# Seismogenic and rheological behaviours from time-dependent analysis of earthquake depth distribution in the Corinth Rift

A Septier, J Déverchère, J Perrot, A Renouard

► To cite this version:

A Septier, J Déverchère, J Perrot, A Renouard. Seismogenic and rheological behaviours from time-dependent analysis of earthquake depth distribution in the Corinth Rift. Tectonophysics, 2026, 919, pp.231008. 10.1016/j.tecto.2025.231008. hal-05442416

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# Seismogenic and rheological behaviours from time-dependent analysis of earthquake depth distribution in the Corinth Rift

A. Septier<sup>a,c ID,*</sup>, J. Déverchère<sup>a ID</sup>, J. Perrot<sup>a ID</sup>, A. Renouard<sup>b ID</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

Dataset link: https://nfo.crlab.eu/earthquakes_catalogs

**Keywords:**

Seismology

Crust yield strength

Rheology

Geology

## ABSTRACT

Understanding earthquake depth distribution is critical for improving seismogenesis models. While the spatiotemporal pattern of earthquakes is well studied, transient changes in depth distribution remain poorly explored. In this study, we investigate how crustal rheological parameters influence the depth of earthquakes through time, focusing on the Corinth rift, a well-monitored region experiencing a high-level seismic activity in a homogeneous extensional stress field.

To calculate crustal yield strength profiles, we compile geophysical and geological data, including heat flow, rock compositions and properties, Moho depth and strain rate. These estimates are then compared to a high-quality 11-year seismic catalogue of the region. An inversion approach is applied to identify crustal layers associated with persistent versus sporadic seismicity defined here instead of the conventional background versus clustered seismicity.

Our time analysis reveals that the persistent seismicity nicely matches the theoretical brittle-ductile transition and allows us to confidently define the seismogenic thickness, while sporadic seismicity is clustered at depths associated with swarm occurrences. Both distributions are subject to kilometre-scale changes after magnitude 4.0 – 5.5 earthquakes, evidencing a relaxation process even after moderate magnitude events. We conclude that in specific case studies aiming to compare depth distribution and yield strength in the crust, the application of declustering methods may not be optimal for examining the potential rheological controls on earthquake depth distribution and their temporal variations. Instead, the analysis of persistent and sporadic seismicity defined in this study is more accurate and reliable than a declustering approach and offers new and valuable insights for this comparison.

## 1. Introduction

It has long been known that the maximum depth of earthquakes can provide a reliable estimate of brittle-ductile transitions within the continental crust (see, e.g., Sibson (1982), Scholz (2019) and Molnar (2020) and references therein). Since the nineties, numerous studies have conducted statistical analyses of earthquake depth distributions to verify whether their shapes correspond to predicted yield rheological profiles (Lamontagne and Ranalli, 1996; Aldersons et al., 2003; Albaric et al., 2009; Muluneh et al., 2021; Rodriguez Piceda et al., 2022). Although the strength of the continental crust may locally change and trigger time-dependent fluid-driven seismicity, for example, within inherited viscous shear zones (Fagereng et al., 2024), this correspondence appears to be good across a wide range of geological settings and deformation fields. However, several factors, such as large

hypocentral uncertainties, shortness of earthquake catalogues, transient seismic activity, and poor knowledge of the thermal and rheological properties of the crust, have often limited, altered, or even hindered this comparison. Nevertheless, recent advances in detection and location techniques (Zhu and Beroza, 2018; Kalkan, 2016; Liu et al., 2020; Zhang et al., 2019), seismic networks (Chouliaras et al., 2013), and understanding of geological processes (Zeng et al., 2022; Chiaraluce et al., 2022; Rodriguez Piceda et al., 2022) are beginning to address some of these limitations, allowing greater accuracy in locating and classifying events such as natural earthquakes, mainshocks and aftershocks.

A standard approach to comparing the depth distribution profile of earthquake hypocentres with the predicted yield strength envelope is to apply a declustering method to earthquake catalogues. This method isolates background seismicity, assumed to reflect the long-term strength

<footnote><sup>a</sup>Geo-Ocean, Univ Brest, CNRS, Ifremer, UMR6538, Plouzané, France</footnote>

<footnote><sup>b</sup>Department of Civil & Environmental Engineering, Imperial College London, UK</footnote>

<footnote><sup>c</sup>Department of Earth, Ocean and Ecological Sciences, University of Liverpool, UK</footnote>

<footnote>* Corresponding author at: Department of Earth, Ocean and Ecological Sciences, University of Liverpool, UK.
E-mail addresses: antoine.septier@univ-brest.fr (A. Septier), jacques.deverchere@univ-brest.fr (J. Déverchère), jperrot@univ-brest.fr (J. Perrot), a.renouard@imperial.ac.uk (A. Renouard).</footnote>
of the crust, from disturbances associated with transient seismic activity (Albaric et al., 2009; Burov, 2011; Radziminovich, 2022) or changes in seismic coupling (Marsan et al., 2017). In practice, the extraction of the background component is based on the assumption that the occurrence of background earthquakes retains the attributes of an independent stationary Poisson model and can be recovered by eliminating dependent transient earthquakes, such as sequences of mainshocks and aftershocks, swarms, or other related phenomena (Mogi, 1963; Parsons, 2008). The potential link between earthquakes is generally estimated by calculating spatial and temporal distances, sometimes modulated by the magnitude (Utsu et al., 1995).

Statistical criteria strongly guide the extraction of background seismicity. Information on earthquake depths, although rarely included due to significant location errors, is used to calculate inter-event distances or probability density functions (Guo et al., 2015). As a result, depth information is not commonly exploited to assess the rheological influence on the depth distribution of earthquakes.

Yet, with good location of hypocentres and integration of location uncertainties into the representation, more in-depth use of earthquake depths could provide interesting information to explore and to complement earthquake-based rheological studies. Assuming that crustal yield strength remains constant under stationary stress and heat, i.e. for a sufficiently long period, yet to be determined, during which no earthquakes occur due to transient variations in stress, we may suggest that earthquakes triggered by rheological and tectonic influences should occur more frequently at certain depths. This hypothesis seems acceptable since regular observation of seismic crises occurring at specific depths and rates suggests the existence of persistent material heterogeneities in the Earth's crust (Miyatake, 2014; Hu et al., 2009).

Based on this premise and a reliable distribution of earthquake depths, this article proposes to identify specific crustal layers based on two types of seismic activity: (1) persistent activity that remains in the same layers over time and (2) sporadic activity that does not occur constantly over a given time span. We then explore their potential link with crustal rheological parameters and mainshock occurrences. It is important to note that the distinction between persistent and sporadic seismic activity differs from the classification of earthquakes resulting from declustering procedures (Console et al., 2010). The declustering algorithms classify individual earthquakes according to their statistical properties. Our objective here is to highlight crustal mechanical weaknesses at a given depth, which we assume to be identified by a rheology-related persistent depth distribution of earthquakes.

