Resolution analysis of finite fault source inversion using one- and three-dimensional Green's functions: 2. Combining seismic and geodetic data

2001 ◽  
Vol 106 (B5) ◽  
pp. 8767-8788 ◽  
Author(s):  
David J. Wald ◽  
Robert W. Graves
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Geodetic Data ◽  
Source Inversion ◽  
Finite Fault ◽  
Resolution Analysis ◽  
Fault Source

A new three-dimensional regularization for finite fault source inversions

2020 ◽  
Author(s):  
Navid Kheirdast ◽  
Anooshiravan Ansari ◽  
Susana Custódio
Keyword(s):  
Source Function ◽  
Three Dimensional ◽  
Dimensional Regularization ◽  
Model Space ◽  
Rupture Process ◽  
Damping Factor ◽  
Source Inversion ◽  
Temporal Dimension ◽  
Finite Fault ◽  
Fault Source

<p>The earthquake rupture process is often represented by a source function that defines slip in space and time. If we assume slip to occur on a planar surface, then the source function becomes a function of three independent variables: two spatial dimensions (slip down-dip and slip along-strike) and one temporal dimension (source time function at each point on the fault). Finite fault inverse problems aim at exploring this model space in order to find the source function that generates synthetic ground motion that best fits the observed data. This inverse problem is severely ill-conditioned. In order both to ensure a regular solution and to avoid over-fitting the data, both physical and mathematical constraints can be imposed.  Common methods of finite fault source inversion typically apply a one-dimensional regularization in time, which gives preference to compacted-support source-time functions, like triangular or trapezoidal functions in time, or to two-dimensional regularizations that ensure smooth variations of slip over the fault plane (Mai et. al, 2016). In this work, we propose an innovative three-dimensional regularization for kinematic source inversions in the frequency domain, which simultaneously requires smooth variations of slip over space (2D) and frequency (1D, smooth spectra) . This new three-dimensional regularization selects the spatial slip distributions that are more similar to those of neighboring frequencies, thus effectively transferring knowledge from one frequency to another. In the framework of Tikhonov regularization, having more than one regularization condition requires more than one damping factor to be inserted in the inversion misfit. Additionally, no orthogonal decomposition (like Generalized Singular Value Decomposition) exists when more than one regularization conditions are imposed. Thus, we investigate a new 3D regularization method using a Bayesian approach with a Markov Chain Monte Carlo (MCMC) simulation. The new method is tested using the SIV-inv1 benchmark exercise. The proposed method is also preliminarily applied to study the rupture process of the 2019 M5.9 Torkamanchay, Iran, earthquake.</p>

Three-Dimensional Green’s Functions in Anisotropic Elastic Bimaterials With Imperfect Interfaces

2003 ◽  
Vol 70 (2) ◽  
pp. 180-190 ◽  
Author(s):  
E. Pan
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Imperfect Interface ◽  
Boundary Integral ◽  
Stress Fields ◽  
Superposition Method ◽  
Interface Conditions ◽  
Complementary Part ◽  
Anisotropic Elastic

In this paper, three-dimensional Green’s functions in anisotropic elastic bimaterials with imperfect interface conditions are derived based on the extended Stroh formalism and the Mindlin’s superposition method. Four different interface models are considered: perfect-bond, smooth-bond, dislocation-like, and force-like. While the first one is for a perfect interface, other three models are for imperfect ones. By introducing certain modified eigenmatrices, it is shown that the bimaterial Green’s functions for the three imperfect interface conditions have mathematically similar concise expressions as those for the perfect-bond interface. That is, the physical-domain bimaterial Green’s functions can be obtained as a sum of a homogeneous full-space Green’s function in an explicit form and a complementary part in terms of simple line-integrals over [0,π] suitable for standard numerical integration. Furthermore, the corresponding two-dimensional bimaterial Green’s functions have been also derived analytically for the three imperfect interface conditions. Based on the bimaterial Green’s functions, the effects of different interface conditions on the displacement and stress fields are discussed. It is shown that only the complementary part of the solution contributes to the difference of the displacement and stress fields due to different interface conditions. Numerical examples are given for the Green’s functions in the bimaterials made of two anisotropic half-spaces. It is observed that different interface conditions can produce substantially different results for some Green’s stress components in the vicinity of the interface, which should be of great interest to the design of interface. Finally, we remark that these bimaterial Green’s functions can be implemented into the boundary integral formulation for the analysis of layered structures where imperfect bond may exist.

THREE‐DIMENSIONAL INDUCED POLARIZATION AND ELECTROMAGNETIC MODELING

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 309-324 ◽  
Author(s):  
Gerald W. Hohmann
Keyword(s):  
Integral Equation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Induced Polarization ◽  
The Body ◽  
Scale Model ◽  
Two Dimensional ◽  
Electromagnetic Modeling ◽  
The Earth

The induced polarization (IP) and electromagnetic (EM) responses of a three‐dimensional body in the earth can be calculated using an integral equation solution. The problem is formulated by replacing the body by a volume of polarization or scattering current. The integral equation is reduced to a matrix equation, which is solved numerically for the electric field in the body. Then the electric and magnetic fields outside the inhomogeneity can be found by integrating the appropriate dyadic Green’s functions over the scattering current. Because half‐space Green’s functions are used, it is only necessary to solve for scattering currents in the body—not throughout the earth. Numerical results for a number of practical cases show, for example, that for moderate conductivity contrasts the dipole‐dipole IP response of a body five units in strike length approximates that of a two‐dimensional body. Moving an IP line off the center of a body produces an effect similar to that of increasing the depth. IP response varies significantly with conductivity contrast; the peak response occurs at higher contrasts for two‐dimensional bodies than for bodies of limited length. Very conductive bodies can produce negative IP response due to EM induction. An electrically polarizable body produces a small magnetic field, so that it is possible to measure IP with a sensitive magnetometer. Calculations show that horizontal loop EM response is enhanced when the background resistivity in the earth is reduced, thus confirming scale model results.

Dynamic Observer Method Based on Modified Green’s Functions for Robust and More Stable Inverse Algorithms

2008 ◽  
Author(s):  
Priscila F. B. Sousa ◽  
Ana P. Fernandes ◽  
Vale´rio Luiz Borges ◽  
George S. Dulikravich ◽  
Gilmar Guimara˜es
Keyword(s):  
Heat Transfer ◽  
Transfer Function ◽  
Green's Functions ◽  
Transfer Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Original Method ◽  
Identification Algorithm ◽  
Transfer Function Model ◽  
Function Model

This work presents a modified procedure to use the concept of dynamic observers based on Green’s functions to solve inverse problems. The original method can be divided in two distinct steps: i) obtaining a transfer function model GH and; ii) obtaining heat transfer functions GQ and GN and building an identification algorithm. The transfer function model, GH, is obtained from the equivalent dynamic systems theory using Green’s functions. The modification presented here proposes two different improvements in the original technique: i) A different method of obtaining the transfer function model, GH, using analytical functions instead of numerical procedures, and ii) Definition of a new concept of GH to allow the use of more than one response temperature. Obtaining the heat transfer functions represents an important role in the observer method and is crucial to allow the technique to be directly applied to two or three-dimensional heat conduction problems. The idea of defining the new GH function is to improve the robustness and stability of the algorithm. A new dynamic equivalent system for the thermal model is then defined in order to allow the use of two or more temperature measurements. Heat transfer function, GH can be obtained numerically or analytically using Green’s function method. The great advantage of deriving GH analytically is to simplify the procedure and minimize the estimative errors.

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