dependent absorption band models $Q^{-1}(f)$.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Absorption band models $Q^{-1}(f)$ considered in the study. The QM18 model is constructed after the work of Mayor et al. (2018b). The alternative QSOB model is constructed after Boatwright et al. (2002) and Sargeant & Ottemöller (2009). The parameters of the models are listed in Table 1.</figcaption></figure>
<table><caption>Table 1
$Q^{-1}(f)$ models used in this study (Equation 5).</caption><thead><tr><th>References</th><th>Q<sub>0</sub></th><th>f<sub>0</sub> (Hz)</th><th>α</th></tr></thead><tbody><tr><td>Boatwright et al. (2002)</td><td>400</td><td>1.5</td><td>0.60</td></tr><tr><td>QSOB model</td><td>350</td><td>1.5</td><td>0.55</td></tr><tr><td>Sargeant and Ottemöller (2009)</td><td>300</td><td>1.5</td><td>0.50</td></tr><tr><td>Mayor et al. (2018b)</td><td>110 - 280</td><td>1.0</td><td>0.90</td></tr><tr><td>QM18 model</td><td>195</td><td>1.0</td><td>0.90</td></tr></tbody></table>

One model, hereafter named “QM18”, based on the work of Mayor et al. (2018b), is characterized by $Q_0 = 195$, $\alpha = 0.90$ and assumes a value of 1 Hz for $f_0$. The alternative model, named “QSOB”, is constructed to test the impact of a higher quality factor ($Q_0 = 350$) and smaller frequency dependence ($\alpha = 0.55$). It is designed after the works of Campillo & Plantet (1991), Chevrot & Cansi (1996), Boatwright et al. (2002), and Sargeant & Ottemöller (2009), which, though defined in different regions (western Europe, southeastern California, and southwestern Great Britain), share similar $Q_0$ ($350 \pm 50$) and $\alpha$ ($0.55 \pm 0.05$) values. The models of Campillo & Plantet (1991) and Chevrot & Cansi (1996) are comprised between the bounds defined by the other two models. Both QM18 and QSOB models are displayed in Figure 6 and their parameters are listed in Table 1. Note that QM18 model induces a weaker (resp. stronger) attenuation at high (resp. low) frequencies than QSOB model.

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Two examples of $M_0$ assessments as a function of hypocentral distance. (a) ($M_{LRéNaSS}$ 2.0) Dinan earthquake, (b) ($M_{LRéNaSS}$ 5.3) La Laigne earthquake. Black arrows point at the crossover distance estimates $r_0$. The orange and red stars represent $M_0$ values at distances smaller or larger than $r_0$, respectively.</figcaption></figure>

We carry out a first inversion for the $Q^{-1}(f)$ corrected spectra, which leads to preliminary estimates of $M_0$ and $f_c$ based on $\omega^{-2}$ fits of the displacement spectrum at each station. These values are then plotted against the hypocentral distance of the associated stations, from closest to farthest from the source (Figure 7). As expected, we observe an increase of $M_0$ at large distances, but we do not observe a systematic variation of $f_c$ assessments with hypocentral distance. We use the median of
$M_0$ values to identify the crossover distance $r_0$, for which $M_0$ starts to shift towards higher values. Working with increasing distances, we define $r^-$ and $r^+$ as the last and first values verifying $M_0(r^-) \le \text{median}(M_0)$, and $M_0(r^+) > \text{median}(M_0)$, respectively, and define $r_0 = (r^- + r^+)/2$ (Figure 7). For our catalog, the values of $r_0$ vary between 60 and 200 km (see Supplemental Materials), depending on the minimum hypocentral distance and distribution of the stations used for each earthquake.

Once the displacement spectra have been corrected for regional attenuation and geometrical spreading, by incorporating the $r_0$ estimates into equation 3, we proceed to a second inversion (step (4) in Figure 3) to carry out the assessment of the site effect parameter $\bar{\kappa}_0$ and then the final inversion of $M_0$ and $f_c$.

