# Multiscale Post-Seismic Deformation Based on cGNSS Time Series Following the 2015 Lefkas (W. Greece) Mw6.5 Earthquake

^{1}

^{2}

^{*}

## Abstract

**:**

_{min}≈ 3.5 days to τ

_{max}≈ 350 days.

## 1. Introduction

## 2. A Thermodynamic Model of Post Seismic Relaxation

_{c}is given from Equation (1) as ${u}_{o}=u\left(t={0}^{+}\right)={u}_{\infty}+S{R}_{O}$, where ${R}_{0}=R\left(t=0\right)$. The scaling factor S of the relaxation constrains the rheological properties of the medium [34,35] and depends on the rheology of the crust and upper mantle, as well as on the local conditions around the site (i.e., crustal heterogeneities). In addition, S depends on the stress released in the medium and reflects the response of the medium (i.e., of the magnitude of the earthquake event) after the co-seismic displacement due to earthquake.

_{B}the Boltzmann constant. If the activation energy is distributed following a distribution function N(E), then the distribution function f(τ) for the relaxation times satisfies the expression f(τ) =N(E)(dE/dτ). Using Expression (4), dτ/dE= (${\tau}_{0}$/k

_{B}T)exp (E/k

_{B}T) = τ/k

_{B}T, hence:

_{min}and maximum E

_{max}activation energy, N(E) has a constant value N

_{o}, which, according to Equation (5) for a temperature T, leads to a distribution of relaxation times $f\left(\tau \right)=({k}_{B}T{{\rm N}}_{0})/\tau ,$ which is proportional to 1/τ. It is evident that the minimum ${\tau}_{min}$ and maximum ${\tau}_{max}$ relaxation times are related to the corresponding minimum and maximum activation energies by the expressions ${\tau}_{min}={\tau}_{0}{e}^{\raisebox{1ex}{${E}_{min}$}\!\left/ \!\raisebox{-1ex}{${k}_{B}T$}\right.}$ and ${\tau}_{max}={\tau}_{0}{e}^{\raisebox{1ex}{${E}_{max}$}\!\left/ \!\raisebox{-1ex}{${k}_{B}T$}\right.}$ and, thus, $\Delta {\rm E}={E}_{max}-{E}_{min}=kT\mathrm{ln}(\frac{{\tau}_{max}}{{\tau}_{min}}$).

_{o}, which is omitted in the calculation merging it with the scaling parameter S. We note that N

_{o}is a constant value for a uniform distribution of energy N(E) between its minimum E

_{min}and maximum E

_{max}value. The weight factor 1/τ in Equation (6) is a result of the application of Arrhenius’ law due to mesoscale relaxation processes operating on different timescales. In Figure 2, we present examples of the relaxation function with time on a logarithmic scale along the horizontal axis with a constant ${\tau}_{min}$ and various values of ${\tau}_{max}$. In a logarithmic scale, the curve in Figure 2 decreases almost linearly, up to a time t ≈ τ

_{max}, and at the end, it flattens out. Thus, the relaxation function R(t) is proportional to ln (t) for t << τ

_{max}. Based on that, the maximum relaxation time τ

_{max}could be estimated as the time where the relaxation curve flattens out.

_{1}(x) [38], which can not be solved analytically but behaves like a negative exponential for very large values of the argument and decays on a logarithmic mode in the intermediate range of the relaxation curve R(t) while deviating from a logarithmic time dependence for t < τ

_{min}. Furthermore, as results from Equation (6) when t = 0, R(t) can be calculated analytically:

_{min}$\approx $ t≪τ

_{max}, expands exponentially and leads to ${\left[\frac{dR\left(t\right)}{dt}\right]}_{t=0}=-\left(\frac{1}{{\tau}_{min}}-\frac{1}{{\tau}_{max}}\right)$ and

_{max}≫ τ

_{min}, the slope of R(t) vs. t at t = 0 is well approximated as ${\left[\frac{dR\left(t\right)}{dt}\right]}_{t=0}=-\frac{1}{{\tau}_{min}}$, while the ratio of the slope to R(0) follows the expression: $\frac{1}{R\left(t=0\right)}{\left[\frac{dR\left(t\right)}{dt}\right]}_{t=0}=\frac{1}{{\tau}_{min}}\mathrm{ln}\left(\frac{{\tau}_{max}}{{\tau}_{min}}\right)=\frac{{R}_{o}}{{\tau}_{min}}$.