We apply our approach to a high-quality decadal catalogue (Corinth Rift Laboratory Team and RESIF Datacenter, 2013) of earthquakes in the Corinth Gulf, Greece, a major active rift system worldwide (Bourouis and Cornet, 2009). We compare the time-depth distribution obtained in the two categories (persistent and sporadic) with a rheological model we constructed for the studied zone. We simplify the rheological modelling to a one-dimensional calculation by carefully selecting crustal geophysical parameters (heat flow, Moho depth, and rock material). We finally compare the results with a declustered catalogue using the Reasenberg method (Reasenberg, 1985). Thanks to this comparison, we discuss potential correlations between our approach and the standard declustering method used. We also discuss promising contributions from a rheological interpretation of earthquake distributions to better understand the link between earthquake concentration at a certain depth and long-term rheological behaviour.

## 2. Corinth rift overview and seismicity dataset

The Corinth Gulf is a continental rift characterised by high seismicity rates and localised extensional deformation (Armijo et al., 1996; Bourouis and Cornet, 2009). It is located between the Peloponnese and continental Greece, is orientated N100° E, and extends over more than 100 km.

The central graben is bounded by young (less than 2 million years) normal fault segments with a roughly E–W orientation (Fig. 1). The rift is thought to be the result of the gravitational instability of a Neogene mountain belt and a relatively fast thinning of the lithosphere. It has often experienced sequences of swarms and aftershocks, particularly after low- to moderate-magnitude earthquakes. Earthquakes of magnitudes greater than 5 are rare in recent catalogues, as documented by Mesimeri et al. (2019), Bernard et al. (2006) and Rohais and Moretti (2016). Deep fluids are suspected to control the space–time evolution of the seismicity through the migration of pore pressure disturbances and a high connectivity between different areas of the rift (Lambotte et al., 2014).

We manually select our dataset geographically by choosing a spatial polygon that minimises horizontal variations in geophysical and geological parameters (e.g. geological depth layers, heat flow, deformation direction) using information from Vallius et al. (2022), Taylor et al. (2011) and Zelt et al. (2005). This involves ensuring that the thickness of geological layers varies by less than 5 km and that the same type of rock is present (see the rock layer model in Fig. 4). The primary purpose of this approach is to isolate and distinguish the vertical rheological influence on the depth distribution of earthquakes over time (Fig. 1).

The depth distribution of earthquakes contained in the selected area of the HUSN catalogue (Evangelidis et al., 2021) is represented using a Gaussian density function (see Fig. 3 for details of the approach and Fig. 2 for the results). Unlike a histogram, this representation gives an absolute maximum, which cannot be modified by shifting the bins or changing their width. It also includes hypocentre uncertainty, providing a representation that is not biased by human decisions regarding bin sizes or edge positions (see toy example in Appendix Figure 14), and which takes into account the value of each data point based on its uncertainty, rather than treating all data points equally, regardless of their uncertainty, as in standard histograms. Appendix Figure 14 illustrates that the size and range of the histogram bars chosen can significantly influence the appearance and position of the maxima, potentially causing bias. Additionally, taking individual uncertainties into account rather than applying a uniform uncertainty to the entire catalogue, such as through bin sizing, reduces information loss. The use of a Gaussian density function in this context is particularly noteworthy, as it provides a unique result without arbitrary choices, ensuring that each measurement is accounted for appropriately.

According to Fig. 2, the depth distribution of earthquakes is mainly (98% of the total) between 0 and 15 km with a maximum depth of 38.9 km, where the average depth uncertainty is less than 2.5 km, for the area selected and the whole HSUN catalogue. The uncertainty regarding depth is represented using the entire HUSN catalogue, as shown in Fig. 1, in order to illustrate the quality of the locations of the entire catalogue. We note that there are too few earthquakes located below 15 km to establish reliable statistics on the uncertainty of their location in the selected area.

We decided to present data up to 40 km, as the rheological envelope can often show a second maximum at a shallower depth before the Moho transition. The fact that no earthquakes appear below 20 km is an important information that supports the rheological model we propose.

Two maxima are identified in the kernel density curve, one at ~5 km and the other at ~10 km, followed by an exponential decrease in earthquake occurrence. Earthquakes are almost absent beyond 20 km.

## 3. Methodology

For this study, we analyse a selected area of the Corinth Rift (see hatched area, Fig. 1) using time and depth windows without incorporating the horizontal dimension (latitude, longitude) into the processing. This means that we assume a laterally constant geological structure after spatially selecting the area under study (Fig. 1).

There are two main reasons for this choice. Firstly, the two-dimensional velocity model proposed by Zelt et al. (2005) shows little
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Maps of the western Corinth Gulf showing: (top) simplified geology of the area superimposed on a shaded topography, after (Skourtsos et al., 2017)) (bottom) seismicity from the HUSN catalogue (2000–2018), with colours and sizes corresponding to magnitudes, after (Evangelidis et al., 2021). The hatched surface surrounded by a blue line is the area selected for this study. Red lines are main active normal faults after (Moretti et al., 2003), the See text for details. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

variation in the crustal rock density model, with a layer thickness varying by a maximum of 2 km, in the area we have chosen for our study (hatched area, Fig. 1). Secondly, we require a sufficient number of earthquakes in each depth layer to establish representative subsets. To increase the number of earthquakes available at each depth, we decided not to discretise the area horizontally, thus neglecting horizontal variations.

This section presents two approaches to extract the “long-term” behaviour of a seismic catalogue: the traditional Reasenberg clustering method and our inverse smoothing method. Then we describe the calculation of the lower seismogenic depth of Zeng et al. (2022) used to complete the graphical representation of spatio-temporal variations in the depth distribution of earthquakes. Finally, we define the equations for building the one-dimensional velocity yield strength model.

### 3.1. Extraction of the “long-term” seismicity

Building on the long-established link between the depth distribution of earthquakes and the yield strength of the crust, a link that has been extensively studied in various settings (Lamontagne and Ranalli, 1996; Aldersons et al., 2003; Albaric et al., 2009; Muluneh et al., 2021; Rodriguez Piceda et al., 2022), we continue to use depth distribution to interpret the modelled rheological behaviour of the crust, this time exploring earthquake depth information in greater detail.

