Aftershocks occurring on faults in the far-field of a large earthquake rupture can generally be accounted for by changes in static stress on these faults caused by the rupture1,2. This implies that faults interact, and that the timing of an earthquake can be affected by previous nearby ruptures3,4,5,6. Here we explore the potential of small earthquakes to act as ‘beacons’ for the mechanical state of the crust. We investigate the static-stress changes resulting from the 1992 Landers earthquake in southern California which occurred in an area of high seismic activity stemming from many faults. We first gauge the response of the regional seismicity to the Landers event with a new technique, and then apply the same method to the inverse problem of determining the slip distribution on the main rupture from the seismicity. Assuming justifiable parameters, we derive credible matches to slip profiles obtained directly from the Landers mainshock7,8. Our results provide a way to monitor mechanical conditions in the upper crust, and to investigate processes leading to fault failure.
The mechanical state of a fault can be characterized by Coulomb stress9 CS = τ - μ(σn - p), a scalar, where τ is shear stress, σn is normal stress, μ is the coefficient of friction, and p is the pore pressure. In general, CS is not known, but a “beacon” fault will experience a static stress change ΔCS = Δτ - μ′Δσn from a known shear dislocation on a causative fault10 (the “rupture”). In a linear isotropic elastic medium ΔCS is a scalar. In this simple formulation, the effective friction parameter μ′ includes a possible change in pore pressure, but is independent of space and time. ΔCS ≪ CS for beacons sufficiently far from the rupture (7.5 km in this case; Fig. 1) and only the component of Δτ parallel to the beacon's slip vector is consequential. ΔCS > 0 is expected to ‘encourage’ the beacon fault toward failure; ΔCS < 0 is expected to ‘discourage’ it from failing. This expectation can be tested for earthquakes that are resolved as shear failures with specific location, strike, dip and rake. In general, ΔCS is significant within a rupture-length of any rupture11. Both deformation and stress changes, however, are dominated by the largest ruptures12. We consider the classical interaction experiment where ΔCS stems from a large rupture and affects a multitude of beacon earthquakes that are assumed to be too small to mutually interact1.
The 1992 Landers sequence in southern California provides the causative rupture in this experiment: it is dominated by the 28 June, M W = 7.2 mainshock and includes the 23 April, ML = 6.1 Joshua Tree ‘foreshock’ and the 28 June, ML = 6.5 Big Bear ‘aftershock’ (Fig. 1). The rupture is within a broad zone of seismicity associated with the plate boundary. A set of relocated and quality-selected focal mechanisms for most M ≥ 1.5 earthquakes in this zone since 1980 have been interpreted for fault structure and preferred nodal planes (data are available at ftp://scec.gps.caltech.edu/pub/focal/focal1.nano). This seismicity stems from many relatively small, scattered faults ranging widely in orientation and kinematics which serve as ideal beacons for stress change13 (Fig. 1). ΔCS is calculated for each beacon as the compound effect of 28 elemental dislocations10 approximating the rupture7.
Earthquakes before the rupture (Fig. 1a) are not affected by it, but their ΔCS provide a reference: the numbers of encouraged (E) and discouraged (D) pre-rupture earthquakes are similar (E/D ≈ 1). After the rupture, E/D is 5–10 times higher, primarily because of an increase in E ( Fig. 2). A suppression of the discouraged pre-rupture earthquakes by the rupture is apparent in the overall response of the far field ( Fig. 2c) and by comparing the spatial distribution of the discouraged and encouraged pre-rupture earthquakes in the near field before and after the rupture14,15 (Fig. 1). Although a broad range of friction values (0.4 < μ′ < 1.0) yields an E/D response qualitatively consistent with the Coulomb failure model, we obtain the strongest E/D response for a high value of friction9, μ′ ≈ 0.8. This optimal friction value seems robust because it remains within ±0.1 of 0.8 for several non-overlapping subsets of beacons during the first 1.5 years after the rupture. The apparent agreement between these and other experimental values is surprising because μ′ includes any Δp effect16.
Earthquakes are very sensitive stress beacons; a significant increase of E/D is detected for 0.15 ≤ ΔCS < 0.25 bar (ref. 2; Fig. 2d). They also retain a memory of the rupture long afterwards. In the far-field, E/D is largest within a few months of the rupture; it decreases gradually afterwards, but it is still about twice the average pre-rupture level several years after the rupture (Fig. 2b). In contrast, the overall rate of seismicity in the far field (for example, E+D in Fig. 2c) returns to pre-rupture levels within a year. The far-field effect of the rupture persists as a change in the kinematics of seismogenic faulting, but not necessarily as a rate increase.