_{min}≪ τ

_{max}and the time t is such that τ

_{min}≪ t ≪ τ

_{max}, we could approximate exponentials such as ${e}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau}_{min}$}\right.}\approx 0\mathrm{and}{e}^{\raisebox{1ex}{$-t$}\!\left/ \!\raisebox{-1ex}{${\tau}_{max}$}\right.}\approx 1$ and Equation (8) results to:

_{1}− ln(t) for τ

_{min}≪τ≪τ

_{max}

_{1}an integration constant. Equation (12) reveals that the relaxation function R(t) exhibits a logarithmic behavior in an intermediate time interval bounded by the relaxation times ${\tau}_{min}$ and ${\tau}_{max}$ with a slope −1 in a graph of R(t) versus ln (t). It becomes evident that when multiplying R(t) with a scale factor S in Equation (1), there is a modification of the slope of the postseismic relaxation versus ln (t) related to the particular characteristics of earthquake event and the seismotectonic settings and the rheology of the deformed zone as well. Substituting Equation (11) into (1), displacement obtains logarithmic behavior:

_{min}≪ τ ≪ τ

_{max}with a slope F

_{2}= S.

_{1}as:

_{1}, we calculate its value for two reasonable values of τ*, as that of ${\tau}^{*}={\tau}_{min}$ and ${\tau}^{*}=2{\tau}_{min}$. For ${\tau}^{*}={\tau}_{min},$(i.e., the transition in the logarithmic range starts in the very early stage of deformation), we have ${R}_{1}=\mathrm{ln}{\tau}_{max}-1+\frac{{\tau}_{min}}{{\tau}_{max}},$ which for $\frac{{\tau}_{min}}{{\tau}_{max}}\ll 1,$ leads to:

_{max}/e ≈ 0.37${\tau}_{max}$. For a more reasonable estimation, we set ${\tau}^{*}=2{\tau}_{min}$, leading to ${R}_{1}=\mathrm{ln}{\tau}_{max}+\mathrm{ln}2-2(1-\frac{{\tau}_{min}}{{\tau}_{max}}),$ which for $\frac{{\tau}_{min}}{{\tau}_{max}}\ll 1,$ leads to:

_{min}≪ τ

_{max}, this scaling can provide an estimation of τ

_{min}, since $\frac{{F}_{2}}{{F}_{1}}={\tau}_{min}$.

_{min}≪ τ

_{max}≪ t, the long time behavior of R(t) can be estimated, replacing the upper limit τ

_{max}with infinity so that [33]

_{min}, the relaxation function is controlled only by τ

_{max}and not τ

_{min}. In that case, the relaxation time τ

_{max}is predominant, and R(t) leads to $R\left(t\right)={e}^{\raisebox{1ex}{$(-t$}\!\left/ \!\raisebox{-1ex}{${\tau}_{max}$}\right.)},$ implying that the term τ

_{max}/t is associated to the relaxation processes with relaxation time τ

_{min}< τ < τ

_{max}. We note that, since the time dependence of 1/t is slow compared to ${e}^{\raisebox{1ex}{$(-t$}\!\left/ \!\raisebox{-1ex}{${\tau}_{max}$}\right.)},$ the relaxation function could be well approximated as an exponential a long time after the main event. Furthermore, the exponential relaxation, as presented in Equation (2), is a special case of the general expression of the relaxation function (3) and results when τ

_{min}and τ

_{max}have similar values, i.e., when τ

_{min}= τ

_{0}− Δ and τ

_{max}= τ

_{0}+ Δ, which leads to $R\left(t\right)\approx \frac{2\Delta}{{\tau}_{o}}{e}^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{${\tau}_{0}$}\right.}$ presenting an exponential decay over time.

## 3. Seismotectonic Setting, Data Selection, and Analysis

_{w}6.3) occurred off-shore of the north-western coast of Lefkas Island [46,47,48,49,50]. In late January–early February 2014, two large-magnitude earthquakes (M

_{w}5.9 and M

_{w}6.1) were located in the western part of Cephalonia [51,52]. However, these events were not linked directly with the CTF zone.

