Treatise on geophysics chapter 6: slip inversion s. Ide

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6.5.1 Characteristics of Slip Models

There are qualitative features frequently pointed out in many slip models, for example, pulse-like slip propagation, few aftershocks in the area with large slipped area, and the tendency of rupture initiation from the bottom of the fault plane. In this section, we review the discussion and physical interpretation of these features.

In many slip models, slip propagates like a pulse with a narrow width compared to the size of whole fault plane. Summarizing seven slip models, Heaton (1990) concluded that the slip pulse is a general feature of seismic rupture process and not an artifact due to the setting of time windows or limited frequency contents of inverted data. One good example among later studies is the Landers earthquake that ruptured 65 km faults, but the width of slip area at any moment is less than 15 km (e.g., Wald and Heaton 1994). Heaton (1990) proposed that the velocity-weakening nature of rock friction is responsible for pulse-like behavior. Cochard and Madariaga (1994) confirmed the generation of slip pulse using velocity-weakening friction law in numerical simulation of dynamic crack propagation. However, this is not the only mechanism of slip pulse. Alternatively Beroza and Mikumo (1996) and Peyrat et al. (2001) demonstrated that spatial heterogeneity of stress can produce slip pulse without a velocity-weakening friction law.

A reliable gross feature of slip inversion is the location and time of the largest slip. The location and time of rupture initiation (hypocenter) is are far more reliable. Therefore, comparing these numbers, we can estimate overall rupture velocity. In 1970s rupture velocity was discussed using macroscopic earthquake models. Geller (1976) compiled rupture velocity of many macroscopic models and gave a typical rupture velocity as 72% of the S wave velocity, with a caution that the value depends on the model assumption. The result of slip inversion is useful to measure model independent rupture velocity. The rupture velocity of 2.2 km/s in the Chi-Chi earthquake (section 6.4) is a well-determined example. It is easy to measure a rupture velocity when there is a slip-pulse in the model. In the model of Wald and Heaton (1994), the Landers earthquake propagated at 2.5-3.0 km/s, with deceleration near fault steps. The rupture velocity of the Sumatra earthquake is also fixed well as 2.8 km/s from the spatio-temporal energy radiation distribution determined by Ishii et al. (2005). In all models, average rupture velocity is less than S wave velocity, but a little faster than 70% suggested from macroscopic source models.

Unlike the average rupture velocity, local rupture velocity can exceeds S wave velocity, although it is not robust because of smoothing introduced in slip inversion. In the slip model of the Imperial Valley earthquake based on trial-and-error, Archuleta (1984) showed that local rupture velocity is faster than S wave velocity. Frankel (2004) reported that the local rupture velocity is as fast as 5 km/s while the average is about 3 km/s in the case of the 2002 Denali fault, Alaska, earthquake (Mw 7.9). Such local super shear rupture is expected when stress is heterogeneous (Day 1982a) or fracture energy is heterogeneous with homogeneous stress (Ide and Aochi 2005). However, the observational evidence is not strong enough to conclude that such super shear rupture propagation is a general phenomenon of earthquake.

Estimation of rupture velocity for small earthquakes is difficult. Ide (2001) determined slip models of 18 moderate (M4-5) earthquakes occurred in the Hida Mountains, Japan, but the estimation of rupture velocity has large uncertainties because of poor resolution. In the smallest slip model ever determined for earthquakes, the model for the micro (M0.8-1.4) earthquake in South Africa gold mine, Yamada et al (2005) found that the rupture velocity is not less than 65% of S wave velocity. Although it is not the results based on slip models, Imanishi et al. (2004) found high rupture velocity for small (M1.3-2.7) earthquakes in western Nagano, Japan, using the method utilizing stopping phases (Imanishi and Takeo 2002). These studies suggests that the rupture velocity is similar for earthquakes from M1 to M9, but each estimation was subject to large errors and obviously further studies are necessary.

