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Fault interactions and seismic hazard
Journal of Geodynamics 29 (2000) 459±467
Fault interactions and seismic hazard C.H. Scholz*, Anupma Gupta Lamont-Doherty Geological Observatory and Department of Earth and Environmental Sciences, Columbia University, Palisades, NY 10964, USA
Abstract Faults usually are not isolated features but exist within a population of faults which may interact through their stress ®elds. This poses two serious problems for seismic hazard analysis. The most severe such problem lies in estimating the likelihood of whether or not a future earthquake will be con®ned to a single fault (or fault segment) or will jump to adjacent faults and result in a larger earthquake. We review recent results which show that it is possible to determine the degree of fault interaction from geological data alone. We propose that the probability of an earthquake jumping from one fault to another will increase with the degree of stress interaction between the faults, and introduce a simple criterion to estimate the degree of interaction based on separation and overlap of echelon normal fault pairs. This statics based criterion for normal faults agrees qualitatively with the limited dynamic modeling of (Harris, R.A., Day, S.M., 1993. Dynamics of fault interaction: parallel strike-slip faults. Journal of Geophysical Research 98, 4461±4472) of the more complex case of strike slip faults and suggests that a more general criterion may be obtainable. The second problem discussed is the hazard associated with earthquakes being triggered by earlier earthquakes on a dierent fault. This phenomena produces seismic hazards distinct from that associated with ordinary aftershocks. We point out that with rapid data acquisition and proper preparation, it is feasible to issue a short-term hazard assessment regarding triggered earthquakes shortly after the occurrence of a potentially triggering event. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The incorporation of geological data on active Quaternary faults: their distribution, lengths, * Corresponding author. Tel.: +1-914-365-8360; fax: +1-914-365-8150. E-mail address: [email protected] (C.H. Scholz). 0264-3707/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 4 - 3 7 0 7 ( 9 9 ) 0 0 0 4 0 - X
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slip direction, and slip rate, has become central to the analysis of geological hazard. Although the basics of this practice has been worked out (Wesnousky et al., 1984), there remain serious problems. These problems arise because faults seldom, if ever, exist as single, isolated structures. They occur within a population of faults and hence are not mechanically isolated but may interact with other faults through their stress ®elds. Furthermore, at a lower hierarchical level, individual faults are found, on closer inspection, to be not single fractures but to be segmented in various ways. For an isolated, non-segmented fault, the seismic hazard evaluation is straightforward. One assumes (with good reason) that the seismic cycle of the fault consists of a single large event that ruptures the entire fault length. A scaling law (e.g., Scholz, 1994; Anderson et al., 1996), then allows one to estimate the slip and hence moment of this earthquake. If one can determine the geological slip rate of the fault then one may also estimate the recurrence time of this earthquake. In the more complex situation in which there is a population of segmented faults, we have the more dicult problem in assessing the possibility that an earthquake will rupture a number of faults or fault segments. Examples of this abound: a particularly well observed example was the 1992 Landers, California, earthquake, which ruptured three major and a number of minor subparallel strike slip faults (Sieh et al., 1993). This possibility poses a quandary for seismic hazard analysis. Consider the simple example of a fault of length L which slips at a known geological rate and contains a single segment boundary as indicated by an oset, say, at midpoint. We may consider two end member hazard models Ð one based on the assumption that both segments break simultaneously and the other on the assumption that each segment breaks independently. In the ®rst case, the expected slip will be double that of the second case (Scholz, 1994), hence maximum earthquake moment is four times larger, but the earthquakes in the second case are expected to occur four times more often. Clearly, the hazard estimate is very dierent for the two scenarios. Is there any way that we can determine if one is more probable than the other? If we had a past history of such earthquakes on this fault that would be very instructive, but for most places the seismic history is too short to provide much guidance on this problem. A second problem in seismic hazard analysis that results from fault interactions is the triggering by one earthquake of an earthquake on another, nearby, fault. The problem with such triggered events is that they do not obey the statistics of aftershock per se, in which they occur on the same fault as the mainshock and the largest aftershock is at least one magnitude unit smaller than the mainshock. In contrast, the triggered event may be as large or larger than the triggering, event. Whereas that hazard posed by aftershocks is contained within the damage region of their mainshock and consists of further damage to already damaged structures, triggered earthquakes can impose major damage to more distant regions. As an example, the M 6.5 Big Bear earthquake was triggered by the Landers earthquake and caused extensive damage in a region which was not itself damaged by the Landers shock (Stein et al., 1992; Jaume and Sykes, 1992). While in that case the triggered earthquake was on a dierent fault, there are also many cases of the triggered event being on the extension of the same fault. In the Umbria-Marche sequence of 1997, a M 5.7 event triggered 9 h later an adjacent M 6.0 event, which ruptured the same fault farther to the north, causing much additional damage (Amato et al., 1998). How can we include this phenomena in seismic hazard analysis?
