We investigate the dynamic rupture process of the 16 January 1995 Kobe earthquake (Mw 6.9) by spontaneous rupture modeling of a 3-D dynamic shear crack that reproduces the slip distribution found from kinematic waveform inversion of strong motion data. We find that using the heterogeneous initial stress field obtained from the kinematic slip model and relatively uniform fracture energy distribution, successfully generates a dynamic model with slip and rupture time distributions that are consistent with the kinematic source inversion. Our results suggest that we may be able to produce realistic dynamic rupture models with simpler assumptions for dynamic source parameters, such as the fracture energy, than have been used in most dynamic rupture models to date.
 The 16 January 1995 Kobe earthquake (Mw 6.9) occurred near the city of Kobe in western Japan, causing a tremendous amount of damage and the loss of many lives. The focal mechanism indicates right-lateral strike slip on a nearly vertical fault with the rupture area extending bilaterally from the hypocenter towards Awaji island to the southwest and towards the city of Kobe to the northeast as shown in Figure 1. Slip models [e.g., Ide and Takeo, 1997] have been obtained by analyzing seismic and geodetic data.
 In this study we construct a dynamic model that generates kinematic motions on the fault plane that are consistent with a kinematic source inversion. We are able to fit the kinematic rupture model with a fairly simple model of dynamic rupture. In our model, the initial stress is strongly variable, but the yield stress and slip weakening distance are almost uniform, which means our model has a relatively uniform fracture energy distribution.
2. Dynamic Modeling
 The final slip distribution of the Kobe earthquake as obtained from kinematic source modeling of Ide and Takeo  and the associated static stress drop distribution are shown in Figure 2. As shown in the slip model, most of the slip occurred beneath Awaji island below the region of greatest surface rupture, while relatively little slip was observed below Kobe. The static stress drop distribution was calculated from the kinematic slip model using the Okada  method, which calculates static stress drop from a distribution of rectangular dislocations in a homogeneous half space. The stress drop distribution reveals two high stress patches separated by a narrow low stress band near Awaji island. Negative stress drop is observed in the relatively low slip area under Kobe.
 Based on the calculated stress drop distribution, we construct a possible dynamic model for the Kobe earthquake. First, we assume a homogeneous shear strength (τs) and final stress level (τf) over the fault plane computed following Byerlee's  law given below.
μs and μf are the static and dynamic frictional coefficient, respectively, and σn is the effective fault normal stress. While depth dependence of the normal stress is likely to occur, we assume a relatively low, and constant, normal stress (∼25 MPa). The lack of a significant increase of earthquake stress parameters with depth are general characteristics of active fault zones in the Earth's crust [e.g., Aki, 1972]. Spudich et al.  also suggest a relatively low normal stress level (25 MPa) for the Kobe earthquake based on observations of temporal rake change. Thus, a uniform low normal stress (25 MPa) is used in our modeling. A heterogeneous initial stress field is obtained by adding the calculated stress drop to the uniform final stress following Olsen et al. . Finally the slip weakening model [Ida, 1972] is used for fault constitutive relation in the dynamic modeling. An appropriate slip weakening distance is chosen by trial and error modeling in order to match the kinematic source inversion results. A larger slip weakening distance (0.5 m) is applied at the shallower depth (h < 5 km) than that (0.1 m) at greater depth (h > 5 km) in order to approximate probable velocity hardening near the Earth's surface. Ide and Takeo  also suggest a larger slip weakening distance near the surface. Our assumed value for τs places an upper bound on the size of the slip weakening distance, which otherwise might be smaller still since the fracture energy is the primary factor controlling the evolution of rupture [Guatteri and Spudich, 2000].
 Several questions arise regarding the uniqueness of our dynamic model. First, the homogeneous shear strength and final stress distribution calculated by the simple Byerlee's law might be unrealistically simple. The heterogeneous stress drop distribution tells us only that some combination of both the initial and final stress is responsible for the stress drop heterogeneity. Here we assume a uniform distribution of all dynamic source parameters except the initial stress, and incorporate the heterogeneous stress drop into the heterogeneous initial stress alone. Dynamic stress drop can also differ from static stress drop. Mikumo et al.  also suggest heterogeneity in the distribution of the slip weakening distance. However, it is difficult to constrain different heterogeneities in dynamic source parameters independently from the given observed ground motions. Our approach is to start with a simple model that includes known heterogeneity and add complexity to the dynamic model as necessary in order to fit the kinematic model.
 We used the 3-D finite difference code developed for the dynamic rupture simulation [Andrews, 1999]. We also used the same layered velocity and density structure as used in the kinematic inversion [Ide and Takeo, 1997]. The grid size and time step used are 0.5 by 0.5 km and 37 ms, respectively.
