where m is a model vector that consists of potency parameters , and G includes theoretical waveforms calculated for every discrete unit of spatio-temporal source volume. This linear equation is solved using standard procedures of the linear least-square inversion (e.g., Menke 1985) on the assumption that the difference between observation and model prediction follows the normal distribution with zero means. However, a flexible model requires a large number of model parameters. In recent analysis the number of parameters is sometimes larger than 3000-4000 (Table 6.1). Usually the number of data points is much larger than the number of parameters, but these data points are almost surely dependent and the problem tends to be underdetermined. The effective rank of matrix G is less than the number of parameters and the best estimates of model parameters are, even if they were obtained, unstable with quite large standard deviations. Such problems require regularization with various constraints, which will be discussed in 6.3.2. Another deficiency of the multi-time-window method is that it limits the timing of slip. When the duration of the temporal basis functions is too short and real slip occurs longer than that, unexpected errors occur in the solution. This problem was demonstrated by Hartzell and Langer (1993) using the models having different number of time windows.
6.2.3.3 Example of nonlinear expression
If the timings of slip are treated as unknown parameters in (2), we can reduce the number of parameters. An example is
, (9)
where is a function which is zero before t = 0 (is 0 before ) and normalized to be unity when integrated with time (Fig. 2b). The parameter is a scaling parameter in time. When is a boxcar function from t = 0 to 1, is called the rise time. A boxcar function (e.g., Yoshida 1986) and a triangle function (e.g., Fukuyama and Irikura 1986; Takeo 1987) are frequently used. A merit of nonlinear expression is that we can use a dynamically plausible function as . Beroza and Spudich (1988) assumed
, (10)
where H(t) is a step (Heaviside) function. This is a truncated version of the slip-rate function in the analytic solution of dynamic crack propagation. We can change the direction of slip vector using unknown parameters as,
. (11)
Substituting (11) into (1) and integrations for space and time yields a predictive data vector from the model parameter vector m consisting of p ^{j}, t ^{j}, r ^{j}, and ^{j},
d^{e} = G(m). (12)
If we assume Gaussian errors of model prediction, linear inversion can provide a unique solution, while nonlinear inversion does not eliminate the problem of nonuniqueness. Instead, the model with nonlinear expression can be extended to include more general features of fault slip.
6.2.4 Calculation of synthetic data
To carry out integration of equation (1) and obtain predictive data vectors (8) or (12), Green’s function must be properly prepared for each point on the fault plane. Unless the spatial basis function in equation (2) is a delta function, we have to calculate Green’s functions from every point source located at a small interval to account for rupture directivity effect in each discrete unit (subfault) as shown in Fig. 6.2.
Theoretical formulation of elastic wave propagations from a point source in layered structure had been developed mainly during the 1970s and 1980s. For global studies, the method using a propagator matrix (Bouchon 1976) and generalized ray theory (Langston and Helmberger 1975) are frequently used to calculate far-field P and S-waves with important contributions from local source and site structures, such as depth phases, sP, sP, and sS. For calculation of near-field full waveforms, the method that uses frequency-wavenumber integration is effective. The reflectivity method (e.g., Fuchs and Müller 1971; Kind 1976) and the Kennett-Bouchon method (e.g., Kennett & Kerry 1979; Bouchon 1981; Yao and Harkrider 1983) are well-known methods and several numerical codes based on these methods are developed and distributed (e.g., Kohketsu 1985; Takeo 1985; Saikia 1994). A general textbook about seismic wave propagation has been published by Kennett (2001). The combination of ray theory and isochrone integration (Bernard and Madariaga 1984; Spudich and Fraser 1984) is also effective in high frequency for near-field stations. For static deformations, the most popular study is Okada (1985) that gives analytic solutions for the surface displacement due to a buried dislocation on a rectangular fault in half space, with a concise summary of previous works.
The seismic velocity structure profiles determined by explosion surveys are helpful to construct a layered structure. Most studies assume a 1D structure from the complex result of 2D or 3D surveys with minor improvements by trial and error. When the structure difference is large among stations, several 1D structures are assumed in one study. Waveforms of small events are useful for to empirically improve better 1D assumption structure as shown by Ichinose et al. (2003) who used compared synthetic waveforms with aftershock records to estimate average structures between the source and the stations.
Though 1D structure is thought to be a fair approximation in many cases, it is obvious that there are many problems for which 1D structure is no longer valid. For example, body waves from submarine earthquakes are followed by large reverberations from water layers and dip angle of sea floor is critical to simulate these waves (Wiens 1987, 1989; Okamoto and Miyatake 1989; Yoshida 1992). Another example is the existence of thick sediment layers recently revealed by structural surveys in urbanized areas. Using a finite difference method for 3D structure, Graves and Wald (2001) and Wald and Graves (2001) demonstrated that the use of 3D Green’s functions can increase the resolution of slip inversion. They also emphasized that this improvement is only possible with the accurate underground structure and careful examination of its validity is required for slip inversion. As demonstrated by Tsuboi et al. (2003) for the 2002 Denali Fault earthquake (Mw 7.9), one of the current fastest computers can calculate seismic waves up to 5 s in a laterally heterogeneous global earth model. Introduction of more accurate 3D Green’s functions into slip inversion is promising.
The acAccurate 3D structure has have beennot been determined except only for small areas on the Earth. Moreover, small earthquakes radiate high-frequency waves, which cannot be modeled theoretically. An alternative of theoretical Green’s function is empirical Green’s function (eGf) which is usually the record of a small earthquake. This concept is originally proposed by Hartzell (1978) for strong-motion simulations. Note that the displacement of small earthquake is not a Green’s function in the strict sense in the equation (1), where Green’s function is defined using an impulsive single force while an earthquake has a double couple force system with a finite duration. Nevertheless, as long as the source duration of the small event is negligible, the difference of the force system is practically insignificant for slip inversion. Fukuyama and Irikura (1986) applied eGf to the 1983 Japan Sea earthquake (Mw 7.7). The large source area of this event requires two aftershocks each of which represents the north and the south subfaults, since an eGf is valid only for the limited space. For a large earthquake which breaks shallow faults, the difference of distance from the free surface is hard to neglect and a number of aftershocks are required to obtain a complete set of empirical Green’s functions, which is almost impossible. Since it is also known that aftershocks are rare in the area where the mainshock slip was large, it is hard to find proper small earthquakes for eGf method. Unfortunately the waves from this most important part of the fault plane is most difficult to simulate using eGf. Nevertheless, in the study with relatively low resolution especially for small earthquakes, eGf is an effective tool as shown by many studies (e.g., Mori and Hartzell 1990; Courboulex et al. 1996; Fletcher and Spudich 1998; Hellweg and Boatwright 1999; Ide 2001; Okada et al. 2001). In these studies seismic waves at a station can resolve the resolution of slip inversion slip distribution mainly in the direction of the ray from the source to the station. To better resolve the slip distribution, a set of various raydepends on the variety of incident angles of seismic waves directions from the sourceare required. Figure 6.3 is an example of the comparison between waveforms from the large event (the 2003 Miyagi-Oki, Japan, earthquake; Mw 7.1) and its possible eGf event of the similar mechanism. The waveforms of eGf events have short durations and site-specific complexity mainly due to local shallow structure. The mainshock records have some distinct phases and the arrivals of the second phases suggest the northward rupture propagation, which is consistent with the slip models presented by Okada and Hasegawa (2003) and Wu and Takeo (2004).