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Ground motions characterized by a multi-scale heterogeneous earthquake model
© Aochi and Ide; licensee Springer. 2014
Received: 22 January 2014
Accepted: 30 April 2014
Published: 29 May 2014
We have carried out numerical simulations of seismic ground motion radiating from a mega-earthquake whose rupture process is governed by a multi-scale heterogeneous distribution of fracture energy. The observed complexity of the Mw 9.0 2011 Tohoku-Oki earthquake can be explained by such heterogeneities with fractal patches (size and number), even without introducing any heterogeneity in the stress state. In our model, scale dependency in fracture energy (i.e., the slip-weakening distance Dc) on patch size is essential. Our results indicate that wave radiation is generally governed by the largest patch at each moment and that the contribution from small patches is minor. We then conducted parametric studies on the frictional parameters of peak (τp) and residual (τr) friction to produce the case where the effect of the small patches is evident during the progress of the main rupture. We found that heterogeneity in τr has a greater influence on the ground motions than does heterogeneity in τp. As such, local heterogeneity in the static stress drop (Δτ) influences the rupture process more than that in the stress excess (Δτexcess). The effect of small patches is particularly evident when these are almost geometrically isolated and not simultaneously involved in the rupture of larger patches. In other cases, the wave radiation from small patches is probably hidden by the major contributions from large patches. Small patches may play a role in strong motion generation areas with low τr (high Δτ), particularly during slow average rupture propagation. This effect can be identified from the differences in the spatial distributions of peak ground velocities for different frequency ranges.
Near-field ground motion is strongly affected by the heterogeneity of earthquake rupture processes, such as multiple ruptures, rupture directivity, and the acceleration and deceleration of the rupture front. To quantitatively estimate the ground motion for seismic hazard assessment, characterization of this heterogeneity is essential. In kinematic earthquake models, an assumed target fault plane is usually decomposed by many rectangular sub-faults on which the spatio-temporal slip distribution is determined by a waveform inversion of seismic data (e.g., Hartzell and Heaton 1983). Numerous slip models have been presented for the Mw 9.0 2011 Tohoku-Oki earthquake (e.g., Ide et al. 2011; Koketsu et al. 2011; Simons et al. 2011; Suzuki et al. 2011). In these slip models, model parameters (e.g., total slip, rupture time, and rise time) have been attributed to each sub-fault.
The seismic waves inverted for slip models are usually displacements or velocities filtered in a low-frequency band, which is typically <1 Hz. There has been debate over whether the source of high-frequency waves is identical to the large slip areas in slip models (Hartzell et al. 1996; Kakehi et al. 1996; Nakahara et al. 1999). Although a general consensus has not yet been reached (e.g., Nakahara 2008), for many subduction earthquakes, high-frequency sources tend to be located deeper than the large slip areas (e.g., Lay et al. 2012). High-frequency sources are typically identified as several small isolated regions referred to as 'strong motion generation areas’ (SMGAs) (e.g., Kamae et al. 1998; Irikura and Miyake 2011). Some SMGA models have also been presented for the Tohoku-Oki earthquake (Kurahashi and Irikura 2011; Asano and Iwata 2012; Satoh 2012). Given that the number of parameters is much smaller when characterizing the kinematic earthquake rupture process in terms of SMGAs than when using a finely discretized finite-fault rupture model, the approach using SMGAs may be a more robust approach for ground motion simulations.
From a dynamic point of view, Ide and Aochi (2005) proposed that the heterogeneity in earthquake faults can be expressed as a superposition of different scales of heterogeneity. This is termed 'multi-scale heterogeneity’ and a seismic source can be represented as cascading ruptures of many circular patches of different sizes on a crack plane. In terms of kinematic finite source description, such heterogeneity may correspond to a class of composite models developed by Frankel (1991) and Zeng et al. (1994), and similar to those of Bernard et al. (1996). Aochi and Ide (2011) and Ide and Aochi (2013) attempted to characterize the Mw 9.0 2011 Tohoku-Oki earthquake based on this source expression and were able to dynamically simulate the growth of the rupture process from a small rupture through to the Mw 9 event. In these studies, a large elliptical patch corresponding to the large slip area of a Mw 9 event, which was unknown prior to this earthquake, was key in explaining the gross seismic moment release, whereas the distributed small circular patches governed the rupture growth before triggering the large elliptical patch. These small patches were introduced to reflect moderate earthquakes that had occurred previously in this region.
