Generalized Bremmer series with rational approximation for the scattering of waves in inhomogeneous media

The Bremmer series solution of the wave equation in generally inhomogeneous media, requires the introduction of pseudodifferential operators. In this paper, sparse matrix representations of these pseudodifferential operators are derived. The authors focus on designing sparse matrices, keeping the accuracy high at the cost of ignoring any critical scattering-angle phenomena. Such matrix representations follow from rational approximations of the vertical slowness and the transverse Laplace operator symbols, and of the vertical derivative, as they appear in the parabolic equation method. Sparse matrix representations lead to a fast algorithm. An optimization procedure is followed to minimize the errors, in the high-frequency limit, for a given discretization rate. The Bremmer series solver consists of three steps: directional decomposition into up- and downgoing waves, one-way propagation, and interaction of the counterpropagating constituents. Each of these steps is represented by a sparse matrix equation. ...

[1]  M. D. Collins.  A two‐way parabolic equation for elastic media , 1993 .

[2]  Gijsbertus J.M. Krijnen,et al.  New formulation of the beam propagation method based on the slowly varying envelope approximation , 1993 .

[3]  Charles Vassallo,et al.  Optical Waveguide Concepts , 1991 .

[4]  Jon F. Claerbout,et al.  Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure , 1970 .

[5]  M. D. Feit,et al.  Time-dependent propagation of high-energy laser beams through the atmosphere: II , 1978 .

[6]  R. Greene.  The rational approximation to the acoustic wave equation with bottom interaction , 1984 .

[7]  G. R. Hadley,et al.  Multistep method for wide-angle beam propagation. , 1992 .

[8]  David Middleton,et al.  Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces , 1962, Inf. Control..

[9]  Michael D. Collins,et al.  Higher-order Padé approximations for accurate and stable elastic parabolic equations with application to interface wave propagation , 1991 .

[10]  Reinhold Pregla,et al.  New beam-propagation algorithm based on the method of lines , 1991, Integrated Photonics Research.

[11]  Richard B. Evans,et al.  A two‐way parabolic equation for acoustic backscattering in the ocean , 1992 .

[12]  Fred D. Tappert,et al.  The parabolic approximation method , 1977 .

[13]  Michael Schoenberg,et al.  Finite-difference modeling of faults and fractures , 1995 .

[14]  Olav Holberg,et al.  COMPUTATIONAL ASPECTS OF THE CHOICE OF OPERATOR AND SAMPLING INTERVAL FOR NUMERICAL DIFFERENTIATION IN LARGE-SCALE SIMULATION OF WAVE PHENOMENA* , 1987 .

[15]  A. R. Mitchell,et al.  The Finite Difference Method in Partial Differential Equations , 1980 .

[16]  H.-P. Nolting,et al.  Results of benchmark tests for different numerical BPM algorithms , 1995 .

[17]  Maarten V. de Hoop,et al.  Generalization of the Bremmer coupling series , 1996 .

[18]  Paul L. Stoffa,et al.  Split-Step Fourier Migration , 1990 .

[19]  Evan K. Westwood,et al.  A higher-order energy-conserving parabolic equqation for range-dependent ocean depth, sound speed, and density , 1991 .

[20]  Michael Wolfe,et al.  J+ = J , 1994, ACM SIGPLAN Notices.

[21]  Roel Baets,et al.  Diversity in Beam Propagation Methods , 1993 .

[22]  G. R. Hadley,et al.  Transparent boundary condition for the beam propagation method , 1992 .

[23]  M. V. Hoop,et al.  Scalar space-time waves in their spectral-domain first- and second-order Thiele approximations , 1992 .

[24]  Piero Sguazzero,et al.  Migration of seismic data by phase-shift plus interpolation: Geophysics , 1984 .

[25]  O. Widlund.  On the Stability of Parabolic Difference Schemes , 1965 .

[26]  O. Holberg,et al.  TOWARDS OPTIMUM ONE‐WAY WAVE PROPAGATION1 , 1988 .

[27]  G. R. Hadley,et al.  Wide-angle beam propagation using Pade approximant operators. , 1992, Optics letters.

[28]  Michael D. Collins,et al.  Applications and time‐domain solution of higher‐order parabolic equations in underwater acoustics , 1989 .

[29]  R. G. Keys.  Absorbing boundary conditions for acoustic media , 1985 .

[30]  William H. Press,et al.  Numerical recipes , 1990 .

[31]  L. Trefethen,et al.  Wide-angle one-way wave equations. , 1988, The Journal of the Acoustical Society of America.

[32]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[33]  M. D. Collins.  Benchmark calculations for higher‐order parabolic equations , 1990 .

[34]  Robert W. Graves,et al.  Modeling acoustic waves with paraxial extrapolators , 1990 .

[35]  Björn Engquist,et al.  Absorbing boundary conditions for wave-equation migration , 1980 .

[36]  Dave Hale,et al.  3-D depth migration via McClellan transformations , 1990 .

[37]  R.M. Mersereau,et al.  The processing of hexagonally sampled two-dimensional signals , 1979, Proceedings of the IEEE.

[38]  Fourier finite-difference migration for steeply dipping reflectors with complex overburden1 , 1995 .

[39]  Ding Lee,et al.  PARABOLIC EQUATION DEVELOPMENT IN RECENT DECADE , 1995 .

[40]  Olivier J. F. Martin,et al.  Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape , 1994 .

[41]  Phase errors of finite-difference migration , 1987 .

[42]  Akihiro Maruta,et al.  Transparent boundary for the finite-element beam-propagation method. , 1993 .

[43]  P. Sen,et al.  Properties of a periodically stratified acoustic half‐space and its relation to a Biot fluid , 1983 .

[44]  Cécile DeWitt-Morette,et al.  Path integration in non-relativistic quantum mechanics , 1979 .

[45]  Myung W. Lee,et al.  Optimization of one-way wave equations. , 1985 .

[46]  Ru-Shan Wu,et al.  Wide-angle elastic wave one-way propagation in heterogeneous media and an elastic wave complex-screen method , 1994 .

[47]  L. Trefethen.  Group velocity in finite difference schemes , 1981 .