S U M M A R YW e developed a new inversion method t o reconstruct static images of seismic sources from geodetic data, using Akaike's Bayesian Information Criterion (ABIC). Coseismic surface displacements are generally related with a slip distribution o n a fault surface by linear integral equations. Parametric expansion of the fault slip distribution by a finite number of known basis functions yields a set of observation equations expressed in a simple vector form. Incorporating prior constraints o n the smoothness of slip distribution with the observation equations, we construct a Bayesian model with unknown hyperparameters.T h e optimal values of the hyperparameters, which control the structure of the Bayesian model, are objectively determined from observed data by using ABIC. Once the values of hyperparameters are determined, we can use the maximum likelihood method t o find the optimal distribution of fault slip. W e examined the validity of this method through a numerical experiment using theoretical data with random noise. We analysed geodetic data associated with the 1946 Nankaido earthquake (Ms = 8.2) by using this method. T h e result shows that the fault slip distribution of this earthquake has two main peaks of 4 and 6 m, located off Kii Peninsula a n d Muroto Promontory. These two high-slip areas are clearly separated by a low-slip zone extending along Kii Strait. Such a slip distribution corresponds with the fact that the rupture process of this earthquake in the western part is notably different from that in the eastern part.
Iterated linear inversion theory can often solve nonlinear inverse problems. Also, linear inversion theory provides convenient error estimates and other interpretive measures. But are these interpretive measures valid for nonlinear problems? We address this question in terms of the joint probability density function (pdf) of the estimated parameters. Briefly, linear inversion theory will be valid if the observations are linear functions of the parameters within a reasonable (say, 95%) confidence region about the optimal estimate, if the optimal estimate is unique. We use Bayes' rule to show how prior information can improve the uniqueness of the optimal estimate, while stabilizing the iterative search for this estimate. We also develop quantitative criteria for the relative importance of prior and observational data and for the effects of nonlinearity. Our method can handle any form of pdf for observational data and prior information. The calculations are much easier (about the same as required for the Marquardt method) when both observational and prior data are Gaussian. We present calculations for some simple one and two‐parameter nonlinear inverse problems. These examples show that the asymptotic statistics (those based on the linear theory) may in some cases be grossly erroneous. In other cases, accurate observations, prior information, or a combination of the two may effectively linearize an otherwise nonlinear problem.
We can regard the occurrence of earthquakes as the partial release of tectonic stress by sudden brittle rupture. In the framework of linear elasticity, any indigenous source including earthquake rupture is represented by a moment tensor. The moment tensor is mathematically equivalent to the volume integral of stress release over the whole elastic region surrounding the source, and so we can quantitatively relate the centroid moment tensor (CMT) of seismic events with an unknown tectonic stress field. On the basis of such an idea and Bayesian statistical inference theory, we developed an inversion method to estimate the 3‐D pattern of tectonic stress from CMT data. Applying the CMT data inversion method to 12,500 seismic events in and around Japan, we obtained precise 3‐D tectonic stress patterns that illuminate the present‐day (Quaternary) complex tectonic motion of Japanese islands. The stress pattern of the Kuril‐Japan‐Nankai arc is basically E–W compression, but the direction of intermediate principal stress changes from N–S (reverse faulting type) in northeast Japan to vertical (strike‐slip faulting type) in southwest Japan. On the other hand, the stress pattern of the Ryukyu and Izu‐Bonin back‐arc regions is basically trench perpendicular tension (normal faulting type). In addition to these basic stress patterns governed by mechanical interaction between the Eurasian, North American, Pacific, and Philippine Sea plates, we can recognize several characteristic local stress patterns corresponding to the horizontal motion of the Kuril fore‐arc sliver, the collision of the Izu Peninsula with the mainland of Japan, and the opening of the Beppu‐Shimabara rift zone.
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