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Nomenclature: Slip distribution in ith direction at (, ). Green’s function : Displacement in ith direction at (x, t) due to unit impulsive force in pth direction at (, ). d^{0}: Data vector d^{e}: Model estimate vector m: Model parameter vector G: Kernel matrix C: Constraint matrix C_{j}: jth Constrant matrix e: Constraint vector e_{j}: jth Constrant vector E_{d}: Data covariance matrix E_{m}: Parameter covariance matrix : Data variance : Variance of jth constraint : Hyper parameter N_{d}: Length of data vector N_{m}: Length of parameter vector N_{c}’: Rank of constraint matrix C p(d^{o}m): Liklihood function for given data vector and unknown parameter vector. : Marginal likelihood, function of hyperparameters, ABIC: Akaike’s Bayesian Information Criterion Moment magnitude, M_{w}: Magnitude scale of earthquake based on seismic moment. M_{w} = (log_{10}(M_{o})9.1)/1.5 in SI unit. Seismic moment, M_{o}: rigidity times seismic potency (Unit Nm). Seismic potency: integration of slip across fault area (Unit m^{3}). Seismic Energy E_{s}: Energy radiated from dynamic rupture and carried to farfield by body waves (J). Slip weakening distance, D_{c}: In slipweakening friction law, the distance where frictional stress reaches the dynamic level. Fracture surface energy, G_{c}: Energy consumed on a unit fault surface before slip frictional stress reaches the dynamic level (Unit J/m^{2}). : Free surface energy, energy to make a new surface. : ijconponent of stress tensor at (x, t) TABLES Table 6.1 Well studied earthquakes and slip models
FIGURES Figure 6.1 Fault models assumed in the studies of slip inversion by different research groups for the 1995 Kobe earthquake. Except for the model of Cho and Nakanishi (2000), colored bold lines with triangles indicating the direction of dip show uppermost edges and colored thin lines show fault planes projected to the surface. Cho and Nakanishi (2000) used 3D blocks instead to the fault planes. Star shows the epicenter location determined by Japan Meteorological Agency (JMA). Black circles are aftershock epicenters determined by JMA within one week of the mainshock. Harvard CMT solution is shown by a red beach ball. Each of the bottom figures shows one of the fault models by a color and the others by grey lines. Figure 6.2 Schematic illustrations showing how to parameterize spatiotemporal slip distribution. The initial time of time function is shown by contours in each of top figures. a) Typical linear expression of multitimewindow analysis. Unknown parameters are amplitude for each window. b) An example of nonlinear expression, in which rupture time is also an unknown parameter. Figure 6.3 Comparison of waveforms of a large earthquake and its aftershocks, the 2003 MiyagiOki, Japan, earthquake. From all available HiNet stations (circles), we showed records at 6 stations (black circles) located in the map and on the focal sphere. For each station, upper and lower traces show the displacement (cm) of the mainshock and the aftershock, respectively. While the records of aftershock are impulsive, the waves of the mainshock last longer. In shaded area, we can identify the second large phases that indicate rupture propagation to the northward. Figure 6.4 Simulated example of application of ABIC to slip inversion. Synthetic displacement at several stations surrounding the input source model are calculated and contaminated with noise of fractional Gaussian type. Slip models are determined for the different set of spatial and temporal hyper parameters, and , and the corresponding values of ABIC are shown in the central panel as a function of and . Since the absolute values are meaningless in this simulation, we set the origin of hyper parameters at the minimum of ABIC. (a) Input model: the slip distribution (upper panels) and time functions (lower panels). There are 10 x 5 subfaults and each subfault has five time windows, consisting 250 parameters. (b) The model that provides the minimum ABIC. (c)(f) Different models determined by stronger or weaker constraints. Fig. 6.5 Comparison of three slip models by Chi et al. (2001), Ma et al. (2001), and Wu et al. (2001). All distribution is shown as the projection to the free surface. (Top left) Location in the map. Surface offsets along the Chelungpu fault is shown by a yellow line. (Top right) Final slip distributions. (Bottom) Slip distributions during each time period. Fig. 6.6 The relation between slip and stress on the fault plane, determined for the 1995 Kobe earthquake (Ide and Takeo, 1995). Each trace is the function calculated at the corresponding location in the middle figure that shows final slip distribution by contours. Fig. 6.7 Comparison between two slip models: the 1995 Kobe earthquake and a small earthquake in South Africa gold mine. Slip distributions during each period are shown. The maximums of slip are about 1.2 m and 6.0 mm for the Kobe and South African earthquakes, respectively. 