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2. Estimating fault interactions When considering the possibility that an earthquake may propagate from one fault to the next it seems reasonable to conclude that this will depend on how strongly the faults are interacting. This will range from faults that are mechanically independent to those that are linked so strongly as to be acting as a single mechanical unit. Similarly, there is probably no simple distinction between interacting faults and segments of a single fault. For example, are the three primary faults ruptured in the Landers earthquake: the Johnson Valley, Emerson, and Camp Rock faults, separate faults or segments of one fault. It is likely that they are considered separate faults only because the geologists that originally mapped them named them individually. It is also likely that those geologists did not have these issues in mind when they did so: they could just as easily have called them the Johnson Valley, Emerson, and Camp Rock segments of the Landers fault system. Such a distinction may therefore be of little value and we ignore it here. Let us consider that the probability of an earthquake rupturing two faults is nil in the case of non-interacting faults and increases monotonically with the degree of interaction. It would, therefore, be of considerable value if there was a method, independent of the record of past earthquakes, of measuring the degree of fault interaction. Faults obey well-de®ned scaling laws relating net geological displacement D and length, L (e.g. Schlische et al., 1996). Isolated faults also have well de®ned alongstrike displacement pro®les (e.g. Dawers et al., 1993) in which the displacement tapers o gradually towards the fault tip. Within a given geological setting, this fault tip taper (FTT) angle is found to vary within fairly small limits. This ®nding can be readily interpreted in terms of `large-scale yielding' crack model in which the fault is considered an elastic crack in which plastic yielding occurs within a volume surrounding the crack tip (Kanninen and Popelar, 1985, pp. 367±372,
Fig. 1. Schematic diagram showing the interaction of two echelon normal faults. A. Fault F1 approaches fault F2 with an overlap O and separation S. It's right-hand tip lies in the stress drop ®eld of fault F2. B. The displacement pro®le of fault F1 is now asymmetric, with a fault tip taper FTT ' at the interaction end steeper than FTT at the non-interacting end. It is this distortion of the displacement pro®le that was used by Gupta and Schozl (1999) to determined the interactions between faults in terms of overlap and separation.
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Fig. 2. Interactions between three normal faults that constitute the main bounding fault system of the northern Lake Malawi rift. Top: Individual displacement pro®les for each fault at three time epochs, 2.5 m yr, 1.5 m yr, and the present. In two way travel time (TWTT) 1 s 0 1 km. Separation of the southern and central faults is approximately 5 km, that for the central and northern faults, 7 km. Bottom: Summed displacement pro®les at the dierent times. By the latest period, the summed pro®le is beginning to approach a standard pro®le for an isolated fault of the same length as the fault system (dashed curve), indicating that the system is beginning to behave as a single fault. (From Contreras et al., 1999).
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this is known as the CTOA, or crack tip opening model in tensile fracture mechanics). This model predicts a linear displacement taper near the fault tip, and that its value is proportional to the (®nite) stress concentration at the crack tip sc sy ÿ sa , where sy is the yield strength of the rock and sa is the regional applied stress. When faults approach one another and begin to interact through their stress ®elds, their displacement pro®les near their interacting tips become distorted in comparison with those of isolated faults (Peacock, 1991; Peacock and Sanderson, 1991, 1994). The usual observation is that the FTT becomes steepened. This phenomenon can be readily explained in terms of the large scale yielding model (Fig. 1). There we show a fault F1 of length L1 approaching a second fault F2 of length L2 with a certain overlap O and separation S (Fig. 1a). Consider, for simplicity, that they are vertically dipping normal faults, since in that case their stress ®elds are symmetric. The stress ®eld for fault 2 is sketched in the ®gure, where solid curves are contours of stress drop and dashed curves are stress increases. Note that although in this schematic ®gure no values are given on the contours, the values of stress changes produced by fault are 102±103 those produced by individual earthquakes (because the D/L ratio for faults, 010ÿ2 , is that much larger than for earthquakes). The right-hand tip of fault 1 is within the stress drop region of fault 2. Hence, at the proximal tip the FTT A sy ÿ sa ÿ Ds, and hence must be greater than at its distal noninteracting end (Fig. 1b). This distortion will increase proportionately to the degree of fault interaction and hence provides a measure of it. Thus, as displacement continues to accumulate on fault 1, it will grow on its distal end, maintaining a constant FTT, but its growth will be impeded at the interacting end because each growth increment will move the tip into a higher stress drop region, hence the FTT will have to continually increase for growth to proceed. This progression is illustrated in Fig. 2 (top), which shows the displacement pro®les for three interacting normal faults that constitute the main boundary system of the central part of the Lake Malawi rift in Africa. A stratigraphic back-stripping method has allowed the reconstruction of the displacement pro®les of these faults at three successive time intervals, showing the progressive growth and interaction