Study of the development phase from homogeneous and heterogeneous dynamic ruptures

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SCARDEC STF database and earthquake development phase

Exhaustive catalogs of STFs (describing the time evolution of the moment rate ˙M ) can be built with two distinct methods which both use teleseismic data from the FDSN (Federation of Digital Seismograph Network). The first approach determines a finite fault model of the seismic source (in general for earthquakes with Mw > 7, (Ye et al., 2016; Hayes, 2017)) from which the absolute STF is computed. On the other hand, in the SCARDEC method (Vallée et al., 2011), seismic moment, focal mechanism, source depth and STFs are more directly obtained through a deconvolution process (see also Tanioka and Ruff (1997)). At each station and for each phase (P or S ), apparent source time functions (ASTFs) are extracted, whose shapes differ due to space-time source effects (Chounet et al., 2017). In order to take into account both this expected distorsion and possible outliers (due to nodal radiation, incorrect instrument response, etc.), SCARDEC database (Vallée and Douet, 2016) provides two representative STFs for each event. A mean STF is first obtained by correlating in time all P-wave ASTFs (less sensitive to space-time source effects than S-wave ASTFs), removing ASTFs far from the beam, and averaging the remaining ASTFs. The optimal STF is then chosen as the P-wave ASTF which is the closest to the mean STF. Such an optimal STF is unlikely to be among the most distorted ASTFs, and its shape is not affected by the smoothing present in the mean STF. The optimal STFs are therefore considered in this study. Deep (> 70 km) and pure strike-slip events are removed from the database due to their specific behavior (Houston, 2001) and the difficulty to robustly extract their P-wave STFs, respectively. The catalog is finally composed of 2221 earthquake STFs (from 1992 to 2017), whose magnitudes range from Mw 5.5 to Mw 9.1 (2011 Tohoku earthquake) and durations from 2 s to 120 s.
We aim here at isolating the development phase, i.e. the time period where STFs grow toward their peak moment rate Fm (that they reach at time Tm). Taking into account that the moment rate always flattens before reaching Fm, we do not consider the highest STF values to be part of the development phase: in the following, we only select the parts of the STF which are before Tm, and whose values are below 0.7 Fm. At low moment rate values, we would ideally track the development phase from its very beginning. However, SCARDEC STFs are retrieved by deconvolving the full P-waveform (under physical constraints such as STF positivity), and the STF fidelity at values much lower than Fm is therefore expected to be relatively low. As a result, we do not analyze here the development phase for STF values lower than 0.07 Fm. The value of these two selected lower and upper limits are not critical and other choices (e.g. starting at 0.05 Fm and stopping at 0.5 Fm) do not affect significantly the following results (see Figure 2.20 in Supplementary Materials).
In order to isolate the development phases in all cases, we consider the two following possible configurations of STFs. The simplest and most common case (representing 62% of the STF catalog) is illustrated by the STFs shown in Figure 2.1a) and 2.1b). Here, even when the STF does not grow monotonically toward its peak, there is a unique monotonic domain connecting the values between 0.07 Fm and 0.7 Fm. This specific section of the STFs, shown in red in Figure 2.1, is selected as the development phase. STFs with complex shapes however do not have such a unique monotonic domain (Figure 2.1c)). In this case, we work on the time interval defined by two times T0 and T1: T0 is the latest time preceding Tm when the STF is as low as 0.07 Fm and T1 is the latest time preceding Tm when the STF is not above 0.7 Fm. In the [T0T1] interval, there may be several local maxima Fp (p = 1, P), around which rupture is not considered to be in a development phase. The development phase is then selected as the combination of monotonic phases preceding each Fp, from the time when they exceed the largest value of all the preceding local maxima (or from T0 if p = 1) to the time where they reach 0.7 Fp. As a consequence, if one of the local peak values before Fp is larger than 0.7 Fp, the monotonic phase preceding Fp is not considered. We finally select the monotonic phase up to T1.

