Scaling of earthquake fault parameters

The long-standing conflict between the predictions of elastic dislocation models and the observation that average coseismic slip increases with rupture length is resolved with application of a simple displacement-depth function and assumption that the base of the seismogenic zone does not result from the onset of viscous relaxation, but rather a transition to stable sliding in a medium that remains stressed at or close to failure. The resulting model maintains the idea of self-similarity for earthquakes across the entire spectrum of earthquake sizes. Introduction Slip and rupture length are the most readily observed parameters used to describe an earthquake source whether determined by direct measurement in the field or from instrumental studies. Although real earthquake rupture is known to be associated with complex slip patterns, the simple parameters of average fault slip (D), fault length (L) and down depth width (W) are commonly at the heart of discussions of the physics of earthquake rupture [e.g., Scholz, 1982a]. Ignoring a shape factor close to unity, elastic models indicate that the slip on a fault in a uniformly stressed elastic medium should be proportional to the smallest fault dimension, and that earthquake stress drops should be scale independent [Figure 1 and e.g., Kanamori and Anderson, 1975]. This is commonly observed for smaller and moderate earthquakes [Hanks, 1977] and led to the proposition that little earthquakes are models for the behaviour of large, less frequent, and more devastating ones [e.g., Scholz, 1990]. However the prediction has been complicated by the inconvenient observation, that for large continental strike-slip earthquakes, coseismic slip steadily increases as a function of earthquake rupture length (Figure 2). Because it is generally assumed that the shortest dimension is the thickness of the seismogenic layer which extends to a relatively constant depth of ~15 km, the increase in slip that –March 5, 2007 submitted BSSA 2 accompanies growth in rupture length L requires that large earthquakes have increasingly higher stress drops than lesser ones and should consequently radiate proportionally more high frequency energy. No evidence for this has been published. We here put forward a model that resolves the conflict within the context of static dislocation theory and a recognition that coseismic slip during large earthquakes may extend below the base of the seismogenic layer. Slip distribution as a function of depth. The general characteristics of observed seismic slip as a function of depth in continental strike-slip regions are illustrated in Figure 3. The histogram in Figure 3a provides an example of the observation that seismic slip of smaller earthquakes is typically concentrated at mid-seismogenic depths of 6 to 10 km and systematically decreases to zero above and below those depths. The same is broadly true when one views the coseismic slip distributions of large earthquakes. The examples of the 1992 Landers and 1999 Hector Mine earthquakes are shown in Figures 3 b & c. Areas of maximum slip are observed to occur at mid-seismogenic depths. While studies of individual earthquakes may differ substantially and the behaviour of one earthquake can differ from another, greatest values of slip are usually in the middle of the seismogenic zone. Over the long term of numerous earthquake cycles slip at greater and lesser depths must be accommodated by aseismic motion. The observed distribution of seismic slip with depth can be explained by changes in frictional behaviour with depth. The simplest view is to consider that earthquakes can initiate only in ‘unstable’ zones where dynamic friction is less than static friction, a condition often referred to as slip weakening [Dieterich, 1972; Scholz et al., 1972]. Rupture once initiated can propagate indefinitely into a stressed region where static and dynamic friction are equal and no dissipation occurs. In practice slip is attenuated in such regions by plastic processes referred to as velocity strengthening [Ruina, 1983]. Except when subject to an abrupt stress increase, slip in these ‘stable’ regions occurs as aseismic creep [Tse and Rice, 1986]. The way in which slip as a function of depth during earthquakes relates the frictional characteristics is shown in Figure 4. Events smaller than M ~ 6 rarely break the surface and slip is limited to mid-seismogenic depths. Larger events with more displacement, commonly propagate to the surface, and also propagate below the depth to which earthquakes can initiate. –March 5, 2007 submitted BSSA 3 Modeling geodetic displacements Two simplified displacement-depth slip profiles are illustrated in Figure 5. The first of these is a simple box. The second, referred to for convenience as tapered, is designed to emulate the character of the displacement-depth profiles shown in Figure 4. The slip is maximum at a depth of 6 km with constant reductions of slip both above and below. The displacement fields at the surface (such as those measured by GPS or InSAR) resulting from the two displacement-depth profiles are