Slip models are frequently compared with aftershock distributions. It is often claimed that aftershocks are rare near or in the areas on the fault where the slip was large. Mendoza and Hartzell (1988b) investigated seven slip models of earthquakes in the western United States that are determined using near-field records and hence relatively reliable. Although it was qualitative, they showed that aftershocks occur mostly outside of or near the edges of the source areas indicated by the patterns of mainshock slip. Das and Henry (2003) investigated various slip models for many events including those in subduction zones and ridges. They found that few, and usually the smaller, aftershocks occur in the high-slip regions of the fault, with only one exception. Wiemer and Katsumata (1999) discussed slip models and b value of Gutenberg-Richter frequency-magnitude relation (Gutenberg and Richter 1944). For the 1984 Morgan Hill (Mw 6.2), Landers, Northridge, and Kobe earthquakes, they found that large slip regions seem to correspond to large b values. However, it should be noted that large slip region has few aftershocks and the estimations of b value in most of these regions were impossible. Although the mechanism of aftershock genesis is not fully understood, the existence of many aftershocks surrounding the areas of large slip indicates the importance of the stress redistribution by the mainshock.

Another often reported behavior is rupture initiation from the bottom of the fault (e.g., Das and Scholz 1983). One may easily compare aftershock distribution and the location of mainshock hypocenter, but the comparison between slip distribution and the initial rupture location is more informative. From the large collection of published slip models, Mai et al. (2005) compared the location of initial rupture and slip distribution and concluded that the rupture started from near the bottom of the assumed fault plane in strike-slip and crustal dip-slip earthquakes. Although Das and Henry (2003) reported that the rupture tends to nucleate in the region of low slip or at the edge of high-slip regions, Mai et al. (2005) did not find such tendency.

Rupture direction of large earthquakes tends to be unidirectional as shown by McGuire et al. (2001) using a 2nd moment inversion method for far-field records. We can estimate the 2nd moment for slip models and determine overall rupture direction. McGuire et al. (2002) investigated slip models for earthquakes larger than M7 and confirmed his earlier conclusion based on far-field observation.

6.5.2 Implication of Slip Models for Fault Dynamics

Spatio-temporal slip distribution is directly connected with spatio-temporal stress distribution. Once spatio-temporal slip distribution is obtained, it is used as a boundary condition for calculation of stress change during the rupture propagation (Ide and Takeo 1997; Bouchon 1997). This yields a slip-stress relation at each point on the fault plane. Figure 6.6 is the first result of such calculation for the Kobe earthquake (Ide and Takeo 1997). Slip-stress relation is basically slip-weakening, similar to the theoretical assumption of Ida (1972) and Andrews (1976). The slip-weakening distance Dc is estimated as 50 cm – 1m. However, it was also shown by Ide and Takeo (1997) that Dc less than 50 cm is not detectable for the Kobe earthquake because of the limited resolution. Olsen et al. (1997) also estimated Dc of 0.8 m for the Landers earthquake using dynamic modeling. However this estimation might suffer the effect of the limited resolution, too. Guatteri and Spudich (2000) demonstrated that similar seismic waves are radiated from dynamic models of different Dc distribution, as far as they have similar distribution of fracture energy Gc, which scales with the product of Dc and breakdown stress drop (= difference between yield stress and dynamic friction level). The estimation of Gc is robust compared to that of Dc because Gc is controlled by rupture velocity and less affected by the limited resolution. Therefore, the most stable dynamic constant derived from slip inversion is not Dc, but Gc.

Beroza and Spudich (1988) first estimated Gc as 2 MJ/m2 from the slip model of the Morgan Hill earthquake. Guatteri et al. (2001) applied a rate- and state-dependent friction law for the model of Ide and Takeo (1997) and obtained a value of Gc as 1.5 MJ/m2. In addition to the fracture energy Gc measured from slip-stress relation, radiated energy from earthquake Es is also calculated using the expression of Kostrov (1974) as


where is the stress on the fault plane, and is the free surface energy to make an infinitesimally small slip on the fault plane. The third term contains the fracture energy derived from slip-stress relation. Summation convention is used. Ide (2002, 2003) applied this equation to slip models of several earthquakes to estimate seismic energy and fracture energy. While the fracture energy is in the order of MJ/m2 for the Kobe earthquake, total radiated energy calculated by this method is smaller by a factor 3 than the values determined using far-field body waves (Boatwright and Choy, 1986). This suggests that the slip model does not account for the total energy radiation because it is determined by seismic waves of relatively lower frequencies with smoothing constraints.