The apparent similarity in both the timing and locations of SMGAs (Kurahashi and Irikura 2011; Asano and Iwata 2012) and the circular patches of Ide and Aochi (2013) suggest that such small-scale heterogeneity corresponds to SMGAs. The dynamic models explored by Aochi and Ide (2011) and Ide and Aochi (2013) were neither constrained nor calibrated directly using observed data. Moreover, the radiation of ground motion based on this multi-scale heterogeneity has not yet been discussed. The purposes of this study are to show the characteristics of ground motions from the simulated fractal patch model for the 2011 Tohoku-Oki earthquake and to discuss the similarities (or differences) with respect to other synthetic heterogeneity models of the earthquake.
Model and method - reference model
Crustal structure used in the ground motion simulations
Upper depth of layer [km]
P wave velocity [m/s]
S wave velocity [m/s]
Quality factor Q
However, according to the SMGAs reported for the 2011 Tohoku-Oki earthquake (Kurahashi and Irikura 2011; Asano and Iwata 2012; Satoh 2012), it is expected that it should be possible to identify certain phases with different frequency contexts. One possible explanation for this is that the difference is due to heterogeneity in stress conditions as our dynamic concept varies fracture energy (keeping stress conditions uniform), whereas the SMGA includes stress heterogeneity. From the point of view of the slip-weakening relationship for fault friction, both approaches imply that small patches may 'locally’ radiate more seismic energy compared with large patches. This study further explores this through parametric studies.
Model parameters of the parametric studies
Large patch (all cases)
Small patches (case 1)
Small patches (case 2)
Small patches (case 3)
Dimension of ellipse (a, b) or circle (r) [km]
a = 57.1, b = 175
r = 25
r = 25
r = 25
Slip-weakening distance Dc [m]
Initial shear stress τ0 [MPa]
Dynamic strength drop Δσ [MPa]
20, 20.5, 21, 22.5
Stress excess Δτexcess [MPa]
15, 15.5, 16, 17.5
Stress drop Δτ [MPa]
where the dynamic strength drop Δσ is defined as Δσ ≡ τp - τr. The fracture energy Gc is defined as Gc ≡ Δσ × Dc/2. We assumed that Dc is proportional to the patch dimension (e.g., Ide and Aochi 2005). In this case, Dc is 3.2 and 0.8 m for a large and small patch, respectively. To initiate rupture on the large ellipse having a large Dc, we assumed that Dc also scales with the distance from the hypocenter around the initiation point and produces a small (but large enough with respect to the simulation element size of 2 km) finite initial crack (radius of 10 km) located at the coordinate origin in Figure 4. This helps to minimize an artificial initial phase of the dynamic rupture process (e.g., Aochi and Ide 2004; Ide and Aochi 2005; Aochi and Douglas 2006).
Given that we did not aim to discuss the effect of the ground surface on the rupture process, all the simulations were carried out in a 3D infinite, homogeneous elastic medium (P and S wave velocities of 6,000 and 3,464 m/s, respectively, and material density = 2,500 kg/m3) using a BIEM (Fukuyama and Madariaga 1995, 1998). In fact, as observed in Figures 2 and 3, the seismic wave radiation originating from the rupture arrival on the ground surface is obvious since the dynamic rupture significantly interacts with the ground surface. This masks the other radiation effects and may be exaggerated, as no shallow, slow layer was included (e.g., Goto et al. 2012; Ide and Aochi 2013). To better identify the radiation from the deep part of the fault as proposed in SMGA models, we ignored the interaction of the dynamic rupture with the ground surface.
We described the derivative stress with respect to the initial stress level (assumed to be zero, i.e., τ0 = 0) rather than to the absolute stress level. Therefore, instead of the two parameters τp and τr we refer to the stress excess Δτexcess(≡τp - τ0) and the static stress drop Δτ(≡τ0 - τr). A large Δτexcess may lead to a delay in the rupture onset. A large Δτ may produce a large fault slip. We firstly kept Δτ constant and changed Δτexcess (case 1 in Table 2). We then varied Δτexcess (case 2), and finally, we changed both parameters simultaneously (case 3). The ground motions were compared at several stations along the fault strike (the same locations as in Figure 1), assuming the same hypocenter location and fault geometry as for the 2011 Tohoku-Oki earthquake. As the focus of this study is the wave radiation from the causal source, we used the simple 1D structure given in Table 1.
Therefore, we gleaned from these simulations that heterogeneity in the static stress drop Δτ (i.e., residual stress level τr) leads to more significant differences in ground motion than those produced by heterogeneity in the stress excess Δτexcess (or peak strength τp). Small patches are visible when they are isolated from a large patch, such that the small patches are in some way separated from the nearby ongoing rupture. Therefore, the locally identified SMGAs may be different from the main rupture front or characterized by a larger stress drop. A more complicated 'inhomogeneous’ fault model (e.g., Ripperger et al. 2008) might be more appropriate in such cases. Stochastically heterogeneous models (e.g., Ide and Aochi 2005) should be used in future ground motion estimation studies.
