Statistical Testing of Theoretical Rupture Models Against Kinematic Inversions

Gilead Wurman, Richard M. Allen, Douglas S. Dreger, and David D. Oglesby (UC Riverside)


The process by which earthquake ruptures initiate and propagate is usually expressed as one of two broadly-defined mechanisms: the cascade model and the preslip model. There is a diversity of modeling results that alternately support either cascade- or preslip-type rupture. At least some of the disagreement between the studies may be due to the high degree of variability among kinematic slip inversions, even for the same event. The recent development of the SRCMOD database makes it possible to examine many fault models in a statistical fashion to suppress the effects of this variability. Although the slip at the beginning of rupture is poorly resolved in kinematic inversions, using such a large number of events allows us to make first-order observations of any relationships.


We examine 152 inversions of 80 different earthquakes in the SRCMOD database ( as well as 7 teleseismic and 8 joint geodetic/teleseismic inversions provided by M. E. Pritchard (Pritchard et al., 2006, 2007; Pritchard and Fielding, 2008; Loveless et al., in review), for a total of 167 inversions and 95 events. We take the final slip distribution for each inversion and reconstruct the time- evolution of slip on the fault from rupture time and rise time information. We calculate the moment release within a given time window by summing the moment based on inferred slip at each grid point. For models with point-wise rupture time or rise time data, we initiate slip on each grid point at the associated rupture time, and increase slip to the final amount in a linear ramp over the associated rise time. In models where one or both of these parameters is not recorded point-wise, we take the reported average rupture velocity or rise time and assume a rupture front expanding isotropically from the hypocenter.

Relationship between early and final moment

The cascading hypothesis implies that at any given time after rupture initiation all earthquakes have the same magnitude. We can approximate this magnitude via the relation $M_{0}=\mu SD=\mu\cdot\pi(V_{R}t)^{2}\cdot\alpha V_{R}t=\pi\mu\alpha (V_{R}t)^{3}$ where $\mu=3\times 10^{11}$ dyne-cm, $S=\pi (V_{R}t)$ is the fault area after $t$ seconds assuming a rupture velocity $V_R=3\times 10^{5}$ cm/s, and $D=\alpha V_{R}t$ is the mean slip with $\alpha=10^{-5}$. We find that a source duration of 1 second corresponds to a moment of approximately $2.5\times 10^{23}$ dyne-cm, or $M_{w}$ 4.9. Thus in the cascading end-member case, any earthquake larger than magnitude 5 should look like a magnitude 5 at one second after nucleation. We can visualize this null hypothesis in Figure 2.30a. The solid diagonal line represents the limit in which the initial magnitude after 1 second is equal to the final magnitude, meaning the rupture has propagated to completion. Since there is no way for the initial magnitude to exceed the final magnitude, no points may lie above this solid line. This can potentially introduce a spurious positive slope to the data, and we minimize this by culling all data points for which the final magnitude is less than the reference magnitude. The hypothesis for a deterministic model is that there will be some positive scaling of the magnitude at 1 second with the final magnitude of the earthquake, yielding a positive slope to the data.

Figure 2.30b-e shows the initial magnitude plotted against the final magnitude for each event for four time windows ranging from 1 second to 8 seconds. We manually pick outliers, and exclude any events for which the initial magnitude is 99% or more of final magnitude, as those events have effectively terminated by the end of the time window. We also exclude all events with a final magnitude less than the reference magnitude for the null hypothesis as described above. The null hypothesis of cascading rupture can be rejected with greater than 95% confidence. The slope of the best-fit lines for all time windows between 1 and 10 seconds are strongly positive, suggesting some degree of non-self-similar behavior for these models. For time windows between 8 and 10 seconds the confidence is not as high as for time windows between 1 and 7 seconds. As rupture evolves, it experiences progressively more of the fault plane's heterogeneities and therefore has progressively more information about the likely final size of the earthquake. We therefore expect the initial magnitude to scale more strongly with final magnitude for longer time windows. One explanation for the degradation in scaling for longer time windows is that more and more events are being excluded due to having completed rupture, thus reducing the number of data points available for analysis. In Figure 2.30b-d, the number of points used for the fit varies between 112 and 128, and by 8 seconds (Figure 2.30e) that number has fallen to 80. Another possibility is that longer time windows afford greater resolution of the slip within the time window, implying that the strong correlation observed for shorter time windows is a spurious result of poorly resolved slip in such short time spans. The influence of poorly resolved slip can be approximated visually by noting the open symbols, which represent models for which the time window was either shorter than the average rise time for the model or for which only one grid element had begun slipping in that time window.

We attempt to reduce the influence of poorly resolved slip by disregarding all of the ``open'' data points from Figure 2.30 which represent cases where the slip is likely to be particularly poorly resolved owing to the time window being too short. In addition, we recalculate both the initial and final magnitude for each point, disregarding any slip which is less than 10% of the peak slip for the model. This is to account for the fact that slip below 10% of peak slip is generally regarded as being poorly resolved in kinematic inversions and thus an unstable component of the slip models. Remarkably, the correlation between early and final magnitude is now even stronger, with the null hypothesis being rejected at greater than 99% confidence for all time windows. This suggests that poor resolution of slip in short time windows is not generating a spurious correlation between early and final magnitude. Rather, the analysis suggests that the decreasing number of data points in longer time windows (owing to more ruptures having run to completion) is primarily responsible for the weaker correlation for 8-10 second time windows seen in Figure 2.30.


Figure 2.30: (a) Graphical expectation for the null hypothesis, that earthquakes are cascading ruptures. Dotted line represents the theoretical magnitude after 1 second. (b-e) Initial vs. final magnitude for time windows of 1, 2, 4 and 8 seconds. Dashed line represents the least-squares best fit to the data.
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We observe a strong scaling of early slip and magnitude with the final magnitude of these events. This result is inconsistent with the hypothesis that earthquakes are cascading rupture phenomena. After filtering the data the scaling remains robust, and in fact is more prominent, indicating that poor resolution of early slip is not the cause of the observed scaling. Given these findings, we must allow for the possibility that earthquakes are not purely cascading phenomena, and that magnitude is at least in part influenced by processes in the early part of the rupture process.


We thank Matt Pritchard for providing 15 of the slip models used in this study.


Loveless, J.P., M.E. Pritchard and N. Kukowski, Testing mechanisms of seismic segmentation with slip distributions from recent earthquakes along the Andean margin, Tectonophysics, in review.

Pritchard, M.E. and E.J. Fielding, A study of the 2006 and 2007 earthquake sequence of Pisco, Peru, with InSAR and teleseismic data, Geophys. Res. Lett., 35, L09308, doi:10.1029/2008GL033374, 2008.

Pritchard, M.E., C. Ji and M. Simons, Distribution of slip from 11 $M_{w} >$ 6 earthquakes in the northern Chile subduction zone, J. Geophys. Res., 111, B10302, doi:10.1029/2005JB004013, 2006.

Pritchard, M.E., E.O. Norabuena, C. Ji, R. Boroschek, D. Comte, M. Simons, T. Dixon and P.A. Rosen, Geodetic, teleseismic, and strong motion constraints on slip from recent southern Peru subduction zone earthquakes, J. Geophys. Res., 112, B03307, doi:10.1029/2006JB04294, 2007.

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