**Site effects assessment and final inversion** Anderson & Hough (1984) pointed out that any amplitude decrease in acceleration spectra at high frequencies highlighted the need to include $\kappa_0$ in the search of a $f_c$ not affected by local structural variations. Considering that the amplitude of the acceleration spectrum of an $\omega^{-2}$ source should tend to a constant at frequencies higher than $f_c$, we attribute any monotonous decrease in this spectrum at high frequency, once corrected from $G(r)$ and $Q^{-1}(f)$, to a site effect. The successive stages in the evaluation of this effect (step (5) in Figure 3) are sketched in Figure 8, with the ($M_{L\text{RéNaSS}}$ 5.3) La Laigne earthquake recorded at station RUFF as an example.

<figure><img src="image_9.png" /><figcaption>**Figure 8.** Steps in the determination and application of $\bar{\kappa}_0$ for the ($M_{L\text{RéNaSS}}$ 5.3) La Laigne earthquake at RUFF station. (a) Observed spectrum corrected for $G(r)$ and the $Q^{-1}(f)$ part of $\kappa(r, f)$ (red curve), and corresponding theoretical spectrum (black curve) obtained from the preliminary inversion. (b) Same as (a), with a purple arrow showing the search for $f_0$, the first frequency below the highest frequency of the spectrum, $f_{max}$, for which the observed spectrum goes above the theoretical spectrum. (c) For each frequency between $f_0$ and $f_{max}$, the difference between the observed and theoretical spectra is calculated. The difference corresponds to $\pi f \kappa_0$, which is used to determine the value of $\bar{\kappa}_0$ for each station (average of $\kappa_0$ values over all these frequencies). (d) Observed spectrum corrected for $G(r)$ and $\kappa(r, f)$, this time including $\bar{\kappa}_0$ (purple curve), and the theoretical spectrum obtained by the final inversion (grey curve).</figcaption></figure>
For a given event and a given station, once the observed spectrum $S_{obs}(f)$ has been corrected for $G(r)$ and $Q^{-1}(f)$, a preliminary inversion is performed and the theoretical source spectrum $S_{theo}(f)$, corresponding to the recovered values of $M_0$ and $f_c$, is calculated (Figure 8a). Starting from the highest frequency $f_{max}$ of the spectrum, we determine a frequency $f_0$ for which $S_{obs}(f_0)$ crosses the theoretical spectrum, and hence the domain where $S_{obs}(f)$ monotonically decreases (Figure 8b). According to equations (2) and (4), $\ln S_{theo}(f) - \ln S_{obs}(f) = \pi f \kappa_0$ (Figure 8c) and, since $\kappa_0$ is frequency-independent, its mean value $\bar{\kappa}_0$ is computed by averaging the quantity $\ln[S_{theo}(f)/S_{obs}(f)]/(\pi f)$ between $f_0$ and $f_{max}$. For a given station, the mean $\bar{\kappa}_0$ parameter corresponds to the average of the values obtained for each event. These mean values are listed in the Supplemental Materials for each station. They range from 0.002 s to 0.01 s, with an average of 0.005 ± 0.002 s.

A final inversion of the observed spectrum, corrected for $G(r)$, $Q^{-1}(f)$ and $\bar{\kappa}_0$ over the entire frequency range (step 3), with $\bar{\kappa}_0$ now included in $\kappa(r, f)$, followed by step (6) in Figure 3), is then performed to recover $M_0$, $f_c$, and their corresponding spectrum $S_{theo}(f)$ (Figure 8d). For a given earthquake $i$, the weight $w_{ij}$ assigned to the results obtained at station $j$ is defined by

$$w_{ij} = \left[ \frac{1}{N_j} \sum (\log_{10} S_{\text{obs}}(f) - \log_{10} S_{\text{theo}}(f))^2 \right]^{-1/2}, \quad (6)$$

where $N_j$ is the number of frequencies used at station $j$. For each earthquake, $M_0$ and $f_c$ are finally defined as the weighted average of the results obtained at the different stations.