Traditionally, to compare the depth distribution of earthquakes and the yield strength of the crust, many studies have filtered their catalogue to extract the long-term behaviour of the Earth's crust using declustering techniques (Albaric et al., 2009; Radziminovich, 2022;
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** (A) Histogram showing the depth distribution of earthquakes within the selected area (see hatched area, Fig. 1) with a Gaussian probability density function in red. The Gaussian density representation is the sum of the Gaussian distribution for each earthquake (see in Appendix Figure 15). The Gaussian distribution is calculated so that its maximum is centred on the depth estimate, and its width is proportional to the estimated location uncertainty. Each Gaussian is normalised so that its integral equals 1. (B) Box plots describing the distributions of uncertainties in the depth of earthquakes for every 2 km for the whole HSUN catalogue. Each box shows the interquartile range, a measure of the dispersion of the median half of the data, equivalent to the difference between the third and first quartile. The vertical orange line in each box corresponds to the median value. Lines, called whiskers, extend from each box and capture the range of the remaining data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Déverchère et al., 2001; Meissner and Strehlau, 1982). These techniques separate transient earthquakes (such as aftershocks and swarms) from background seismicity, which is supposed to reflect the desired long-term effect.

### 3.1.1. Reasenberg declustering method

There are many declustering methods, each with its assumptions and advantages. However, as our study does not focus on declustering methods per se or on comparing them with each other, we decided to use a declustering method commonly employed to compare DDEs and YSEs.

The Reasenberg method is one of the most frequently used declustering methods (Reasenberg, 1985). This approach creates clusters of earthquakes based on interevent spatial and temporal distances, whose connection is defined by adaptive space–time interaction zones.

The principle of this method is first to define a spatial relationship between earthquakes using their magnitude and then apply Omori's exponential law to delimit the temporal distribution of related earthquakes with a hypothetical b-value. These two criteria define the space–time interaction zone. This zone, explained by the cracking radius of the clustered seismicity, implies that no background earthquake can occur there.

The spatial interaction threshold *d* depends on the magnitude of the earthquake **M** and is expressed by the following law:

$$ \log d(\text{km}) = 0.4M - 1.943 + k \quad (1) $$

with

$$ \mathbf{k} = \begin{cases} 1 & \text{for distance from the largest earthquakes} \\ 0 & \text{for the last one} \end{cases} \quad [\text{days}] $$

The temporal interaction zone is based on Omori's law with a decay rate of 1. Thus, the time required to observe the next earthquake with probability p1 is expressed by τ:

$$ \tau = \frac{-\ln(1 - p_1)t}{10^{2(M-M_c)/3}} \quad (2) $$

where **M** and **Mc** are respectively the magnitude of the earthquake and the magnitude of completeness. In addition, **Mc** increases in the above equation by a certain factor $x_k$ in the case of clustered seismicity.

For a new earthquake to be added to a cluster, it must occur within the radius of the crack ($r_f$act) as defined by Kanamori and Anderson (1975).

To obtain the clustered earthquakes and extract the background seismicity, we apply the Reasenberg code to the HSUN catalogue using

<caption>**Table 1**</caption>

Original input parameters for the Reasenberg declustering algorithm (Reasenberg, 1985) described in Section 3.1.1.

<table><thead><tr><th>Parameter</th><th>τ<sub>min</sub> (days)</th><th>τ<sub>max</sub> (days)</th><th>p<sub>1</sub></th><th>x<sub>k</sub></th><th>M<sub>c</sub></th><th>r<sub>f</sub> act</th></tr></thead><tbody><tr><td>Value</td><td>1</td><td>10</td><td>0.95</td><td>0.5</td><td>1.5</td><td>10</td></tr></tbody></table>

the set of values expressed in Table 1, as suggested by van Stiphout (2012).

We apply the declustering method to the entire catalogue and not just to the selected area (hatched area, Fig. 1), as this method does not take into account depth or geology and, therefore, does not require any special selection. Moreover, earthquakes close to the selected area may have triggered aftershocks within it, so it may be beneficial to include all nearby seismicity to distinguish between background events.

### 3.1.2. Inverse smoothing method

The Reasenberg method is based solely on statistical observations that enable the classification of two types of activity: one in clusters and the other one following a Poisson distribution. Like most declustering methods, the Reasenberg method does not incorporate the depth of earthquakes (see a list of methods in van Stiphout (2012)). However, this criterion is the main observable when comparing the yield strength envelope and the depth distribution of earthquakes. In the following paragraphs, we explain the principle of the smoothing method we developed to highlight potential mechanical weaknesses related to rheological properties in depth information of long-term earthquake distributions.

To design the smoothing method, we assume that at any depth in the crust, the deviatoric stress is close to the frictional strength (Townend and Zoback, 2000) and that the effective stress released by seismicity is proportional to the deviatoric stress (Scholz, 2019). Consequently, earthquakes should be more frequent at some depths close to the brittle–ductile transition than at other ones. Based on this postulate, we can expect a “persistent” distribution of seismic activity at depth in the crust, indicating a “stable” seismicity behaviour proportional to the yield strength of the “critically stressed brittle crust” (Townend and Zoback, 2000). By contrast, we can define sporadic patches of activity at depth in the crust with earthquakes that do not occur regularly at the same depth over several time windows.

To extract these two types of distribution (persistent and sporadic), we use a smoothing process following a minimisation approach (Eq.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Graphical illustration of the minimisation equation for 14 layers at depth and two time windows (t and t+1). The probability distribution $P_{yt}$ (in green) explains most of the earthquake distribution at depth at each time windows t, without varying by more than a factor $\epsilon$ between layers of depth y over consecutive time windows. The probability distribution $S_{yt}$ (in purple) is the difference between the persistent distribution and total seismicity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

(4)), which we solve using the HiGHS simplex algorithm (Huangfu and Hall, 2018).

The total seismicity of the earthquake catalogue contained in a time window t, called $T_t$, is decomposed as the sum of the seismicity occurring in the sporadic distribution $S_t$ and in the persistent distribution $P_t$ for each time $T_t$, with $T_t = \Delta t * t$ ($\Delta t$ and t being the time step and the time index, respectively; see Eq. (3)).

$$T_t = S_t + P_t, \text{ with } S_t \in M_{Y,T}(\mathbb{R}_+), P_t \in M_{Y,T}(\mathbb{R}_+^*) \quad (3)$$

As we expect the persistent depth distribution of earthquakes to explain most of the earthquake occurrence, we want to minimise the number of sporadic earthquakes in our minimisation problem (Eq. (4)).

$$\forall t, \min S_t \quad (4)$$

In addition, the persistent depth distribution of earthquakes must show minimal variations throughout the catalogue. Therefore, we introduce a constraint for $P_t$ in all time windows and at all depths. We discretise the depth into several layers, with a step $\Delta y$ such that the depth D of the layers follows $D = \Delta y * y$ and $P_t = \sum_{y=0}^{y} P_{y,t}$. Each persistent layer must retain the same probability of earthquake occurrence throughout the time windows, with a maximum variation of $\epsilon$ (see Eq. (5)). This means that the ratio between the number of earthquakes following the persistent distribution and the total number of earthquakes at different times must be less than $\epsilon$.

$$\forall (y, t), \left\| \sum_{k=-n}^{n} \frac{P_{y+k,t}}{T_t} - \frac{P_{y+k,t-1}}{T_{t-1}} \right\| \le \epsilon \quad (5)$$

To account for the uncertainty in depth location and to reduce the binning effect of our discretised depth, we compare $2n + 1$ layers at a time. To do this, we sum the number of earthquakes that follow the persistent distribution in the y layer with the earthquakes in the n layers above and below.

