Importance of direct and indirect triggered seismicity

Using the simple ETAS branching model of seismicity, which assumes that each earthquake can trigger other earthquakes, we quantify the role played by the cascade of triggered seismicity in controlling the rate of aftershock decay as well as the overall level of seismicity in the presence of a constant external seismicity source. We show that, in this model, the fraction of earthquakes in the population that are aftershocks is equal to the fraction of aftershocks that are indirectly triggered and is given by the average number of triggered events per earthquake. Previous observations that a significant fraction of earthquakes are triggered earthquakes therefore imply that most aftershocks are indirectly triggered by the mainshock.Comment: Latex document of 17 pages + 2 postscript figure

Statistical analysis of rockfall volume distributions: implications for rockfall dynamics.

International audienceWe analyze the volume distribution of natural rockfalls on different geological settings (i.e., calcareous cliffs in the French Alps, Grenoble area, and granite Yosemite cliffs, California Sierra) and different volume ranges (i.e., regional and worldwide catalogs). Contrary to previous studies that included several types of landslides, we restrict our analysis to rockfall sources which originated on subvertical cliffs. For the three data sets, we find that the rockfall volumes follow a power law distribution with a similar exponent value, within error bars. This power law distribution was also proposed for rockfall volumes that occurred along road cuts. All these results argue for a recurrent power law distribution of rockfall volumes on subvertical cliffs, for a large range of rockfall sizes (102–1010 m3), regardless of the geological settings and of the preexisting geometry of fracture patterns that are drastically different on the three studied areas. The power law distribution for rockfall volumes could emerge from two types of processes. First, the observed power law distribution of rockfall volumes is similar to the one reported for both fragmentation experiments and fragmentation models. This argues for the geometry of rock mass fragment sizes to possibly control the rockfall volumes. This way neither cascade nor avalanche processes would influence the rockfall volume distribution. Second, without any requirement of scale-invariant quenched heterogeneity patterns, the rock mass dynamics can arise from avalanche processes driven by fluctuations of the rock mass properties, e.g., cohesion or friction angle. This model may also explain the power law distribution reported for landslides involving unconsolidated materials. We find that the exponent values of rockfall volume on subvertical cliffs, 0.5 ± 0.2, is significantly smaller than the 1.2 ± 0.3 value reported for mixed landslide types. This change of exponents can be driven by the material strength, which controls the in situ topographic slope values, as simulated in numerical models of landslides [Densmore et al., 1998; Champel et al., 2002]. INDEX TERMS: 5104 Physical Properties of Rocks: Fracture and flow; 1815 Hydrology: Erosion and sedimentation; 8122 Tectonophysics: Dynamics, gravity and tectonics

Bath's law Derived from the Gutenberg-Richter law and from Aftershock Properties

The empirical Bath's law states that the average difference in magnitude between a mainshock and its largest aftershock is 1.2, regardless of the mainshock magnitude. Following Vere-Jones [1969] and Console et al. [2003], we show that the origin of Bath's law is to be found in the selection procedure used to define mainshocks and aftershocks rather than in any difference in the mechanisms controlling the magnitude of the mainshock and of the aftershocks. We use the ETAS model of seismicity, which provides a more realistic model of aftershocks, based on (i) a universal Gutenberg-Richter (GR) law for all earthquakes, and on (ii) the increase of the number of aftershocks with the mainshock magnitude. Using numerical simulations of the ETAS model, we show that this model is in good agreement with Bath's law in a certain range of the model parameters.Comment: major revisions, in press in Geophys. Res. Let

Are Aftershocks of Large Californian Earthquakes Diffusing?

We analyze 21 aftershock sequences of California to test for evidence of space-time diffusion. Aftershock diffusion may result from stress diffusion and is also predicted by any mechanism of stress weakening. Here, we test an alternative mechanism to explain aftershock diffusion, based on multiple cascades of triggering. In order to characterize aftershock diffusion, we develop two methods, one based on a suitable time and space windowing, the other using a wavelet transform adapted to the removal of background seismicity. Both methods confirm that diffusion of seismic activity is very weak, much weaker than reported in previous studies. A possible mechanism explaining the weakness of observed diffusion is the effect of geometry, including the localization of aftershocks on a fractal fault network and the impact of extended rupture lengths which control the typical distances of interaction between earthquakes.Comment: latex file of 34 pages, 15 postscript figures, minor revision. In press in J. Geophys. Re

Relation between stress heterogeneity and aftershock rate in the rate-and-state model