## RESULTS

The seismic moment, associated $M_W$ (Equation 1), and corner frequency are computed for all 108 earthquakes using the spectral analysis described above. All results are listed in the Supplemental Materials (see Data and Resources).

### $M_L - M_W$ relationship

The moment magnitudes determined in this study vary between 1.7 and 5.0. They are compared with the corresponding $M_L$ computed by CEA-LDG and RéNaSS in Figure 9. The local magnitude determinations by CEA-LDG and BCSF-RéNaSS rely on procedures and models standardized for France, but they do not take into account the high microseismic noise, low attenuation and near site effects of northwestern France. This leads to an overestimation of the local magnitudes by both agencies in this region (Cara et al., 2015; Beucler et al., 2021). Taking these regional specificities into account for the determination of $M_W$ enables us to characterize the regional seismicity more accurately and to establish new $M_L - M_W$ conversion laws.

The $M_L - M_W$ distributions display a greater dispersion for $M_{L\text{RéNaSS}}$ than for $M_{L\text{LDG}}$. Until 2019, before the improvement of the seismic network in northwestern France, $M_{L\text{RéNaSS}}$ estimates are hardly correlated with and almost systematically higher than $M_W$ (Figure 9). On the other hand, whatever the attenuation model, the distribution of events occurring between 2020 and 2024 shows a linear correlation between $M_{L\text{RéNaSS}}$ and $M_W$.
$$M_{L\text{RéNaSS}} = 1.13 (\pm 0.1) M_{W} - 0.21 (\pm 0.1), \qquad (7)$$

with a regression coefficient greater than 0.96.

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Left: $M_L - M_W$ relationship for the BCSF-RéNaSS local magnitude, compared to the moment magnitude $M_W$ determined in this study, for both (a) QM18 and (c) QSOB attenuation models. Right: $M_L - M_W$ relationship for the CEA-LDG local magnitude, compared to the moment magnitude $M_W$ determined in this study, for both (b) QM18 and (d) QSOB models. The green and red triangles differentiate the earthquakes before and after 2019, respectively. The orange lines represent the $M_L - M_W$ conversion laws from this study determined using (a) and (c) only the red triangles and (b) and (d) both red and green triangles. The correlation coefficient $r$ is specified in the top left-hand corner for each conversion law. On the right, the dashed green lines correspond to the $M_{L\text{LDG}} - M_W$ conversion law from Cara et al. (2015). All results are listed in the Supplemental Materials (see Data and Resources).</figcaption></figure>
In the case of $M_{LLDG}$, the distributions of the magnitudes assessed before and after 2019 do not differ significantly, because their estimates rely on a procedure and on station locations which remained almost unchanged in the last 30 years (Duverger et al., 2021). Consequently, the $M_{LLDG}-M_W$ relationship displays a constant linear correlation from 2015 to 2024 for the entire observed magnitude range for both attenuation models:

$$M_{LLDG} = 1.20 (\pm 0.1) M_W - 0.19 (\pm 0.2), \qquad (8)$$

with a regression coefficient also greater than 0.96.

Cara et al. (2015) proposed that, at the scale of the French territory, the $M_{LLDG}-M_W$ relationship is subdivided into three different conversion laws depending on the range of magnitude, represented by the green lines in panels (b) and (d) of Figure 9. According to our results, such different laws are not necessary for northwestern France, where the single linear regression (8) holds for $M_W$ in the range [1.7 – 5.0], or, equivalently, [1.9 – 5.8] for $M_{LLDG}$.

