Cocco¹ M., King² G.C.P. and C. Nostro¹

Observations of faulting episodes clearly indicate that the major fault zones consist of different fault segments, which are singly able to generate large magnitude earthquakes. When an earthquake occurs on a particular segment, it perturbs the stress state on adjacent faults and may favor or inhibit subsequent earthquake ruptures. Because these changes in the state of stress affect the likelihood for future earthquakes, their determination is important for the assessment of earthquake hazard.

Fault interaction and earthquake triggering caused by coseismic ruptures have been recently studied either with dynamic or static stress changes. Dislocation models have been currently used to calculate Coulomb stress variations caused by shear dislocations (Stein et al., 1992 and 1994; Harris and Simpson, 1992; King et al., 1994; Harris et al., 1995; Hodgkinson et al., 1996; Stein et al., 1997; Nostro et al., 1997; Toda et al., 1998; Nalbant et al., 1998; Nostro et al., 1998). Static stress changes have been proposed to explain variations of seismicity rates (Reasemberg and Simpson, 1992; Simpson and Reasemberg, 1994; Toda et al., 1998), off-fault aftershocks (Das and Scholz, 1981; Stein et al., 1992 and 1994; Dieterich, 1972 and 1994; Dieterich and Kilgore, 1996) and probability changes for the occurrence of impending earthquakes (see Toda et al., 1998, and references therein). Dynamic models have been proposed to study fault interaction and earthquake triggering in terms of dynamic stress variations (Harris and Day, 1993; Gomberg et al., 1997 and 1998). The Landers 1992 earthquake provided several episodes of short range fault interactions (Stein et al., 1992; Harris and Simpson, 1992; Cotton and Coutant, 1997) and long range interactions and remotely triggered seismicity (Hill et al., 1993). Spudich et al. (1995) discussed the problem related to the triggering of the Big Bear aftershock caused by the 1992 Landers earthquake, pointing out that a time delay of 3.5 hours is difficult to be explained either with dynamic or static stress changes. Gomberg et al. (1998) explained the main characteristics of both short (static and dynamic) and long range (dynamic) triggering in terms of a frictional instability model. These authors discuss how static and dynamic loads can alter the timing of impending earthquakes.

Most of the aforementioned papers rely on the computation of the stress field outside a rupturing fault. Several other authors discuss the spatio-temporal evolution of the stress over the fault plane (see Quin, 1990, Miyatake, 1992, Mikumo and Miyatake, 1995, Bouchon, 1997, among several others). In these studies the stress on the fault plane is inferred either by finding the crack model that best fits the slip distribution or by deriving the spatio-temporal stress distribution directly from kinematic rupture models. Boatwright and Cocco (1996) interpreted the heterogeneous distributions of slip, the interevent seismicity and the aftershocks distribution on the fault plane in terms of lateral variations of frictional properties. They modeled these variations using a rate- and state-dependent friction law (see Dieterich, 1978, 1979a and b, 1980, 1986; Ruina, 1980, 1983; Rice and Tze, 1986; Scholz, 1990, and references therein).

All these studies emphasize that the stress evolves in time and in space over and around a rupturing master fault. However, in order to determine the onset of an earthquake rupture (i.e. the time of earthquake initiation) the absolute stress level and the constitutive laws on the neighboring faults should be known. If the absolute stress level is known, the computation of the Coulomb stress provides information on where on the neighboring faults the static friction level is exceeded, and does not determine whether or not the stress concentration is sufficient to induce propagation on the surrounding fault segments. The overcoming of the static friction level is only a necessary condition for the rupture initiation. If the stress level is sufficient to allow the rupture to propagate on the secondary fault, thus we have a triggering of a subevent, an aftershock or a separate earthquake depending on the delay of the induced rupture onset. The time delay depends on the rheological state of the secondary fault. Gomberg et al. (1998) point out that the triggering response (the time advance) to a transient or a static load depends on when in the loading cycle the load is applied. However, even if the rupture does not propagate on the secondary fault, the prestress conditions on this fault are modified in any case by the earthquake occurred nearby (Perfettini et al., 1998). The prestress conditions are determined by the stress on the fault plane as resulting from shear and normal stress components given both by the tectonic load and previous earthquakes occurred on the surrounding faults. The Coulomb stress analysis allows to infer where the earthquake is favored or inhibited, and nothing can be said on the exact timing for earthquake triggering. If the induced Coulomb stress is consistent with the remote tectonic loading, an increase of Coulomb stress will act as several years of tectonic load depending on the stress rate of the area. If this is true, an increase of CFF will change the probability for future earthquake ruptures.

