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Frechet kernels

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Journal Article
Journal: Geophysics
Published: 06 December 2021
Geophysics (2022) 87 (1): T61–T70.
...Guangchi Xing; Tieyuan Zhu ABSTRACT We have formulated the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we...
FIGURES | View All (11)
Journal Article
Published: 01 October 2008
Bulletin of the Seismological Society of America (2008) 98 (5): 2517–2525.
... where the scales of velocity heterogeneities are comparable or smaller than the width of the Fresnel zone. We calculate the Fréchet sensitivity kernels using the scattering-integral method in conjunction with finite-difference wave simulation in three-dimensional media. The differences...
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Cross sections of normalized 2D and 3D <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for the traveltime d...
Published: 03 November 2006
Figure 2. Cross sections of normalized 2D and 3D Fréchet kernels for the traveltime delay and amplitude variations. (a) 2D sensitivity kernel for traveltime shifts; (b) 3D sensitivity kernel for traveltime residuals; (c) 2D Fréchet kernel for amplitude variations; (d) 3D Fréchet kernel
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Velocity <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> (     K  c     ) for (a, d, g, and j) waveform, (b...
Published: 06 December 2021
Figure 3. Velocity Fréchet kernels ( K c ) for (a, d, g, and j) waveform, (b, e, h, and k) traveltime, and (c, f, i, and l) amplitude objective functions: (a-c)  K 0 , the contribution from the lossless propagation ( L 0 ); (d-f)  K 1 , the contribution from
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Attenuation <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> (     K  γ     ) for (a, d, g, and j) waveform,...
Published: 06 December 2021
Figure 4. Attenuation Fréchet kernels ( K γ ) for (a, d, g, and j) waveform, (b, e, h, and k) traveltime, and (c, f, i, and l) amplitude objective functions: (a-c)  K 0 , the contribution from the lossless propagation ( L 0 ); (d-f)  K 1 , the contribution from
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The comparison between <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> computed using the AD method and the...
Published: 06 December 2021
Figure 7. The comparison between Fréchet kernels computed using the AD method and the FD method: (a, d, g, and j) AD kernels, (b, e, h, and k) FD kernels, and (c, f, i, and l) residual between the AD and the FD kernels. (a-c) Waveform velocity kernel ( K c , W and K ˜ c
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Single-source <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for test 1 (the velocity-anomaly model). (a, ...
Published: 06 December 2021
Figure 8. Single-source Fréchet kernels for test 1 (the velocity-anomaly model). (a, c, and e) Velocity kernels ( K c ) and (b, d, and f) attenuation kernels ( K γ ). (a and b) Waveform, (c and d) traveltime, and (e and f) amplitude.
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Stacked <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for test 1 (the velocity-anomaly model). (a, c, and...
Published: 06 December 2021
Figure 9. Stacked Fréchet kernels for test 1 (the velocity-anomaly model). (a, c, and e) Velocity kernels ( K c ) and (b, d, and f) attenuation kernels ( K γ ). (a and b) Waveform, (c and d) traveltime, and (e and f) amplitude.
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Single-source <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for test 2 (the attenuation-anomaly model). (...
Published: 06 December 2021
Figure 10. Single-source Fréchet kernels for test 2 (the attenuation-anomaly model). (a, c, and e) Velocity kernels ( K c ) and (b, d, and f) attenuation kernels ( K γ ). (a and b) Waveform, (c and d) traveltime, and (e and f) amplitude.
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Stacked <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for test 2 (attenuation-anomaly model). (a, c, and ...
Published: 06 December 2021
Figure 11. Stacked Fréchet kernels for test 2 (attenuation-anomaly model). (a, c, and e) Velocity kernels ( K c ) and (b, d, and f) attenuation kernels ( K γ ). (a and b) Waveform, (c and d) traveltime, and (e and f) amplitude.
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Cross-sections of transmission <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for 2D and 3D wave experimen...
Published: 01 January 2004
Figure 5. Cross-sections of transmission Fréchet kernels for 2D and 3D wave experiments. The cross-sections are obtained at the halfway between the source and receiver, where the width of the Fresnel zone is largest. (a) Cross-sections (single frequency, thin solid line; frequency averaged, thick
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Validation of the finite-frequency <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> in Figure  7  using a sy...
Published: 01 January 2004
Figure 8. Validation of the finite-frequency Fréchet kernels in Figure 7 using a synthetic forward modelling experiment. The “true” velocity medium is a combination of a heterogeneous reference medium and two small-scale anomalies located at depths 68 m and 83 m, respectively. (a) Finite
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Finite-frequency traveltime delay <span class="search-highlight">Fréchet</span> <span class="search-highlight">kernels</span> for a transmitted and ref...
Published: 01 January 2004
Figure 4. Finite-frequency traveltime delay Fréchet kernels for a transmitted and reflected wave in a homogeneous medium. The ray path between the source and receiver is shown with the white line, and the Fresnel zone boundary for the central frequency is plotted with the white dashed line
Journal Article
Published: 11 August 2021
Seismological Research Letters (2021) 92 (6): 3832–3844.
...Yujiang Xie; Catherine A. Rychert; Nicholas Harmon; Qinya Liu; Dirk Gajewski Abstract Full waveform inversion or adjoint tomography has routinely been performed to image the internal structure of the Earth at high resolution. This is typically done using the Fréchet kernels and the approximate...
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Journal Article
Journal: Geophysics
Published: 01 May 1984
Geophysics (1984) 49 (5): 566–576.
... field and the transmitted current. The depth of penetration and the resolution of mean resistivity and anisotropy are presented in terms of Frechet kernels and resolving kernels. The kernels are analytic for the special case of a uniform crust. The shape of the Frechet kernels for resistivity...
Journal Article
Journal: Geophysics
Published: 01 November 1993
Geophysics (1993) 58 (11): 1646–1654.
...Steve Cardimona; Jan Garmany Abstract The analytic kernel in the space-time domain for the Frechet derivative of acoustic waveform data with respect to changes in the slowness model is given by the Born approximation solution to the integral equation of waveform scattering. Preconditioning...
Journal Article
Journal: Geophysics
Published: 31 October 2022
Geophysics (2022) 87 (6): R465–R486.
... elastic P-wave data, making this approach in this respect equivalent to elastic waveform inversion. We also develop a method to analyze model constraint by examining the energy of the wavefield Fréchet derivative thereby avoiding the influence of the data residuals in traditional Fréchet kernels...
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Journal Article
Journal: Geophysics
Published: 01 September 1987
Geophysics (1987) 52 (9): 1297–1302.
... a normalization factor, to the Frechèt kernels of the conventional equations obtained by linearization. Accordingly, the sensitivity functions play the role of weighting functions for electrical conductivity despite the nonlinear dependence of the model and the data. In terms of the integral equations...
Journal Article
Journal: Geophysics
Published: 18 March 2016
Geophysics (2016) 81 (2): C53–C68.
... FWI of multicomponent transmission data for two different model parameterizations. To analyze the sensitivity of the objective function to the model parameters, the Fréchet kernel of FWI is obtained by linearizing the elastic wave equation using the Born approximation and employing the asymptotic...
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Journal Article
Journal: Geophysics
Published: 01 June 1995
Geophysics (1995) 60 (3): 899–911.
.... Updating the fields and Green's functions relates small changes in the predicted data to small changes in the model parameters through Frechet kernels. These fields and Green functions are calculated with an efficient 3-D finite-difference solver. Since the resistivity is invariant along strike, the 3-D...