constructed with the formulas of Okada [1992] and provided in Figure 5c. Displacements from the tapered slip function are displayed by circles while those from the box are provided as a line. The two very different slip functions produce the same displacement field. In other words the same displacement field can be produced by distinctly differing slip functions. Differences between the displacement fields do occur very close to the fault and result from near surface differences in the two slip functions, but are scarcely measurable in the field and do not concern establishing the form of the slip distribution at depth. We take the same approach in Figure 6 where a series of tapered displacement functions are shown and labeled a-g, respectively. As in Figure 5, the maximum slip for each is placed at 6 km and decreases at a constant rate above and below, and in this sense the tapered slip functions are self-similar and, as explained in Figure 1, have the same stress drop. The maximum slip for each range from about 2 to 8m and surface slip from 0 to 6 m. Associated with each tapered slip function is a box slip distribution like that shown in Figure 5 that produces the same surface deformation field, and similarly labeled a through g. It is not necessary that the tapered slip function is a particularly accurate depiction of the slip as a function of depth. We only suggest that it is much more reasonable than a box. Independent of details the exercise illustrates how poorly slip as a function of depth is constrained and that more plausible profiles can fit the same data. A number of other observations related to Figure 6 have a bearing on the question of whether or not large earthquakes differ in a fundamental sense from small ones. The surface slip for the box functions is consistently ~1 m greater than for the respective tapered function (Figure 6). The depth to which slip extends ranges from ~ 12 to ~25 km for the tapered function while the box models extend only to ~ 11 to ~ 17 km (Figure 7a). While the average value of slip for the box and tapered functions –March 5, 2007 submitted BSSA 4 is about the same at lesser magnitudes, the average value of slip of the box functions systematically increases over the respective tapered functions as the magnitude of the pairs increases (Figure 7b). Finally, although the respective box and tapered distributions are of distinctly different form, they produce the same surface deformation field at the surface but are characterized by almost the same Geometric Moment (Potency)(Figure 7c). From this it appears that while geodesy may be robust in estimating earthquake size (Moment) from a particular deformation field, the shape of the causative slip function is poorly resolved. Each the box and tapered slip distribution pairs in Figure 6 may also be characterized by the average strain and, hence, stress drop (Figure 8). The strain drop is proportional to Dmx/W for the box and tapered slip functions when slip is confined to the subsurface (small earthquakes). In the case of slip functions which break the surface the strain drop is Dmx/W’ for the tapered function and Dmx/2W for the box. The factor of 2 in the expression for the box includes the effect of the free surface. The value of W’ for the tapered slip function approaches 2W as displacement Dmx increases. The calculation of strain drop for each slip function pair shows that the box function implies a steady increase of stress drop with increasing slip whereas strain drop remains constant for the tapered model. Discussion In 1982 Scholz proposed that for large earthquakes rupture would end at the base of the seismogenic layer which could be determined from the aftershock depth distribution. Thus in the context of elastic dislocation theory earthquakes above some magnitude should have the same surface slip independent of length. This has been termed the W model and, in light of independent observations suggesting earthquakes share an approximately constant stress drop, is in conflict with the observed increasing slip with length for large strike-slip earthquakes. In response he posed an alternative referred to as the L model assuming the base of the seismogenic zone was unconstrained such that length became effectively the shortest dimension. The mechanical explanation of such a model has not been straightforward and conflicts with elastic dislocation theory. The conundrum has been addressed by evoking dynamic explanations which subsequently formed the basis of discussion over many years [Bodin and Brune, 1996; Heaton, 1990; Manighetti et al., 2007; Romanowicz, 1994; Scholz, 1982a; Shaw and Scholz, 2001]. –March 5, 2007 submitted BSSA 5 More recently it has become clear that neither a W nor and L model is appropriate. Slip does not increase linearly with rupture length for large earthquakes, nor does it saturate for rupture lengths greater than ~15 km. Rather, the rate of increase of strike-slip offset appears to continually decrease with rupture length without reaching a plateau (Figure 2). The observation removes the premise of either the L or W model and the contradictions