Figure 3 compares the dynamic modeling results to the results obtained from waveform inversion [Ide and Takeo, 1997]. Column (a), (b), (c) are snapshot images of the rupture propagation for the Kobe earthquake obtained from our dynamic modeling in terms of traction (MPa), slip (m), and slip velocity (m/s), respectively. The rupture is initiated by artificially forcing the initial stress drop to the final stress level (τf) in a circular nucleation zone (radius = 3 km). The forced stress drop starts at the hypocenter of the earthquake and continues to propagate inside the nucleation zone with the fixed rupture velocity (3 km/s). The artificially initiated rupture near the high stress patch located at the center of the fault plane, thereafter, propagates as a self-sustaining (spontaneous) rupture throughout the first high stress patch as shown in the figure. It then traverses a narrow low stress band between the two high stress patches and propagates to the surface of Awaji island producing a large amount of slip beneath the island. Rupture on the farthest northeast reaches of the fault, near Kobe, occur after most of the southwest part of the rupture has completed. The relatively low initial stress beneath the city of Kobe and large slip weakening distance at shallow depth prevents surface rupture in this area. Column (b′) and (c′) are snapshot images of the rupture propagation obtained from waveform inversion [Ide and Takeo, 1997].
 Since we took only the final slip distribution from the kinematic model to build the dynamic model, it is interesting to compare the spatio-temporal evolution of source parameters obtained from the dynamic modeling and kinematic inversion. The general trend of the rupture propagation as well as the final slip distribution shows a very good agreement. Figure 4 shows the comparison of the temporal accumulation of slip as obtained by the kinematic source inversion (solid) and the dynamic modeling (dashed), respectively. Even though we did not take any information except the final slip distribution from the kinematic model (i.e., no rupture time information), the rupture time distribution shows a clear similarity. The temporal evolution of slip is for the most part consistent between the two models. The most notable exception is immediately beneath the city of Kobe. It is surprising that we can model the behavior of this event with such a simple dynamic model that assumes largely uniform dynamic properties, with an especially simple fracture energy distribution.
 Our very simple dynamic model of the Kobe earthquake produces kinematic motions on the fault consistent with the finite source inversion in terms of the slip and rupture time distribution even though the dynamic model is constructed using only the final slip distribution of the kinematic model. This supports the notion that reasonable estimates of the temporal evolution of slip in an earthquake can be inferred from the slip distribution alone [Guatteri et al., 2003]. There are several other slip models for the Kobe earthquake [Sekiguchi et al., 1996; Wald, 1996; Yoshida et al., 1996] and the details of those slip distributions are different, even though large-scale features, such as the location of asperities, agree with one another. We applied the same dynamic modeling approach to the slip model of Sekiguchi et al. , which has a more strongly variable slip distribution and failed to generate kinematic motions consistent with the temporal evolution of rupture in their model. Our modeling approach may not be capable of producing highly variable dynamic models, unless we allow quantities, such as the fracture energy to be strongly variable.
 Although there are strong tradeoffs between the slip weakening distance and the yield stress [Guatteri and Spudich, 2000], our assumed value for τs, which is just above the peak initial stress, places an upper bound on the size of the slip weakening distance, which otherwise would trade off with the peak stress and might be much smaller. Our analysis indicates that the smaller slip weakening distance (∼0.1 m) than the estimate (0.5 ∼ 1 m) of Ide and Takeo  is required, especially near the nucleation area, for spontaneous rupture propagation. Our approach has some limitations for constraining the slip weakening distance of the Kobe earthquake due to the simplified assumptions we have made. In contrast to our model, heterogeneity in the distribution of the slip weakening distance has been suggested for this earthquake [Mikumo et al., 2003]. However, allowing the slip weakening distance to be heterogeneous greatly increases the degrees of freedom in the rupture modeling. We show both that the behavior of the slip weakening distance may be simple and that the slip weakening distance of the Kobe earthquake might be smaller than the estimate of Ide and Takeo , as they mentioned based on their resolution analysis.
 Our dynamic modeling approach can not uniquely constrain the distribution of dynamic parameters, however, simple and plausible dynamic models can be developed provided the slip model is relatively smooth.
 We thank Joe Andrews for allowing us to use his dynamic simulation code (dynelf) and Satoshi Ide for giving us his kinematic inversion results for the Kobe earthquake. We thank Takashi Miyatake, Eiichi Fukuyama, David Oglesby, and Kim Olsen for helpful discussions. This research was sponsored by NSF grant CMS-0200436.