of the faults (Contreras et al., 1999). The main features of this sequence is that the interacting fault tips become relatively pinned, propagating little while their proximal displacement gradients progressively steepen, while the distal fault tips propagate freely, maintaining constant displacement gradients. This case was analyzed by Gupta and Scholz (1999), who showed that the interacting tips were, in fact, propagating into progressively higher stress drop regions, while developing progressively more deformed displacement pro®les. They also analyzed many other cases and showed that these systematics are general. They also found that there is a critical stress drop at which the tips become fully pinned. At about that point, minor fractures begin to develop in the interaction region, progressively linking the two faults. The summed displacement pro®les of the two main faults plus the minor faults begin to approach the pro®le expected from a single fault of length equal to that of the combined lengths of the interacting faults, indicating that they are starting to act as a single mechanical unit (e.g. Dawers and Anders, 1995). The dashed curve in Fig. 2 (bottom) is an example. These results illustrate that one can, in fact, de®ne the degree of fault interaction, which forms a continuum from non-interacting to fully linked. As a practical matter it is very dicult if not impossible in most cases to obtain the high resolution displacement pro®les that were used as a diagnostic tool in the above discussion. It is relatively easy to measure overlap and
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separation between adjacent faults, however, and the Gupta and Scholz (1999) analysis allows one to portray the degree of interaction in terms of those parameters. Fig. 3 summarizes their results. Plotted are the separation and overlap of one fault pair, each normalized to the fault length of the second fault (There will be two such values for each fault pair Ð which may dier signi®cantly if the faults have very dierent lengths. Interaction will thus depend on the direction of rupture propagation.). The region of interaction is shaded, with the degree of interaction increasing monotonically from the minimum interaction curve Ds 50 MPa) to the maximum Ds 250 MPa). The interaction criterion described above was determined for normal faults, which only involve the interaction of shear stresses. The situation is more complicated for strike slip faults, because in that case slip on one fault will induce changes in both shear and normal stress on the other, and those changes are asymmetric with respect to the fault tip (Segall and Pollard, 1980). In that case the interaction will depend on the sign of the jog (compressional or extensional). The open circle and square in Fig. 3, for example, are the maximum compressional and extensional jogs, respectively, that could be jumped by a 10 MPa stress drop earthquake in the simulations of Harris and Day (1993). The compressional case corresponds fortuitously to the limiting case for normal faults given by our criterion. The extensional case shows that it is easier for an extensional jog to be jumped Ð the limiting separation is greater. Harris and Day discuss some observations that corroborate that ®nding. As examples, the solid circle and square are the value for the Johnson Valley±Emerson and Emerson±Camp Rock segments, respectively, of the Landers (strike-slip) rupture. As shown, they are both strongly interacting. The important point though, is that the Harris and Day results, using dynamic models, demonstrate the basics of our static analysis: that for small separations earthquakes can easily jump the jog, but there is some critical separation which no earthquake can jump. A related observation is that segment boundaries for strike slip faults
Fig. 3. A simple criterion for the interaction of echelon normal faults pairs in terms of overlap and separation, based on Gupta and Scholz (1999). Solid circle and square are the Johnson Valley±Emerson and Emerson±Camp Rock segments, respectively, of the Landers rupture. Open circle and square are the maximum compressional and extensional jogs of a strike slip fault, respectively that could be jumped by a 10 MPa stress drop earthquake in the modeling of Harris and Day (1993). The line is the scale invarient O/S ratio observed for segment boundaries of strike-slip faults by Aydin and Nur (1982).
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such as pull-apart basins and compressional horsts have been found to have a scale invariant O/S03 (Aydin and Nur, 1982), which is shown as the line in Fig. 3. This standard O/S ratio indicates that such segment boundaries are strongly interacting Ð it is this interaction that results in this scale invariance (Gupta and Scholz, 1999). That these results for strike-slip faults are approximately in accord with the model derived for normal faults indicates that the normal stress changes are secondary with respect shear stress changes in promoting earthquake jumping across segment boundaries. Whether or not an earthquake will jump between faults depends also upon the initial conditions, namely the stress state on the two faults at the time of the earthquake. The probability that two adjacent faults are in synchroneity in their seismic cycles can be expected to increase with the degree of fault interaction because that increases the chance that the two faults ruptured together in the previous earthquake. This reinforces the idea that jumping probability increases with degree of interaction. We have, of course, no way of assigning jumping probability values to interaction levels. At the present, we only oer this criterion as a guide for those who have to face this problem in the course of making a seismic hazard evaluation. Much more work needs to be done on this problem, particularly for the case of strike-slip faults.