It emerges from the considerations discussed above that there exist different processes responsible for fault interaction depending on the analyzed spatial and temporal scales. Static stress changes, for instance, provide information about interactions in a determined time scale between earthquakes: that is, from several tens of seconds (when the dynamic stress field reaches the static configuration) to years (before that viscoelastic diffusion processes become dominant). Other time dependent processes can modify the static stress distribution, owing to fluids migration, aseismic sliding, or viscoelastic pervasive relaxation of rocks. Dynamic stress variations are expected to be responsible for subevent triggering as well as for the heterogeneities of crustal faulting episodes.

The interaction between fault segments can be investigated using static analyses if the time-dependent stress concentrations generated during the propagation of the coseismic rupture are not significant. In contrast, coseismic interactions between faults belonging to the same segment and rupturing during the same earthquake are controlled by the stress field produced by the propagation of the dynamic rupture.

When an earthquake is in progress dynamic processes are clearly important in the epicentral region and the progress of rupture cannot be studied without considering dynamic effects. On the other hand when an event initiates independently of other events only static stress conditions need be considered. Between these two extremes there is room for debate about the relative role of static and dynamic stresses. The question can be posed from two perspectives. First, theoretically dynamic stresses at a distance from an event are greater than static stresses and thus one might imagine that they should prove more important. However, in most cases aftershocks and later earthquakes occur long after the dynamic stresses have passed and correlate well with the static stress field. Thus it seems that some time dependent process must be important. Simply raising the stress above some triggering threshold is not enough.

Second, in a few cases, notably after the Loma Prieta earthquake, however events have been triggered at distances so great that static stresses would appear too small to have a triggering effect. It must either be concluded that these events resulted from dynamic stresses or that static stresses have been locally amplified. The former explanation has received the most attention and permits relatively straight forward numerical modeling of the amplitude of the dynamic stresses that must have been involved. But some questions remain. Similar or larger stresses are associated with other earthquakes and great earthquakes can produce equally large transient stress world wide. Distant triggering, however is exceptional rather than common place suggesting that such triggering may reflect special conditions at the target areas rather than a general process. Some role of fluids seems to be favored to explain such distant triggering and may either enhance dynamic or static effects. After Landers earthquake some of the triggered events of the Little Skull Mountain (LSM) zone are known to be associated with fluids. The LSM events occurred at the base of the seismogenic zone where the role of fluids in a normal faulting environment may be important (Sibson, 1992; Sibson, 1998) even though there is no local surface expression of contemporary hydrothermal or volcanic activity.

A number of possible mechanisms can be proposed for the role of fluids. Shaking may trigger the exsolution of gases and thus create large changes of pressure as these force their way through the crust creating seismicity in the process. Fluids in volcanic or hydrothermal regions occur in fissure, dikes or sills. Unlike faults, such elongate features have essentially no friction between their surfaces and will change form under even the smallest stresses. Stress amplifications at the ends of such features can be of the order of the length to width ratio which can be 100:1 or more. Thus even static stresses of 0.01 bars can be amplified to 1 bar, recognized to be sufficient to trigger events. Little other evidence can be adduced to contribute to resolving the problem although we note that tidal triggering of earthquakes commonly occurs in volcanic regions but not in purely tectonic environments. Thus very small stress changes are significant in such regions. Furthermore, tidal periods are much too long to allow exsolution of cases by shaking to be the mechanism.