According to the Coulomb failure model, some encouraged faults rise above critical stress and are expected to fail immediately, whereas discouraged faults are pushed away from failure and will take longer to reach it. The post-rupture deficit in discouraged earthquakes (Fig. 2c) is expected to persist until tectonics has re-loaded all negative ΔCS faults to their pre-rupture stress level. Tectonic loading is slow, particularly for secondary faults. Even for the San Andreas fault, the encouraging ΔCS effect southwest of the Landers rupture is equivalent to a decade of tectonic loading4,5. Thus, a simple Coulomb formulation can account for a prolonged E/D response. The high rate of encouraged earthquakes, however, is also persistent after the mainshock (Fig. 2c). Thus failure is delayed for encouraged as well as discouraged earthquakes, in violation of a simple Coulomb model9. A delay in the nucleation of earthquakes can be accounted for by fluid flow in response to Δ p (poroelastic response)15 and/or by rate-and-state-dependent friction17.
In spite of data and model limitations, Fig. 2 and many similar tests demonstrate some success in modelling the beacon response to a known large rupture. We expand on these experiments by turning the problem around and extracting information on the rupture from the seismicity. We parametrize the approximately 75 km long Landers rupture (Fig. 1 ; overlapping segments are combined) as 26 segments, each 3 km long, 15 km deep, and with uniform slip. Total moment is fixed and starting slip values are the same for all segments. The Big Bear and Joshua ruptures are assigned fixed uniform slip9. We then iterate for the slip distribution that yields the optimal response, as predicted by elastic interaction and Coulomb failure. E/D tends to be unstable where data are scarce (Fig. 2a). This optimal value is therefore chosen as the maximum value of (E - D) integrated over a post-rupture period (3 years in Fig. 3). Beacons are weighted positively with ΔCS. Each step in an iteration maximizes the score by redistributing slip on the main rupture while keeping total moment constant and computes a new ΔCS for each beacon. This process is repeated until the score cannot be further improved. We have tested many iterations covering a range of starting, weighting and declustering parameters, spatial and temporal constraints on the data, and iteration schemes. We could clearly differentiate iteration schemes that yielded inconsistent spiky results from others that yielded relatively consistent and credible results.
In one class of iterations, slip change ΔS on one segment is traded with opposite change -ΔS/25 on each of the other 25 segments, in each step. ΔS = K (0, or 1, or -1) is determined by a grid search for the highest score. After all segments have had this role, the procedure is repeated until the score has reached the maximum for a particular value of K. Then the process can be repeated for a smaller K, and so on. We generally start these iterations with ΔS = 2 m and stop them at ΔS = 0.02 m. Even when K is held constant, iterations of this kind tend to converge rapidly on permissible slip distributions with persistent features (see ‘grid search’, dotted line in Fig. 3). In another class of iterations, one segment trades slip with only one of the others in each step, and then with each of the others in turn. As in the previous case, this procedure is repeated for all segments, and then again in subsequent sweeps until improvements in the score for each sweep become vanishingly small. These iterations are particularly laborious because they sample the solution space in great detail over slip distributions with extreme variations; they may yield high scores, but they tend to converge on unrealistically spiky and inconsistent slip distributions. The need for imposing a degree of smoothing is a common feature of inversion problems18; in this case it may derive in part from clustering in the field of beacons. In a third class of iterations, all segments are treated equally in each step. The change in slip for a segment Δ Si is proportional to how much it affects the score: Δ Si = C[d(score)/dSi]. The factor C is chosen from a set of values by grid-search for the optimal score. The final task in the nth iteration step is to normalize the absolute values of slip on each segment to maintain the total moment invariant: Sin=[S i(n-1) + Δ Sin][Σ|S i(n-1)|/Σ|S i(n-1) + ΔS in]. Iterations of this kind (see ‘derivatives’, solid and dashed lines in Fig. 3) tend to converge more rapidly than iterations by ‘grid search’. In spite of fundamental differences, these methods tend to yield similar slip distributions (for example, Fig. 3).