3. Triggered earthquakes That the static stress change brought about by an earthquake can trigger earthquakes on other nearby faults was ®rst pointed out by Das and Scholz (1982). Since then a considerable literature has been devoted to this topic, which is now reasonably well understood (e.g. King et al., 1994). Such triggered events occur with time delays of hours to a few months following the triggering event Ð they are not simultaneous even though the stresses in the seismic waves may be several times the residual static stresses (Cotton and Coutant, 1997). This is because the rock friction laws are insensitive to transient stresses (Dieterich, 1987; Scholz, 1998). These pose a seismic hazard distinct from that of aftershocks. Aftershocks are restricted to the rupture dimension of the mainshock, hence they pose an additional hazard only to the mainshock damage area. This is well known and the authorities are accustomed to warning the populace of the continued danger due to aftershocks. Furthermore, aftershocks in this strict sense are of secondary importance: the sum of seismic moment in an aftershock sequence is typically only about 5% of that of the main shock and the largest aftershock is at least one magnitude unit less than the mainshock. They generally only cause damage to structures already damaged by the mainshock. None of this applies to triggered events. They may be as large or larger than the triggering event, and they may cause extensive damage in a region not damaged by the triggering event. How might we deal with the hazard associated with this phenomena? Triggering occurs only on faults on which the triggering earthquake has produced a small but ®nite increase in the Coulomb stress t ÿ s ÿ p resolved on the fault in its slip direction. It seems reasonable to suppose that the probability of this happening will increase with the magnitude of that Coulomb stress increase. Furthermore, because the mechanism of this triggering must be the same as the triggering of aftershocks, sensu strictu, we can suppose that
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the probability of a triggered event occurring will decrease hyperbolically with time following the triggering event, in the same manner as the decay of aftershocks. Let us suppose that we have the proper infrastructure in place: we have a local seismic network, telemetered so that we can quickly determine, in the event of a large earthquake, its moment, focal mechanism and rupture length. If we also have a digitized map of all Quaternary active faults in the same district, we can then quickly determine the Coulomb stress change that has occurred on all those faults. We are then in a position to determine what other regions in the district have experienced an increase in short-term hazard. A warning based on such an analysis could have a number of meritorious eects, depending on the circumstances. Toda et al. (1998) conducted an ex post facto exercise of this type following the 1995 Kobe, Japan, earthquake, with encouraging results.
4. Conclusions We have introduce approaches to two outstanding problems in seismic hazard analysis. Both have to do with fault interactions. The ®rst problem is how to estimate the likelihood of an earthquake jumping from one fault to another. It is proposed that this is related to the degree to which the two faults interact through their stress ®eld. In the case of an echelon arrangement of normal faults, a related study (Gupta and Scholz, 1999) shows that the interaction between a pair of such faults de®nes a continuum between non-interacting and fully linked. The degree of interaction can be estimated by the overlap and separation of the faults, normalized to fault length. For echelon strike-slip faults the relationship is more complex, being asymmetric because of normal stress interactions, as shown by the dynamic simulations of Harris and Day (1993). However, both their analysis and observations for strike-slip fault arrays are roughly consistent with the normal fault model. Much more work needs to be done, however, before quantitative estimates of probabilities can be made for the purpose of hazard analysis. The second problem concerns the triggering of earthquakes on nearby faults by coseismic stress changes, which results in a short term increase in seismic hazard following large earthquakes. If one can obtain in timely manner the source characteristics of that event and one has a regional map of quaternary active faults, it is feasible to estimate the short term increase of seismic hazard for the other faults in the district. This is done by calculating Coulomb stress changes on all nearby faults and recalculating short-term probabilities in the manner used by Toda et al. (1998).
Acknowledgements I thank Prof. G. Cello for inviting me to the workshop `The Resolution of Geological Analysis and Models for Earthquake Faulting Studies' in Camerino, which ®rst prompted me to think about this problem. This work was partially supported by NSF grant EAR 97-06475. Lamont-Doherty Earth Observatory contribution number 5926.
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References Amato, A., Azzara, R., Chiarabba, C., Cimini, G.B., Cocco, M., DiBona, M., Margheriti, L., Mazza, S., Mele, F., Selvaggi, G., Basili, E., Boschi, E., Courboulex, F., Deschamps, A., Gaet, S., Bittarelli, G., Chiaraluce, L., Piccinini, D., Ripepe, M., 1998. The 1997 Umbria-Marche, Italy, earthquake sequence: a ®rst look at the main shocks and aftershocks. Geophysical Research Letters 25, 2861±2864. Anderson, J.G., Wesnousky, S.G., Stirling, M., 1996. Earthquake size as a function of fault slip rate. Bulletin Seismological Society of America 86, 683±690. Aydin, A., Nur, A., 1982. Evolution of pull-apart basins and their scale independence. Tectonics 1, 91±105. Contreras, J., Anders, M.H., Scholz, C.H., 1999. Kinematics of normal fault growth and fault interaction in the central part of the Malawi rift. Journal of Structural Geology, in press. Cotton, F., Coutant, O., 1997. Dynamic stress variations due to shear faults in a plane layered media. Geophysical Journal International 128, 676±688. Das, S., Scholz, C.H., 1982. O-fault aftershocks