Finally the concept of valving proposed by Sibson (1992) could play a role. This combines aspects of the foregoing concepts but suggests that at the base of the seismogenic zone a transition occurs from essentially hydrostatic to near lithostatic stresses. The barrier between the two is created by the clogging of pathways by hydrothermal mineralisation. Dynamic or static stress changes may act to re-open pathways creating a rapid increase of pore pressure in the seismogenic zone.

While the foregoing possibilities are inviting little, data seems available to distinguish between them.

Possible evidence for a transition from dynamic to static stress triggering (Cotton and Coutant, 1997) has been suggested for the Kobe earthquake by Toda et al. (1998). While nearer aftershocks seem to correlate best with static stresses, more distant ones are perhaps better explained by dynamic effects (Cotton and Coutant, 1997). Belardinelli et al. (1998) have studied the temporal evolution of dynamic stress and its evolution to the static stress level for the 1980 Irpinia earthquake. These authors investigated the subevent triggering of these earthquake and they interpreted the delayed triggering in terms of fault frictional properties. Several studies (Harris and Day, 1997; Cotton and Coutant, 1997; Belardinelli et al., 1998) have clearly shown by means of numerical modeling of stress outside a rupturing fault that after few tens of seconds the dynamic stress reaches the static configuration. Scholz (1998) suggests that faults (and their frictional properties) are quite insensitive to transient dynamic stress, while they respond to static stress changes.

Although it would be pleasing to be able to understand fully the relation between static and dynamic triggering, it is important to realize that from the perspective of establishing hazard dynamic effects do not appear to be important.

The motions that occur at the time of an earthquake are the most dramatic part of the seismic cycle and produce the most dramatic stress changes. The crust then responds to these abrupt changes with the most dramatic manifestation being the aftershock sequence. The aftershock distribution can be modeled in terms of Coulomb Stress as a consequence of the main shock slip alone (as we will discuss in the following sections), provided none of the aftershocks are large. Large aftershocks however, must be specifically included because they can substantially modify the distribution of Coulomb stress. An example is the Big Bear aftershock of the Landers earthquake (e.g King et al., 1994; Stein et al., 1992). Big aftershocks are not the only post-earthquake deformation that may modify the stress distribution, non-seismic creep or fluid flow may modify the stress distribution in the period following a main event.

Non-seismic creep may occur in two ways. Faults may continue to slip aseismically or the lower crust may deform in a viscous fashion. Near surface creep or creep within that mimics the slip distribution in the main event will tend to slightly augment the co-seismic stresses. Such an effect will be largely undetectable since only changes in the distribution of Coulomb stress and not small amplitude changes are likely to be detectable. If however parts of the fault did not move in the main event slip, then the Coulomb distribution pattern will change and the aftershock distribution or subsequent interactions with later large events will be not be explained by the mainshock slip function.

One likely form of post-earthquake slip is for slip to continue on a down-dip part of the main fault. This again does not greatly change the distribution of Coulomb stress lobes, but very greatly increases their magnitude and the area affected by significant stress changes. Much the same occurs if there is viscous relaxation in the lower crust and for strike-slip it is identical. This has been modeled for the Landers earthquake in a simple way by Stein et al. (1992) and for dip slip events by Stein et al. (1994). Although the magnitude of the effect, if it occurs, should be large, its effects have not yet been observed.

Earthquakes perturb fluid pressures in the crust and it has been shown that over a period of 6 to 12 months these re-equilibrate. This can take the form of simple relaxation of pressures (Muir-Wood and King, 1993) or fluid pressures can be transmitted to a distance (Nur and Booker, 1972; Hudnut et al., 1989). There have been no claims to observe the former effect, which should operate to modify effective friction with time. Since the effects are likely to be similar to effects attributed to "Rate-State Friction" the effects may occur, but have not been correctly identified. Miller et al. .(1996) modeled the earthquake cycle on a fault as a coupled shear stress-high pore pressure dynamic system. In such models fluid flow and pore pressure changes are the dominant factors controlling the stress state of the fault and the time dependent friction. Although these models provide a successful explanation for complex slip patterns on the fault plane and distribution of interevent seismicity, the state of stress of a seismogenic fault is also determined by the redistribution of shear and normal stress caused by the earthquakes occurred nearby. The developing of a poro-elastic model, where fluid flow and elastic stress changes are coupled, is an important task for future investigations.

