5:15 PM - 7:15 PM
[SSS10-P07] Analytical approaches for simultaneous inversion of distributed slip, Green's function, and fault geometry
Keywords:Simultaneous inversion of distributed slip, Green's function, and fault geometry
Slip inversion is an inverse analysis of the dislocation problem that describes the motion of a medium subject to the displacement discontinuity boundary condition on faults. The current de facto standard is to calculate fault slip by assuming the shape of the fault and the solution (Green's function) of the displacement response to unit fault slip (Yabuki & Matsu'ura, 1992; Ide, 2007), but it is plagued by a solution bias derived from model assumptions (Dutta et al., 2021), which becomes an obstacle for the secondary use of estimates (Agata et al., 2021). Such model biases in slip inversions can be expressed in a unified form as an error of the integral kernel in the observation equation (Green's function of the inverse problem) (Yagi & Fukahata, 2011). Yagi & Fukahata (2011) recognized that Green's function is also an unknown unless the true model is a known, revealing that the error propagation from Green's function uncertainty can be a dominant error factor in estimating slip. Agata et al. (2021) then argue a simultaneous inversion of slip and the Green's function.
However, it should be noticed that the error in Green's function in the inverse dislocation problem is not equal to the error in Green's function in the forward dislocation problem, i.e., the fundamental solution that expresses the medium response to the source force (Hori et al., 2021), due to the presence of the error in assumed fault geometry (Matsu'ura, 1977). In conventional inversion analysis, the fault geometry works as a hyperparameter that prescribes the model to relate slip and data (Fukahata & Wright, 2008), and three-dimensional geometry estimation suffers from accuracy insufficiency (e.g., Jolivet et al., 2014). On this issue, Shimizu et al. (2021) showed that the inversion (Kikuchi & Kanamori, 1991) of inelastic strain, termed potency, can be a breakthrough that replaces the use of slip, the orientation of which is prescribed by the assumed fault shape: they clarified that the fault shape estimate as a potency-consistent boundary has the same level of accuracy as the potency estimate. However, the fault shape in Shimizu et al. (2021) is expressed by B-spline functions, so a technical difficulty remains in solving methods as B-spline expresses only a quasi-two-dimensional shape in which the dip/strike angle changes in a uniaxial direction.
We report on a formulation for simultaneously estimating the three-dimensional shape and slip of a fault plane and associate analytical expressions of the solutions. We also report that analytic representations for Green's function estimate within our formulation following Shimizu et al. (2021), where Green's function uncertainty in the inverse problem separates into Green's function uncertainty in the sense of the forward problem (the medium response uncertainty) and geometrical uncertainty of faults. This poster thus presents a method for simultaneously estimating the slip, Green's function as medium responses, and fault geometry, with analytical solutions.
In our problem setting, the model parameters are the potencies and normal and slip vectors at each coordinate on the fault plane, as well as Green's function of the observation equation. The distribution of observed data is expressed as a Gaussian distribution, and the following prior distributions are set for those model parameters: an arbitrary Gaussian distribution for the prior of the potencies, a Gaussian distribution that peaks at the reference Green's function for the prior distribution of the Green's function, and for the normal and slip directions at each coordinate, the von Mises distribution such that the fault plane is determined as the macroscopic crack face that best explains the double-couple component of shear inelastic strain (the opening component for mode I); as a smooth surface consistent with linear continuum mechanics (Romanet et al., 2024), the crack face is then calculated. In the above problem setting, the posterior distributions of the slip, Green's function, and fault geometry are evaluated, and the optimal solution with the maximum posterior and the associated posterior covariances are calculated from the fault normals. In an advanced approach, the covariance matrix traces (variance scale factors) for the data distribution and the prior distributions of slip and Green's function are simultaneously estimated as hyperparameters, as in Yagi & Fukahata (2011).
In our poster, we will report on the verification of this formulation using synthetic tests, as well as on its application to the geodetic slip deficit inversion for the Nankai subduction zone and the teleseismic slip inversion for the 2013 Balochistan earthquake.
However, it should be noticed that the error in Green's function in the inverse dislocation problem is not equal to the error in Green's function in the forward dislocation problem, i.e., the fundamental solution that expresses the medium response to the source force (Hori et al., 2021), due to the presence of the error in assumed fault geometry (Matsu'ura, 1977). In conventional inversion analysis, the fault geometry works as a hyperparameter that prescribes the model to relate slip and data (Fukahata & Wright, 2008), and three-dimensional geometry estimation suffers from accuracy insufficiency (e.g., Jolivet et al., 2014). On this issue, Shimizu et al. (2021) showed that the inversion (Kikuchi & Kanamori, 1991) of inelastic strain, termed potency, can be a breakthrough that replaces the use of slip, the orientation of which is prescribed by the assumed fault shape: they clarified that the fault shape estimate as a potency-consistent boundary has the same level of accuracy as the potency estimate. However, the fault shape in Shimizu et al. (2021) is expressed by B-spline functions, so a technical difficulty remains in solving methods as B-spline expresses only a quasi-two-dimensional shape in which the dip/strike angle changes in a uniaxial direction.
We report on a formulation for simultaneously estimating the three-dimensional shape and slip of a fault plane and associate analytical expressions of the solutions. We also report that analytic representations for Green's function estimate within our formulation following Shimizu et al. (2021), where Green's function uncertainty in the inverse problem separates into Green's function uncertainty in the sense of the forward problem (the medium response uncertainty) and geometrical uncertainty of faults. This poster thus presents a method for simultaneously estimating the slip, Green's function as medium responses, and fault geometry, with analytical solutions.
In our problem setting, the model parameters are the potencies and normal and slip vectors at each coordinate on the fault plane, as well as Green's function of the observation equation. The distribution of observed data is expressed as a Gaussian distribution, and the following prior distributions are set for those model parameters: an arbitrary Gaussian distribution for the prior of the potencies, a Gaussian distribution that peaks at the reference Green's function for the prior distribution of the Green's function, and for the normal and slip directions at each coordinate, the von Mises distribution such that the fault plane is determined as the macroscopic crack face that best explains the double-couple component of shear inelastic strain (the opening component for mode I); as a smooth surface consistent with linear continuum mechanics (Romanet et al., 2024), the crack face is then calculated. In the above problem setting, the posterior distributions of the slip, Green's function, and fault geometry are evaluated, and the optimal solution with the maximum posterior and the associated posterior covariances are calculated from the fault normals. In an advanced approach, the covariance matrix traces (variance scale factors) for the data distribution and the prior distributions of slip and Green's function are simultaneously estimated as hyperparameters, as in Yagi & Fukahata (2011).
In our poster, we will report on the verification of this formulation using synthetic tests, as well as on its application to the geodetic slip deficit inversion for the Nankai subduction zone and the teleseismic slip inversion for the 2013 Balochistan earthquake.