clusters caused by shear stress increase? . Bulletin of the Seismology Society American 72, 1669±1675. Dawers, N.H., Anders, M.H., Scholz, C.H., 1993. Growth of normal faults: displacement-length scaling. Geology 21, 1107±1110. Dawers, N.H., Anders, M.H., 1995. Displacement length scaling and fault linkage. Journal of Structural Geology 17, 604±614. Dieterich, J.H., 1987. Nucleation and triggering of earthquake slip: eect of periodic stresses. Tectonophysics 144, 127±139. Gupta, A., Scholz, C.H., 1999. A model of normal fault interaction based on observations and theory. Journal of Geophysical Research, in press. Harris, R.A., Day, S.M., 1993. Dynamics of fault interaction: parallel strike-slip faults. Journal of Geophysical Research 98, 4461±4472. Jaume, S.C., Sykes, L.R., 1992. Changes in state of stress on the southern San Andreas fault resulting from the california earthquake sequence of April to June, 1992. Science 258, 1325±1328. Kanninen, M.F., Popelar, C.H., 1985. Advanced Fracture Mechanics. Oxford University Press, New York. King, G.C.P., Stein, R.S., Lin, J., 1994. Static stress changes and the triggering of earthquakes. Bulletin of the Seismology Society America 84, 935±953. Peacock, D.C.P., 1991. Displacements and segment linkage in strike-slip fault zones. Journal of Structural Geology 13, 1025±1035. Peacock, D.C.P., Sanderson, D.J., 1991. Displacements, segment linkage and relay ramps in normal fault zones. Journal of Structural Geology 13, 721±733. Peacock, D.C.P., Sanderson, D.J., 1994. Geometry and development of relay ramps in normal fault systems. Bulletin of the American Association of Petroleum Geologists 78, 147±165. Schlische, R.V., Young, S.S., Ackermann, R.V., Gupta, A., 1996. Geometry and scaling relations of a population of very small rift related normal faults. Geology 24, 683±686. Scholz, C.H., 1994. A reappraisal of large earthquake scaling. Bulletin of the Seismology Society America 84, 215± 218. Scholz, C.H., 1998. Earthquakes and friction laws. Nature 391, 37±42. Segall, P., Pollard, D.D., 1980. Mechanics of discontinuous faults. Journal of Geophysical Research 85, 4337±4350. Sieh, K., et al., 1993. Near-®eld investigations of the Landers earthquake sequence, April to July 1992. Science 260, 171±176. Stein, R.S., King, G.C.P., Lin, J., 1992. Change in failure stress on the southern San Andreas fault system caused by the 1992 magnitude 7.4 Landers earthquake. Science 258, 1328±1332. Toda, S., Stein, R.S., Reasenberg, P.A., Dieterich, J.H., Yoshida, A., 1998. Stress transferred by, the 1995 M ÿ w 6:9 Kobe, Japan, shock: Eect on aftershocks and future earthquake probabilities. Journal of Geophysical Research 103, 24543±24565. Wesnousky, S.G., Scholz, C.H., Shimazaki, K., Matsuda, T., 1984. Integration of geological and seismological data for the analysis of seismic hazard: a case study of Japan. Bulletin of the Seismology Society America 74, 687± 708.
Fault interactions and seismic hazard C.H. Scholz*, Anupma Gupta Lamont-Doherty Geological Observatory and Department of Earth and Environmental Sciences, Columbia University, Palisades, NY 10964, USA
Abstract Faults usually are not isolated features but exist within a population of faults which may interact through their stress ®elds. This poses two serious problems for seismic hazard analysis. The most severe such problem lies in estimating the likelihood of whether or not a future earthquake will be con®ned to a single fault (or fault segment) or will jump to adjacent faults and result in a larger earthquake. We review recent results which show that it is possible to determine the degree of fault interaction from geological data alone. We propose that the probability of an earthquake jumping from one fault to another will increase with the degree of stress interaction between the faults, and introduce a simple criterion to estimate the degree of interaction based on separation and overlap of echelon normal fault pairs. This statics based criterion for normal faults agrees qualitatively with the limited dynamic modeling of (Harris, R.A., Day, S.M., 1993. Dynamics of fault interaction: parallel strike-slip faults. Journal of Geophysical Research 98, 4461±4472) of the more complex case of strike slip faults and suggests that a more general criterion may be obtainable. The second problem discussed is the hazard associated with earthquakes being triggered by earlier earthquakes on a dierent fault. This phenomena produces seismic hazards distinct from that associated with ordinary aftershocks. We point out that with rapid data acquisition and proper preparation, it is feasible to issue a short-term hazard assessment regarding triggered earthquakes shortly after the occurrence of a potentially triggering event. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The incorporation of geological data on active Quaternary faults: their distribution, lengths, * Corresponding author. Tel.: +1-914-365-8360; fax: +1-914-365-8150. E-mail address: [email protected] (C.H. Scholz). 0264-3707/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 4 - 3 7 0 7 ( 9 9 ) 0 0 0 4 0 - X
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slip direction, and slip rate, has become central to the analysis of geological hazard. Although the basics of this practice has been worked out (Wesnousky et al., 1984), there remain serious problems. These problems arise because faults seldom, if ever, exist as single, isolated structures. They occur within a population of faults and hence are not mechanically isolated but may interact with other faults through their stress ®elds. Furthermore, at a lower hierarchical level, individual faults are found, on closer inspection, to be not single fractures but to be segmented in various ways. For an isolated, non-segmented fault, the seismic hazard evaluation is straightforward. One assumes (with good reason) that the seismic cycle of the fault consists of a single large event that ruptures the entire fault length. A scaling law (e.g., Scholz, 1994; Anderson et al., 1996), then allows one to estimate the slip and hence moment of this earthquake. If one can determine the geological slip rate of the fault then one may also estimate the