The trade-off between the smoothing parameter $\epsilon$ (Eq. (5)) and the neighbourhood size n (Eq. (5)) determines the effective smoothing

applied in the optimisation. More specifically, increasing n widens the range of depth values considered when evaluating smoothing, effectively allowing for greater depth variability while respecting the constraint defined by $\epsilon$. In our implementation, n is selected to correspond to a depth window of 3 km. For example, if depth cells are spaced 1 km apart, then setting n = 2 includes the current cell and its two immediate neighbours above and below, forming a depth window of 5 km centred on the cell.

However, it is important to remember that in our method we do not work with raw counts. Instead, we consider the relative distribution, i.e. the proportion of earthquakes in each depth bin for a given time window. The smoothing constraint ensures that these proportions do not change too abruptly with depth and time, within a tolerance defined by $\epsilon$. This approach allows us to balance fidelity to the observed spatio-temporal distribution of seismicity with the requirement for a smooth and interpretable solution.

As the minimisation process looks for the best solution of $P_{y,t}$ and $S_{y,t}$ for all time windows and depths simultaneously, the persistent depth distribution is the best distribution that should explain most earthquakes. In contrast, the sporadic depth distribution should represent all outliers of this distribution across time windows.

To summarise, the inversion approach follows these five steps:

1. We compute the Gaussian distribution in regular time windows.

2. We discretise the continuous depth distribution at regular depth intervals. We now have a matrix where the coefficient $(i, j)$ is the number of earthquakes in the ith time window at the jth depth layer.

3. Each column is divided by the sum of earthquakes to obtain the density of earthquakes.

4. Then, rows are smoothed along side the n row below and above, with the previous and next columns so that the difference in density is less than $\epsilon$. We now have a smoothed matrix of earthquake density ($P_{y,t}$). The difference with the input matrix gives the gap in earthquake occurrences at each depth ($C_{y,t}$).
5. To convert from density back to the "number of earthquakes", we multiply by the average number of earthquakes per column.

## 3.2. Representation of the seismicity depth profile

To better visualise the temporal variations in the distribution of earthquake depths, we chose to represent them using the maximum, average, and lower seismogenic depths for each time window using the formula given by Zeng et al. (2022):

$$D_{seis} = \frac{1}{N} \sum_{i} [d(xi; |x - x_i| < r_{samp}) > D_{90}] + \sigma \quad (6)$$

with,

* $D_{seis}$ the lower seismogenic depth

* d the earthquake focal depth.

* $x_i$ the focal location of the ith earthquake.

* $r_{samp}$ the sampling distance.

* $D_{90}$ the depth of the 90% quantile.

* N the total count of earthquakes for earthquakes below $D_{90}$ within $r_{samp}$.

* $\sigma$ one standard deviation of the mean seismogenic depth of earthquakes below $D_{90}$ within $r_{samp}$.

As we expect the distribution of earthquakes to be centred around one or two major peaks (Fig. 10), we illustrated, for each period, the maximum occurrence and the average depth of the earthquakes.

## 3.3. Modelling of the crustal yield strength as a function of depth

As we have already selected the study area to have negligible horizontal variation using available geophysical and geological information (Zelt et al., 2005; Bernard et al., 2006; Vallius et al., 2022), we can simplify the model to a one-dimensional depth yield strength of the crust.

The yield strength of the crust in the first 100 km can be modelled by two components: brittle and ductile. The brittle component (Sibson, 1977; King hubbert and Rubey, 1959; Zhang et al., 2013) is formulated as follows:

$$\sigma_{br} = \sigma_1 - \sigma_3 = \gamma g (1 - \delta) \int_0^z \rho(z) dz \quad (7)$$

with,

* $\sigma_1 - \sigma_3$ [Pa] is the differential stress between the maximum and minimum stresses, which in our case are the vertical stress and the stress parallel to the main tectonic strain rate.

* $\gamma$ is a factor that depends on the fault regime. In our case, it is mostly normal; therefore $\gamma = 0.75$ since we are in an extensional stress field.

* $\delta$ is the pore fluid factor. Under hydrostatic conditions, we assume a constant value of 0.4.

* $\rho(z)$ [kg m$^{-3}$] is the density of the rock layer at depth z.

* z [m] the depth of the rock layer.

We can model the ductile behaviour (Viti et al., 1997; Kirby and Kronenberg, 1987; Reber et al., 2020) of each rock layer as follows:

$$\sigma_{du} = \sigma_1 - \sigma_3 = \left(\frac{\epsilon}{A}\right)^{\frac{1}{n(z)}} \exp\left(-\frac{Q(z)}{nRT(z)}\right) \quad (8)$$

with,

* Q(z) [J] the activation energy of the rock

* R [J K$^{-1}$ mol$^{-1}$] is the gas constant

* $\epsilon$ [s$^{-1}$] is the strain-rate

* T(z) [K] is the temperature of the rock layer at depth z

* A(z) [Pa$^{-n}$s$^{-1}$] and n(z) are empirical constants that depend on the type of rock

To obtain Eq. (8), we need to model the depth evolution of temperature (Lachenbruch, 1970) which can be determined using a finite element method:

$$T(z) = T_{top} + \frac{qT_{top}}{k} z \quad (9)$$

with,

* $T_{top}$ [K] the temperature on top of the layer

* q [J s$^{-1}$] the heat flow

* k [W m$^{-1}$ K] the thermal conductivity of the rock layer

By combining the Eqs. (7), (8), and (9), we can express the yield strength of the crust at each depth z according to the following equation:

$$yse(z) = \min(\sigma_{du}(z), \sigma_{br}(z)) \quad (10)$$

# 4. Results

## 4.1. Description of the yield strength model

The yield strength model we obtained, shown in Fig. 4, is calculated using Eq. (10). The parameters used for the calculation are detailed in Table 2 with a stratification of 5 km of quartzite, 14 km of diorite, 11 km of diabase and then dunite, according to the observations by Zelt et al. (2005) and Tiberi et al. (2000, 2001). Heat flow is a critical parameter for the model, but the database provided by Global Heat Flow Data Assessment Group et al. (2024) does not contain direct information on the Corinth Gulf. Local convection in the karstic sediments of the footwalls of the Corinth rift is suspected to be responsible for the thermal anomalies observed near some active faults (Doan and Cornet, 2007).