Our parametric study confirms the observations of Ide and Aochi (2005). We have determined that the role of small patches during rupture propagation makes the rupture front heterogeneous, not by delaying it, but through advancing it according to the low fracture energy given that the rupture propagation is governed by the energy balance. In our model, the rupture propagation velocity is close to the S wave speed, and it is difficult to identify the effect of further advancing the rupture front on the seismic wave radiation. Nevertheless, such an effect would be evident if the rupture velocity were slower, and in fact, we observed a slight difference between case 1 of Figure 5 and case (a) of Figure 9. However, the influence on the ground motion is limited (Figure 10). Ulrich and Aochi (2014) attempted to identify different sizes of patch by inversion, but any secondary small patches were difficult to identify whereas a large patch was able to be easily characterized. This is because the radiation from the small patches tends to be hidden by the waves from the greater area of a large patch. This effect should be important when the main rupture triggers remote small patches. In other cases, the rupture of small patches should be significantly delayed. However, we cannot rule out the possibility of delayed rupture of small patches for more realistic friction laws. For example, in the case when fault strength τp is time-dependent, the rupture may be delayed according to the accumulation of strain and some relaxation process. This is in effect the same mechanism as aftershock occurrence (e.g., Dieterich 1994; Helmstetter and Shaw 2009).
However, the differences in Δτ demonstrate the influence of small patches on ground motions. The mechanical features dynamically investigated in this study should be consistent with kinematic interpretations. In the original concept of SMGAs, Irikura and Miyake (2011) proposed a high stress drop on the SMGAs to distinguish it from background rupture propagation. This appears to make sense, given that Aochi and Dupros (2011) reconstructed a fault constitutive relationship from the SMGA source model (Irikura 2008) for the 2007 Mw 6.6 Niigata-Chuetsu-Oki earthquake in Japan and obtained a difference in Δτ (and Δτexcess) that was about double between SMGA area and the rest of the fault plane. The studies of Das and Aki (1977) and Mikumo and Miyatake (1978) have shown that the rupture process is very sensitive to stress parameters such as Δτ and Δτexcess (e.g., Madariaga and Olsen 2000; Gabriel et al. 2012). Even if our study were to take into account differences in fracture energy, the sensitivity to these parameters remains valid and coherent.
We have simulated the wave radiation process from multi-scale heterogeneous models of fracture energy for a mega-earthquake such as the 2011 Tohoku-Oki earthquake. Based on our previous research, we started by assuming heterogeneity only in the slip-weakening distance Dc under uniform stress conditions and frictional levels τp and τr (i.e., uniform stress excess Δτexcess and stress drop Δτ). However, in this case, once a rupture is initiated on a large patch, the influence of nearby small patches on the ground motions is insignificant and hidden by the dominant wave radiation from the large patch, which has a rupture propagation velocity similar to the S wave speed. We then introduced additional heterogeneity in Δτexcess and/or Δτ. Finally, we showed that heterogeneity in Δτ has a greater influence on ground motions than the same level of heterogeneity just in Δτexcess. In fact, small patches can be easily ruptured by the use of the multi-scale concept in fracture energy, but these are difficult to rupture behind a large patch even if high Δτexcess is assumed. Small patches exert an influence on SMGAs with low τr (i.e., those with a higher static stress drop Δτ). This effect can be identified from differences in the spatial distribution of peak ground velocities at different frequency ranges. Our models indicate that the distribution of large patches may be inferred from past seismicity when earthquakes recur frequently. However, uncertainties on the stress heterogeneity need to be treated stochastically. Our multi-scale heterogeneity earthquake model is consistent with inferred earthquake models, such as SMGA models, and may be a useful approach for probabilistically introducing stress state heterogeneity when predicting ground motion for quantitative seismic hazard studies.
We thank Hiroe Miyake and Nobuki Kame for inviting us to the working group on 'Dynamic Source Models for the Next Generation Ground Motion Prediction’ in 2013-2014, which was funded by the Earthquake Research Institute of the University of Tokyo. We also thank John Douglas for offering comments that improved this paper. The comments from Martin Mai and an anonymous reviewer were very helpful. This research began under the framework of the French-Japanese ANR-JST joint program DYNTOHOKU (2011-2013) and was then funded by the French national project S4 (Subduction: Slow and Standard Seismology, 2012-2014, ANR-2011-BS56-017) of the Agence National de la Recherche. Most of the calculations were made at the French national supercomputing center GENCI-CINES (grant 2013/2014-c46700). This work was also partially supported by JSPS KAKENHI (23244090) and MEXT KAKENHI (21107007) grants.
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