recurrence time of this earthquake. In the more complex situation in which there is a population of segmented faults, we have the more dicult problem in assessing the possibility that an earthquake will rupture a number of faults or fault segments. Examples of this abound: a particularly well observed example was the 1992 Landers, California, earthquake, which ruptured three major and a number of minor subparallel strike slip faults (Sieh et al., 1993). This possibility poses a quandary for seismic hazard analysis. Consider the simple example of a fault of length L which slips at a known geological rate and contains a single segment boundary as indicated by an oset, say, at midpoint. We may consider two end member hazard models Ð one based on the assumption that both segments break simultaneously and the other on the assumption that each segment breaks independently. In the ®rst case, the expected slip will be double that of the second case (Scholz, 1994), hence maximum earthquake moment is four times larger, but the earthquakes in the second case are expected to occur four times more often. Clearly, the hazard estimate is very dierent for the two scenarios. Is there any way that we can determine if one is more probable than the other? If we had a past history of such earthquakes on this fault that would be very instructive, but for most places the seismic history is too short to provide much guidance on this problem. A second problem in seismic hazard analysis that results from fault interactions is the triggering by one earthquake of an earthquake on another, nearby, fault. The problem with such triggered events is that they do not obey the statistics of aftershock per se, in which they occur on the same fault as the mainshock and the largest aftershock is at least one magnitude unit smaller than the mainshock. In contrast, the triggered event may be as large or larger than the triggering, event. Whereas that hazard posed by aftershocks is contained within the damage region of their mainshock and consists of further damage to already damaged structures, triggered earthquakes can impose major damage to more distant regions. As an example, the M 6.5 Big Bear earthquake was triggered by the Landers earthquake and caused extensive damage in a region which was not itself damaged by the Landers shock (Stein et al., 1992; Jaume and Sykes, 1992). While in that case the triggered earthquake was on a dierent fault, there are also many cases of the triggered event being on the extension of the same fault. In the Umbria-Marche sequence of 1997, a M 5.7 event triggered 9 h later an adjacent M 6.0 event, which ruptured the same fault farther to the north, causing much additional damage (Amato et al., 1998). How can we include this phenomena in seismic hazard analysis?
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2. Estimating fault interactions When considering the possibility that an earthquake may propagate from one fault to the next it seems reasonable to conclude that this will depend on how strongly the faults are interacting. This will range from faults that are mechanically independent to those that are linked so strongly as to be acting as a single mechanical unit. Similarly, there is probably no simple distinction between interacting faults and segments of a single fault. For example, are the three primary faults ruptured in the Landers earthquake: the Johnson Valley, Emerson, and Camp Rock faults, separate faults or segments of one fault. It is likely that they are considered separate faults only because the geologists that originally mapped them named them individually. It is also likely that those geologists did not have these issues in mind when they did so: they could just as easily have called them the Johnson Valley, Emerson, and Camp Rock segments of the Landers fault system. Such a distinction may therefore be of little value and we ignore it here. Let us consider that the probability of an earthquake rupturing two faults is nil in the case of non-interacting faults and increases monotonically with the degree of interaction. It would, therefore, be of considerable value if there was a method, independent of the record of past earthquakes, of measuring the degree of fault interaction. Faults obey well-de®ned scaling laws relating net geological displacement D and length, L (e.g. Schlische et al., 1996). Isolated faults also have well de®ned alongstrike displacement pro®les (e.g. Dawers et al., 1993) in which the displacement tapers o gradually towards the fault tip. Within a given geological setting, this fault tip taper (FTT) angle is found to vary within fairly small limits. This ®nding can be readily interpreted in terms of `large-scale yielding' crack model in which the fault is considered an elastic crack in which plastic yielding occurs within a volume surrounding the crack tip (Kanninen and Popelar, 1985, pp. 367±372,
Fig. 1. Schematic diagram showing the interaction of two echelon normal faults. A. Fault F1 approaches fault F2 with an overlap O and separation S. It's right-hand tip lies in the stress drop ®eld of fault F2. B. The displacement pro®le of fault F1 is now asymmetric, with a fault tip taper FTT ' at the interaction end steeper than FTT at the non-interacting end. It is this distortion of the displacement pro®le that was used by Gupta and Schozl (1999) to determined the interactions between faults in terms of overlap and separation.
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Fig. 2. Interactions between three normal faults that constitute the main bounding fault system of the northern Lake Malawi rift. Top: Individual displacement pro®les for each fault at three time epochs, 2.5 m yr, 1.5 m yr, and the present. In two way travel time (TWTT) 1 s 0 1 km. Separation of the southern and central faults is approximately 5 km, that for the central and northern faults, 7 km. Bottom: Summed displacement pro®les at the dierent times. By the latest period, the summed pro®le is beginning to approach a standard pro®le for an isolated fault of the same length as the fault system (dashed curve), indicating that the system is beginning to behave as a single fault. (From Contreras et al., 1999).