Therefore, we use a commonly accepted average value of 53 mW/m$^2$ (Doan and Cornet, 2007). In addition, the thickness and existence of the diorite/diabase layer are open to debate as we lack precise information at this depth. Without constraint, we have positioned it at an intermediate depth between the top of the diorite and the base of the diabase.

The yield strength obtained in Fig. 4 increases linearly in the first brittle part of the crust between 0 km and 10 km. The brittle-ductile transition occurs at ca. 10 km at 200 °C. Then, the ductile part of the crust shows a decrease in yield strength down to 20 km.

## 4.2. Impact of depth uncertainty on the results

The parameter that most influences the quality and reliability of the results is undoubtedly the number of earthquakes and the uncertainty of their depth. However, the number of earthquakes can be easily increased by lengthening the time window, reducing the size of the depth cells (e.g., the bin size of the histograms), or enlarging the study area. Of course, each of these solutions involves a compromise in terms of loss of resolution in the results. This means that the smallest variations in depth over time or in geological settings will be equal to the size of the corresponding parameters, which is a common limitation when comparing the depth distribution of earthquakes and the yield strength of the crust.

To estimate the impact of earthquake depth uncertainty on the results obtained, we propose using a Gaussian distribution-based approach to mitigate potential biases, even if this does not improve overall accuracy. The following toy example (Fig. 5) illustrates the impact on visual representation and the smallest variations that can be detected.

This toy example illustrates how the depth distribution of earthquakes (equal on the left and unequal on the right) can be reconstructed at different levels of uncertainty, taking into account other nearby concentrations (e.g., two maximum peaks that may merge or disappear
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Range of models of yield strength of the crust for a strain rate ranging from 1e-17 s⁻¹ to 1e-15 s⁻¹ normalised by their integral in *blue* and the temperature in *black* as a function of depth for the Corinth Gulf. See text for details. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 2**</caption>

Parameters used to calculate the crustal yield strength of the Corinth rift. All information was extracted from Kirby and Kronenberg (1987), Viti et al. (1997), Burov (2011) and Anikiev et al. (2020).

<table><thead><tr><th>Parameters</th><th>Quartzite</th><th>Diorite</th><th>Diabase</th><th>Dunite</th></tr></thead><tbody><tr><td>Power Law Exponent (n)</td><td>2.4</td><td>2.4</td><td>3.4</td><td>4</td></tr><tr><td>Rheological constant (A [MPa<sup>-n</sup>s<sup>-1</sup>])</td><td>6.3e-6</td><td>1e-3</td><td>3e-1</td><td>9.5e2</td></tr><tr><td>Density (ρ [kg/m<sup>3</sup>])</td><td>2700</td><td>2900</td><td>2800</td><td>2900</td></tr><tr><td>Activation enthalpy (Q [kJ/mol])</td><td>165</td><td>219</td><td>430</td><td>430</td></tr><tr><td>Thermal conductivity (k [Wm<sup>-1</sup>C<sup>-1</sup>])</td><td>2.4</td><td>2.7</td><td>2.1</td><td>3.95</td></tr><tr><td>Thickness ([Km])</td><td>5</td><td>14</td><td>11</td><td>inf</td></tr></tbody></table>

into one). We chose to increase the distribution range between each reference earthquake repartition (blue bars) by 2 km each time (thus, the gap between the first two bars is 2 km, then 4 km, and so on). This gives us a corresponding overall distribution in depth using a Gaussian approach for different depth uncertainties (1, 2, 3, 5, 7, and 10 km).

The left-hand graph in Fig. 5 shows that all seismic distribution peaks are identified in the Gaussian distribution for uncertainties less than 2 km. Beyond this value, the first two earthquake peaks, spaced 2 km apart, are no longer apparent. The second and third peaks, spaced 4 km apart, remain identified until the uncertainty reaches 5 km, after which the first, second, and third peak merge. The fourth peak, separated from the third peak by 8 km, remains distinguishable even for an uncertainty of 10 km.

We conclude from this simplified example that two peaks in the earthquake distribution can be correctly identified if the value separating them is at least as large as the depth uncertainty. Otherwise, these peaks may reflect artefacts of the method used to produce the catalogue or effects of its representation. In some cases, even larger uncertainties still allow the peaks in the earthquake distribution to be separated, provided that they are not too uneven. Given the overall depth uncertainty, which is less than 2.5 km in our case study, we should be able to distinguish groups of events separated by a distance close to 2.5 km, a value compatible with most local or regional seismic station networks of reasonable density.

### 4.3. Depth trends for background and clustered seismicity

The Reasenberg clustering algorithm has been applied to the whole HUSN catalogue presented in Section 2. The histogram shape of the

depth distribution of background and clustered seismicity (Fig. 6) looks similar, with slight variations between bins. Both reveal a maximum peak at around 10 km. The clustered seismicity presents another clear peak at around 23 km.

The density kernels broadly follow the histogram distribution, except that at around 10 km, two distinct peaks can be established for the clustered seismicity instead of just one with the histogram (see Fig. 6B). One appears to have a maximum at 9 km and the other at 10 km.

In both cases, the maximum peak at around 10 km corresponds to the brittle-ductile transition observed in the yield strength model. Regarding clustered seismicity, the other maximum at 23 km is close to the diabase-diorite transition, which we have set at 21 km in our geological model.

In addition, the temporal analysis of the depth distribution of the two categories of events (Fig. 7) shows that the background events are more numerous (proportionally) than the clustered events at a depth greater than 20 km, which corresponds to the apparent maximum seismogenic depth (Fig. 2).

As shown in the histogram of the depth distribution of events in the catalogue (Fig. 2), this observation is mainly due to the fact that the density of events decreases with depth after the maximum occurrence at 10 km, greatly affecting the degree of completeness as a function of depth (Fig. 8).

Although the magnitude of completeness (determined using the method illustrated in Appendix Figure 13, following Wiemer (2000)) of the entire catalogue is less than 2, we chose to represent temporal variations in the depth distribution of background and transient events using a cut-off magnitude of 2. This choice ensures that the results are interpretable down to 23 km (see Fig. 8), when it increases by 0.5.
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Toy example illustrating the impact of earthquakes concentration (distributed evenly on the (A) and unevenly on the (B)) at depths of 3, 5, 9, and 17 km (represented as blue bars) on the visual representation obtained using a Gaussian distribution approach. Different colours and markers represent different levels of earthquake depth uncertainty (from 1 to 10 km). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Depth distribution histogram and kernel density representation for background seismicity (A) and clustered seismicity (B) using the Reasenberg method on the entire HUSN seismic catalogue.</figcaption></figure>

Like most other declustering methods, the Reasenberg method uses the magnitude of events and the magnitude of catalogue completeness as classification criteria. As a result, the magnitude of completeness and the magnitude distribution vary with depth, limiting an exhaustive temporal study.

