Chapter 5 Chapter 5 Static Stress Change Triggering of Earthquakes
5.2 Method
The percent of aftershocks consistent with Coulomb stress triggering, also called the Coulomb Index (CI), is computed for the first month of aftershocks of each sequence.
This provides a simple quantitative measurement of the performance of the Coulomb stress triggering model. The CI of the observed sequence is also compared to the
distribution of CI values for a suite of random, synthetic sequences. If the CI of the observed sequence is large enough to be sufficiently unlikely to have come from the distribution for random sequences, the success of the model is considered statistically significant.
5.2.1 Observed Sequences
The aftershock data sets for Landers and Northridge comprise all events in the fo- cal mechanism catalog, described in Chapter 1, that occurred within the first month following the mainshock, including events which are not strictly aftershocks. Stress change due to the mainshock at the location of each aftershock is calculated by mod- eling the mainshock slip as a dislocation in an elastic halfspace, using the program ELFPOINT by Robert W. Simpson, based on the subroutines of Okada [1992]. The thrust faulting Mw=6. 7 Northridge mainshock is modeled using the slip distribu- tion of Wald et al. [1996]; and the right-lateral Mw=7.3 Landers mainshock using the slip distribution of Wald and Heaton [1994]. Two other events of the Landers sequence, the Mw=6.1 Joshua Tree preshock and the Mw=6.2 Big Bear aftershock were modeled as square dislocations with 0.35 m slip and 12-15 km diameter.
The computed stress change tensor is resolved on to both nodal planes of each focal mechanism, and the Coulomb stress change is found using Equation 5.1. The change in pore pressure is found from the relationship [Rice and Cleary, 1976]:
(5.3)
This equation assumes that the system is undrained. Since pore pressure diffusion is very slow, on a length scale of about 1 m over 5 years, the undrained assumption seems reasonable when studying the first few years of an aftershock sequence. I use the parameter values {3=0.7, and J..L=0.6 [Byerlee, 1978].
The CI is defined as the percent of events consistent with triggering. This includes all events which experienced a positive 6.CS on both nodal planes, and one-half of the events with a positive 6.CS on only one plane, since there is a 50% chance that the
nodal plane with positive D.CS is the true nodal plane. Error bars for the CI are found using a bootstrap technique. The data set is resampled 100 times, with replacement, and the CI for the resampled sequences found. The mechanisms are also varied within their formal 10" confidence regions. The fraction of events with only one positive D.C S plane that is counted towards the CI is chosen from a binomial distribution for the data set size and an expected value of 0.5.
The CI is found for the entire data set, and also for subsets based on event magnitude, distance from the mainshock fault, magnitude of stress change, time, and, for the Northridge sequence, hypocentral depth. The magnitude of the stress change is defined as !:lTmax, the maximum shear stress change on a plane of any orientation.
5.2.2 S y nthetic Sequences
The choice of random synthetic sequences, which are to be compared with the ob- served sequences, is inevitably subjective. Many studies use synthetic sequences in which the locations of the earthquakes are the same as in the observed sequence, but the mechanisms are redistributed. However, these synthetic sequences don't sam- ple regions where no aftershocks occurred, and a truly random sequence should be allowed to sample all of the relevant space. Other studies use pre-event seismicity as a comparison data set. However, the geometry of pre-event seismicity is usually very different from the geometry of the aftershock sequence, and these geometrical differences may map into differences in CI which are unrelated to triggering. Also, pre-event seismicity may be affected by processes leading up to the mainshock.
My solution is to create synthetic sequences with the basic geometry of an af- tershock sequence (i.e., clustered around the mainshock fault plane), but without limiting events to the regions in which aftershocks occurred. The three spatial co- ordinates for each synthetic event are chosen randomly and independently from the probability distributions shown in Figure 5.1. The probability distributions were designed to produce sequences with plausible depth distribution for the region and
substantial clustering along the mainshock fault plane. The locations of rv25% of the events in each sequence are chosen by adding small perturbations to the locations of previously chosen events to introduce clustering into the sequences.
The three mechanism parameters are chosen randomly and independently from the probability distributions in Figure 5.1. These distributions are smoothed versions of the parameters for the observed sequences. The magnitude of each event is ran- domly chosen from a Gutenberg-Richter distribution, and the time chosen randomly assuming an Omori Law decay in seismicity rate. One hundred synthetic sequences are created for Landers and for Northridge. The CI for each sequence is found the same way as for the observed sequences, and the distribution of CI values for random sequences is determined.
5.2.3 Statistical Test
The distribution of CI values for the synthetic sequences is used to determine how confident one can be that the CI of the observed sequence is too large to have occurred simply by chance. This is equivalent to the confidence level at which one can reject the null hypothesis that the observed sequence CI is indistinguishable from that of the synthetic sequences. For example, if the CI of the observed sequence is at the 95th percentile of the distribution of CI values for the synthetic sequences, the confidence level is 95%. A weighted average of the confidence level, over the error bar of the CI of the observed sequence, is also found, and this is the confidence level reported.