A new estimation of the decay of aftershock density with distance to the mainshock

Abstract

[1] We investigate how aftershocks are spatially distributed relative to the mainshock. Compared to previous studies, ours focuses on earthquakes causally related to the mainshock rather than on aftershocks of previous aftershocks. We show that this distinction can be made objectively but becomes uncertain at long time scales and large distances. Analyzing a regional earthquake data set, it is found that, at time t following a mainshock of magnitude m, the probability of finding an aftershock at distance r relative to the mainshock fault decays as r −g , where g is typically between 1.7 and 2.1 for 3 ≤ m < 6 and is independent of m, for r less than 10 to 20 km and t less than 1 day. Uncertainties on this probability at larger r and t do not allow for a correct estimation of this spatial decay. We further show that a static stress model coupled with a rate-and-state friction model predicts a similar decay, with an exponent g = 2.2, in the same space and time intervals. This suggests that static stress changes could explain the repartition of aftershocks around the mainshock even at distances larger than 10 times the rupture length.Citation: Marsan, D., and O. Lengliné (2010), A new estimation of the decay of aftershock density with distance to the mainshock,

Triggering cascades and statistical properties of aftershocks

“…In addition to the temporal characteristics of aftershock sequences discussed in the sections above, we can enrich the description of the aftershock statistics by studying the spatial distribution of aftershocks relative to their main shock. This distribution is a signature of the triggering process and, thus, plays an important role in discriminating between potential candidates for this mechanism [ Felzer and Brodsky , ; Gomberg and Felzer , ; Lippiello et al , ; Marsan and Lengliné , ; Richards‐Dinger et al , ; Powers and Jordan , ].…”

Reply to “Comment on ‘A model of earthquake triggering probabilities and application to dynamic deformations constrained by ground motion observations’ by Ross Stein”

“…Since the main shock selection criteria only eliminated larger main shocks within 100 km (see Methods), there is contamination from aftershocks of larger mainshocks beyond 100 km, as well as contamination from other sources of background seismicity.” Careful inspection of Figure S5 reveals that in Figure 1 no points from the M5–6 data set were excluded within the 0–12 km distance range (chosen for reasons discussed above and in the caption) and the density range plotted. Stein states that the selection criteria applied to the M5–6 data set render “the M5–6 portion of the Gomberg and Felzer [2008] Figure 1 irreproducible.” We note that this does not appear to be supported by the fact that Marsan and Lengliné [2010] had no difficulty duplicating the results of Felzer and Brodsky [2006]. The critical point is being able to determine accurate locations for the main shock fault planes; Marsan and Lengliné [2010] developed an automated algorithm for fault plane location and found that the aftershocks of a larger set of M5–6 main shocks displayed a very similar aftershock density decay with distance to that shown in our figure.…”

“…Stein states that the selection criteria applied to the M5–6 data set render “the M5–6 portion of the Gomberg and Felzer [2008] Figure 1 irreproducible.” We note that this does not appear to be supported by the fact that Marsan and Lengliné [2010] had no difficulty duplicating the results of Felzer and Brodsky [2006]. The critical point is being able to determine accurate locations for the main shock fault planes; Marsan and Lengliné [2010] developed an automated algorithm for fault plane location and found that the aftershocks of a larger set of M5–6 main shocks displayed a very similar aftershock density decay with distance to that shown in our figure. We emphasize again that the point of Gomberg and Felzer [2008] was not to reconfirm or debate the validity of Felzer and Brodsky 's [2006] approach and results, but rather to interpret them.…”

An “Estimate & Score Algorithm” for simultaneous parameter estimation and reconstruction of incomplete data on social networks

“…In the presence of no unknown events, there are both parametric [12] and nonparametric [25][26][27][28] ways to model the underlying stochastic process on each edge of the social network. For this work, we chose a parametric form for the triggering density to validate the model but the results could easily be extended to the nonparametric case.…”