### 4.4. Results obtained using conventional smoothing methods

There are numerous methods available for smoothing data; however, no single standard method is universally applied in comparative studies. To ensure the integrity of our result, we propose comparing our smoothing approach with other conventional methods. We decided to use our own method because we wanted more control, as we discussed

in the previous section (Section 3.1.2). However, this remains a form of "smoothing", and therefore the results do not differ significantly from other smoothing methods.

Here we present two additional smoothed representations of the time and depth of earthquakes, which should correspond to the representation of persistent seismicity. We have chosen two simple approaches for clarity and to provide a comprehensive understanding of the data processing methods. The first is a time moving average (Fig. 9A), meaning that each coefficient of the matrix is the average of the coefficients from the previous and next columns in the same row. The second approach is a 2D low-pass filter (Fig. 9B), which involves applying a two-dimensional fast Fourier transform to the dataset and retaining only the low-frequency components. We then keep the
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Depth changes through time using the whole HUSN catalogue shown on Fig. 1: (A) background seismicity and (B) clustered (crisis) seismicity in time and depth without smoothing using the Reasenberg method with a cut-off magnitude of 1.6. As shown in Fig. 8, the interpretation after 25 km can be “ambiguous” due to the change in the magnitude of completeness.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Representation of the magnitude of completeness determined using the maximum number of events in each magnitude range (see example in Appendix Figure 13) at each depth layer every 3 km for the whole HUSN catalogue.</figcaption></figure>

coefficients in the low-frequency range before calculating the inverse transform. We arbitrarily chose to keep 0.25% of the frequency content. However, between 0 and 50%, the visual representation did not show significant changes in the results.

Both methods exhibit similar variations over time with deeper earthquake during the same time window, with the same scale of deepening. However, the number of earthquakes differs. The low-pass filter, while not influenced by a higher total number of earthquakes, does produce “circular” artefacts. These artefacts can lead to the appearance of earthquakes in areas where none were initially present. Also the fast Fourier transform (FFT) can introduce oscillations in the number of earthquakes, potentially leading to negative numbers of earthquakes that do not make sense in nature.

The moving average method offers a more straightforward approach, resulting in fewer factors that can influence the outcome. The number of earthquakes differs between the two smoothing approaches because the moving window average is only a local smoothing. This means that the data in the first column does not affect the results in the last column. We therefore obtain a series of smooth variations, but not an overall distribution that explains the density in all time windows. Furthermore, in this case, we just smooth the depth “cell” with the

same one, on the next days, which means that uncertainties in depth may have a greater impact.

In summary, although the two conventional methods produce distributions that are usable and comparable to our method, each has its drawbacks. Our method (§3.1.2) is adaptable, as it combines the advantages of both approaches without their disadvantages: global smoothing and the use of the depth dimension, as in FFT, without introducing artefacts and without increasing the number of earthquakes beyond the initial count, as in time windows. This adaptability makes our smoothing approach relevant in this specific case of study.

### 4.5. Depth trends and temporal evolution of persistent activity layers

We now compare the modelled crustal yield strength envelope with the depth of persistent layers obtained (Figs. 4 and 10), as defined in Section 3.1. To do this, we chose a time step and a depth step, which were determined by the density of the earthquake dataset and the quality of the locations at 60 days and 1 km, respectively. If we had increased the size of the windows, we would have obtained more average results with no visible details. On the other hand, if we had reduced the size of the windows, the data would have been too discretised, and empty time windows without earthquakes would have been created. An equivalent figure without any processing can be found in the Appendix (see Appendix Figure 16), where the visual improvement obtained by the smoothing method can be appreciated. The average and maximum depths are more or less the same, but the deepening of earthquakes, although guessable in Appendix Figure 16, is highlighted in Fig. 10.

The yield strength (Fig. 4, right) is close to zero after 15 km and indicates ductile deformation. This aligns with the decrease in seismic activity below 15 km, as highlighted by the lower seismogenic depth (Fig. 10, A), which is rarely lower than 15 km outside of the activity of a major crisis. In addition, the maximum and mean depths (Fig. 10) fluctuate around 10 km, which corresponds to the depth of the brittle-ductile transition (Fig. 4).

To better understand the depth variations observed over time (Fig. 10), we examine the temporal distribution of each type of seismicity (i.e. persistent and sporadic), as shown in Fig. 11. In this figure, we observe that the trend of the curve corresponding to earthquakes that occur in the sporadic activity layers increases in stages (see $S_{yt}$, in green). These steps are related to the transient occurrence of clusters of earthquakes. The curve corresponding to earthquakes occurring in persistent activity layers remains stable (see $P_{yt}$, in red), while being related to the rate of occurrence of earthquakes in the raw
<figure><img src="image_12.png" /><figcaption>Fig. 9. In both figures, the colour represents the number of earthquakes at each depth for different time windows: every 1 km and 60 days. We applied two conventional smoothing method to the depth distribution of earthquakes (hatched area of the HUSN Corinth catalogue) the same than for our smoothing method. (A) The time windows moving average is applied to the Gaussian distribution matrix, (B) A 2D low-pass filter is applied to the Gaussian distribution matrix.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. (A) Gaussian kernel estimates for the occurrence of earthquakes that follow a recurrent depth distribution ($P_{yr}$) over the selected time windows (i.e. 60 days), and (B) for the occurrence of earthquakes extracted from the raw catalogue over the same selected time windows.</figcaption></figure>

catalogue, as shown by the evolution of the directrix of the black curve corresponding to the whole catalogue.

The strong spikes in sporadic activity that we observe in Fig. 11, for example between November and December 2015 (see the green curve $S_{yr}$), are correlated with the spatiotemporal clusters identified by the Reasenberg method and shown in dashed green. These activity jumps correspond to the temporal occurrence of clustered seismicity highlighted by the skewness and kurtosis approach of Mesimeri et al. (2019) (see the grey bands in Fig. 11). We note that the variations observed in persistent activity (see the red curve $P_{yr}$) still contain clustered activity. This can be seen in Fig. 10, where an increase in the thickness of the seismogenic layer occurs close to a sufficiently large crisis, as in January 2012 and November 2014.

## 4.6. Average envelope for depth distribution of sporadic and persistent activity

We complete our observations by analysing the 11-year average density of earthquakes at depth by type of activity (i.e., persistent and sporadic; see Fig. 12). The distribution of the sporadic earthquakes (Syt, in green) shows two local maxima at 10 km and 15 km. The maximum at 10 km can be correlated with the predicted brittle-ductile transition (see Fig. 4). The maximum at 15 km corresponds to the maximum seismogenic depth, a depth that is only exceeded during a sufficiently large crisis.