Transmission of fluid has been identified in a volcanic zone in Afar (Noir et al., 1997) where a swarm of events propagate over a distance of 50 km over a period of 50 days. They show that this is compatible with plausible fluid conductivities along a fissure system. Similar results have been found by Cocco et al. (1998) to explain the sequence of moderate magnitude earthquakes during the 1997 Umbria-Marche seismic sequence.

An important question that has to be answered for the interpretation of Coulomb stress changes over time intervals ranging between several years to centuries is the determination of the contribution of viscoelastic relaxation of the lower crust and the upper mantle. In other words, it is necessary to understand how the viscoelastic relaxation modifies the static stress patterns resulting from coseismic dislocations. This problem is particularly important to study sequences of historical earthquakes.

Overall, although post-seismic effects are generally thought to be important, so far they have rarely been observed.

The Coulomb Failure Function, CFF (see Harris and Simpson, 1992; Reasenberg and Simpson, 1992; Stein et al., 1992 and 1994; Simpson and Reasenberg, 1994; King et al., 1994; Harris et al., 1995; Nostro et al., 1997), is defined as

.

B is used to take into account the modifications of the effective normal stress caused by pore fluid pressure. Harris (1998) discusses the validity of this assumptions which is motivated by the lack of knowledge about the role of pore fluids. Although it is a convenient assumption, Harris (1998) points out the risk of missing some important clues in the modeling and interpreting data.

The CFF function is based on the Coulomb criterion for shear failure (e. g. Jaeger and Cook, 1979). Changes in the CFF are given by

(2)

Once the static stress changes produced by a dislocation have been evaluated, the induced stress on the surrounding faults can be easily computed if the geometry as well as the faulting mechanism of the secondary faults are known. By this approach Harris and Simpson (1992) and Simpson and Reasenberg (1994) evaluated earthquake-induced changes in static stress on southern and central California faults produced by the 1989 Loma Prieta and the 1992 Landers earthquakes. However, if the geometry and mechanism of the secondary faults are unknown, the contribution of the regional stress field must be taken into account (see King et al., 1994, for a detailed discussion). In fact, the maximum changes in the Coulomb failure stress caused by an earthquake dislocation occur on secondary planes optimally oriented with respect to the total stress field. These planes are those where aftershocks are expected to occur (Stein et al., 1992; King et al., 1994). Following King et al. (1994) and Nostro et al. (1997), the optimally oriented planes can be determined both by the coseismic static stress change , and by the pre-existing regional stresses (that must be known). Thus, the total stress is

. (3)

In those cases where the geometry and the mechanism of the secondary faults are unknown or uncertain, it is possible to use a grid-search procedure to find the orientation of the optimally oriented fault planes where the Coulomb stress is maximum (Nostro et al., 1997). In this case, the CFF is computed using the total stress tensor, as shown by equation (3). Once the orientation of the optimally oriented planes is found in any node of the 3D spatial grid, we can compute the Coulomb stress changes on each of them, as a result of slip on the master fault, using equation (2).

Following this procedure, if the orientation and the magnitude of the regional stress are known, it is possible to compute the orientation of the planes optimally oriented for failure and therefore to calculate the change of coseismic Coulomb stress on them. The application to vertical strike slip faults is easier because it is possible to solve a two-dimensional problem (plane stress problem), where the vertical components of the regional stress tensor can be neglected (King et al., 1994). However, the application to dip slip faults requires the solution of a 3D problem where the ratio between vertical and horizontal components of regional stress tensor must be known.

