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Hessian matrix

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Journal Article
Journal: Geophysics
Published: 19 June 2019
Geophysics (2019) 84 (4): R625–R640.
...Bowu Jiang; Jianfeng Zhang ABSTRACT We have developed an explicit inverse approach with a Hessian matrix for the least-squares (LS) implementation of prestack time migration (PSTM). A full Hessian matrix is divided into a series of computationally tractable small-sized matrices using a localized...
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Journal Article
Journal: Interpretation
Published: 18 October 2016
Interpretation (2016) 4 (4): T627–T635.
... strategy is used to overcome the local minima problem and the cycle-skipping phenomenon. Another obstacle in this application is the slow convergence rate. The inverse Hessian can enhance the poorly blurred gradient in FWI, but obtaining the full Hessian matrix needs intensive computation cost; thus, we...
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Journal Article
Journal: Geophysics
Published: 08 August 2012
Geophysics (2012) 77 (5): R207–R216.
...Yong Ma; Dave Hale ABSTRACT We present a method, in realistic-size full-waveform inversion (FWI), to explicitly construct a projected Hessian matrix and its inverse matrix, which we subsequently used to solve FWI with a quasi-Newton method. Newton’s method is practically unfeasible in solving...
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Journal Article
Published: 01 October 2008
Bulletin of the Seismological Society of America (2008) 98 (5): 2402–2415.
... a new pseudo-Hessian matrix by combining the conventional pseudo-Hessian matrix with amplitude fields. Because the conventional pseudo-Hessian matrix is assumed to neglect the zero-lag autocorrelation terms of impulse responses in the approximate Hessian matrix of the Gauss–Newton method, it has certain...
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Image
(a) The initial <b>Hessian</b>         H    ˜      0      , (b) the <b>Hessian</b> <b>matrix</b>...
Published: 08 August 2012
Figure 4. (a) The initial Hessian H ˜ 0 , (b) the Hessian matrix H ˜ 1 after the first iteration, (c) the Hessian update H ˜ 1 − H ˜ 0 in the first iteration, and (d) the inverse Hessian H ˜ 1 − 1 .
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Comparison between the (a) blockwise <b>Hessian</b> <b>matrix</b> and (b) modified blockw...
Published: 19 June 2019
Figure 2. Comparison between the (a) blockwise Hessian matrix and (b) modified blockwise Hessian matrix with a reflector-oriented localized approach. Notice the spreading of energy away from the diagonal of Hessian matrix and the high amplitudes on some columns of the modified blockwise Hessian
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Block-band structure of the <b>Hessian</b> <b>matrix</b>. (a) Upper left crop of the full...
Published: 12 February 2020
Figure 1. Block-band structure of the Hessian matrix. (a) Upper left crop of the full matrix ( 0.2 n z · n x × 0.2 n z · n x ). (b) Windowed portion of the Hessian ( 5 n z × 5 n z , dashed red square in [a]). (c) Single n z × n z block
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(a) Sensitivity analysis of the <b>Hessian</b> <b>matrix</b>     H    . (b) Sensitivity a...
Published: 22 November 2019
Figure 2. (a) Sensitivity analysis of the Hessian matrix H . (b) Sensitivity analysis of the multidamped Hessian matrix H damp . In each subfigure, the top shows the eigenvalues and the corresponding eigenvectors and the bottom shows the resolution matrix.
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Diagonal components of (a) the pseudo-<b>Hessian</b> <b>matrix</b>, and (b) of the modifi...
Published: 01 December 2010
Figure 1. Diagonal components of (a) the pseudo-Hessian matrix, and (b) of the modified pseudo-Hessian matrix.
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Illustration of the derivation of the offset-dependent <b>Hessian</b> <b>matrix</b>. Poin...
Published: 19 June 2019
Figure 1. Illustration of the derivation of the offset-dependent Hessian matrix. Points p and q denote the imaging point and scattering point, respectively. The solid line denotes the course of modeling, where seismic waves emanate from the source, pass through scattering point
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<b>Hessian</b> <b>matrix</b>     H     eigenvalue distributions. Blue curve E0: Gauss-New...
