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Frechet matrix

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Journal Article
Journal: Geophysics
Published: 01 November 1997
Geophysics (1997) 62 (6): 1884–1895.
... of the Frechét matrix determines the influence of various model parameters on common-source and common-midpoint traveltimes. The singular-value analysis shows that common-source traveltimes depend mainly on the reflector position and shape. The common-midpoint traveltimes also contain additional information...
Journal Article
Journal: Geophysics
Published: 01 December 1999
Geophysics (1999) 64 (6): 1920–1927.
... with offset (AVO) analysis, which can invert for only two elastic parameters. Numerical analysis with the second-order approximation shows that the condition number of the Frechet matrix for three elastic parameters is improved significantly from using a linear approximation. Therefore, those quadratic...
Image
The influence of the dip  β  of the fracture normal on the condition number...
Published: 01 January 2003
F IG . 3. The influence of the dip β of the fracture normal on the condition number κ of the Frechét matrix. The model includes three rotationally invariant fracture sets with the same β embedded in isotropic rock with V P / V S = 2 ( β = 0° corresponds to vertical fractures
Series: Geophysical Developments Series
Published: 01 January 1999
DOI: 10.1190/1.9781560802150.ch15
EISBN: 9781560802150
.... 1 . The solid line shows isolines of the misfit functional. It has a minimum at the solution point ( x = 1, y = 2). Iteration starts from the point x = 0.4, y = 1, which is marked by the asterisk in Fig. 1 . At this point the Frechet matrix is estimated using a finite-difference method...
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Journal Article
Journal: Geophysics
Published: 01 March 2007
Geophysics (2007) 72 (2): WA73–WA84.
... challenging problem. We present a new approach to 3D inversion of MCSEM data based on rigorous integral-equation (IE) forward modeling and a new IE representation of the sensitivity (Fréchet derivative matrix) of observed data to variations in sea-bottom conductivity. We develop a new form of the quasi...
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Journal Article
Journal: Geophysics
Published: 28 September 2009
Geophysics (2009) 74 (5): WB153–WB163.
...Bing Zhou; Stewart Greenhalgh Abstract 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...
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Journal Article
Journal: Geophysics
Published: 27 December 2021
Geophysics (2021) U29–U41.
... the misfit function, which is achieved by a matrix-free adjoint-state method without cumbersome ray tracing and explicit estimation of the Fréchet derivative matrix in the curvilinear coordinate system. The new tomographic scheme is evaluated through numerical examples with different seismic structures...
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Image
(a) Logarithms of normalized singular values       s    t       (black dots...
Published: 10 January 2012
Figure 14. (a) Logarithms of normalized singular values s t (black dots) and absolute values of elements of eigenvector matrix w t (gray squares) of the Frechét-derivative matrix F t (equation  31 ) for the geometry shown in Figure  10 and (b) RMS
Image
Sample wavepaths from the 2D Sigsbee model case (Figure  1 ): (a) wavepath ...
Published: 01 October 2008
Figure 4. Sample wavepaths from the 2D Sigsbee model case (Figure 1 ): (a) wavepath fans (normal-incidence and offset wavepath from the same reflection point) in the sedimentary section and through the salt body; (b) normal-incidence wavepaths (color values are proportional to inversion matrix
Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (4): 1399–1407.
...F IG . 3. The influence of the dip β of the fracture normal on the condition number κ of the Frechét matrix. The model includes three rotationally invariant fracture sets with the same β embedded in isotropic rock with V P / V S = 2 ( β = 0° corresponds to vertical fractures...
FIGURES | View All (4)
Image
Wavepaths involved in the constitution of the Hessian operators for a singl...
Published: 07 September 2021
Figure 2. Wavepaths involved in the constitution of the Hessian operators for a single source-receiver pair. The crosscorrelation of the Frechét derivative wavefields associated with two parameters at x i and x j forms the element of the approximate Hessian matrix
Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (6): 2026–2039.
... approach to compute plane wave seismograms and the Fréchet derivative matrix as a by-product of forward modeling. We then demonstrate that in a gradient-descent–type optimization scheme, regularization is critical for obtaining stable and geologically realistic solutions. Although, in most applications...
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Journal Article
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (3): 800–808.
... remains a nontrivial computational task. We present a 2-D one-pass inversion procedure formulated as a multichannel deconvolution. It is based on the equation for the electrical potential linearized under the Born approximation, and it makes use of the 2-D form of the Fréchet derivatives evaluated...
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Journal Article
Journal: Geophysics
Published: 27 May 2016
Geophysics (2016) 81 (4): R147–R155.
.... By coupling the gradient computation to nonlinear optimization, it avoids the need for explicit computation of the Fréchet derivative matrix. Furthermore, its cost is equivalent to twice the solution of the forward-modeling problem, irrespective of the size of the input data. The presence of anisotropy...
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Image
Matching upgoing and downgoing wavefields to construct a normal-incidence w...
Published: 01 October 2008
wavepath superimposed on velocity model (color values are proportional to inversion matrix coefficients or Fréchet derivatives in the absence of a layer freezing). The equivalent normal raypath is marked in orange.
Journal Article
Journal: Geophysics
Published: 08 December 2009
Geophysics (2009) 74 (6): WCB1–WCB10.
... tomography was revisited with the proposal of a simple gradient-based approach that avoids ray tracing and estimation of the Fréchet derivative matrix. The key point becomes the derivation of the gradient of the misfit function obtained by the adjoint-state technique. The adjoint-state method is very...
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Journal Article
Journal: Geophysics
Published: 31 December 2008
Geophysics (2009) 74 (1): E75–E91.
... validated a new inversion method to estimate 2D parametric spatial distributions of electrical resistivity from array-induction measurements acquired in a vertical well. The central component of the method is an efficient approximation to Fréchet derivatives where both the incident and adjoint fields...
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Journal Article
Journal: Geophysics
Published: 31 October 2006
Geophysics (2006) 71 (6): G321–G332.
... of the coefficient matrix, thus reducing storage costs from 22 N to 10 N complex, double-precision words with no decrease in computational performance. The disk model serves as a generalized representation of any number of resistive targets in the marine environment, including basaltic sills, carbonates...
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Journal Article
Journal: Geophysics
Published: 01 January 2004
Geophysics (2004) 69 (1): 265–274.
... forward-modeled quantities are obtained during each iteration by dynamic ray tracing along normal rays pertaining to the input data points. Fréchet derivatives for the tomographic matrix are calculated by ray perturbation theory. The inversion procedure is demonstrated on a 2D synthetic prestack data set...
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Series: Geophysical Developments Series
Published: 01 January 1999
DOI: 10.1190/1.9781560802150.ch14
EISBN: 9781560802150
... method requires the computation of the Fréchet derivative (matrix of partial derivatives) operating on a vector of residuals, which is equivalent to solving another forward problem. Simulations show the efficiency of this algorithm, especially when the number of illumination angles is limited. BCG-FFT...
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