## Source scaling relationship

The files that contain the values of $M_0$ and $f_c$ obtained for each earthquake using the QM18 and QSOB attenuation models are provided as Supplemental Material. The corresponding $M_0-f_c$ relationships are displayed in panels (a) and (b) of Figure 10. Power law distributions $M_0 \propto f_c^{-d}$ are observed in both cases for $1.7 \le M_W \le 5.0$. The coefficient $d$ is estimated through a regression analysis in which the weight assigned to each event is the number of stations used for its source determination. Several regressions are performed, with an acceptation of events based on the standard deviation $\epsilon_{f_c}$. The regressions represented by black lines in Figure 10 (a) and (b) were obtained using events with $\epsilon_{f_c}/f_c \le 0.6$, which covers at least two thirds of the studied earthquakes. They correspond to the power laws

$$M_0(f_c) \propto \begin{cases} f_c^{-4.0} & ; \text{ (QM18 model)} \\ f_c^{-3.3} & ; \text{ (QSOB model)} \end{cases} \qquad (9a)$$

with errors on the coefficients lower than 0.5%. In order to interpret the results of $M_0$ and $f_c$ in term of stress drops $\Delta\sigma$ and source size, we consider a circular earthquake rupture, for which (Eshelby, 1957)

$$\Delta\sigma = \frac{7 M_0}{16 a^3}, \qquad (10)$$

where the source radius $a$ relates to the corner frequency $f_c$ through

$$a = k \frac{V_S}{f_c}. \qquad (11)$$

The value of $k$ depends on the rupture geometry and velocity. It may vary between 0.21 (Madariaga, 1976) and 0.37 (Brune, 1970). The following results are presented and discussed using $k = 0.37$ (Brune's model). They may be scaled to other $k$ values
248

if necessary, for instance by factors of 0.57 for *a* and 5.47 for Δσ when translated to Madariaga's model. The source radius values of the studied earthquakes are shown as a function of *M*₀ in panels (c) and (d) of Figure 10.

249

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

**Figure 10.** Top: seismic moment *M*₀ and corresponding moment magnitude *M*<sub>w</sub> as a function of the corner frequency *f*<sub>c</sub> for both (a) QM18 and (b) QSOB attenuation models. Bottom: associated source radius *a* as a function of *M*₀ for both (c) QM18 and (d) QSOB models (note that *a* is increasing downwards and that *M*₀ is on the y-axis and x-axis for panels (a) and (b) and panels (c) and (d), respectively). The color of the symbols represents the relative error on *f*<sub>c</sub> while their size corresponds to the number of stations used for the inversion. The stars highlight the *M*<sub>w</sub> 1.7 Dinan (D) and the *M*<sub>w</sub> 5.0 La Laigne (L) earthquakes and the triangles show the aftershocks of the 2019-06-21 Layon (*M*<sub>w</sub> 4.2) earthquake. The events with less than 3 stations or *e*<sub>f<sub>c</sub></sub>/f<sub>c</sub> > 1.0, which were not accepted for the regression, are represented by open black circles. The black and red lines correspond to the data correlation and the *a*-*M*<sub>0</sub> relationships according to Equation 10 for different stress drop values, respectively. The two grey areas between the dashed lines (lighter) and the dotted lines (darker) represent the empirical scaling laws constructed by Leonard (2014) for interplate regions and SCRs, respectively. Panels (c) and (d) display the distribution of stress drop values for QM18 and QSOB models.
In the case of $f_c = 1$ Hz, we obtain $M_0$ values in the range [2.8 - 4.9] $\times 10^{16}$ Nm, corresponding to $M_W = 5.0 \pm 0.1$. These values coincide with the results obtained for the La Laigne earthquake ($M_W$ 5.0) for both the QM18 and QSOB models. Based on this event, for which the results are not significantly affected by the choice of attenuation model, and introducing the value of $V_S$ determined previously ($V_S$ = 3.56 km/s) into equation (11), we find $a(f_c = 1 \text{ Hz}) = 1317$ m. Then, using the corresponding $M_0$ values and equation (10), we obtain stress drops in the range $5.4 \le \Delta\sigma \le 9.4$ MPa.

