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Frequency-space explicit schemes

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Proceedings Papers

Paper presented at the 2016 SEG International Exposition and Annual Meeting, October 16–21, 2016
Paper Number: SEG-2016-13972629
... of high frequencies in industrial RTM. However, stencils for the spatial derivative in the conventional ISFD schemes are designed only in the space domain. When these stencils are directly used to solve the wave equations, the dispersion always exists since it ignores the dispersion in time direction. Liu...
Proceedings Papers

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015
Paper Number: SEG-2015-5813150
...-squares. Chu and Stoffa (2012) proposed an implicit scheme in space and time domain for the scalar wave equation, showing that the implicit scheme significantly improved the precision compared with the explicit FD schemes. However, the implicit FD methods described above are only applied in acoustic wave...
Proceedings Papers

Paper presented at the The 27th International Ocean and Polar Engineering Conference, June 25–30, 2017
Paper Number: ISOPE-I-17-702
... ABSTRACT An equal norm multiple scale Trefftz method (MSTM) associated with the Lie-group scheme GL(n, R) in Euclidean space is developed to describe two dimensional nonlinear sloshing behaviors. In this paper, the explicit and implicit GL(n, R) in the Euclidean space are used for time...
Proceedings Papers

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008
Paper Number: SEG-2008-2201
...: The Sigsbee2A ve- locity model To demonstrate the power of inversion, I applied the model-space inversion approach to the Sigsbee2A model. The explicit Hessian operator is computed by using the random phase-encoding method with the frequency band from 5 Hz to 35 Hz, which is equivalent to the frequency band...
Proceedings Papers

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007
Paper Number: SEG-2007-2240
... velocity reference extrapolated wavefields. This method has proven popular and, although it is no longer unconditionally stable (e.g., Etgen, 1994), it is more stable than a typical explicit space-frequency method (Margrave et al., 2006). Other ways of extending the concept of a phase-shift include...
Proceedings Papers

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008
Paper Number: SEG-2008-2097
... FD or staggered-grid FD methods can further reduce the numerical dispersion, but they still suffer from the numerical dispersion when too few samples per wavelength are used (Sei and Symes, 1994). The PSM is attractive as the space operators are exact up to the Nyquist frequency. In other words...
Proceedings Papers

Paper presented at the 2013 SEG Annual Meeting, September 22–27, 2013
Paper Number: SEG-2013-1299
... sparse matrices, with a maximum number of nonzeros per row that depends on the spatial accuracy of the finite difference ap- proximation (e.g., 5 for fourth-order accuracy). To understand the range of grid spacing x and frequency max at which the LOD method would provide an advantage over traditional...
Proceedings Papers

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012
Paper Number: SEG-2012-0749
... of the impedance matrix. NUMERICAL EXAMPLES Figure 2 displays the real parts of the 10 Hz frequency re- sponses for a homogeneous VTI medium (vpz = 1500m/s, = 0.25, and = 0) computed by the explicit second-order finite difference operators. The grid spacings are 12.5 m in both the horizontal and the vertical...
Proceedings Papers

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008
Paper Number: SEG-2008-2077
... of order 2 and 4 (in both space and time). gustafsson staggered grid scheme simulation upstream oil & gas wave equation requirement difference method dispersion misalignment error geophysics frequency wavelength grid order scheme wave propagation reservoir characterization...
Proceedings Papers

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007
Paper Number: SEG-2007-2325
... and the receiver wavefield backward in time (Baysal, et al, 1983; Whitmore, 1983; McMechan, 1983). RTM simulates wave propagation in space domain using explicit finite-difference scheme without any approximation to the wave equation. It has no dip limitation and images most of wave types including turning waves...
Proceedings Papers

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010
Paper Number: SEG-2010-3081
... This work describes extending the existing multiscale paradigm using stable explicit and implicit FD. Solutions of stable FD are produced in the original nondecimated time-space domain. The scale of solutions is determined by the time-step parame- ter and requires neither decimation nor low-pass filtering...
Journal Articles
SPE Form Eval 4 (03): 349–358.
Paper Number: SPE-16554-PA
Published: 01 September 1989
... to shales in that discontinuous and continuous (i.e., continuous between two or more wells) shales coexist. The occurrence of discontinuous shales, however, clearly becomes more prevalent at large well spacings. With reference to Fig. 3, which shows a plot of cumulative frequency vs. shaIe length for two...
Proceedings Papers

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007
Paper Number: SEG-2007-2305
... comparisons for those situations not covered by one-way methods. Velocity sensitivity analysis The acoustic homogeneous wave equation is generally solved in the frequency domain using a depth extrapolation scheme or in the space-time domain as a boundary value problem where source or receiver...
Proceedings Papers

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010
Paper Number: SEG-2010-3044
... a significant portion of fluid. Finite-Difference Solution Scheme For FD solution, the dependent variables in the VS and VP systems are stored on uniformly-spaced, staggered, spatial and temporal grids (figures 1 and 2). The primary advantage of staggered storage schemes over alternative non-staggered...
Proceedings Papers

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010
Paper Number: SEG-2010-3076
... 1296, doi:10.1111/j.1365-246X.2004.02289.x. Jo, C.-H., C. Shin, and J. H. Suh, 1996, An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator : Geophysics, 61, 529 537, doi:10.1190/1.1443979. Kosloff, D., R. Pestana, and H. Tal-Ezer, 2008, Numerical solution of the constant...
Proceedings Papers

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015
Paper Number: SEG-2015-5799892
... migration approach, where the diagonal Hessian operator is computed explicitly by using Green’s function as a Gaussian-beam summation and then its inverse is applied to migrated images. One important advantage of the modelspace inversion, compared to iterative data-space inversion schemes, is that most...
Journal Articles
SPE J. 23 (04): 623–632.
Paper Number: SPE-10492-PA
Published: 01 August 1983
... spacing. However, relaxation is very inefficient at eliminating long wavelength error. This is usually manifested by an acceptable initial decrease in the residual after which the convergence rate becomes painfully slow. The multigrid method uses different grids to eliminate the different frequencies...
Proceedings Papers

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009
Paper Number: SEG-2009-3010
... the wavenumber dispersion relationship: sin li l x xvk r and , where cos lilz xvk r )( lixv r is the localized space average velocity of point ix r and can be replaced by the velocity )( ixv r . Then we can get, ll xvzxi llii dzdxe zxxxHxwxH ill cossin( + vrrr (11) (11) is the frequency and dip angle dependent...
Proceedings Papers

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010
Paper Number: SEG-2010-3226
... the equation (1) explicitly, using a finite-difference scheme, the stability condition sets a limit on the size of the time step. This time step should satisfy the Courant-Friedrichs-Lewy condi- tion (Courant et al., 1967), which for a 3-D problem, using a 2nd order scheme in time and space, gives: t < min( x...
Proceedings Papers

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008
Paper Number: SEG-2008-2057
... and one horizontal segment. The model had a constant velocity of 4500 m/sec. The peak frequency in this example was 25Hz, the horizontal and vertical grid spacing was 27m, and the time step size was 2msec. The input time section was obtained by numerical modeling using the program Susylnv from the SU...

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