The distribution of persistent activity (Pyt, in red) closely matches a Gaussian distribution, which is consistent with the distribution profile
<figure><img src="image_14.png" /></figure>

**Fig. 11.** Cumulative number of earthquakes over time for the HUSN catalogue (black), persistent seismic activity (Pyt, red) and sporadic seismic activity (Syt, green). The dashed green curve shows the cumulative number of clustered seismicity derived from the Reasenberg Clustering analysis (Reasenberg, 1985). The vertical grey bands indicate the occurrence of seismic earthquakes clusters determined by Mesimeri et al. (2019) using a skewness and kurtosis approach. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

<figure><img src="image_15.png" /></figure>

**Fig. 12.** Depth distribution of earthquakes for the raw catalogue selected area, persistent seismic activity (Pyt) and sporadic seismic activity (Syt) in the Corinth Gulf. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of the raw catalogue and should reflect the effect of long-term deformation. This observation agrees with the depth distribution of background earthquakes identified by the Reasenberg algorithm (Fig. 6, A).

# 5. Discussion

## 5.1. *Depth distribution of earthquakes and yield strength*

Previous studies of the depth distribution of earthquakes have highlighted the shape correspondence between long-duration catalogues and the yield strength of the crust (Lamontagne and Ranalli, 1996; Aldersons et al., 2003; Albaric et al., 2009; Muluneh et al., 2021; Rodriguez Piceda et al., 2022). From Fig. 10, we can add that even with short-duration datasets (at least a few months), the yield strength of the crust seems to influence the distribution of persistent earthquakes occurring outside of large seismic crises, with the maximum number of

earthquakes observed near the brittle-ductile transition at 10 km. In addition, the average lower seismogenic depth of the entire catalogue is close to 15 km, a depth at which the yield strength of the crust is close to zero and in the ductile regime, making it less likely to generate earthquakes (Scholz, 2019).

The distribution of sporadic earthquakes at depth throughout the catalogue (Fig. 12, Syt in green) shows two local maxima that may correspond to potential geological variations at depth. The first maximum close to 10 km is near the brittle-ductile transition, at about the same depth as the persistent activity peak. As earthquakes occur continuously in layers of persistent activity, earthquakes that occur in layers of sporadic activity in this same depth range can also be observed and correspond to a type of clustered seismicity (Fig. 6, B). Indeed, authors such as Mesimeri and Karakostas (2018) have also revealed repetitive continuous-type earthquakes at a depth of 10 km along the two major north-dipping western rift faults, the Psathopyrgos and Aigion faults.
Based on this observation, they surmised that creep may occur along the brittle-ductile transition zone. This continuous seismic behaviour may be evidence for an aseismic slip at the boundary between the brittle and ductile layers.

The other maximum at approximately 15 km (Fig. 12, Syt and Fig. 6 B) could correspond to fluid-induced activity. At this depth, the yield strength (Fig. 4) is close to zero and is interpreted to reveal a deficit in the seismic moment released by rifting, possibly related to overpressurized fluids rising from the deeper crust (Lambotte et al., 2014). Seismicity at this depth is thus likely due to transient changes in the local stress field. We note that a maximum at 23 km can also be seen in Fig. 6 B, which could be related to a geological transition between diorite and diabase. However, owing to the lack of geological and geophysical information at this depth, this conclusion remains hypothetical.

## 5.2. Distinction between persistent and sporadic seismicity and differences from background and clustered events

Firstly, it is important to note that, according to the hypothesis we developed earlier, the distinction between persistent and sporadic seismicity is primarily based on physical criteria. This classification aims to define a persistent depth distribution of events over multiple time windows, which can be interpreted as representing the yield strength envelope of the crust, in line with the conclusions of Lamontagne and Ranalli (1996), Aldersons et al. (2003), Albaric et al. (2009), Muluneh et al. (2021) and Rodriguez Piceda et al. (2022).

The method we have developed is based on the percentage of total earthquakes occurring in each depth cell over several time windows. From this data, we search for a “stable” depth distribution of earthquakes over the considered time windows. The result is not a classification or statistic of individual events but rather the depth distribution of earthquakes reflecting as closely as possible the yield strength of the crust.

On the other hand, the classification of background seismicity and clustered seismicity is based on statistical data. For instance, the Reasenberg clustering method relies on empirical relationships between the temporal and spatial distances of earthquakes and their magnitude, derived from statistical observations. Similarly, the ETAS model (Ogata, 1998; Zhuang et al., 2002) uses a set of parameters to model seismicity in detail and probabilistically distinguish background earthquakes from clustered earthquakes, such as aftershocks. The ETAS parameters modelled can often be linked to physical processes (Kamranzad et al., 2025), but they are not directly physical parameters.

It is also important to note that the data used in declustering methods, such as ETAS, include the spatial distance between earthquakes, time intervals, and earthquake magnitudes. These variables are used to identify a constant underlying earthquake rate (the background rate), which in turn is assumed to generate the remaining seismicity through cascading triggering effects, such as aftershocks. The result of these methods is a classification of each earthquake as background or clustered, as well as a statistical model of the spatio-temporal evolution of seismicity in the studied region.

## 5.3. Transient deepening of the seismicity

During periods of high seismic activity, earthquakes occur deeper than the maximum seismogenic depth estimated during less active periods (Figs. 4 and 10). For example, there is a clear deepening of seismicity of 5 and 8 km in December 2011 and November 2014, respectively. The observed deepening can be attributed to a local change in the stress field or to the reactivation of a “mature” fault propagating in the crust (Manighetti et al., 2007). The faults described by these authors are made up of broken segments separated by bulk material. So, depending on the thickness and strength of the inter-segments, an earthquake of a sufficient magnitude may initiate a cascade of

earthquakes along the fault. The more previous earthquakes there are, the more mature a fault becomes as it weakens its inter-segments.

In 2012, the maximum seismogenic depth deepened by 1 to 2 kilometres. It became shallower gradually until reaching the previous depth of 15 km, in the 4 to 6 months following a magnitude 4 earthquake December 2011 (Fig. 10). In November 2014, a magnitude 4.8 earthquake caused the maximum seismogenic depth to deepen much more, by approximately 5 to 6 km, but it returned to the mean over a shorter period (around 2 to 3 months; see Fig. 10). This difference can be attributed to the depth of the earthquake: the 2012 magnitude 4 earthquake occurred at a depth of 14 km, while the 2014 magnitude 4.8 earthquake occurred at 18.5 km, near the diorite-diabase transition.

This type of “instantaneous” deepening followed by a slower return to the previous value has already been highlighted by Cheng and Ben-Zion (2020) for the Ridgecrest earthquake of magnitude 7.2 and by Rolandone et al. (2004) for the Landers earthquake of magnitude 7.3, for instance. In both cases, at least 3 km deepening was observed, which persisted for several years. In our study, we show that this deepening can occur for earthquakes of much lower magnitude (i.e., Mw 4.0–4.8).

The fact that we can observe a deepening of roughly the same order for earthquakes of much lower magnitude suggests a different depth limit for brittle failure after mainshocks than during more “quiet” periods, which could be due to the maximum depth of fault rooting. However, in the case of smaller earthquakes, the “recovery time” is much shorter since it only takes a few months for the maximum seismogenic depth to return to its previous value.

