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Frechet derivative

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Journal Article

Source-independent seismic envelope inversion based on the direct envelope Fréchet derivative Available to Purchase

Pan Zhang, Ru-Shan Wu, Liguo Han
Journal: Geophysics
Published: 01 October 2018
Geophysics (2018) 83 (6): R581–R595.
DOI: https://doi.org/10.1190/geo2017-0360.1
...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

Fast 2D inversion of large borehole EM induction data sets with an efficient Fréchet-derivative approximation Available to Purchase

Gong Li Wang, Carlos Torres-Verdín, Jesús M. Salazar, Benjamin Voss
Journal: Geophysics
Published: 31 December 2008
Geophysics (2009) 74 (1): E75–E91.
DOI: https://doi.org/10.1190/1.3033213
... 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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View PDF for article title, Fast 2D inversion of large borehole EM induction data sets with an efficient <span class="search-highlight">Fréchet</span>-<span class="search-highlight">derivative</span> approximation
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Journal Article

Fast Frechet derivative calculation using an auxiliary source array method: An application to array resistivity measurements Available to Purchase

Tsili Wang, Alberto Mezzatesta
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (5): 1364–1371.
DOI: https://doi.org/10.1190/1.1487082
...Tsili Wang; Alberto Mezzatesta Abstract Frechet derivatives provide the vital information for parametric resistivity inversion, but the calculation for a multidimensional problem is often computer intensive. This paper presents a new technique for fast calculation of the Frechet derivatives...
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Journal Article

Approximate Frechet derivatives in inductive electromagnetic soundings Available to Purchase

David E. Boerner, J. S. Holladay
Journal: Geophysics
Published: 01 December 1990
Geophysics (1990) 55 (12): 1589–1595.
DOI: https://doi.org/10.1190/1.1442810
...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...
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Journal Article

Explicit expressions for the Fréchet derivatives in 3D anisotropic resistivity inversion Available to Purchase

S. A. Greenhalgh, B. Zhou, M. Greenhalgh, L. Marescot, T. Wiese
Journal: Geophysics
Published: 10 April 2009
Geophysics (2009) 74 (3): F31–F43.
DOI: https://doi.org/10.1190/1.3111114
...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

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

Bing Zhou, Stewart Greenhalgh
Journal: Geophysics
Published: 28 September 2009
Geophysics (2009) 74 (5): WB153–WB163.
DOI: https://doi.org/10.1190/1.3123766
...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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Image
The rms sum of the Fréchet derivative using the (a) TT model (Figure 18a) and (b and c) EAWI-MF-DU final model (Figure 21g). The maximum time window T used in Fréchet derivative calculation is depicted in the text boxes. Higher values indicate better constraint of the velocity model in that location. The Fréchet derivative suggests significantly better constraint of the model in the upper 10 m.

The rms sum of the Fréchet derivative using the (a) TT model (Figure  18a )... Available to Purchase

Published: 31 October 2022
Figure 24. The rms sum of the Fréchet derivative using the (a) TT model (Figure  18a ) and (b and c) EAWI-MF-DU final model (Figure  21g ). The maximum time window T used in Fréchet derivative calculation is depicted in the text boxes. Higher values indicate better constraint of the velocity
Image
Longitudinal conductivity Fréchet derivative dG∕dσl variations in the subsurface for an anisotropic model having longitudinal conductivity of 0.1S∕m, transverse conductivity of 0.025S∕m, and strike of the symmetry axis of 0°. The various sensitivity plots are for different dips of the symmetry axis θ0 for 0°, 15°, 45°, 75°, and 90°. The profiles are perpendicular to the strike of the plane of stratification. The electrodes are again at (5,0,0) and (10,0,0).

Longitudinal conductivity Fréchet derivative d G ∕ d σ l var... Available to Purchase

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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Fréchet derivative dG∕dσt 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.

Fréchet derivative d G ∕ d σ t plots at the surface ( z... Available to Purchase

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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Fréchet derivative dG∕dσ for an isotropic model having a conductivity of 0.1S∕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 and (c) z=0.5 depicting sensitivity variations in plan view are shown. For the z=0.5 depth slice, the electrode positions (white diamonds) have been projected onto this plane.

