Abstract
Wave processes occurring in a fractured rock material in seismic exploration are numerically simulated and studied. The grid-characteristic method on hexahedral grids is extended to construct a seismic image of a fractured elastic medium (i.e., the positions of the reflecting layers under the daytime surface). The underlying idea is that the method is applied to both direct and adjoint problems (in the latter problem, seismometer data recorded on the daytime surface are transformed into corresponding seismic sources). In this case, a seismic image is obtained as the convolution of the direct and adjoint fields. Numerical computations are used to produce synthetic seismograms (time dependences of point oscillations on the daytime surface) for rock materials with gas-saturated and liquid-saturated cracks. The inhomogeneities are explicitly described (at the stage of hexahedral grid generation) and the stressed state on their boundaries is updated at every time step (by solving a contact discontinuity problem). The algorithm is used to construct seismic images for media. An analysis of the results yields information not only on the locations of the cracks, but also on their spatial orientations.
Similar content being viewed by others
REFERENCES
J. F. Claerbout, “Toward a unified theory of reflector mapping,” Geophysics 36, 467–481 (1971).
J. F. Claerbout and S. M. Doherty, “Downward continuation of moveout corrected seismograms,” Geophysics 37, 741–768 (1972).
J. F. Claerbout, “Coarse grid calculations of wave in inhomogeneous media with application to delineation of complicated seismic structure,” Geophysics 35, 407–418 (1970).
W. S. French, “Computer migration of oblique seismic reflection profiles,” Geophysics 40, 961–980 (1975).
W. A. Schneider, “Integral formulation for migration in two and three dimensions,” Geophysics 43, 49–76 (1978).
R. W. Clayton and R. H. Stolt, “A Born–WKBJ inversion method for acoustic reflection data,” Geophysics 46 (11), 1559–1567 (1981).
J. K. Cohen and N. Bleistein, “Velocity inversion procedure for acoustic waves,” Geophysics 44 (6), 1077–1087 (1979).
W. Beydoun and A. Tarantola, “First Born and Rytov approximations: Modeling and inversion conditions in a canonical example,” J. Acoust. Soc. Am. 83, 1045–1055 (1988).
O. Ya. Voinov, V. I. Golubev, and I. B. Petrov, “Elastic imaging using multiprocessor computer systems,” CEUR Workshop Proc. 1787, 491–495 (2016).
G. McMechan, “Determination of source parameters by wavefield extrapolation,” Geophys. J. Int. 71, 613–628 (1982).
G. McMechan, “Migration by extrapolation of time-dependent boundary values,” Geophys. Prospect. 31, 413–420 (1983).
E. Baysal, D. Kosloff, and J. Sherwood, “Reverse time migration,” Geophysics 48, 1514–1524 (1983).
A. N. Danilin, “Detection of diffractors in complex acoustic media on the basis of the CSP-RTD method,” Vestn. Balt. Fed. Univ. Ser. Fiz.-Mat. 4, 143–147 (2015).
J. Virieux, H. Calandra, and R. É. Plessix, “A review of the spectral, pseudo-spectral, finite-difference and finite-element modeling techniques for geophysical imaging,” Geophys. Prospect. 59 (5), 794–813 (2011).
J. M. Carcione, C. Herman Gérard, and P. E. Kroode, “Y2K review article: Seismic modeling,” Rev. Literature Arts Am. 67 (4), 1304–1325 (2002).
E. Etienne, J. Chaljub, J. Virieux, et al., “An hp-adaptive discontinuous Galerkin finite-element method for 3D elastic wave modeling,” Geophys. J. Int. 183 (2), 941–962 (2010).
V. Hermann, M. Käser, and E. C. Castro, “Nonconforming hybrid meshes for efficient 2-D wave propagation using the discontinuous Galerkin method,” Geophys. J. Int. 184 (2), 746–758 (2011).
V. I. Golubev, I. B. Petrov, and N. I. Khokhlov, “Simulation of seismic processes inside the planet using the hybrid grid-characteristic method,” Math. Model. Comput. Simul. 7 (5), 439–445 (2015).
V. I. Golubev, I. B. Petrov, N. I. Khokhlov, and K. I. Shul’ts, “Numerical computation of wave propagation in fractured media by applying the grid-characteristic method on hexahedral meshes,” Comput. Math. Math. Phys. 55 (3), 509–518 (2015).
I. E. Kvasov, S. A. Pankratov, and I. B. Petrov, “Numerical simulation of seismic responses in multilayer geologic media by the grid-characteristic method,” Math. Models Comput. Simul. 3 (2), 196–204 (2011).
A. S. Kholodov and Ya. A. Kholodov, “Monotonicity criteria for difference schemes designed for hyperbolic equations,” Comput. Math. Math. Phys. 46 (9), 1560–1588 (2006).
I. B. Petrov and A. S. Kholodov, “Numerical study of some dynamic problems of the mechanics of a deformable rigid body by the mesh-characteristic method,” USSR Comput. Math. Math. Phys. 24 (3), 61–73 (1984).
Y. Luo, J. Tromp, B. Denel, and H. Calandra, “3D coupled acoustic-elastic migration with topography and bathymetry based on spectral-element and adjoint methods,” Geophysics 78 (4), 193–202 (2013).
V. I. Golubev, “Method for visualizing and interpreting results of full-wave seismic computations,” Tr. Mosk. Fiz.-Tekh. Inst. 6 (1), 154–161 (2014).
ACKNOWLEDGMENTS
This work was supported by the Russian Science Foundation, project no. 14-11-00263, at the Moscow Institute of Physics and Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Golubev, V.I., Voinov, O.Y. & Petrov, I.B. Seismic Imaging of Fractured Elastic Media on the Basis of the Grid-Characteristic Method. Comput. Math. and Math. Phys. 58, 1309–1315 (2018). https://doi.org/10.1134/S0965542518080080
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542518080080