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Frechet derivatives

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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: 10 April 2009
Geophysics (2009) 74 (3): F31–F43.
...S. A. Greenhalgh; B. Zhou; M. Greenhalgh; L. Marescot; T. Wiese Abstract We have developed explicit expressions for the Fréchet derivatives or sensitivity functions in resistivity imaging of a heterogeneous and fully anisotropic earth. The formulation involves the Green's functions...
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Journal Article
Journal: Geophysics
Published: 01 December 1990
Geophysics (1990) 55 (12): 1589–1595.
...David E. Boerner; J. S. Holladay Abstract Frechet derivatives play dual roles in electromagnetic (EM) methods as averaging functions relating conductivity to EM fields and as sensitivity functions relating conductivity perturbations to changes in these fields. For one-dimensional EM inductive...
Image
<b>Fréchet</b> <b>derivatives</b> computed from the Green’s functions in Figure  8 .
in > Geophysics
Published: 19 April 2016
Figure 9. Fréchet derivatives computed from the Green’s functions in Figure  8 .
Image
Schematic of <b>Frechet</b> <b>derivatives</b> (or sensitivities) at subsurface point  k ...
in > Geophysics
Published: 10 April 2009
Figure 2. Schematic of Frechet derivatives (or sensitivities) at subsurface point k for a true current source at point s and an adjoint source at potential electrode position p . The Green's functions from both sources and/or their gradients are required in the formulation.
Image
The <b>Frechet</b> <b>derivatives</b> with respect to  R   xo   in the layer between 60.9...
in > Geophysics
Published: 01 January 2001
F IG . 4. The Frechet derivatives with respect to R xo in the layer between 60.96 and 63.40 m. The first track shows the model resistivity variations in depth. The shaded area in the second track shows the location of the invasion zone for which the Frechet derivatives are shown. The third
Image
The <b>Frechet</b> <b>derivatives</b> with respect to  R   t   in the layer between 60.96...
in > Geophysics
Published: 01 January 2001
F IG . 5. The Frechet derivatives with respect to R t in the layer between 60.96 and 63.40 m.
Image
The <b>Frechet</b> <b>derivatives</b> with respect to  R   xo   in the layer between 65.8...
in > Geophysics
Published: 01 January 2001
F IG . 6. The Frechet derivatives with respect to R xo in the layer between 65.84 and 66.14 m.
Image
The <b>Frechet</b> <b>derivatives</b> with respect to  R   t   in the layer between 65.84...
in > Geophysics
Published: 01 January 2001
F IG . 7. The Frechet derivatives with respect to R t in the layer between 65.84 and 66.14 m.
Journal Article
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (5): 1364–1371.
...F IG . 4. The Frechet derivatives with respect to R xo in the layer between 60.96 and 63.40 m. The first track shows the model resistivity variations in depth. The shaded area in the second track shows the location of the invasion zone for which the Frechet derivatives are shown. The third...
FIGURES | View All (9)
Journal Article
Journal: Geophysics
Published: 01 October 2018
Geophysics (2018) 83 (6): R581–R595.
...Pan Zhang; Ru-Shan Wu; Liguo Han ABSTRACT Seismic envelope inversion (EI) uses low-frequency envelope data to recover long-wavelength components of the subsurface media. Conventional EI uses the same waveform Fréchet derivative as conventional full-waveform inversion. Due to linearization...
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Journal Article
Journal: Geophysics
Published: 22 March 2005
Geophysics (2005) 70 (2): U1–U7.
... 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...
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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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<b>Fréchet</b> <b>derivative</b>     d  G  ∕  d  σ     for an isotropic model having a co...
in > Geophysics
Published: 10 April 2009
Figure 3. Fréchet derivative d G ∕ d σ for an isotropic model having a conductivity of 0.1 S ∕ m . The current source is at (5,0,0) and the potential electrode is at (10,0,0). (a) The cross-sectional view at y = 0 and two horizontal depth slices at (b) z = 0
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Longitudinal conductivity <b>Fréchet</b> <b>derivative</b>     d  G  ∕  d   σ  l      var...
in > Geophysics
Published: 10 April 2009
Figure 4. Longitudinal conductivity Fréchet derivative d G ∕ d σ l variations in the subsurface for an anisotropic model having longitudinal conductivity of 0.1 S ∕ m , transverse conductivity of 0.025 S ∕ m , and strike of the symmetry axis of 0°. The various
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<b>Fréchet</b> <b>derivative</b>     d  G  ∕  d   σ  l      plots at the surface     (  z...
in > Geophysics
Published: 10 April 2009
Figure 5. Fréchet derivative d G ∕ d σ l plots at the surface ( z = 0 ) for the same model and electrode configuration as Figure 4 . Note the symmetry of the plots and the decrease in sensitivity with increasing dip.
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Transverse conductivity <b>Fréchet</b> <b>derivative</b>     d  G  ∕  d   σ  t      varia...
in > Geophysics
Published: 10 April 2009
Figure 6. Transverse conductivity Fréchet derivative d G ∕ d σ t variations for the same anisotropic model and electrode configuration as in Figure 4 . The various sensitivity plots are for differing dips of the symmetry axis, as indicated. The profiles are perpendicular
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<b>Fréchet</b> <b>derivative</b>     d  G  ∕  d   σ  t      plots at the surface     (  z...
in > Geophysics
Published: 10 April 2009
Figure 7. Fréchet derivative d G ∕ d σ t plots at the surface ( z = 0 ) for the same model and electrode configuration as in Figure 6 . All patterns are symmetrical about the x -axis. In the case of zero dip, the sensitivity is zero.
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Velocity model for comparing the analytic <b>Fréchet</b> <b>derivative</b> with the numer...
in > Geophysics
Published: 22 March 2005
Figure 1. Velocity model for comparing the analytic Fréchet derivative with the numerical Fréchet derivative.
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Comparison of the analytic and numerical <b>Fréchet</b> <b>derivative</b> of traveltime, ...
in > Geophysics
Published: 22 March 2005
Figure 2. Comparison of the analytic and numerical Fréchet derivative of traveltime, where the shot is located at (a) 1.5 km, (b) 2.5 km, and (c) 4.5 km, respectively. FDM = finite-difference method.