Fréchet derivative d G ∕ d σ for an isotropic model having a co... Available to Purchase

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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Transverse conductivity Fréchet derivative dG∕dσt 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 to the strike of the plane of stratification.

Transverse conductivity Fréchet derivative d G ∕ d σ t varia... Available to Purchase

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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Fréchet derivative dG∕dσl 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.

Fréchet derivative d G ∕ d σ l plots at the surface ( z... Available to Purchase

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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Velocity model for comparing the analytic Fréchet derivative with the numerical Fréchet derivative.

Velocity model for comparing the analytic Fréchet derivative with the numer... Available to Purchase

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 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.

Comparison of the analytic and numerical Fréchet derivative of traveltime, ... Available to Purchase

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.
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Comparison of the analytic and numerical Fréchet derivative of amplitude where the shot is located at (a) 1.5 km, (b) 2.5 km, and (c) 4.5 km, respectively. FDM = finite-difference method.

Comparison of the analytic and numerical Fréchet derivative of amplitude wh... Available to Purchase

Published: 22 March 2005
Figure 3. Comparison of the analytic and numerical Fréchet derivative of amplitude where the shot is located at (a) 1.5 km, (b) 2.5 km, and (c) 4.5 km, respectively. FDM = finite-difference method.
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The 3-D sensitivity function (Frechét derivative) of normalised apparent resistivity for the pole-pole array with a pole-pole distance of 1 m presented as horisontal depth slices at z = 0.01, 0.25, 0.5, 0.75, 1.0, and 1.5 m. The electrodes are located at (−0.5, 0, 0) m and (0.5, 0, 0) m. Each slice is shown as a surface plot and a contoured picture. The same logarithmic scale with three intervals per decade is used in the contouring of all slices, but the vertical axis in the surface plots is changed as the maximum of the Frechét derivative decreases rapidly with depth. All distances are in meters. Figure from Møller (1996).

The 3-D sensitivity function (Frechét derivative) of normalised apparent re... Available to Purchase

Published: 01 January 2002
F IG . 12. The 3-D sensitivity function (Frechét derivative) of normalised apparent resistivity for the pole-pole array with a pole-pole distance of 1 m presented as horisontal depth slices at z = 0.01, 0.25, 0.5, 0.75, 1.0, and 1.5 m. The electrodes are located at (−0.5, 0, 0) m and (0.5, 0, 0
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2-D form of the Fréchet derivative for an electrode configuration with current and potential electrodes placed at x = −15 m (A), x = 15 m (B), x = −25 m (M), and x = 45 m (N) (configuration s6 in Figure 4). Light gray indicates negative values; dark gray, positive values. Contour intervals are −0.010, −0.003, −0.001, 0, 0.001, 0.003, 0.010.

2-D form of the Fréchet derivative for an electrode configuration with curr... Available to Purchase

Published: 01 January 2001
F IG . 2. 2-D form of the Fréchet derivative for an electrode configuration with current and potential electrodes placed at x = −15 m ( A ), x = 15 m ( B ), x = −25 m ( M ), and x = 45 m ( N ) (configuration s6 in Figure 4 ). Light gray indicates negative values; dark gray, positive
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The Frechet derivatives with respect to Rxo 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 to fifth tracks show the Frechet derivatives of the potential (V4, VA, and VG), the first potential difference (F23, F56, and FFG), and the second potential difference (S234, S789, and SEFG) data, respectively. The solid lines show the results calculated by the ASAM, and the shaded lines show the results calculated by the PPM with 5% perturbation.

The Frechet derivatives with respect to R xo in the layer between 60.9... Available to Purchase

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
Journal Article

Refraction traveltime tomography using damped monochromatic wavefield Available to Purchase

Sukjoon Pyun, Changsoo Shin, Dong-Joo Min, Taeyoung Ha
Journal: Geophysics
Published: 22 March 2005
Geophysics (2005) 70 (2): U1–U7.
DOI: https://doi.org/10.1190/1.1884829
...Figure 1. Velocity model for comparing the analytic Fréchet derivative with the numerical Fréchet derivative. ...
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Fréchet derivatives computed from the Green’s functions in Figure 8.

Fréchet derivatives computed from the Green’s functions in Figure  8 . Available to Purchase

Published: 19 April 2016
Figure 9. Fréchet derivatives computed from the Green’s functions in Figure  8 .