We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate-and-state model of Dieterich [1994].We show that an exponential stress distribution Pt(au) ~exp(-tautau_0) gives an Omori law decay of aftershocks with time ~1/t^p, with an exponent p=1-A sigma_n/tau_0, where A is a parameter of the rate-and-state friction law, and \sigma_n the normal stress. Omori exponent p thus decreases if the stress "heterogeneity" tau_0 decreases. We also invert the stress distribution P(tau) from the seismicity rate R(t), assuming that the stress does not change with time. We apply this method to a synthetic stress map, using the (modified) scale invariant "k^2" slip model [Herrero and Bernard, 1994]. We generate synthetic aftershock catalogs from this stress change.The seismicity rate on the rupture area shows a huge increase at short times, even if the stress decreases on average. Aftershocks are clustered in the regions of low slip, but the spatial distribution is more diffuse than for a simple slip dislocation. Because the stress field is very heterogeneous, there are many patches of positive stress changes everywhere on the fault.This stochastic slip model gives a Gaussian stress distribution, but nevertheless produces an aftershock rate which is very close to Omori's law, with an effective p<=1, which increases slowly with time. We obtain a good estimation of the stress distribution for realistic catalogs, when we constrain the shape of the distribution. However, there are probably other factors which also affect the temporal decay of aftershocks with time. In particular, heterogeneity of A\sigma_n can also modify the parameters p and c of Omori's law. Finally, we show that stress shadows are very difficult to observe in a heterogeneous stress context.Comment: In press in JG

Importance of small earthquakes for stress transfers and earthquake triggering

We estimate the relative importance of small and large earthquakes for static stress changes and for earthquake triggering, assuming that earthquakes are triggered by static stress changes and that earthquakes are located on a fractal network of dimension D. This model predicts that both the number of events triggered by an earthquake of magnitude m and the stress change induced by this earthquake at the location of other earthquakes increase with m as \~10^(Dm/2). The stronger the spatial clustering, the larger the influence of small earthquakes on stress changes at the location of a future event as well as earthquake triggering. If earthquake magnitudes follow the Gutenberg-Richter law with b>D/2, small earthquakes collectively dominate stress transfer and earthquake triggering, because their greater frequency overcomes their smaller individual triggering potential. Using a Southern-California catalog, we observe that the rate of seismicity triggered by an earthquake of magnitude m increases with m as 10^(alpha m), where alpha=1.00+-0.05. We also find that the magnitude distribution of triggered earthquakes is independent of the triggering earthquake magnitude m. When alpha=b, small earthquakes are roughly as important to earthquake triggering as larger ones. We evaluate the fractal correlation dimension of hypocenters D=2 using two relocated catalogs for Southern California, and removing the effect of short-term clustering. Thus D=2alpha as predicted by assuming that earthquake triggering is due to static stress. The value D=2 implies that small earthquakes are as important as larger ones for stress transfers between earthquakes.Comment: 14 pages, 7 eps figures, latex. In press in J. Geophys. Re

Rupture et instabilités : sismicité et mouvements de terrain

We analyze the rupture associated with two natural phenomena, earthquakes and landslides. In the first part, we study a simple stochastic model of seismicity, based on the two best-established empirical laws for earthquakes, the power law decay of seismicity after an earthquake and the power law distribution of earthquake energies. This model assumes that each earthquake can trigger aftershocks, with a rate increasing with its magnitude. The seismicity rate is in this model the result of the whole cascade of direct and secondary aftershocks. We analyze the space-time organization of the seismic activity in the different sub- and super-critical regimes of the model. We show that this simple model can reproduce many properties of real seismicity, such as the variability of the aftershocks decay law, the acceleration of the seismic activity before large earthquakes, the diffusion of aftershocks, the migration of foreshocks, and the modification of the magnitude distribution before large earthquakes. We find that this model provides a good predictability for a fraction of earthquakes that are triggered by a previous large event. We demonstrate the essential role played by the cascades of earthquake triggering at all scales in controlling the seismic activity. The second part is devoted to the analysis of landslides. A study of several catalogs of rock falls shows that the distribution of rockfall volumes follows a power-law distribution, arising either from the scale invariant heterogeneity of the rock-mass, or from the dynamics of a self-organized critical system. We propose that the precursory acceleration of the displacement before some catastrophic landslides can be reproduced using a slider block model with a rate-and-state dependent friction law. Application of this model to two landslide slip histories suggests that we can distinguish an acceleration of the sliding velocity in the stable regime from an unstable acceleration leading to a catastrophic collapse.On s'intéresse à la rupture associée à deux classes de phénomènes naturels, les séismes et les instabilités gravitaires. Pour les séismes, on étudie un modèle stochastique de sismicité, basé sur les deux lois les mieux établies pour la sismicité, la décroissance en loi de puissance du taux de sismicité après un séisme, et la distribution en loi de puissance des énergies des séismes. Dans ce modèle, on suppose que chaque séisme déclenche d'autres séismes, dont le nombre augmente avec l'énergie du choc principal. Le taux de sismicité global résulte de la cascade de déclenchements de séismes directs et indirects. On analyse l'organisation spatiale et temporelle de la sismicité dans les différents régimes sous- et sur-critiques du modèle. Ce modèle permet de reproduire un grand nombre de propriétés de l'activité sismique, telles que la variabilité de la décroissance des séquences d'aftershocks, l'augmentation de l'activité sismique avant un séisme, la diffusion des aftershocks, la migration des foreshocks et la modification de la distribution des magnitudes avant un séisme. On obtient avec ce modèle une bonne prédictabilité d'une fraction des séismes qui sont déclenchés à court terme après un grand séisme. Nos résultats démontrent le rôle essentiel des cascades de déclenchement de séismes a toutes les échelles dans l'organisation de l'activité sismique. Concernant l'étude des instabilités gravitaires, une étude statistique de plusieurs catalogues d'éboulements rocheux montre que la distribution des volumes de roches suit une loi de puissance. On propose que cette distribution en loi de puissance résulte soit de l'hétérogénéité initiale de la matrice rocheuse, soit de la dynamique d'un système critique auto-organisé. Certains glissements de terrains sont précédés par une accélération de la vitesse de glissement avant la rupture finale. On peut reproduire l'évolution temporelle du glissement a l'aide d'un modèle de bloc rigide avec une loi de friction dépendante de la vitesse de glissement et de l'état de contact entre le bloc et sa surface de glissement. L'analyse de deux glissements de terrains avec ce modèle permet de distinguer une accélération du glissement dans le régime stable, d'une accélération instable qui évolue vers une rupture catastrophique

Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws

The inverse Omori law for foreshocks discovered in the 1970s states that the rate of earthquakes prior to a mainshock increases on average as a power law ~ 1/(t_c-t)^p' of the time to the mainshock occurring at t_c. Here, we show that this law results from the direct Omori law for aftershocks describing the power law decay ~ 1/(t-t_c)^p of seismicity after an earthquake, provided that any earthquake can trigger its suit of aftershocks. In this picture, the seismic activity at any time is the sum of the spontaneous tectonic loading and of the activity triggered by all preceding events weighted by their corresponding Omori law. The inverse Omori law then emerges as the expected (in a statistical sense) trajectory of seismicity, conditioned on the fact that it leads to the burst of seismic activity accompanying the mainshock. The often documented apparent decrease of the b-value of the GR law at the approach to the main shock results straightforwardly from the conditioning of the path of seismic activity culminating at the mainshock. In the space domain, we predict that the phenomenon of aftershock diffusion must have its mirror process reflected into an inward migration of foreshocks towards the mainshock. In this model, foreshock sequences are special aftershock sequences which are modified by the condition to end up in a burst of seismicity associated with the mainshock.Comment: Latex document of 35 pages, 10 figure

Sub-critical and Super-critical Regimes in Epidemic Models of Earthquake Aftershocks

We present an analytical solution and numerical tests of the epidemic-type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks and mainshocks on the same footing. The occurrence rate of aftershocks triggered by a single mainshock decreases with the time from the mainshock according to the modified Omori law K/(t+c)^p with p=1+theta. A mainshock at time t=0 triggers aftershocks according to the local Omori law, that in turn trigger their own aftershocks and so on. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a sub-critical regime n<1 to a super-critical regime n>1. In the sub-critical regime, we recover and document the crossover from an Omori exponent 1-theta for t<t* to 1+theta for t<t* found previously in [Sornette and Sornette, 1999a] for a special case of the ETAS model. In the super-critical regime n>1 and theta>0, we find a novel transition from an Omori decay law with exponent 1-theta fot t<t* to an explosive exponential increase of the seismicity rate fot t>t*. The case theta<0 yields an infinite n-value. In this case, we find another characteristic time tau controlling the crossover from an Omori law with exponent 1-theta for t<tau, similar to the local law, to an exponential increase at large times. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (i) the suggestion that a small p-value may be a precursor of a large earthquake, (ii) the relative seismic quiescence sometimes observed before large aftershocks, (iii) the positive correlation between b and p-values, (iv) the observation that great earthquakes are sometimes preceded by a decrease of b-value and (v) the acceleration of the seismicity preceding great earthquakes.Comment: Latex document of 41 pages + 6 eps figures + 1 tabl

Sea ice thickness and elastic properties from the analysis of multimodal guided wave propagation measured with a passive seismic array

Field data are needed for a better understanding of sea ice decline in the context of climate change. The rapid technological and methodological advances of the last decade have led to a reconsideration of seismic methods in this matter. In particular, passive seismology has filled an important gap by removing the need to use active sources. We present a seismic experiment where an array of 247 geophones was deployed on sea ice, in the Van Mijen fjord near Sveagruva (Svalbard). The array is a mix of 1C and 3C stations with sampling frequencies of 500 and 1000 Hz. They recorded continuously the ambient seismic field in sea ice between 28 February and 26 March 2019. Data also include active acquisitions on 1 and 26 March with a radar antenna, a shaker unit, impulsive sources, and artificial sources of seismic noise. This data set is of unprecedented quality regarding sea ice seismic monitoring, as it also includes thousands of microseismic events recorded each day. By combining passive seismology approaches with specific array processing methods, we demonstrate that the multimodal dispersion curves of sea ice can be calculated without an active source and then used to infer sea ice properties. We calculated an ice thickness, Young's modulus, and Poisson's ratio with values h=543 cm, E=3.90.15 GPa, and nu=0.340.02 on 1 March, and h=583 cm, E=4.4 +/- 0.15 GPa, and nu=0.32 +/- 0.02 on 5 March. These values are consistent with in situ field measurements and observations.Peer reviewe