imply a reduction in the scale length of fracture, as indicated above; the best-documented example of self-healing patches of slip is the inversion for the slip in the Landers earthquake (60). It can be shown that self-healing slip pulses of constant duration require a critical scaling dimension for their generation. Cochard and Madariaga (30) have argued persuasively that the critical scale size is given by the dimensions of a nucleation zone, which we equate with an asperity. In this model, the pulse generated at the asperity breaks into a region of lower stress drop, wherein its velocity of growth attenuates, slip begins to diminish, and ultimately a velocity-dependent friction that is intrinsic to the healing process causes ultimate cessation of motion.
(v) The earthquakes studied by Heaton (28, 60) show significant irregularities of the slip distributions in the rupture plane, which indicate the presence of significant inhomogeneity of stress drop and/or fracture threshold in the plane. The slip at the surface in the Landers earthquake of 1992 has been studied in detail in the field (60–62) and over the entire plane by inversion of the seismic signal (60, 63, 64). Of particular interest is the presence of time delays of about 1–3 sec between termination of rupture on one strand of the fault and initiation of rupture on a neighboring strand (60, 63, 64). These time delays between rupture of successive strands of the large fracture can be accounted for by inhomogeneity of the fracture threshold or the stress drop, or both, as well as by a slip-weakening in slow time, over the time interval of the time delay, of the strength in the high-stress drop barriers.
We summarize these observations on the seismicity of Southern California as follows. Not all faults in Southern California behave in the same way statistically. For example, the SAF today supports large earthquakes only and does not support small ones. Hence an effort to apply a universal model that yields the G-R statistical law is doomed to failure.
The G-R law is valid for small earthquakes but not for large ones. The G-R law is a manifestation of seismicity over the entire area of Southern California. Thus efforts to model the power-law character through a process of self-organization on a single fault is misdirected. The power law is probably a manifestation of the distribution of little faults and/or aftershocks in two dimensions—i.e., it is a manifestation of the geometry of faulting in Southern California.
Although we do not know what the distribution for large earthquakes is, the distribution for large earthquakes on faults other than the SAF is likely to show large spatial fluctuations. Large earthquakes on the SAF occur more frequently than large earthquakes on any other major fault in Southern California; the SAF is the most rapidly slipping element of the fault network but may be slipping under conditions of low average stress drop—i.e., a small energy-to-moment ratio.
The temporal distribution of earthquakes must be dependent on the limitations of fracture size. What stops the growth of a fracture is the encounter of a growing crack with a barrier region, which is a zone of large stress drop. If the large stress drops are localized, the earthquake ruptures are confined, and the characteristic earthquake model is appropriate. Localization is possible where significant changes in the geometry of faulting are encountered, such as at sites of step-overs (echeloning) or bifurcations (fault junctions). Under the constraint of uniform average rate of moment release, strong barrier sites must themselves break or the stresses at these sites must relax. Not only is geometry on a network of faults likely to be important for the modeling enterprise, but also the geometry of individual faults is going to lead to fluctuations in strength, in rapidity of restoration of strength (suturing) after a big earthquake. These geometrical fluctuations are likely to lead to nonuniformity of slip. Temporal fluctuations in seismicity imply the importance of time-dependent stress transmission processes, such as those associated with creep. Probably the most alluring proposal for intermediate-term earthquake prediction is the idea that local quiescence of small earthquakes develops near the site of a future strong earthquake and that intermediate magnitude activity increases at distance from the future earthquake. The most likely candidate for modeling the fluctuations in stress drops and fracture thresholds is an asperity model for individual earthquakes and a barrier model that accounts for the complexity of the fault network.
This research was supported by a grant from the Southern California Earthquake Center. This paper is publication number 4621 of the Institute of Geophysics and Planetary Physics, University of California, Los Angeles, and is publication number 322 of the Southern California Earthquake Center.