A noticeable difference is observed between the slopes of the regressions between $f_c$ – or equivalent source radius – and $M_0$, obtained for models QM18 and QSOB (Figure 10 and Equations (9)). Indeed, the attenuation models, by controlling the slopes of the observed spectra whether in displacement or acceleration (Figure 5c and 5d), have a significant impact on the determination of the corner frequency $f_c$, equivalent source radius, and stress drop (Abercrombie, 2021). For a given seismic moment, the QM18 model, with a power $d = 4.0$, leads to lower corner frequencies than the QSOB model ($d = 3.3$). Assigning the superscripts QM18 and QSOB to the results obtained with the respective models, we observe that, on the average, $f_c^{QM18}/f_c^{QSOB} \sim 0.6$, or, equivalently, $a^{QM18}/a^{QSOB} \sim 1.7$, leading to $\Delta\sigma^{QM18}/\Delta\sigma^{QSOB} \sim 0.2$.

## DISCUSSION

The uncertainties of the spectral analysis applied in this article stem from the modeling of the source and from the uncertainties on site effects and attenuation properties of the region.

### Regional parameters

We considered a window length of 60 s to acknowledge for the largest epicentral distances (up to 500 km), which led to the inclusion of $L_g$ and surface waves in the considered signal. Incidentally, the increase of $M_0$ after the crossover distance $r_0$, due to the presence of these phases, was corrected by incorporating the geometrical factor $G(r)$ in the inversion (Figure 7). Despite this correction, an increase is still apparent, which can imply that the frequency-dependent factor $\gamma(f)$, defined by Boatwright et al. (2002), is not adapted to the studied region. When we attempted to consider the $M_0(r)$ values only for $r \le r_0$, only a few stations were retained for the inversion, giving inconclusive results with no noticeable information on the $M_0-f_c$ relationship. Further analysis is thus necessary to determine whether adapting $\gamma(f)$ would allow us to overcome this difficulty.

For the first time in the region, we provide a comprehensive estimate of $\overline{\kappa}_0$ for each station of the network. Our estimates, in the range [0.002 - 0.01], coincide with the results of Pilz et al. (2019, Figure 5) and Mayor et al. (2018a, Figure 3) in this region. However, we still observe a decrease of the $\overline{\kappa}_0$ corrected acceleration spectra at high frequencies (Figure 8), mainly for borehole stations, that we attribute to unmodeled local site effects (Hollender et al., 2020; Castellaro et al., 2022). Ongoing studies aim to improve site characterization in the region, including a systematic assessment of the uppermost 30 m beneath the stations ($V_{S30}$).
Similarly, the attenuation model strongly influences the determination of $f_c$. In this study, we consider two attenuation models. While the results obtained with the QSOB model are close to a constant stress drop source model, i.e. $M_0 \propto f_c^{-3}$ (Madariaga, 2011), the QM18 model would imply a strong increase of $\Delta\sigma$ with $M_0$. Given the large uncertainties on $Q^{-1}(f)$, a plausible interpretation can be that the QM18 model was developed based on research conducted before 2018, prior to the network densification (Mayor et al., 2018b). The resulting limited data set may have hindered an accurate modeling of attenuation, especially at high frequencies. The construction of a regional attenuation model is thus mandatory to refine the $M_0 - f_c$ relations obtained in this study.

## Earthquake source scaling

The data set used in this study is strongly constrained by the density of the seismological network and by the available bandwidth of the instruments. Firstly, only a few near-field stations (hypocentral distances smaller than 50 km) were available, requiring the use of far-field stations. Furthermore, the sensors within the permanent network, which constrain the studied frequency range (Boore, 1986), mainly operate at a sampling rate of 100 sps, thereby limiting the useful frequency band to [0.01–50] Hz at most, regardless of the microseismic noise. Considering that a corner frequency might be accurately determined down to ~80% of the Nyquist frequency (50 Hz), we could theoretically go down to $M_W \sim 1.5$. However, our attempts to include events with $M_W < 1.7$ failed due to the level of the microseismic noise, which could not be overcome by the lack of near-field stations.