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this is known as the CTOA, or crack tip opening model in tensile fracture mechanics). This model predicts a linear displacement taper near the fault tip, and that its value is proportional to the (®nite) stress concentration at the crack tip sc sy ÿ sa , where sy is the yield strength of the rock and sa is the regional applied stress. When faults approach one another and begin to interact through their stress ®elds, their displacement pro®les near their interacting tips become distorted in comparison with those of isolated faults (Peacock, 1991; Peacock and Sanderson, 1991, 1994). The usual observation is that the FTT becomes steepened. This phenomenon can be readily explained in terms of the large scale yielding model (Fig. 1). There we show a fault F1 of length L1 approaching a second fault F2 of length L2 with a certain overlap O and separation S (Fig. 1a). Consider, for simplicity, that they are vertically dipping normal faults, since in that case their stress ®elds are symmetric. The stress ®eld for fault 2 is sketched in the ®gure, where solid curves are contours of stress drop and dashed curves are stress increases. Note that although in this schematic ®gure no values are given on the contours, the values of stress changes produced by fault are 102±103 those produced by individual earthquakes (because the D/L ratio for faults, 010ÿ2 , is that much larger than for earthquakes). The right-hand tip of fault 1 is within the stress drop region of fault 2. Hence, at the proximal tip the FTT A sy ÿ sa ÿ Ds, and hence must be greater than at its distal noninteracting end (Fig. 1b). This distortion will increase proportionately to the degree of fault interaction and hence provides a measure of it. Thus, as displacement continues to accumulate on fault 1, it will grow on its distal end, maintaining a constant FTT, but its growth will be impeded at the interacting end because each growth increment will move the tip into a higher stress drop region, hence the FTT will have to continually increase for growth to proceed. This progression is illustrated in Fig. 2 (top), which shows the displacement pro®les for three interacting normal faults that constitute the main boundary system of the central part of the Lake Malawi rift in Africa. A stratigraphic back-stripping method has allowed the reconstruction of the displacement pro®les of these faults at three successive time intervals, showing the progressive growth and interaction of the faults (Contreras et al., 1999). The main features of this sequence is that the interacting fault tips become relatively pinned, propagating little while their proximal displacement gradients progressively steepen, while the distal fault tips propagate freely, maintaining constant displacement gradients. This case was analyzed by Gupta and Scholz (1999), who showed that the interacting tips were, in fact, propagating into progressively higher stress drop regions, while developing progressively more deformed displacement pro®les. They also analyzed many other cases and showed that these systematics are general. They also found that there is a critical stress drop at which the tips become fully pinned. At about that point, minor fractures begin to develop in the interaction region, progressively linking the two faults. The summed displacement pro®les of the two main faults plus the minor faults begin to approach the pro®le expected from a single fault of length equal to that of the combined lengths of the interacting faults, indicating that they are starting to act as a single mechanical unit (e.g. Dawers and Anders, 1995). The dashed curve in Fig. 2 (bottom) is an example. These results illustrate that one can, in fact, de®ne the degree of fault interaction, which forms a continuum from non-interacting to fully linked. As a practical matter it is very dicult if not impossible in most cases to obtain the high resolution displacement pro®les that were used as a diagnostic tool in the above discussion. It is relatively easy to measure overlap and
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separation between adjacent faults, however, and the Gupta and Scholz (1999) analysis allows one to portray the degree of interaction in terms of those parameters. Fig. 3 summarizes their results. Plotted are the separation and overlap of one fault pair, each normalized to the fault length of the second fault (There will be two such values for each fault pair Ð which may dier signi®cantly if the faults have very dierent lengths. Interaction will thus depend on the direction of rupture propagation.). The region of interaction is shaded, with the degree of interaction increasing monotonically from the minimum interaction curve Ds 50 MPa) to the maximum Ds 250 MPa). The interaction criterion described above was determined for normal faults, which only involve the interaction of shear stresses. The situation is more complicated for strike slip faults, because in that case slip on one fault will induce changes in both shear and normal stress on the other, and those changes are asymmetric with respect to the fault tip (Segall and Pollard, 1980). In that case the interaction will depend on the sign of the jog (compressional or extensional). The open circle and square in Fig. 3, for example, are the maximum compressional and extensional jogs, respectively, that could be jumped by a 10 MPa stress drop earthquake in the simulations of Harris and Day (1993). The compressional case corresponds fortuitously to the limiting case for normal faults given by our criterion. The extensional case shows that it is easier for an extensional jog to be jumped Ð the limiting separation is greater. Harris and Day discuss some observations that corroborate that ®nding. As examples, the solid circle and square are the value for the Johnson Valley±Emerson and Emerson±Camp Rock segments, respectively, of the Landers (strike-slip) rupture. As shown, they are both strongly interacting. The important point though, is that the Harris and Day results, using dynamic models, demonstrate the basics of our static analysis: that for small separations earthquakes can easily jump the jog, but there is some critical separation which no earthquake can jump. A related observation is that segment boundaries for strike slip faults
Fig. 3. A simple criterion for the interaction of echelon normal faults pairs in terms of overlap and separation, based on Gupta and Scholz (1999). Solid circle and square are the Johnson Valley±Emerson and Emerson±Camp Rock segments, respectively, of the Landers rupture. Open circle and square are the maximum compressional and extensional jogs of a strike slip fault, respectively that could be jumped by a 10 MPa stress drop earthquake in the modeling of Harris and Day (1993). The line is the scale invarient O/S ratio observed for segment boundaries of strike-slip faults by Aydin and Nur (1982).