According to the discussion presented above, it is well accepted that when an earthquake occurs it modifies the state of stress on nearby faults. As previously described, this perturbation can be evaluated by the static stress changes under the assumption that aseismic slip, fluid migration and viscoelastic stress diffusion processes do not modify the induced stress field. In other words, the perturbation to the state of stress remains unchanged if these other processes can be neglected.

Within the validity of these assumptions, the analysis of static stress is valid at time scales longer than those characteristic of the dynamic stress release (few tens of seconds) and shorter than those characteristic of viscoelastic stress diffusion (i.e. postseismic) processes (tens of years). These conditions have to be taken into account when Coulomb stress changes are used to interpret the migration of seismic activity as well as earthquake triggering.

It is well accepted that the static stress changes induced by a main shock can explain the spatial distribution and the fault plane solutions of aftershock sequences. This has been observed in many mainshock-aftershock sequences (Das and Scholz,, 1982; Stein et al., 1992, 1994; Nostro et al., 1997, among several others; see Harris 1998 for a detailed reference list). King et al. (1994) emphasized that mainshock induced stress changes are dominant when the earthquake stress drop has a similar magnitude than the regional tectonic stress. In fact, if the amplitude of the regional stress drop (which is commonly unknown) is larger than the perturbation to the stress state caused by the earthquake, the off-fault aftershock should not be affected by the static stress changes. On the contrary, in the cases where the stress drop of the mainshock is larger enough to overcome the regional stress, the mainshock induced stress changes control the spatial distribution and the faulting mechanisms of the aftershocks. It is important to emphasize that the correlation between the area of static stress increase and the aftershock distribution is not enough to establish a causal relationship: if mainshock stress changes control the aftershock behavior, thus their fault plane solutions are highly variable and the slip directions on the favorably oriented planes are consistent with the induced traction changes.

There exist several good examples of such interaction caused by static stress changes; however, we also discuss some difficult observational test such as the 1989 Loma Prieta event and the 1997 Colfiorito earthquake. In particular Beroza and Zoback (1993) showed that aftershock fault plane solutions of the 1989 Loma Prieta earthquake are not consistent with the mainshock induced stress changes: only 52% of the aftershocks agree with the traction changes caused by the Loma Prieta earthquake. They concluded that mainshock induced strength changes are a likely explanation for the observed aftershock pattern. This result reinforces the requirements of comparing mainshock induced stress changes on the fault plane of the aftershocks and observed slip directions.

Dieterich and Kilgore (1994) have shown that the duration of aftershock sequences depends on the absolute stress level. There have been several attempts to estimate the frictional properties and the state of background stress from aftershock sequences (Gross and Kisslinger, 1997; Gross and Burgman, 1998).

The most straight forward way to look for the effects of Coulomb Stress Changes is to correlate total seismicity following an event with region of increased or decreased Coulomb Stress. It is plots of this sort that have proved to be the most convincing. There is however, reason to believe that changes of stress operate through some sort of time dependent friction law to change the rate of seismicity. If this is the case then changes of seismicity rate are the most appropriate to compare with Coulomb stress Changes. This has been carried out for the 1989 Loma Prieta earthquake by Reasemberg and Simpson (1994) and for the 1995 Kobe earthquake by Toda et al. (1998); a clear correlation is found in both these studies. The biggest problem with the method lies in the need to compare pre-event seismicity (which is not always available) with post-event seismicity. For large events the pre-event activity can be very low so that while increases are statistically significant, reductions are not.