It can be assumed that earthquakes sometimes occur in potentially mature fault systems, generating a cascade of earthquakes that propagate to the root of the fault because the gaps between fault segments are small (Manighetti et al., 2007). Theoretically, this phenomenon should occur where the brittle behaviour of the crust is predominant (Scholz, 2019). Over time, the crust heats the area damaged by earthquakes (Vidale and Li, 2003). This healing process increases the density of the rock and makes it more resistant. This hypothetical mechanism may explain why the seismogenic depth initially increases and then gradually returns to its previous level. The healing process also influences the permeability of the rock (Xue et al., 2013). As the rock compacts, fluids can be expelled or trapped, which can dry out the rock and increase the fluid pore pressure, thus altering the rheological behaviour.

## 5.4. Definition of long-term seismic activity for rheological analyses

Using our decomposition of persistent and sporadic seismic depth layers, we show that the depth distribution of earthquakes differs according to the type of activity. A much less clear-cut difference in depth distribution can be observed between the background and the clustered seismicity obtained after applying the Reasenberg clustering method. This may be due to how background and clustered seismicity are commonly defined. Traditionally, background earthquakes are expected to occur continuously according to a Poisson process (Gardner and Knopoff, 1974) and have no identified precursors, while clustered seismicity groups transient earthquakes induced by precursors and obeying the Gutenberg-Richter and Omori laws (Utsu et al., 1995; Console et al., 2010; Helmstetter and Sornette, 2003).

This definition does not account for geological or depth information and is based purely on statistical approaches. The classical use of background seismicity to define “long-term” tectonic phenomena to compare the depth distribution of earthquakes and the yield strength envelopes of the Earth’s crust (Albaric et al., 2009; Déverchère et al., 2001; Radziminovich, 2022; Meissner and Strehlau, 1982) may only capture some of this information and could potentially distort it. Furthermore, the long-term behaviour of the background seismicity is often obtained by averaging the depth distribution over the whole catalogue. This operation has its flaws. The depth of earthquakes
can undergo significant changes during periods of high seismic activity, leading to possible misinterpretation of long-term behaviour. Even during “healing” periods (Rolandone et al., 2004; Vidale and Li, 2003), earthquake depths can fluctuate, further complicating the analysis of long-term behaviour. Lastly, the notion that rheological behaviour is constant over time can also be called into question, given the constantly changing stress load within the crust and the transient redistribution of deformation linked to a wide range of rheological heterogeneities (Fagereng et al., 2024) and frictional properties in the continental crust (Wang et al. 2023).

### 5.5. Sporadic distribution and physical interpretation

Although our primary objective is to study the yield strength of the crust, it is nevertheless interesting to examine the possible physical origins of this sporadic distribution. We do not have sufficient data to create a meaningful model that could explain this variation from the persistent distribution. However, it is important to note that this distribution shows the occurrence of earthquakes outside of the long-term earthquake distribution.

The first reason that could explain such variation is that sporadic distribution often appears near or during aftershocks or swarms. This suggests that events deviating from the persistent distribution belong to this transient category. It is known that aftershocks can result from local stress variations or other transient changes in the medium, inducing a cascade of events.

Strong earthquakes can produce local variations in pore pressure (Ali, 2015), allowing deeper earthquakes or weakening of the crust beyond the expected yield strength we have calculated. In addition, after mainshocks, the crust may be locally subjected to relaxation processes (Meade et al., 2022; Barbot and Fialko, 2010; Bascou and Jouanne, 2023) that would increase the deformation rate at specific depths, thereby shifting the earthquake distribution from the long-term distribution.

We may also consider the fault weakening (Tse et al., 1986) caused by shear stress concentration and pore pressure, which could influence seismic activity and potentially trigger slip pulses on mature faults (Manighetti et al., 2007) with significantly lower strength than predicted by Byerlee's law (NYSE) (Hirakawa and Ma, 2016). This means that the changes in the depth of sporadic distribution could highlight specific layers subject to higher stress values (e.g., a brittle layer transmitting the maximum horizontal compressive stress in the crust) or layers subject to overpressure due to variations in pore pressure or, more generally, fluid intrusions.

## 6. Conclusion

In this study, we have introduced a new way to examine the relationship between earthquake depth distribution and crustal yield strength profiles thanks to a monitoring of persistent and sporadic seismicity through depth and time. When applied to the comprehensive and reliable HUSN catalogue of the central and western Corinth Gulf, this approach has significant implications for our understanding of this relationship.

Our analysis shows that persistent seismicity exhibits a quasi-Gaussian distribution centred on the brittle-ductile transition at around 10 km depth, reflecting a robust correlation with crustal rheology. Conversely, sporadic earthquakes deviate from this distribution, peaking at 9 km and 15 km depths, often close to discontinuities in material properties or rupture behaviour, reflecting transient processes likely related to fluid dynamics in the crust.

Persistent seismicity also evidences that magnitude 4.0–5.5 earthquakes can induce tectonic stress perturbations sufficient to deepen the maximum depths of earthquakes beyond the modelled seismic thickness. Until now, this phenomenon has mainly been observed following larger earthquakes (M > 7). However, the fact that the

deepening is roughly of the same order for lower magnitudes indicates that a limit corresponding to the root of the fault restricts the maximum depth of earthquakes (extent of brittle failure) to a different value from that observed during less active seismic periods.

Using our representation and the conventional background and clustered seismicity patterns identified by the Reasenberg algorithm, we show that the integration of crustal yield strength analysis with persistent and sporadic distributions provides a new and accurate way for understanding the link between the depth distribution of earthquakes and rheology. This approach appears quite useful for better assessing the brittle-ductile transitions and for defining transient changes following moderate magnitude earthquakes. Our study opens a promising avenue for refining seismic hazard models and advancing our understanding of earthquake dynamics.

## CRediT authorship contribution statement

**A. Septier:** Writing – review & editing, Writing – original draft, Conceptualization. **J. Déverchère:** Writing – review & editing, Validation, Supervision, Project administration, Funding acquisition. **J. Perrot:** Writing – review & editing, Validation, Supervision. **A. Renouard:** Writing – review & editing, Validation, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work was supported by ISblue project, Interdisciplinary graduate school for the blue planet (ANR-17-EURE-0015) and co-funded by a grant from the French government under the program "Investissements d'Avenir" embedded in France 2030. We would also like to thank Chastity Aiken for the time she took to help.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.tecto.2025.231008.

## Data availability

The catalogues from the Corinth Gulf can be downloaded via the CRLAB website (Corinth Rift Laboratory Team and RESIF Datacenter, 2013 https://nfo.crlab.eu/earthquakes catalogs (last access 11/2023).

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