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such as pull-apart basins and compressional horsts have been found to have a scale invariant O/S03 (Aydin and Nur, 1982), which is shown as the line in Fig. 3. This standard O/S ratio indicates that such segment boundaries are strongly interacting Ð it is this interaction that results in this scale invariance (Gupta and Scholz, 1999). That these results for strike-slip faults are approximately in accord with the model derived for normal faults indicates that the normal stress changes are secondary with respect shear stress changes in promoting earthquake jumping across segment boundaries. Whether or not an earthquake will jump between faults depends also upon the initial conditions, namely the stress state on the two faults at the time of the earthquake. The probability that two adjacent faults are in synchroneity in their seismic cycles can be expected to increase with the degree of fault interaction because that increases the chance that the two faults ruptured together in the previous earthquake. This reinforces the idea that jumping probability increases with degree of interaction. We have, of course, no way of assigning jumping probability values to interaction levels. At the present, we only oer this criterion as a guide for those who have to face this problem in the course of making a seismic hazard evaluation. Much more work needs to be done on this problem, particularly for the case of strike-slip faults.
3. Triggered earthquakes That the static stress change brought about by an earthquake can trigger earthquakes on other nearby faults was ®rst pointed out by Das and Scholz (1982). Since then a considerable literature has been devoted to this topic, which is now reasonably well understood (e.g. King et al., 1994). Such triggered events occur with time delays of hours to a few months following the triggering event Ð they are not simultaneous even though the stresses in the seismic waves may be several times the residual static stresses (Cotton and Coutant, 1997). This is because the rock friction laws are insensitive to transient stresses (Dieterich, 1987; Scholz, 1998). These pose a seismic hazard distinct from that of aftershocks. Aftershocks are restricted to the rupture dimension of the mainshock, hence they pose an additional hazard only to the mainshock damage area. This is well known and the authorities are accustomed to warning the populace of the continued danger due to aftershocks. Furthermore, aftershocks in this strict sense are of secondary importance: the sum of seismic moment in an aftershock sequence is typically only about 5% of that of the main shock and the largest aftershock is at least one magnitude unit less than the mainshock. They generally only cause damage to structures already damaged by the mainshock. None of this applies to triggered events. They may be as large or larger than the triggering event, and they may cause extensive damage in a region not damaged by the triggering event. How might we deal with the hazard associated with this phenomena? Triggering occurs only on faults on which the triggering earthquake has produced a small but ®nite increase in the Coulomb stress t ÿ s ÿ p resolved on the fault in its slip direction. It seems reasonable to suppose that the probability of this happening will increase with the magnitude of that Coulomb stress increase. Furthermore, because the mechanism of this triggering must be the same as the triggering of aftershocks, sensu strictu, we can suppose that
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the probability of a triggered event occurring will decrease hyperbolically with time following the triggering event, in the same manner as the decay of aftershocks. Let us suppose that we have the proper infrastructure in place: we have a local seismic network, telemetered so that we can quickly determine, in the event of a large earthquake, its moment, focal mechanism and rupture length. If we also have a digitized map of all Quaternary active faults in the same district, we can then quickly determine the Coulomb stress change that has occurred on all those faults. We are then in a position to determine what other regions in the district have experienced an increase in short-term hazard. A warning based on such an analysis could have a number of meritorious eects, depending on the circumstances. Toda et al. (1998) conducted an ex post facto exercise of this type following the 1995 Kobe, Japan, earthquake, with encouraging results.
4. Conclusions We have introduce approaches to two outstanding problems in seismic hazard analysis. Both have to do with fault interactions. The ®rst problem is how to estimate the likelihood of an earthquake jumping from one fault to another. It is proposed that this is related to the degree to which the two faults interact through their stress ®eld. In the case of an echelon arrangement of normal faults, a related study (Gupta and Scholz, 1999) shows that the interaction between a pair of such faults de®nes a continuum between non-interacting and fully linked. The degree of interaction can be estimated by the overlap and separation of the faults, normalized to fault length. For echelon strike-slip faults the relationship is more complex, being asymmetric because of normal stress interactions, as shown by the dynamic simulations of Harris and Day (1993). However, both their analysis and observations for strike-slip fault arrays are roughly consistent with the normal fault model. Much more work needs to be done, however, before quantitative estimates of probabilities can be made for the purpose of hazard analysis. The second problem concerns the triggering of earthquakes on nearby faults by coseismic stress changes, which results in a short term increase in seismic hazard following large earthquakes. If one can obtain in timely manner the source characteristics of that event and one has a regional map of quaternary active faults, it is feasible to estimate the short term increase of seismic hazard for the other faults in the district. This is done by calculating Coulomb stress changes on all nearby faults and recalculating short-term probabilities in the manner used by Toda et al. (1998).
Acknowledgements I thank Prof. G. Cello for inviting me to the workshop `The Resolution of Geological Analysis and Models for Earthquake Faulting Studies' in Camerino, which ®rst prompted me to think about this problem. This work was partially supported by NSF grant EAR 97-06475. Lamont-Doherty Earth Observatory contribution number 5926.
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