Seismic moment acceleration within the development phase

Once the development phase is extracted for each STF, we aim at characterizing it without using hypocentral time information, in order to quantify how rupture develops independently of when rupture develops. Formally, we look for the moment evolution of the development phase Md where Md(t) = M(t + Td), Td being the unknown time at which the development phase starts. A way to characterize Md is to consider a discrete sampling of prescribed moment rates ( ˙Md)i, and to compute the seismic moment acceleration (STF slope) each time that the development phase crosses ( ˙Md)i. To do so, we consider 4 different values of ( ˙Md)i (i = 1, 40), from 1017 to 1019 Nm.s−1, in order to sample the development phase of most earthquakes. Outside of this range, moment rates are either mostly below 0.07 Fm or above 0.7 Fm, and cross only a few development phases. As further documented later, the maximum considered moment rate (1019 Nm.s−1) is typically reached 6s after the beginning of the development phase for monotonically growing STFs. In terms of magnitude, the smallest earthquakes of the SCARDEC database (Mw = 5.5) can be analyzed by this sampling, and only the largest earthquakes (Mw > 8.4) are systematically excluded. Figure 2.1 illustrates the method for three STFs and four moment rates ( ˙Md)i (green dashed lines). Low values of moment rate are mostly sampled by small events (as they will lie below 0.07 Fm for large ones) and high values of moment rate are mostly sampled by large events (as they will lie above 0.7 Fm for small ones). However, this general behavior does not prevent us from sampling a large range of magnitudes at a given moment rate. As shown in the example of Figure 2.1, the moment acceleration of the development phase at the ( ˙Md)15 level can be computed from Mw = 6.2 to Mw = 6.8.

Variability and magnitude-independent behavior

Such slope measurements can be first used to detect a potential magnitude-dependent behavior, in which the slope measured when the development phase crosses prescribed moment rates would be for instance steeper for larger events. For the Ni development phases crossing ( ˙Md)i, we compute the slope values ( ¨Md)ij(j = 1,Ni) as a function of Mw, to observe whether or not a magnitude-dependent signal appears. Figure 2.2 shows an example of the 892 ( ¨Md)15j values for ( ˙Md)15 = 5.2×1017 Nm.s−1. The following analysis of ¨Md values with respect to Mw has to be done with care, because a given ( ˙Md)i value does not sample equally well all magnitude ranges (Section 2.2.2), as also illustrated in value decreases both towards low Mw (only impulsive STFs reach ( ˙Md)i) and towards high Mw (only STFs with relatively low Fm have ( ˙Md)i in their development phase).
As a consequence, ( ¨Md)ij values are expected to be biased toward high values for small magnitude events, as confirmed by Figure 2.2. We thus focus on the Mw domain where most of the development phases cross the chosen ( ˙Md)i (for example between Mw = 6.3 and Mw = 7.0 in the case shown in Figure 2.2).
same behavior is observed for all the other prescribed ( ˙Md)i (Figures 2.10 to 2.19 of the Supplementary Materials). This shows that if a magnitude-dependent signal exists, it is fully dominated by the intrinsic variability of the development phase. This means that when an earthquake develops and reaches a given moment rate ( ˙Md)i, moment acceleration cannot be used as an indicator of the final magnitude (only a lower bound can of course be estimated based on the seismic moment already released). This observation may appear different from the recent results of Melgar and Hayes (2019), who extracted a magnitude-dependent signal from STF accelerations (using also the SCARDEC catalog).
Their approach is however fundamentally different as they simply computed an averaged moment acceleration by dividing the moment rate from the rupture time, at several prescribed rupture times ( = 2, 5, 10, 20 s). Using this definition, they observe an increase of the moment acceleration with the final event magnitude, clearly appearing for equal to 10 s and 20 s. In such an analysis, there is however no guarantee that the earthquake at is still in its development phase, particularly when is a significant fraction of the global earthquake duration. As an example, 20 s is a significant fraction of the global duration of an Mw = 8 earthquake (whose average global duration is about 60 s, e.g. Vallée (2013)). It is therefore not uncommon, at 20 s, that Mw = 8 earthquakes STFs flatten as they approach their peak moment rate (and some of them may have already passed it). As a result, on average, acceleration can be understood to be statistically lower than for a Mw = 9 earthquake, for which the peak always occurs far after 20 s. Melgar and Hayes (2019) results likely reflect the magnitude-dependent shape of the earthquake STFs, at a macroscopic scale, while we are here specifically studying their fast growing parts.