_{w}6.4) occurred in the southern part of Lefkas Island [53,54] just north of Cephalonia. The main event was followed by a significant post-seismic activity that extended southwards towards Cephalonia. The hypocenter of this earthquake was located at a depth of ~10 km, and the fault plane solutions demonstrated a NNE–SSW striking dextral strike-slip seismic fault with a reverse component that dipped east at a high angle of about 70° ± 5°. The after-shock seismic activity extended along three distinctive clusters north and south of the main event [54]. The northern cluster seemed to extends towards the epicentral area of the 2003 strong earthquake, the central one was located in the vicinity of the main event, while the southern one extended towards Cephalonia Island and about 15 km southwest of the epicenter. The spatial coverage of the aftershocks indicated less dense distribution close to the epicentral area, while away from the main shock, there was higher concentration of post-seismic activity. The latter may indicate that, during the main event, higher energy was released in the main ruptured fault, which comprised part of the major CTF zone, while the post-seismic activity was triggered in nearby secondary and adjacent local faults, away from the main rupture plain [54]. The post seismic events extended in a NE–SW direction for approximately 50 km, extending north towards the 2003 earthquake and south towards the northern Cephalonia. The southern seismic cluster occurred in the same area that was also activated during all the previous intense seismic periods in 2003 and in 2014 [48,49].

#### 3.1. GNSS Data Analysis

_{East}= (20.45 ± 0.03) mm/yr, V

_{North}= (6.81 ± 0.02) mm/yr, and V

_{Up}= (−0.71 ± 0.08) mm/yr with respect to IGb08 reference frame (http://igscb.jpl.nasa.gov/network/refframe.html) (accessed on 30 December 2020). The earthquake of 17 November 2015 caused a co-seismic displacement of the PONT station of 360 mm to south, 201 mm to the west, and a subsidence of 73 mm (Figure 4).

#### 3.2. Post Seismic Time Series Analysis

_{pst}t, where V

_{pst}is the long term loading velocity a long time after the earthquake when the system has reached a new equilibrium state. In our case, V

_{pstEast}= (17.48 ± 0.07) mm/yr, V

_{pstNorth}= (3.72 ± 0.07) mm/yr, and V

_{pstUp}= (−0.44 ± 0.18) mm/yr, which was removed from the observed postseismic data. To apply our approach, we analyzed the post seismic relaxation observed after the 17 November 2015 Mw6.5, Lefkas island earthquake. In Figure 5, the North–South displacement, which was the event’s predominant post seismic displacement, presented as a function of time for a period 1000 days after the event. For energy reasons, the minimum relaxation time τ

_{min}cannot be equal to zero in order to have a converged relaxation function. From Figure 5, the transition from the linear to logarithmic part is observed, permitting the estimation of τ

_{min}≈ 3–5 days. Since the longest relaxation time τ

_{max}is the time where the relaxation stops, from Figure 5, we estimate τ

_{max}≈ 350 days. The aforementioned estimations of τ

_{min}and τ

_{max}result in a value R

_{o}≈ 4.2 to 4.7. Using the observed values of $u\left(t\right),$ we estimate that an average value of displacement in the range from 300 to 600 days is about −26 mm, a value that could be used to approximate ${u}_{\infty}.$ In Figure 5, the observed displacement, along with the theoretical one, calculated using Equation (1), with S = 4.8 mm, ${u}_{\infty}=-26\mathrm{mm}$, τ

_{min}≈ 3.5 days, and τ

_{max}≈ 350 days (i.e., R

_{o}≈ 4.6), is presented as a very good agreement. Furthermore, the time evolution of displacement suggests a slope F

_{2}≈ −4.21 for the logarithmic part and an approximate value F

_{1}≈ −1.09 for the short linear early part of deformation. Comparing the above estimation with Equations (10) and (13), we conclude that, since τ

_{min}<< τ

_{max}, the scaling of the slopes leads to τ

_{min}≈ 3.8 days, in agreement with our estimation from Figure 5.

_{min}≈ 3.5 days and τ

_{max}≈ 350 days, respectively. A comparison of the observed and calculated values is given in Figure 7, along with a dichotomous line.

## 4. Discussion

^{α}, where α is an exponent related macroscopically with the fractal geometry of the subvolume distribution and the dynamical nature of relaxation [67,68,69]. This type of hierarchically constrained dynamics is able to derive the logarithmic relaxation frequently observed and associate it with thermally activated processes. We note that this behavior can result either from the parallel relaxation of subvolumes or by a sequential relaxation series of strongly correlated complex events [28].

