A reappraisal of large earthquake scaling

Twelve years ago I pointed out that observations indicate that the slip in large earthquakes scales with their length, rather than their width, as expected from the canonical model. Romanowicz (1992) has recently argued that more recent data show that the opposite is true, which has obliged me to re-examine the question. I point out here a defining flaw in her analysis, in that she allowed Mo*, the moment at the cross-over from small to large earthquakes, to be a free variable in her curve fitting, whereas this parameter can be defined independently. When this parameter is independently fixed, I find that the up- dated data set confirms my earlier conclusion that for large crustal earthquakes M0 scales with L 2. A little over a decade ago I pointed out that obser- vations indicate that the slip in large earthquakes scales with their rupture length (Scholz, 1982). Several sub- sequent papers provided additional data supportive of this view (Chen and Molnar, 1983; Shimazaki, 1986; Scholz et al., 1986). Large earthquakes are defined in this con- text as those that rupture the entire seismogenic thick- ness W0 and are thus constrained to propagate farther in a horizontal direction only. For crustal earthquakes, W0 is nearly constant -15 km; so this scaling conflicts with conventional models of earthquakes, such as dislocation models, because they predict that slip scales with width, which is invariant for these earthquakes. I offered two alternative ways to interpret these observations. One is that such conventional W models are correct but that stress- drop could no longer be regarded as scale-invariant but must scale with fault length. As an alternative, I sug- gested the possibility of what I termed an L model, in which slip intrinsically scales with fault length. This created a minor crisis in seismology, which has yet to be resolved. It is no exaggeration to say that my L model suggestion has not attracted a noticeable coterie of admirers, nor has my other suggestion, and there has been no other model proposed in the interim that ex- plains the observed L scaling [although Heaton (1990) has suggested a way in which length may scale with stress- drop, a variation of my W model scaling alternative]. In the face of this, an alternative is to question the obser- vations, which is what Romanowicz (1992) has recently chosen to do. In her interpretation of an updated data set, she concluded that seismic moment, M0, scales lin- early with rupture length L for large earthquakes, which is consistent with W models and implies that slip, u, is scale invariant for large crustal earthquakes. This great discrepancy with the earlier result, she implies, is due to the presence of superior modern measurements in the updated data set. This issue has major implications, both practical and theoretical, and merits some debate. Here I present my own reappraisal of the data, in which the conclusions are very different. Reappraisal of the Data Romanowicz's conclusions were based on a log-log Plot of M0 versus L for a data set of strike-slip earth- quakes. She showed that the data in this figure could be better fit by two straight lines, one with slope 3 in the lower scale range and one with slope 1 in the upper scale range, than with a single line of slope 2. For small earth- quakes, which grow in both dimensions with L W, M0 should scale with L 3. For large earthquakes, where W has saturated at W0 and growth only occurs in the L di- rection, M0 should scale with L according to conven- tional models; hence the problem produced by my ob- servation that M0 scales with L 2 for those events. Superficially, then, it appears that she has resolved the issue. The problem, however, is that she has allowed the cross-over length, L*, which separates small from large earthquakes, to be a free variable in her curve-fitting ex- ercise. She thus obtains a cross-over moment of 0.6 to 0.8 x 1020 N-m, which is appropriate for a rupture length of 60 to 70 km. By adding an additional free parameter to the curve fitting, one would expect a much better fit to be achieved; the question is whether such a step is justified. If small earthquakes grow with L W, then the transition to large earthquakes (defined as an event that ruptures the entire seismogenic thickness) would be L* W0 ~ 10 to 20 km for the data being studied. This may be checked observationally; because large earth- quakes almost invariably nucleate near the base of the seismogenic layer (Das and Scholz, 1983), it follows that the transition to large earthquakes is at the length at which one begins to consistently see surface ruptures. Inspec- tion of the earthquakes in Romanowicz's (1992) Figure 215