Published: 19 April 2016
Figure 5. Hessian matrix H eigenvalue distributions. Blue curve E0: Gauss-Newton data Hessian H D only. Green curve E1: Gauss-Newton data Hessian and regularization H D + H ap   mod . Red curve E2: Gauss-Newton data Hessian and regularization H D
Image
Normalized eigenvalue spectra of the <b>Hessian</b> <b>matrix</b>       G    T    G     f...
Published: 14 January 2015
Figure 9. Normalized eigenvalue spectra of the Hessian matrix G T G for a complete pole-pole (solid gray curve) and a comprehensive four-point data set (dashed black curve).
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(a) Eigenvalues of the <b>Hessian</b> <b>matrix</b>. (b) Components of the eigenvectors (...
Published: 25 July 2013
Figure 2. (a) Eigenvalues of the Hessian matrix. (b) Components of the eigenvectors (numbered 1 to 4) associated with the four largest eigenvalues of the Hessian. The input data include PP and PS reflections for the model in Figure  1 for X / Z = 1.5 . The superscript (2) denotes
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(a) Eigenvalues of the <b>Hessian</b> <b>matrix</b> and (b) the components of the eigenve...
Published: 25 July 2013
Figure 4. (a) Eigenvalues of the Hessian matrix and (b) the components of the eigenvectors (numbered 1 to 4) associated with the four largest eigenvalues of the Hessian. The input data include PP reflections for the model in Figure  1 for X / Z = 1 .
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(a) Eigenvalues of the <b>Hessian</b> <b>matrix</b> and (b) the components of the eigenve...
Published: 25 July 2013
Figure 5. (a) Eigenvalues of the Hessian matrix and (b) the components of the eigenvectors (numbered 1 to 4) associated with the four largest eigenvalues of the Hessian. The input data include PP reflections for the model in Figure  1 for X / Z = 2 . The data are contaminated
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The <b>Hessian</b> <b>matrix</b> for the target region. The top panel shows the diagonal ...
Published: 28 January 2013
Figure 3. The Hessian matrix for the target region. The top panel shows the diagonal components of the matrix; the bottom left panel shows the row of the matrix (or point spread function) taken from the image point at inline 7.45 km, crossline 2.50 km, and relative depth 0.30 km (the intersection
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The <b>Hessian</b> <b>matrix</b> for the target region. The top panel shows the diagonal ...
Published: 28 January 2013
Figure 4. The Hessian matrix for the target region. The top panel shows the diagonal components of the matrix; the bottom left panel shows the row of the matrix (or point spread function) taken from the image point at inline 8.65 km, crossline 1.60 km, and relative depth 0.80 km (the intersection
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Eigenvalue spectra of the <b>Hessian</b> <b>matrix</b> for four-point (4P) (bipole-bipole...
Published: 04 January 2011
Figure 2. Eigenvalue spectra of the Hessian matrix for four-point (4P) (bipole-bipole) configurations, normalized with respect to the dominant eigenvalue. Diagram (a) shows eigenvalues versus eigenvalue number (index) computed for complete and comprehensive (compr.) data sets. Diagram (b
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Eigenvalue spectra of the <b>Hessian</b> <b>matrix</b> for three-point (pole-bipole [PB])...
Published: 04 January 2011
Figure 6. Eigenvalue spectra of the Hessian matrix for three-point (pole-bipole [PB]) configurations, normalized with respect to the dominant eigenvalue. Diagram (a) shows eigenvalues versus eigenvalue number (index) computed for complete and comprehensive (compr.) data sets. Diagram (b
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Representation of two normalized eigenvalue spectra of the <b>Hessian</b> <b>matrix</b>  ...
Published: 14 September 2010
Figure 2. Representation of two normalized eigenvalue spectra of the Hessian matrix F T C D − 1 F (equation 9 ). The horizontal axis is normalized with respect to the total number of model parameters, and the vertical axis is normalized with respect to the largest eigenvalues