Although the relations between large events and subsequent smaller events is of importance, interactions between major faults is of greater practical significance, at least for seismic hazard assessment. Some such relations have been demonstrated for California by Stein et al. (1992, 1994) and by Deng and Sykes (1997b) but the number of events to examine is limited. A much larger set of events greater than magnitude 6 can be found in Turkey and the Aegean Sea. These have been studied by Stein et al. (1992) and Nalbant et al. (1998). Combined these studies produce a remarkable result. Among the many events only one has occurred in a region of Coulomb shadow produced by earlier events. Of the remaining events several others occurred in regions where stress had been enhanced by greater then 2 bars. If smaller stress increases are also regarded as significant, many others occurred in regions of increased Coulomb stress. The events which showed no correlation with stress increases all occurred at the beginning of the period. In the absence of data we cannot say whether correlations would have existed.

Whatever, the detailed physics involved in stress interactions, these empirical results strongly suggest that simple stress calculations provide a powerful tool for predicting regions of increased seismic hazard. One important question resulting from recent studies is the prediction of possible earthquake triggering as well as the definition of the variation of the earthquake probabilities caused by moderate and large earthquakes. In particular, because the triggering mechanisms depend on the absolute stress state and on the frictional properties of the faults, the answer to such question is not an easy task.

Because in these applications fault interaction is investigated over long time interval, the contributions of viscoelastic relaxation processes and tectonic loading have to be taken into account. We have already discussed the former process in the previous sections, and we will discuss the latter in the following one. It is important to emphasize that many other study cases of fault interaction between historical events exist in the world and that this is an important topic for future applications.

The simplest method for Calculating Coulomb Stress changes is to examine only the changes due to known earthquakes and ignore immediate post-seismic effects and the effects of long-term tectonic loading. For relatively short time spans (short compared to earthquake repeat times) this may be a reasonable approximation. However for longer periods loading must inevitably become more important.

Over long enough time spans the problem reduces to seismic gap theory. In the absence of creep, future events will occur in slip gaps where there has been an absence of events. Thus if really good data are available, stress concentrations resulting from seismic moment deficits can be identified. This is rarely possible. Trying to work back in time using Coulomb interactions is more difficult.

For strike-slip earthquakes there is some consensus about how loading occurs. The San Andreas fault for example, is thought to extend as a narrow feature to considerable depth. Below a "locking depth" above which earthquakes are considered to occur the fault is thought to slip aseismically at a uniform rate. This allows tectonic loading to be calculated in a straightforward way. The only problem arises because the locking depth is not, in general well known and the slip rate at depth must be correctly estimated. This approach has been applied by Deng and Sykes (1996) to the SAF and by Stein et al. (1997) to the North Anatolian Fault. A slight variation on Coulomb modeling that incorporates loading has been adopted by Harris and Simpson (1998). They examine the time taken for "Stress Shadows" (regions where earlier events have created negative stresses) to disappear by the increasing stress loaded by the regional tectonic stress. This latter approach provides an interesting tool to constrain the stress rate.

However, for normal faulting regions the crust below seismogenic depths is thought to deform as a viscous fluid. Certainly modeling of geological structures excludes the possibility that normal faults simply extend to depth as been assumed for strike-slip faults. Modeling such loading is not straightforward. Deng and Sykes (1996) following Savage (1976) estimated loading by applying negative earthquakes where reverse faulting events have occurred. This approach has been applied in California, where comparatively few dip-slip events have occurred, but at best it is a poor approximation.

The radical difference between accepted lower crustal models in different regions is disturbing. Both cannot be correct suggesting that some important understanding, required to model long term loading, is lacking. This is one of the most important topics over which scientific research in this field should be focused.

It emerges from the discussion presented above that the stress redistribution caused by an earthquake rupture can hasten, delay or trigger a subsequent event. However, because the determination of the time of occurrence of the impending earthquake cannot be determined without considering the frictional response to the sudden change in stress, these induced stress are used to compute the variations of the probability of occurrence. Toda et al. (1998) have presented calculations of Earthquake Probability changes caused by static stress perturbations. Although these results represent a useful task for seismic risk assessment, their interpretation is not straighforward. In other words, mature faults can respond to the induced stress changes in different ways depending on their frictional constitutive properties. In these circumstances, the computed earthquake probability changes are not uniquely determined by the static stress perturbations.