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Observational evidence of a power law between ¨M and ˙M

The magnitude independency derived in the previous section justifies the combined use of ( ¨Md)ij for all values of i, in order to determine a generic behavior of the rupture development. Figure 2.3 represents (in log-log scale) all the moment acceleration values as a function of the moment rate values (yellow dots). Direct observation in Figure 2.3 reveals that ¨Md grows with ˙Md, which first implies that the time evolution of the moment rate in the development phase cannot be linear. In order to quantify the general behavior, we try to fit our observations with a power law of the type ¨Md = ˙M m d . Using the method detailed in Section 2.7.1 of the Supplementary Materials, a linear fit (in log-log scale) leads to values of m = 0.63±0.015 and log() = 6.7±0.28 at the 90% confidence interval, with a correlation coefficient of 0.8 (Figure 2.3).

Table of contents :

1 L’apport des fonctions source : de la description globale des séismes à celle du détail de leurs processus 
1.1 Généralités sur la source sismique
1.2 La méthode SCARDEC : accès à un catalogue de fonctions source pour une description complète de la rupture
1.3 Propriétés globales et transitoires extraites des catalogues de fonctions source et leur lien avec les modèles de rupture
1.3.1 Paramètres de source macroscopiques
1.3.2 Propriétés transitoires et déterminisme
1.3.3 Modèles cinématiques du développement de la rupture
1.3.4 Modèles en cascade et implications pour le développement de la rupture
2 How does seismic rupture accelerate? Observational insights from earthquake source time functions 
2.1 Introduction
2.2 Moment acceleration in the development phase
2.2.1 SCARDEC STF database and earthquake development phase
2.2.2 Seismic moment acceleration within the development phase
2.2.3 Variability and magnitude-independent behavior
2.3 Time evolution of the development phase
2.3.1 Observational evidence of a power law between ¨M and ˙M
2.3.2 Power-law time exponent of the development phase
2.3.3 Implications for earthquake source physics
2.4 Different behaviors between development phase and early rupture stage
2.5 Conclusion
2.6 Acknowledgments
2.7 Supplementary materials
2.7.1 Statistical analysis for m, , nd and d values
2.7.2 Setting-up of the synthetic STFs catalog
3 Analysis of rupture parameters during the development phase in kinematic source models 
3.1 Stress drop analysis with SRC kinematic inversions catalog
3.2 Source characteristics of a circular crack model with rupture velocity variability .
3.2.1 Analytical slip solution and generation of a random temporal evolution of the rupture velocity
3.2.2 Kinematics of the modified crack model
3.2.3 Effect of rupture velocity variability on synthetic Source Time Functions
3.2.4 Discussion and conclusion
3.3 Rupture properties of a kinematic fractal k−2 source model
3.3.1 Ruiz Integral Kinematic (RIK) model setup
3.3.2 Fault parametrization and global source properties
3.3.3 Rise-time evolution and its effect on synthetic Source Time Functions.
3.3.4 Correlation between slip velocity and rupture velocity
4 Study of the development phase from homogeneous and heterogeneous dynamic ruptures 
4.1 Dynamic view of an earthquake rupture and multi-scaling numerical model .
4.1.1 Stress and energy budget of an earthquake
4.1.2 Fracture surface energy and slip-weakening law
4.1.3 Formulation of the dynamic problem and numerical method
4.2 Rupture propagation on continuous and discontinuous growing fracture surface energy
4.2.1 Continuous fracture surface energy
4.2.2 Discontinuous fracture surface energy
4.3 Effect of heterogeneous distribution of fracture surface energy on the development phase
4.3.1 Multiscale fractal Dc distribution of circular patches
4.3.2 Complex rupture propagation of largest events
4.3.3 Simulated STFs extracted from dynamic simulations for heterogeneous Dc distribution
4.4 Combination of heterogeneities from random initial stress field and fractal fracture surface energy
4.4.1 Random spatial initial stress field
4.4.2 Properties of the development phase for models combining heterogeneous Dc and 0 maps
4.4.3 Discussion and conclusion
Conclusions et Perspectives 
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