1. Bak, P. & Tang, C. (1989) J. Geophys. Res. 94, 15635–15637.
2. Nakanishi, H. (1990) Phys. Rev. A 41, 7086–7089.
3. Nakanishi, H. (1991) Phys. Rev. A 43, 6613–6621.
4. Christensen, K. & Olami, Z. (1992) J. Geophys. Res. 97, 8729– 8735.
5. Christensen, K. & Olami, Z. (1992) Phys. Rev. A 46, 1829–1838.
6. Christensen, K., Olami, Z. & Bak, P. (1992) Phys. Rev. Lett. 68, 2417–2420.
7. Olami, Z., Feder, H.J.S. & Christensen, K. (1992) Phys. Rev. Lett. 68, 1244–1247.
8. Olami, Z. & Christensen, K. (1992) Phys. Rev. A 46, 1720–1723.
9. Gutenberg, B. & Richter, C.F. (1944) Bull. Seismol Soc. Am. 34, 185–188.
10. Gutenberg, B. & Richter, C.F. (1954) Seismicity of the Earth (Princeton Univ. Press, Princeton, NJ).
11. Gutenberg, B. & Richter, C.F. (1956) Ann. Geofis. 9, 1–15.
12. Gutenberg, B. (1956) Q.J.Geol. Soc. London 112, 1–14.
13. Kanamori, H. & Anderson, D.L. (1975) Bull. Seismol. Soc. Am. 65, 1073–1095.
14. Knopoff, L. & Kagan, Y. (1977) J. Geophys. Res. 82, 5647–5657.
15. Kanamori, H. (1977) J. Geophys. Res. 82, 2981–2987.
16. Hanks, T.C. & Kanamori, H. (1979) J. Geophys. Res. 84, 2348– 2350.
17. Turcotte, D.L. (1990) Global Planet. Change 3, 301–308.
18. Rundle, J.B. (1989) J. Geophys. Res. 94, 12337–12342.
19. Romanowicz, B. (1992) Geophys. Res. Lett. 19, 481–484.
20. Romanowicz, B. & Rundle, J.B. (1993) Bull. Seismol. Soc. Am. 83, 1294–1297.
21. Pacheco, J.F., Scholz, C.H. & Sykes, L.R. (1992) Nature (London) 355, 71–73.
22. Okal, R. & Romanowicz, B.A. (1994) Phys. Earth Planet. Inter. 87, 55–76.
23. Sornette, D., Knopoff, L., Kagan, Y. & Vanneste, C. (1996) J. Geophys. Res., in press.
24. Gardner, J.K. & Knopoff, L. (1974) Bull. Seismol. Soc. Am. 64, 1363–1367.
25. Sieh, K.E. (1978) J. Geophys. Res. 83, 3907–3939.
26. Sieh, K., Stuiver, M. & Brillinger, D. (1989) J. Geophys. Res. 94, 603–623.
27. Schwartz, D.P. & Coppersmith, K.J. (1984) J. Geophys. Res. 89, 5681–5698.
28. Heaton, T.H. (1990) Phys. Earth Planet. Inter. 64, 1–20.
29. Knopoff, L. (1996) Proc. Natl. Acad. Sci. USA 93, 3830–3837.
30. Cochard, A. & Madariaga, R. (1994) Pure Appl Geophys. 142, 419–445.
31. Sieh, K.E. & Williams, P.L. (1990) J. Geophys. Res. 95, 6629– 6645.
32. Jones, L.M. (1988) J. Geophys. Res. 93, 8869–8891.
33. Raleigh, C.B., Sieh, K.E., Sykes, L.R. & Anderson, D.L. (1982) Science 217, 1097–1104.
34. Lindh, A.G. (1983) U.S. Geol. Survey Open File Rep. 83–63.
35. Sykes, L.R. & Nishenko, S.P. (1984) J. Geophys. Res. 89, 5905–5927.
36. Press, F. & Allen, C.R. (1995) J. Geophys. Res. 100, 6421–6430.
37. Knopoff, L. (1993) Proc. Am. Philos. Soc. 137, 339–349.
38. Knopoff, L. & Ni, X.X. (1995) in Impact, Waves and Fracture, eds. Batra, R.C., Mal, A.K. & MacSiphigh, G.P. (Am. Soc. Mech. Eng., New York), pp. 175–187.