The weighting function w(ΔCS) is one of the most important factors in the outcome of the iterations. In most iterations the weight was antisymmetric about ΔCS = 0. Generally, the weight drops gradually with |ΔCS| (in Fig. 3: dotted line, by 0.01 per bar; solid and dashed lines, decreasing to 0.01 per bar) until |ΔCS| ≈ 0.15 bar (in Fig. 3: dotted line, w = 0.1 at |ΔCS| = 0.1 bar; solid and dashed lines, w = 0.2 at |ΔCS| = 0.2 bar) and then drops rapidly to zero for smaller ∣ΔCS∣ (in Fig. 3: dotted line, 1 per bar; solid and dashed lines, increasing to 2 per bar). This kink in the weight function correlates with the threshold value of ΔCS (Fig. 2d). The weight may change during the iteration (only dashed and solid in Fig. 3). At the beginning of the iteration the kink at the stress threshold is subtle because the slip distribution is uniform and the ΔCS values are far from realistic. At this stage, all beacons need to play a role in properly directing the convergence. As the iteration converges to more realistic ΔCS values, the weighting function is progressively more kinked. The shape of the weight function at low ΔCS affects the distance range over which beacons are effective; it is particularly important for the northern part of the slip distribution which is controlled by a few clusters (Fig. 1).
Near-field beacons (7.5–27 km) are critical for resolving the main features of the slip distribution (10–20 km wavelengths), but they have a relatively weak E/D response early after the rupture because of dense clusters of discouraged earthquakes (Figs 1 and 2a versus Fig. 2b). The data were declustered by eliminating beacons sufficiently close in time, space and fault kinematics (and reduced by 1/5 in Figs 1, 2 and 3). Nevertheless, local interactions may still dominate the response for earthquakes in the Yucaipa, Gold Mountain, Eureka Peak and Iron Ridge clusters (Y, G, E and I in Fig. 1) which coincide with foci of early post-rupture moment release neglected in our rupture model (ML = 4.7 at Y; ML = 5.3 at G; ML = 5.6 at E and 23 cm of left-lateral slip at I). Local interactions may be more prevalent where ΔCS is large and beacons are dense. In general, changes in declustering parameters have small effects on slip distribution. But when clusters of beacons suspected of violating the assumption of mutual independence are excluded, slip distributions tend to be less peaked (Fig. 3, dashed versus solid), probably because these clusters are relatively close to the rupture (Fig. 1). Temporal data selection is less critical. Progressively later batches of seismicity, as late as a year after the rupture, tend to yield comparable slip distributions.
Postseismic deformation related to the rupture, such as post-rupture creep8 and viscous response below the rupture20, may significantly affect the beacon response. Tectonic loading is probably a relatively minor factor within a few years of the mainshock. Focal mechanisms are selected according to quality14,15 and are expected to introduce noise, but little systematic bias. Our structural interpretation of focal mechanisms for fault planes14,15 is another obvious source of noise, but it is substantially better than random, as demonstrated by the much subdued beacon effect from the opposite planes (Fig. 2b). Future experiments will benefit from continuing improvements in data quality and quantity. Beacons are more abundant around the southern part of the rupture where, not surprisingly, slip distribution features appear to be more robust (Fig. 3). Although μ′ = 0.8 yields optimal E/D, a uniform friction value is unrealistic in view of possible dependency on fault slip rate17, on presence of fluid, and on depth (temperature and pressure)16. Some of the early post-rupture earthquakes may have been discouraged and reflect inadequacy of the Coulomb formulation. Earthquakes sufficiently close to failure may occur in spite of a negative ΔCS, according to state-rate friction17,19 and an initial discouraging stress change may become encouraging as a result of fluid flow in poroelastic conditions16. Finally, alternative models of the rupture derived from the mainshock are significantly different8 (Fig. 3), so some of the post-rupture discouraged earthquakes may reflect inaccuracies with the rupture model (for example, Fig. 2).
We have obtained slip distributions of the Landers 1992 rupture from off-rupture aftershocks assuming a simple Coulomb interaction model and using different methods and assumptions. They compare with each other and with distributions obtained directly from the mainshock nearly as well as these direct determinations compare with each other (Fig. 3). Although small earthquakes contribute little to total deformation, their distribution in space, time, and fault kinematics is very sensitive to stress changes and offer unique opportunities to monitor stress. When applied to abundant high-resolution earthquake data, the approach outlined here is expected to provide new insight into a variety of phenomena affecting the mechanical state of the crust. Examples might include natural phenomena, such as fault creep or magma injection, or large engineering operations, such as impounding of water reservoirs or flooding of oil fields. Delay in the response of seismicity to known stress changes can be revealing about the nucleation of small earthquake ruptures. Conversely, mechanical changes resolved from seismicity may reveal processes preparatory to large earthquakes.
We are grateful for helpful suggestions and/or discussions from J. Hardebeck, P. Reasenberg, J. Vermilye, C. Scholz, D. Simpson, L. Sykes, R. Stein, B. Shaw, B. Menke and K. Jacob. Financial support was provided by the Southern California Earthquake Center, the US Geological Survey and the NSF.
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Study on displacement field generated by aftershocks in Landers earthquake fault zone and its adjacent areas
Acta Seismologica Sinica (2005)