The Brune stress drops assessed in this study range between [0.1–10.3] MPa and [0.3–54.3] MPa for the QM18 and QSOB attenuation models, respectively, with average values $\overline{\Delta\sigma}^{QM18} = 1.8$ MPa and $\overline{\Delta\sigma}^{QSOB} = 10.0$ MPa. The values obtained with the QM18 model are consistent with those observed in an interplate context by, e.g., Abercrombie & Rice (2005) and Boyd et al. (2017), while the higher values obtained with the QSOB model are more representative of intraplate regions (e.g., Yamada et al., 2005; Viegas et al., 2010; Boyd et al., 2017), in particular the Central United States (Huang et al., 2017).

Numerous papers on earthquake scaling relations have been published during the last decades (e.g., Wells & Coppersmith, 1994; Mai & Beroza, 2000; Leonard, 2014; Thingbaijam et al., 2017). In this study, we select Leonard (2014)'s empirical scaling laws between seismic moment, fault dimensions and average slip, for earthquakes occuring in both interplate regions and SCRs. Considering a relation of the form $\log_{10}M_0 = c + 1.5 \log_{10}a$ between seismic moment and the ruptured area $a$ of the events, he obtains $5.69 \le c \le 6.47$, mean value $\bar{c} = 6.08$, and $6.21 \le c \le 6.51$, $\bar{c} = 6.37$, for interplate and SCRs, respectively. These values are valid for a strike-slip motion, which is the mostly representative mechanism in the Armorican Massif (Mazzotti et al., 2021; Alloncle, 2024). For a given seismic moment, they indicate a smaller rupture area in SCRs, leading, on average, to a stress drop about twice higher than in interplate regions. The domains of Brune's stress drops corresponding to Leonard (2014)'s relations are plotted in Figure 10 under the hypothesis of a circular fault. The vast majority of the stress drops derived for the QM18 model correspond to the lower range of Leonard (2014)'s values, while a better agreement is
observed between the latter and the values obtained for the QSOB model. The correlation coefficients of the bi-logarithmic $M_0-f_c$ regressions determined for the QM18 and QSOB models are both equal to 0.86. However, as noted previously, there is a noticeable difference between the slopes of the $M_0-f_c$ relations for both attenuation models. The $M_0 \propto f_c^{-3.3}$ relation observed for the QSOB model differs by only 3% from the -3 coefficient expected for self-similar source processes. Conversely, the -4 value obtained with the QM18 model would imply that the stress drops can vary by several orders of magnitude between $M_W \sim 2$ and $M_W \sim 5$ events. Given the large uncertainties on the attenuation properties of the region, we initially attributed this apparent $M_0-f_c$ behavior to the stronger $Q^{-1}$ dependence of the QM18 model at high frequencies (Figure 6 and Table 1), but the following discussion highlights another possible interpretation.

## Link between $M_L - M_W$ and $M_0 - f_c$ relations

Here, we compare the relations obtained between local magnitudes and $M_W$ with the $M_0 - f_c$ power laws, and discuss their link in terms of attenuation and stress drop. For self-similar events and a purely elastic propagation medium between sources and stations, local and moment magnitudes should be equal (e.g., Deichmann, 2017). However, a number of studies (e.g., Hanks & Boore, 1984; Cara et al., 2015; Munafó et al., 2016; Deichmann, 2017) found that the ratio between local magnitudes and $M_W$ increases in the case of small earthquakes, tending to ~1.5. Synthetic tests and theoretical developments performed by Munafó et al. (2016) and Deichmann (2017) demonstrated that this 1.5 proportionality for small earthquakes comes from the definitions of magnitudes in terms of signal amplitude and duration in an attenuating medium. Deichmann (2017) also demonstrated that, even in the case of a purely elastic medium, a variation of stress drop with seismic moment translates into a proportionality between local and moment magnitudes which differs from unity.