Another method to estimate the slip weakening distance Dc was proposed by Mikumo et al. (2003) and Fukuyama and Mikumo (2003). Instead of usual definition of Dc in a slip-stress diagram, they measure the slip amount when slip rate reaches the maximum as Dc’. Applying this method to the 2000 Tottori and the Kobe earthquake, they concluded that Dc’ scales with the final slip. However, Spudich and Guatteri (2004) suggested that it can be an artifact due to low pass filters. In fact, when the frequency contents of the data have an upper limit, that limit affects the maximum value of Dc’. Therefore, as Spudich and Guatteri (2004) pointed out, it is important to compare data and model prediction in the frequency domain as reviewed in 6.3.4.

If fault plane is flat, absolute stress level is usually irrelevant to seismic wave radiation. However, when the rake of fault slip rotates during the rupture process and frictional traction is collinear with the instantaneous slip motion, there is a unique absolute stress consistent with the motion (Spudich 1992). Some slip models allow rake angles to vary during slip, which can be used to estimate the absolute stress level (Spudich 1992; Guatteri and Spudich 1998). Since the details of slip distribution from slip inversion depend on researcher’s assumptions, we should be careful about judging whether the observed rake rotation is constrained directly by the data, and not an artifact due to model assumptions.

6.5.3 Dynamic Modeling and Slip Models

Most slip models are kinematic models and some of the features are not physically realistic. For example stress diverges to infinity around subfault boundaries with a constant slip and dynamically feasible slip function is usually asymmetric unlike the frequently used triangle function (Fig. 2a). In the previous subsection, we reviewed dynamic implication directly deduced from such kinematic slip models. On the other hand, dynamic modeling is performed to construct an elastodynamically reasonable source model using only some aspects of slip inversion, for example, final slip distribution or rupture timing. A dynamic source model is constructed based on fracture mechanics, with the assumption of stress state and friction/fracture law. This is one practical way to discuss source behavior and seismic wave radiation at high frequencies where kinematic slip model is no longer reliable.

The problem of crack propagation in elastic medium has been solved numerically since the late 1970s (Andrews 1976; Das and Aki 1977; Day 1982b). While early studies solved some ideal cases, the first application to the complex rupture process of real earthquake is Quin (1990) who carried out dynamic modeling of the Imperial earthquake based on the slip model of Archuleta (1984). He calculated the stress drop from the slip distribution of the slip model, assumed the distribution of strength, and simulated spontaneous crack propagation at prescribed rupture velocity of the original model satisfying the fracture criterion of the maximum stress (strength). Miyatake (1992) applied a similar method to four inland earthquakes in Japan. Since these procedures refer only final slip distribution and rupture timing of the original slip model, seismic waves radiated from the dynamic model cannot explain the observation. Fukuyama and Mikumo (1993) and Ide and Takeo (1996) developed dynamic modeling methods that iteratively improve dynamic models to explain observed waveforms. The above-mentioned studies of dynamic modeling assume the maximum stress fracture criterion (Das and Aki 1977), which depends on the grid size. Rather, natural fault slips must obey a friction law. Olsen et al. (1997) carried out the first dynamic modeling to simulate the Landers earthquake, in which slip occurs following a slip-weakening friction law. This dynamic model is later improved by Peyrat et al. (2001) to determine heterogeneous stress distribution on a flat fault plane.

