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Chebyshev expansion

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Journal Article
Journal: Geophysics
Published: 15 February 2022
Geophysics (2022) 87 (2): S63–S73.
.... We have adopted a new Fourier method based on the Chebyshev expansion of the second kind. Theoretical analyses and numerical experiments indicate that our method is comprehensively superior to a similar method based on the Chebyshev expansion of the first kind in terms of balanced amplitude and error...
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Journal Article
Journal: Geophysics
Published: 01 December 1996
Geophysics (1996) 61 (6): 1758–1768.
...Vladimir Y. Grechka; George A. McMechan; Vitaly A. Volovodenko Abstract Seismic wave propagation described by differential equations with variable coefficients may be solved by the Chebyshev polynomial expansion method (CPEM). This method approximates a model and forward solutions by orthonormal...
Journal Article
Journal: Geophysics
Published: 19 June 2012
Geophysics (2012) 77 (4): T117–T123.
... used explicit time integration methods correspond to the truncated Taylor series approximations that can cause significant errors for large time steps. The rapid expansion method (REM) uses the Chebyshev expansion and offers an optimal solution to the second-order-in-time wave equations. When applying...
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Journal Article
Journal: Geophysics
Published: 04 January 2006
Geophysics (2006) 71 (1): I1–I9.
... on a Chebyshev expansion of the evolution operator, with the spatial derivatives computed via a staggered Fourier pseudospectral technique. The results match analytical solutions of the initial-value problem and the Green's function. An example of the EM field produced by a set of magnetic sources...
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Image
Snapshot for the four-layer model in Figure  1a  based on the <span class="search-highlight">Chebyshev</span> exp...
Published: 03 August 2016
Figure 2. Snapshot for the four-layer model in Figure  1a based on the Chebyshev expansion method for the OSE matrix. (a-d) Modeling using time-step values of 0.001, 0.004, 0.008, and 0.010 s, respectively.
Image
Plot of polynomials (a)     Q  200    , (b)     Q  600    , (c)     Q  400 ...
Published: 06 August 2010
Figure 2. Plot of polynomials (a) Q 200 , (b) Q 600 , (c) Q 400 , and (d) Q 800 of the Chebyshev expansion for the EAGE salt model calculated as part of the one-step REM. The source is located at the top of the model and at x = 8.0 km . The timing
Journal Article
Journal: Geophysics
Published: 03 August 2016
Geophysics (2016) 81 (5): S359–S366.
...Figure 2. Snapshot for the four-layer model in Figure  1a based on the Chebyshev expansion method for the OSE matrix. (a-d) Modeling using time-step values of 0.001, 0.004, 0.008, and 0.010 s, respectively. ...
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Journal Article
Journal: Geophysics
Published: 22 November 2019
Geophysics (2020) 85 (1): E15–E26.
... approximations of temporal derivatives have finite accuracy and may introduce numerical errors and dispersion (e.g., Adhidjaja and Hohmann, 1989 ). To address this issue, Tal-Ezer (1986) proposes a spectral method for hyperbolic equations based on a Chebyshev expansion of the time operator. The time evolution...
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Journal Article
Journal: Geophysics
Published: 21 July 2014
Geophysics (2014) 79 (5): S163–S172.
... there are the following: finite differences (FDs), which makes use of the Taylor expansion ( Etgen, 1986 ); the rapid expansion method (REM), which makes use of the Chebyshev expansion ( Kosloff et al., 1989 ; Pestana and Stoffa, 2010 ); and the two-step explicit marching method, which makes use of a polynomial...
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Journal Article
Journal: Geophysics
Published: 01 January 2002
Geophysics (2002) 67 (6): 1983–1990.
... response. The filtering can be approximated in the wavenumber domain by, for example, power series, Chebyshev polynonomial fitting, Pade approximants, or continued fraction expansions. The filters must then be Fourier transformed back to the frequency–space domain, where the resulting filter is convolved...
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Journal Article
Journal: Geophysics
Published: 02 July 2019
Geophysics (2019) 84 (4): T299–T311.
... with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We...
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Journal Article
Journal: Geophysics
Published: 14 April 2020
Geophysics (2020) 85 (3): T165–T178.
... method (REM) that uses Chebyshev matrix polynomials to approximate the continuous solution operator of the discrete wave equation. The second technique is a Lax-Wendroff method (LWM) that replaces time derivatives in the Taylor expansion of the solution wavefield with their equivalent spatial PS...
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Journal Article
Journal: Geophysics
Published: 13 September 2019
Geophysics (2019) 84 (5): E311–E321.
... = 0 , we may use the wavefield at t = 0 + , just after the impulse, as an initial field condition. Using the Chebyshev expansion of exp ( x ) , in the absence of sources, the discrete solution ( Carcione 2006 ) is w N M ( t ) = ∑ k = 0 M b k ( t...
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Journal Article
Journal: Geophysics
Published: 23 March 2010
Geophysics (2010) 75 (2): S23–S34.
... the first-kind Chebyshev polynomials to economize the results of the Taylor expansion. Finally, we optimize the constant coefficients using the globally optimized scheme, which are fixed and feasible for arbitrary velocity models. Theoretical analysis and nu-merical experiments have demonstrated...
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Journal Article
Journal: Geophysics
Published: 06 August 2010
Geophysics (2010) 75 (4): T121–T131.
...Figure 2. Plot of polynomials (a) Q 200 , (b) Q 600 , (c) Q 400 , and (d) Q 800 of the Chebyshev expansion for the EAGE salt model calculated as part of the one-step REM. The source is located at the top of the model and at x = 8.0 km . The timing...
FIGURES | View All (10)
Image
Wavefield time modeling snapshots at     t  =  0.8        s     using the g...
Published: 21 November 2013
Figure 4. Wavefield time modeling snapshots at t = 0.8     s using the gradient velocity model of Figure  2 . The four images show different RITE methods: (a) Chebyshev-series expansion, (b) WLSQ-series expansion, (c) one-step PSPI, and (d) one-step operator interpolation. For all
Journal Article
Journal: Geophysics
Published: 16 June 2011
Geophysics (2011) 76 (4): S177–S185.
... by comparing migrated images. In the present paper, I compare modeling results to analytic solutions for simple geometries to obtain a more solid basis for the number of expansion terms needed for the REM. Moreover, the computation of Chebyshev polynomials appears unnecessarily complicated in Pestana...
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Journal Article
Journal: Geophysics
Published: 22 June 2015
Geophysics (2015) 80 (4): R203–R216.
... propagation and the inversion scheme. To mitigate this problem, we have developed a rapid expansion method (REM) for numerical wavefield extrapolation inside the FWI workflow. This method approximated the partial time derivative of the wave equation using Chebyshev polynomials instead of the conventional...
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Series: Society of Exploration Geophysicists Geophysics Reprint Series
Published: 01 January 2012
EISBN: 9781560803089
... into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated...
Journal Article
Published: 01 October 1986
Bulletin of the Seismological Society of America (1986) 76 (5): 1367–1383.
... on the nonlinear least-squares technique with Chebyshev polynomials as the expansion functions. The degree of the fitting polynomial is determined by the F test, and estimates of the standard errors for the smoothed data are provided. The procedure is illustrated by its application to synthetic and actual data...