that were the most difficult to explain. We pose here a model that is neither W nor L but retains the simplicity of the constant stress drop of the original W model being based on simple dislocation theory and is consistent with current understanding of fault behavior with depth. Constant stress drop is achieved by assuming a slip function that tapers with depth and can extend below the seismogenic depth. It is consistent with geodetic observation and does not appear to violate applications of inverse methods to waveform data used in estimating seismic slip distributions on faults. The latter permit considerable latitude in the depth to which slip can extend [e.g., Beresnev, 2003] and some authors simply limit or clip the depth of the grid used for the inversions and thus require that slip does not extend below the seismogenic zone [Somerville, 2006], effectively dismissing the possibility of the latter as unimportant. The effect is evident in the slip distribution for the 1992 Landers earthquake shown in Figure 3c where the abrupt cutoff of large values of slip at the base of the model are likely to be artificial and actual slip extends deeper. Consequently it seems that, within the resolution of the data available to us at present, static models adequately describe in a general way the geometric character of the earthquake rupture surface and slip distribution for events of all sizes. This is not to suggest that dynamic processes are not important. They obviously are and models such as Shaw and Scholz [2001] imply the extension of slip below the seismogenic depth that we propose. However, neither dynamic processes nor assumptions that large earthquakes are mechanically different from smaller ones need be invoked to explain the characteristics of surface slip associated with large earthquakes. While the analysis obviates the need for dynamics to explain the characteristics of surface slip with rupture length, it raises the question of the process that leads to a systematic increase in the depth to which rupture extends as rupture length grows. For this we allude to the idea that seismological characteristics of a fault evolve in concert with the accumulation of slip and structural characteristics of a fault [Wesnousky, 1988]. Active strike-slip faults of low displacement and small earthquakes tend to be structurally more complex than those of greater displacement –March 5, 2007 submitted BSSA 6 and host larger events. With time however, the accumulation of slip leads to a geometrically simpler fault zone and a closer geometric relation with the underlying shear zone. For such faults rupture in large events can extend well below the depths where earthquakes can initiate (Figure 9). Conclusions An old contradiction is resolved and the behavior of small earthquakes can supply a model for large earthquakes. This is important for earthquake engineers and seismic hazard analysts. The model requires that a modest amount of rupture extends well below the seismogenic depth and that the base of the seismogenic zone does not result from the onset of viscous relaxation but rather a transition to stable sliding in a medium that remains stressed at or close to failure at all times. References Beresnev, I. A. (2003), Uncertainities in finite-fault slip inversions: To what extent to believe? (A critical review), Bulletin of the Seismological Society of America, 93, 2445-2458. Bodin, P., and J. N. Brune (1996), On the scaling of slip with rupture length for shallow strike-slip earthquakes: Quasi-static models and dynamic rupture propagation, Bulletin of the Seismological Society of America, 86, 1292-1299. Dieterich, J. H. (1972), Time dependent friction in rocks, Journal of Geophysical Research, 77, 3690-3697. Hanks, T. H. (1977), Earthquake stress-drops, ambient tectonic stresses, and the stresses that drive plates, Pure and Applied Geophysics, 115, 441-558. Heaton, T. H. (1990), Evidence for and implication of self-healing pulses of slip in earthquake rupture, Physics of the Earth and Planetary Interiors, 64, 1-20. Kanamori, H., and D. Anderson (1975), Theoretical basis of some empirical relations in seismology, Bulletin of Seismological Society of America, 65, 1073-1096. Kaverina, A., et al. (2002), The combined inversion of seismic and geodetic data for the source process of the 16 October 1999 Mw 7.1 Hector Mine, California, earthquake, Bulletin of Seismological Society of America, 92, 1266-1280. King, G. C. P., et al. (1994), Block versus continuum deformation in the western United States, Earth and Planetary Science Letters, 128, 55-64. Manighetti, I., et al. (2007), Earthquake scaling, fault segmentation, and structural maturity, Earth and Planetary Science Letters, 253, 429-438. Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space, Bulletin of Seismological Society of America, 82, 1018-1040. Romanowicz, B. (1994), Comment on "A reappraisal of large earthquake scaling", Bulletin of the Seismological Society of America, 84. –March 5, 2007 submitted BSSA 7 Ruina, A. L. (1983), Slip instability and state variable friction laws, Journal of Geophysical Research, 88, 10,359-310,370. Scholz, C. (1982a), Scaling laws for large earthquakes: consequences for physical models, Bulletin of Seismological Society of America, 72, 1-14. Scholz, C. (1990), The Mechanics of Earthquakes and Faulting, 439 pp., Cambridge University Press. Scholz, C. H., et al. (1972), Detailed studies of frictional sliding of granite and implications for the earthquake mechanism, Journal of Geophysical Research, 77, 6392-6406. Shaw, B. E., and C. H. Scholz (2001), Slip-length scaling in large earthquakes: Observations and theory and implications for earthquake physics, Geophysical Research Letters, 28, 2991-2994. Somerville, P. (2006), Review of magnitude-rupture area scaling of crustal earthquakes, Report (personal communication), 22 pp, URS Corporation. Tse, S. T., and J. R. Rice (1986), Crustal earthquake instability in relation to the depth variation of frictional slip properties., Journal of Geophysical Research, 91, 94529472. Wald, D. J., and T. H. Heaton (1994), Spatial and temporal distribution of slip for the 1992 Landers, California, earthquake, Bulletin of Seismological Society of America, 84, 668-691. Wesnousky, S. (1988), Seismological and structural evolution of strike-slip faults, Nature, 335, 340-342. Wesnousky, S. G. (2007), Displacement and geometrical characteristics of earthquake surface ruptures: Issues and implications for seismic hazard analysis and dynamics of the earthquake rupture process, Bulletin of the Seismological Society of America, in prep and to be submitted soon after this. –March 5, 2007 submitted BSSA 8 Figure Captions Figure 1. Strain profiles for dislocation surfaces (A) entirely and (B) partially in an elastic medium (shaded) on which an elliptical slip distribution has been imposed for each. Surfaces are of infinite length into page. Strain drop Δε for each is almost identical and equal to d/W’ where d is maximum displacement and W’ is width (shortest dimension) of the surface. Small differences in constant strain drop (dashed line) will occur near the surface for case (B). For case (B) where the crack intersects the free surface W’ is greater than the portion W of crack embedded in the medium. By analogy W is the depth to which a fault extends into the earth’s crust and the endmember cases W=W’ and W’=2W correspond to the definition of small and large earthquakes by Scholz [1982a]. Figure 2. Average surface displacement is an increasing function of rupture length for continental strike-slip earthquakes. Adapted from Wesnousky [2007]. Figure 3. a) Histogram of total slip contributed by small earthquakes as a function of depth for background seismicity recorded between 36.4°N and 38.0°N and 121.0°E and 123.0°E in California during period of 1969 to 1994 (adapted from King et al. [1994]). b) Coseismic slip distributions on (B) 1999 Hector Mine and (C) 1992 Landers earthquake fault plane determined from teleseismic body waves and displacement waveforms, respectively, modified from Wald and Heaton [1994] and Kaverina et al. [2002]. Contours in meters. Figure 4. A) Conceptual model for frictional characteristics with depth. Earthquakes may only initiate in mid-crustal depths in the ‘unstable’ zone where dynamic friction is less than static friction (velocity weakening). Slip may not initiate above or below where static-friction is equal to or greater than dynamic friction (velocity strengthening). B) Generalized depiction of expected coseismic slip distribution with depth for a moderate M~6 and large M~8 earthquake. Maximum slip for each is at mid-crustal level. Only the largest events propagate far into the ‘stable’ zones. Figure 5. The (A) box and (B) tapered displacement functions produce (C) the same deformation field (circles and lines) at the earth’s surface. Vertical axis is meters of surface displacement and horizontal is distance in kilometers from fault. Figure 6. Paired box and tapered displacement-depth profiles are labeled a through g. The respective pairs produce the same deformation field at the Earth’s surface. Figure 7. Comparison of (A) depth extent, (B) average displacement, and (C) the geometric moment for the paired box and tapered displacement-depth profiles shown in Figure 6. –March 5, 2007 submitted BSSA 9 Figure 8. Comparison of values of strain drop for the paired box and tapered displacement-depth profile models in Figure 6. Strain drop increases with earthquake size for the box function but remains constant for tapered function. See text for further discussion. Figure 9. Conceptual model for the interaction of earthquake slip in the seismogenic zone and deformation in the lower crust. (A) Active strike-slip faults of low displacement tend to be structurally more complex and associated with relatively smaller earthquakes than faults of greater displacement. The complexity hinders rupture propagtion both horizontally and vertically. (B) The accumulation of slip leads to a geometrically simpler fault zones. The increasing simplification leads to a simpler geometric relation with the underlying shear zone and increasing capability for rupture to extend below the depth to which earthquakes can initiate.