One of the recent major topics in dynamic modeling is the geometry of fault planes. Unlike a strike slip fault, a shallow dipping fault does not have symmetry because of the existence of the free surface. Oglesby et al. (1998) carried out dynamic modeling of a 2D shallow thrust fault and showed that the change of fault normal stress has an important effect on the fracture criterion at the rupture front. This result suggests that a complex slip distribution that implies stress heterogeneity can be due to complex fault geometry. Actually, Oglesby and Day (2001) showed that homogeneous stress and dipping fault plane can explain the gross feature of observed near-source ground motion for the Chi-Chi earthquake, for example, the larger movement in the hanging wall than the footwall. Dynamic rupture propagation along more complex fault system has become feasible. Aochi and Fukuyama (2002) showed that homogeneous background stress and change of strike angles in a fault system are essential to explain spontaneous rupture transfer between three major faults during the Landers earthquake. Aochi et al. (2003) also showed that the stress distribution of Peyrat et al. (2001) improves the details of dynamic slip distribution.

6.5.4 Scaling of Earthquake Heterogeneity

Most scaling relations between macroscopic parameters of earthquake source are derived from a geometrical similarity, i.e., constant stress drop scaling (e.g., Kanamori and Anderson 1975). However, the breakdown of the scaling relations is not well understood. There are still debatable issues, such as the effect of finite width of seismic layer on the scaling between fault length and slip amount (e.g., Matsu’ura and Sato 1997; Scholz 2002), and size dependence of the ratio between seismic energy and seismic moment, especially for small earthquakes (e.g., Ide and Beroza 2001; Kanamori and Brodsky 2001).

On the other hand, studies have recently started for the scaling of the complex earthquake source. It is not straightforward to extract meaningful parameters from various slip models. Using many published slip models, Somerville et al. (1999) determined the size of fault plane excluding the regions of little slip near the edges. Furthermore they define an asperity as “a region on the fault rupture surface that has large slip relative to the average slip on the fault”, measured the total area of rectangular asperities in each model, and concluded that the asperities occupy 17.5% of the fault plane. The fault size of slip model is also measured by Mai and Beroza (2000) using autocorrelation function. They obtained a nonlinear relation between slip and fault length for large strike slip events. Mai and Beroza (2002) calculated autocorrelation of slip distribution and showed that they are explained by von Karman autocorrelation function or functions with power-law dependence at high frequencies. Such scaling about complexity of earthquake is useful to constrain input source models for strong-motion prediction. However, it should be noted that slip inversion includes many assumptions and smoothing constraints for regularization that greatly affect high frequency components of seismic waves. Moreover, although previous studies dealt with only final slip distribution, time history of slip will be also important for such purpose in the future studies.

Slip models have been obtained for moderate earthquakes of M3-4 (Courboulex et al. 1996; Ide 2001). Although the resolution is worse than that of large events like the events summarized in Table 1, these models can be used to compare slip area and aftershock distribution and to distinguish a simple rupture process from a sequence of discrete subevents. So far the slip models of the smallest size were determined for five microearthquakes in South African gold mine (Yamada et al. 2005). Figure 6.7 compares one of these models (Mw 1.4) with the slip models of the Kobe earthquake (Ide and Takeo 1997). It is interesting that all scales differ about 1000 times, which implies a geometrical similarity roughly consistent to the 5.4 unit difference in moment magnitude scale. Although the resolution is poor, this small event certainly consists of subevents that are also identified directly in observed waves. Some small earthquakes do have complex rupture processes comparable to large events. However, it is still unclear how general such complexity is.

Seismic waves from an earthquake generally have so-called initial phases before the large main phase (e.g., Umeda 1990; Iio 1992, 1995; Abercrombie and Mori 1994; Ellsworth and Beroza 1995). Sometimes initial phases are quite different from the main phase. Nakayama and Takeo (1997) determined a slip model for the 1994 Sanriku-Haruka-Oki earthquake (Mw 7.7). This earthquake began with slow propagation at about 2.0 km/s with small slip during the first 20 s and later it was accelerated up to 3.0 km/s and radiated large seismic waves. The construction of slip models only for initial phases is possible. Shibazaki et al. (2002) determined a slip model for the initial 0.6 s of the Kobe earthquake whose total duration is about 10 s. Comparing with another slip model of an M4 aftershock in similar scale, they found that the rupture propagation of the mainshock is slightly slower than that of the aftershock. Such comparison is still rare, but it is an interesting future topic.

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