# Treatise on geophysics chapter 6: slip inversion s. Ide

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## Nomenclature

: Slip distribution in i-th direction at (, ).

Green’s function : Displacement in i-th direction at (x, t) due to unit impulsive force in p-th direction at (, ).

d0: Data vector

de: Model estimate vector

m: Model parameter vector

G: Kernel matrix

C: Constraint matrix

Cj: j-th Constrant matrix

e: Constraint vector

ej: j-th Constrant vector

Ed: Data covariance matrix

Em: Parameter covariance matrix

: Data variance

: Variance of j-th constraint

: Hyper parameter

Nd: Length of data vector

Nm: Length of parameter vector

Nc’: Rank of constraint matrix C

p(do|m): Liklihood function for given data vector and unknown parameter vector.

: Marginal likelihood, function of hyperparameters,

ABIC: Akaike’s Bayesian Information Criterion

Moment magnitude, Mw: Magnitude scale of earthquake based on seismic moment. Mw = (log10(Mo)-9.1)/1.5 in SI unit.

Seismic moment, Mo: rigidity times seismic potency (Unit Nm).

Seismic potency: integration of slip across fault area (Unit m3).

Seismic Energy Es: Energy radiated from dynamic rupture and carried to far-field by body waves (J).

Slip weakening distance, Dc: In slip-weakening friction law, the distance where frictional stress reaches the dynamic level.

Fracture surface energy, Gc: Energy consumed on a unit fault surface before slip frictional stress reaches the dynamic level (Unit J/m2).

: Free surface energy, energy to make a new surface.

: ij-conponent of stress tensor at (x, t)

TABLES

Table 6.1 Well studied earthquakes and slip models

 Earthquake Date Mw Reference Near-field Data Other Data Model year/month/day Type1) Freq. (Hz) Type2) Type3) San Fernando, USA 1971/02/09 6.7 Trifunac (1974) D (5) 0.11- ― L (22) Imperial Valley, 1979/10/15 6.4 Olson & Apsel (1982) A (26) 0.1-0.33 ― L (320) USA Hartzell & Heaton (1983) V (12) 0.1-1.0 FD （19） L (336) Archuleta (1984) D (18) -0.5 ― N (840) Loma Prieta, USA 1989/10/18 6.9 Beroza (1991) D (20) 0.3-2.0 ― N (1680) Steidl et al. (1991) V (38) 0.05-1.0 ― N (456) Wald et al. (1991) V (16) 0.1-1.0 FV (16) L (288) Liu and Archuleta (2004) V (16) 0.05-1.0 ― N (480) Landers, USA 1992/06/28 7.2 Cohee & Beroza (1994) D (18) 0.018-0.25 ― L (612) Wald & Heaton (1994) D (16) 0.077-0.5 FD (11), G L (1116) Cotton & Campillo (1995) D (11) 0.05-0.5 ― N (144) Northridge, USA 1994/01/17 6.7 Hartzell et al. (1996) V (35) 0.1-1.0 ― N (784) Wald et al. (1996) V (38) 0.1-1.0 FV (13), G L (1176) Hyogoken-Nanbu 1995/01/17 6.9 Horikawa et al. (1996) V (18) 0.05-0.5 G L (480) (Kobe), Japan Sekiguchi et al. (1996) D (19) 0.1-1.0 ― L (2240) Wald (1996) V (19) 0.05-0.5 FV (13), G L (1728) Yoshida et al. (1996) D (18) -1.0 FD (17), G L (360) Ide & Takeo (1997) D (18) 0.025-0.5 ― L (3344) Chi-Chi, Taiwan 1999/09/21 7.6 Chi et al. (2001) V (21) 0.02-0.5 ― L (8320) Ma et al. (2001) D (21) no filtering FD(22), G L (4032) Wu et al. (2001) V (47) 0.016-0.5 G L (4840) Zeng et al. (2001) V (15) 0.05-0.25 G N (960) Ji et al. (2003) V (36) -1.6 ― N (1620) Tottori, Japan 2000/10/06 6.7 Iwata and Sekiguchi (2002) V (12) 0.1-1.0 G L (1632) Semmane et al. (2005) D (23) 0.1-1.0 G N (480) 1) A: acceleration, V: velocity, D: displacement, the number of stations in parenthesis 2) FD: far-field displacement, FV: far-field velocity, the number of stations in parenthesis, G: geodetic data 3) L: linear presentation, N: non-linear presentation, the number of parameters in parenthesis

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 spatio-temporal slip distribution. The initial time of time function is shown by contours in each of top figures. a) Typical linear expression of multi-time-window 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 Miyagi-Oki, Japan, earthquake. From all available Hi-Net 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.

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