DC resistivity Frechet derivatives for a uniform anisotropic medium with a tilted axis of symmetry

2009 ◽  
Vol 2009 (1) ◽  
pp. 1
Author(s):  
T. Wiese ◽  
S. Greenhalgh ◽  
B. Zhou ◽  
L. Marescot ◽  
M. Greenhalgh
Keyword(s):  
Anisotropic Medium ◽  
Dc Resistivity ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Axis Of Symmetry

scholarly journals Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

2009 ◽  
Vol 166 (4) ◽  
pp. 673-699 ◽  
Author(s):  
S. A. Greenhalgh ◽  
L. Marescot ◽  
B. Zhou ◽  
M. Greenhalgh ◽  
T. Wiese
Keyword(s):  
Anisotropic Medium ◽  
Electric Potential ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Axis Of Symmetry

On the computation of the Fréchet derivatives for seismic waveform inversion in 3D general anisotropic, heterogeneous media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB153-WB163 ◽  
Author(s):  
Bing Zhou ◽  
Stewart Greenhalgh
Keyword(s):  
Anisotropic Medium ◽  
Heterogeneous Media ◽  
Jacobian Matrix ◽  
Displacement Vector ◽  
Waveform Inversion ◽  
Model Parameters ◽  
Seismic Waveform ◽  
Model Parameterization ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

We present a perturbation method and a matrix method for formulating the explicit Fréchet derivatives for seismic body-wave waveform inversion in 3D general anisotropic, heterogeneous media. Theoretically, the two methods yield the same explicit formula valid for any class of anisotropy and are completely equivalent if the model parameterization in the inversion is the same as that used in the discretization scheme (unstructured or structured mesh) for forward modeling. Explicit formulas allow various model parameterization schemes that try to match the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization, offer effective and efficient means of forming the Jacobian matrix from the explicit Fréchet derivatives. The sensitivity patterns of the displacement vector to the independent model parameters in a weakly anisotropic medium clearly convey the imaging capability possible with seismic waveform inversion in such an anisotropic medium.

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

scholarly journals Refraction traveltime tomography using damped monochromatic wavefield

Geophysics ◽  
2005 ◽  
Vol 70 (2) ◽  
pp. U1-U7 ◽  
Author(s):  
Sukjoon Pyun ◽  
Changsoo Shin ◽  
Dong-Joo Min ◽  
Taeyoung Ha
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Computational Effort ◽  
Fréchet Derivative ◽  
Reciprocity Theorem ◽  
Damped Wave Equation ◽  
Traveltime Tomography ◽  
Frechet Derivatives ◽  
First Arrival ◽  
Fréchet Derivatives

For complicated earth models, wave-equation–based refraction-traveltime tomography is more accurate than ray-based tomography but requires more computational effort. Most of the computational effort in traveltime tomography comes from computing traveltimes and their Fréchet derivatives, which for ray-based methods can be computed directly. However, in most wave-equation traveltime-tomography algorithms, the steepest descent direction of the objective function is computed by the backprojection algorithm, without computing a Fréchet derivative directly. We propose a new wave-based refraction-traveltime–tomography procedure that computes Fréchet derivatives directly and efficiently. Our method involves solving a damped-wave equation using a frequency-domain, finite-element modeling algorithm at a single frequency and invoking the reciprocity theorem. A damping factor, which is commonly used to suppress wraparound effects in frequency-domain modeling, plays the role of suppressing multievent wavefields. By limiting the wavefield to a single first arrival, we are able to extract the first-arrival traveltime from the phase term without applying a time window. Computing the partial derivative of the damped wave-equation solution using the reciprocity theorem enables us to compute the Fréchet derivative of amplitude, as well as that of traveltime, with respect to subsurface parameters. Using the Marmousi-2 model, we demonstrate numerically that refraction traveltime tomography with large-offset data can be used to provide the smooth initial velocity model necessary for prestack depth migration.

Application of GDQ method in nonlinear analysis of a flexible manipulator undergoing large deformation

2013 ◽  
Vol 227 (12) ◽  
pp. 2671-2685 ◽  
Author(s):  
Mohammad R Fazel ◽  
Majid M Moghaddam ◽  
Javad Poshtan
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equation ◽  
Flexible Manipulator ◽  
Runge Kutta Method ◽  
Highly Nonlinear ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Generalized Differential

Analysis of a flexible manipulator as an initial value problem, due to its large deformations, involves nonlinear ordinary differential equations of motion. In the present work, these equations are solved through the general Frechet derivatives and the generalized differential quadrature (GDQ) method directly. The results so obtained are compared with those of the fourth-order Runge–Kutta method. It is seen that both the results match each other well. Further considering the same manipulator as a boundary value problem, its governing equation is a highly nonlinear partial differential equation. Again applying the general Frechet derivatives and the GDQ method, it is seen that the results are in good match with the linear theory. In both cases, the general Frechet derivatives are introduced and successfully used for linearization. The results of the present study indicate that the GDQ method combined with the general Frechet derivatives can be successfully used for the solution of nonlinear differential equations.

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