Comment of "The Gutenberg-Richter or Characteristic Earthquake Distribution, Which is it?" by S. G. Wesnousky

Yan Y. Kagan

Published February 1996, SCEC Contribution #219

In a recent article, Wesnousky (1994) (henceforth referred to as Wesnousky) analyzes seismicity and geologic deformation rates for five faults in southern California and arrives at the conclusion that the characteristic model yields a better description of the earthquake size statistical distribution than the Gutenberg-Richter (G-R) magnitude- frequency law. To analyze earthquake occurrences in the fault zones, Wesnousky compares seismicity levels during the past 50 to 60 yr with the return rate of characteristic earthquakes. The characteristic recurrence time is calculated on the assumption that all of the seismic moment rate evaluated on the basis of geologic slip-rate measurements is released by the characteristic events (equations 2 and 3 in Wesnousky). Wesnousky finds that for four out of five zones, the characteristic earthquake rate thus estimated is larger by a factor of more than 10 compared to the rate obtained by the extrapolation of instrumental seismic records. Although the results of paleoseismic studies are sometimes used to corroborate the recurrence periods, the major proof for the characteristic rate discrepancy is based on a comparison of magnitude-frequency plots and the return periods calculated using equations (2) and (3) of Wesnousky.

Thus, Wesnousky’ s results depend crucially on whether the seismic activity (productivity) of small and intermediate earthquakes in fault zones can be effectively correlated with the geologic deformation rate, as well as whether the above equations correctly evaluate the rate of occurrence for large earthquakes. Even if we accept the estimates of geologic slip-rate deformation rate proposed by Wesnousky, two factors can significantly change his conclusions: (1) parameters of the earthquake size distribution measured for each of the faults and (2) calculations of the characteristic recurrence rate. The latter depends on the contribution from earthquakes that are smaller or larger than the characteristic limit. Wesnousky briefly discusses smaller earthquakes and finds that their contribution is negligible; the possibility of earthquakes larger than the characteristic limit is not considered.

The guiding principle of this commentary is the idea that the simple null hypothesis (model) needs to be first fully and critically investigated, and only if it can be shown that the null hypothesis should be rejected, we should formulate an alternative model. The null hypothesis for the earthquake size statistical distribution is that it follows the G-R law. The difference between the G-R law that approximates the earthquake size distribution for large regions and the size distribution for geologically defined faults is explained as the consequence of the distribution of fault sizes and slip rates. Wesnousky et al. (1983) argue that whereas earthquake size distribution for “individual” faults follows the characteristic model, as the faults themselves are distributed according to a power law, the resulting magnitude-frequency relation for a large region is the G-R law. However, for southern California, application of the characteristic model yields cumulative curves that are significantly different from the historical/instrumental magnitude-frequency relation (Working Group, 1995, see their Fig. 14 and discussion on pp. 399 and 417). The “characteristic” curves also are less resembling the linear G-R relation than the experimental curve. Only by invoking the “cascade” procedure in which an earthquake ruptures through several characteristic fault segments, the discrepancy can be reduced to a manageable factor of 2. The cascade hypothesis is not a part of the original characteristic model (Schwartz and Coppersmith, 1984); in the strict interpretation of the model, no earthquake larger than the characteristic one is allowed.

As a measure of earthquake size, I use the scalar seismic moment of an earthquake, which I denote by symbol M. Occasionally, I also use an empirical magnitude of earthquakes denoted by m. To transform the seismic moment into magnitude, I use the standard relation m = 2/3 (log10 M — 16), (1) where M is measured in dyne cm. In this commentary, I first remark on the statistical problems of verifying the characteristic hypothesis using relatively small seismic zones, then I analyze how earthquakes that are smaller and larger than the characteristic size contribute to the seismic moment rate. To quantify the latter comparison, I introduce two null hypotheses that can be tested against the characteristic model as shown in Figure 1b of Wesnousky. The first null hypothesis, H00, assumes that the size of tectonic earthquakes is distributed according to the gamma distribution (see equation 5 below) with the universal values for its parameters over the world. The hypothesis tests whether very large events (m ≥ 8) may occur everywhere in the world; these earthquakes carry most of the tectonic deformation (Kagan, 1993, 1994, 1995b). The second null hypothesis, H01, assumes that earthquakes follow the cumulative G-R relation up to a value of mmax (maximum magnitude) and no earthquake can occur with m > mmax The maximum magnitude is specific to each fault and is allowed to vary from region to region. In Kagan (1993), this relation is called the “maximum moment (magnitude) distribution.”

Kagan, Y. Y. (1996). Comment of "The Gutenberg-Richter or Characteristic Earthquake Distribution, Which is it?" by S. G. Wesnousky. Bulletin of the Seismological Society of America, 86(1), 274-285.