Building on the work of Bethmann et al. (2011, 2012) and Deichmann (2017), it is possible to relate our $M_0 - f_c$ relationship to the $M_L - M_W$ relationships (see Appendix A) for each attenuation model. Knowing that $M_0 \propto f_c^{-d}$, one can establish $M_L = aM_W + \text{cst}$. For the QM18 model, $d=4$, and for the QSOB model, $d=3.3$. For QSOB, where d is close to 3, we should observe self-similarity, in which case $M_L = M_W + \text{cst}$, and the slope should be equal to 1. However, this does not correspond to our observations (1.12 for $M_{L\text{RéNaSS}}$ and 1.21 for $M_{L\text{LDG}}$).

Thus, if QSOB is an appropriate attenuation model for the region and self-similarity is indeed observed, this variation in slope is due to the attenuation considered in the calculation of $M_L$. On the other hand, if the QM18 model is correct, with $d=4$, it would imply a strong variation in $\Delta\sigma$ with $M_W$, which should give a slope of $M_L = 1.125M_W + \text{cst}$. This tends to indicate that the $M_0 - f_c$ and $M_L - M_W$ regressions are consistent in the case of the RéNaSS.

Regarding the $M_{L\text{LDG}} - M_W$ relationships, the slope (1.19 for QM18 and 1.21 for QSOB) can only be explained by the use of a strong attenuation to compute $M_{L\text{LDG}}$. Indeed, the LDG uses $Q_0 = 128$, which is 30% lower than the QM18 model ($Q_0 = 195$) and 60% lower than the QSOB model ($Q_0 = 350$), corresponding to an attenuation 30% stronger than the QM18 model (and
60% stronger than QSOB). Additionally, the $M_{LLDG}$ determination includes stations more than 95 km away, which captures signals from $L_g$ and surface waves, which is not always the case for the RéNaSS.

While we previously discussed that, in the context of self-similarity and comparison with stress drop from other SCR studies, the QSOB model returns results closer to expectations, this contrasts with observations when considering the link between the $M_L - M_W$ relationships and the $M_0 - f_c$ power laws based on the work of Deichmann (2017). Specifically, we observe more consistent results here for the $M_{LRéNaSS} - M_W$ relationship for the QM18 model.

Among these two hypotheses, and given that the $M_{LRéNaSS} - M_W$ conversion law has only been applicable since 2020, more data is needed to draw definitive conclusions regarding self-similarity.

## CONCLUSION

The objective of this study was to contribute to the determination of the moment magnitude, $M_W$, corner frequency $f_c$, and associated stress drop $\Delta\sigma$ of low-to-moderate earthquakes, in order to better understand the seismicity of SCRs, such as northwestern France. In agreement with previous studies, our results highlight the influence of the regional characteristics on the determination of the moment magnitude and source parameters. Although the seismological network significantly improved in northwestern France since the end of 2019, the S/N ratio associated with the available seismic sensors remains a critical constraint on the assessment of source parameters in the context of low magnitude earthquakes occurring in a passive margin region, by limiting the accepted bandwidth in the inversion process. With the current network, any magnitude greater than 1.7 can be accurately determined.

The $M_{LLDG} - M_W$ conversion law established in this study for $1.7 \le M_W \le 5.0$: $M_{LLDG} \propto 1.20 M_W - 0.19$ ($r = 0.97$), will be useful in future hazard assessments. When taking into account the low attenuation properties of the region in a model based on a sufficiently dense data network, such as the QSOB model, we obtain satisfactory constraints on $M_0$, $f_c$ and associated stress drop. The $M_0-f_c$ relationship, determined for the first time in northwestern France, is well described by a power law within the $1.7 - 5.0 M_W$ range, with a Brune stress drop of the order of 10 MPa on the average, which compares similarly to the results obtained in other intraplate regions. When combined with the $M_0 - f_c$ relations, the regressions obtained between local magnitudes and $M_W$ indicate that the slope of the $M_{LLDG} - M_W$ relation is governed by attenuation effects, while the $M_{LRéNaSS} - M_W$ relation may be explained either by attenuation effects or by an increase of $\Delta\sigma$ with $M_0$. However, a more precise description of the attenuation is required before concluding about a possible deviation from self-similarity between low and large magnitude earthquakes.