_{m}is the maximum size of energy barrier, ν is an attempt frequency to jump over the energy barrier, and β = 1/k

_{B}T. Approximation of (17) leads to the following behavior in short, intermediate, and long time periods, respectively:

_{min}dominates, while for late time periods, the slow relaxation mechanism, related to the relaxation of volumes farther from the epicenter and present relaxation time close to τ

_{max}, contributes most. It is reasonable to assume that the maximum relaxation time depends on the perturbation that triggers the relaxation and, thus, on the deformed volume related with the main event, along with the geotectonic conditions in the deformed region, such as stress, temperature, and fluids.

_{min}could give information on the fastest relaxation mechanism, which is scaled with the smallest deformed subvolume, even if it is sometimes not clearly observed due to the immediate transition of the deformation process into the logarithmic range, resulting in an unreliable estimation of τ

_{min}.

_{min}≈ 3.5 days and τ

_{max}≈ 350 days, respectively.

## 5. Concluding Remarks

- A multiscale post-seismic relaxation mechanism based on the existence of a distribution in relaxation time is presented.
- Assuming an Arrhenius dependence of the relaxation time with the uniform distributed activation energy in a mesoscopic scale, we conclude there is a generic logarithmic type relaxation on a macroscopic scale.
- The hierarchically constrained dynamics model could be used to understand the evolution of post seismic relaxation.
- The model was applied in the case of 2015 Lefkas (Greece) Mw6.5 earthquake, where cGNSS time series were recorded in a station located in the vicinity of the epicentral area. The application of the present approach to the Lefkas event fits the observed displacements, implying a distribution of relaxation times in the range τ
_{min}≈ 3.5 days to τ_{max}≈ 350 days.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of exponential relaxation (see Equation (1)) with S = 5 mm, u

_{∞}= −25 mm and τ = 300 days.

**Figure 2.**The relaxation function R(t) (see Equation (6) in the text) for τ

_{min}= 5 days and τ

_{max}= 200, 400, 600 days.

**Figure 3.**Map of the central Ionian islands of Lefkas, Cephalonia, and Ithaca, showing the main faulting zones and marking the continuous GNSS stations (triangles) PONT and SPAN in Lefkas and VLSM in Cephalonia islands. Circles represent the earthquake epicenters (M > 3) after the 17 November 2015 main event. Red and blue vectors represent the cGNSS horizontal velocities before and after the 2015 earthquake, respectively. Black arrows show the co-seismic deformation. CTF: Cephalonia Transform Fault.

**Figure 4.**Time series of PONT continuous GNSS stations located in Lefkas island. Red line indicates the date of the 2015 seismic event.

**Figure 5.**The observed displacement, along with the theoretical calculated using Equation (1) (see text), with S = 4.8 mm, ${u}_{\infty}=-26\mathrm{mm}$, τ

_{min}≈ 3.5 days, and τ

_{max}≈ 350 days (i.e., R

_{o}≈ 4.6) is presented as a very good agreement.

**Figure 6.**The experimentally estimated Ω(t) function for the first 400 days of the post-seismic relaxation after the Lefkas main event, along with the calculated one using Equation (16) for τ

_{min}≈ 3.5 days and τ

_{max}≈ 350 days, respectively.

**Figure 7.**A comparison of the experimentally estimated Ω(t) function with the calculated values R(t)/Ro for the post-seismic relaxation period after the Lefkas main event for τ

_{min}≈ 3.5 days and τ

_{max}≈ 350 days, respectively (see text).

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Vallianatos, F.; Sakkas, V.
Multiscale Post-Seismic Deformation Based on cGNSS Time Series Following the 2015 Lefkas (W. Greece) Mw6.5 Earthquake. *Appl. Sci.* **2021**, *11*, 4817.
https://doi.org/10.3390/app11114817

**AMA Style**

Vallianatos F, Sakkas V.
Multiscale Post-Seismic Deformation Based on cGNSS Time Series Following the 2015 Lefkas (W. Greece) Mw6.5 Earthquake. *Applied Sciences*. 2021; 11(11):4817.
https://doi.org/10.3390/app11114817

**Chicago/Turabian Style**

Vallianatos, Filippos, and Vassilis Sakkas.
2021. "Multiscale Post-Seismic Deformation Based on cGNSS Time Series Following the 2015 Lefkas (W. Greece) Mw6.5 Earthquake" *Applied Sciences* 11, no. 11: 4817.
https://doi.org/10.3390/app11114817