## DATA AND RESOURCES

The EPOS-France data used in the work presented here, can be downloaded from FDSN webservices through https://seismology.resif.fr, under the network code FR (RESIF, 1995a), RA (RESIF, 1995b) and RD (RESIF, 2018) (last accessed March 2025). Data from the BGS (British Geological Survey, 1970) can be downloaded from the IRIS Wilber 3 system (http://ds.iris.edu/)
wilber3/find_event, last accessed March 2025). The information on the stations can be found on seismology.resif.fr and earthquakes.bgs.ac.uk.

Supplemental Material for this article includes information on the networks and stations, the relocation process, and the results of the source inversions for each earthquake in the studied catalog.

## DECLARATION OF COMPETING INTERESTS

The authors acknowledge there are no conflicts of interest recorded.

## ACKNOWLEDGMENTS

The authors acknowledge the Epos-France group and the British Geological Survey for the data distribution. Epos-France is a Research Infrastructure (RI) managed by the CNRS-INSU. It is a consortium of nineteen French research organisations and institutions, included in the roadmap of the Ministry of Higher Education, Research and Innovation. Epos-France RI is also supported by the Ministry of Ecological Transition. This work was made possible by the manual discrimination performed by BCSF-RéNaSS (R. Dretzen, S. Lambotte, M. Grunberg and C. Perrin https://renass.unistra.fr/en/zones/france/) and CEA/DASE (https://www-dase.cea.fr/evenement/syntheses_resultat.php?lang=en) operators. Thanks to Claudio Satriano for providing to the community an open access to SourceSpec code. In depth reviews and constructive comments by the guest editor Annemarie Baltay and two anonymous reviewers were very helpful to improve a first version of the manuscript.

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## APPENDIX A : Link between $M_0 - f_c$ and $M_L - M_W$ relations in the elastic case

Deichmann (2017) showed that, in the purely elastic case and considering a constant rupture velocity, the definitions of magnitudes $M_L$ and $M_W$ imply

$$M_L = \frac{2}{3} \log_{10} M_0 + \frac{1}{3} \log_{10} \Delta\sigma + Cst. \quad (A1)$$

In the following expressions, the value of $Cst$ varies from one equation to another, but it is always independent of $M_L$, $M_0$, $\Delta\sigma$, $f_c$, and $M_W$. Using the expressions (10) and (11) for $\Delta\sigma$,

$$\log_{10} \Delta\sigma = \log_{10} M_0 + 3 \log_{10} f_c + Cst, \quad (A2)$$

and equation (A1) becomes

$$M_L = \log_{10} M_0 + \log_{10} f_c + Cst. \quad (A3)$$

If the seismic moment and the corner frequency are related through a scaling law

$$M_0 \propto f_c^{-d}, \quad \text{i.e.,} \quad \log_{10} f_c = -\frac{1}{d} \log_{10} M_0 + Cst,$$

equation (A3) may be rewritten

$$\begin{aligned} M_L &= \left(1 - \frac{1}{d}\right) \log_{10} M_0 + Cst \\ &= \frac{3}{2} \left(1 - \frac{1}{d}\right) M_W + Cst, \end{aligned} \quad (A4)$$

and, according to (A2), the variations of $\Delta\sigma$ with seismic moment and magnitude take the equivalent forms

$$\Delta\sigma \propto M_0^{(1-3/d)}$$

$$\begin{aligned} \log_{10} \Delta\sigma &= \frac{3}{2} (1 - 3/d) M_W + Cst \\ &= \frac{d-3}{d-1} M_L + Cst. \end{aligned} \quad (A5)$$

The coefficient $\frac{3}{2}(1 - 3/d)$ is the parameter $m$ in Deichmann (2017, equation 20). In the case of self-similarity, $d = 3$, $\Delta\sigma$ is equal to a constant, and $M_L = M_W$.