Generalization of the Bremmer coupling series

Journal of Mathematical Physics

Volumn 37, Issue 7, 1996, Pages 3246-3282

  De Hoop, Maarten V ^a,b  

^a SCHLUMBERGER CAMBRIDGE RESEARCH   (United Kingdom)

^b Department of Geology and Geological Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030559551     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531566     Document Type: Article

References (69)

1. J. F. Claerbout, "Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure," Geophysics 35, 407-418 (1970).
2. M. V. de Hoop and A. T. de Hoop, "Scalar space-time waves in their spectral domain first- and second-order Thiele approximations," Wave Motion 15, 229-265 (1992).
3. M. V. de Hoop and A. T. de Hoop, "Interface reflection of spherical acoustic waves in the first- and second-order rational parabolic approximations and their artifacts," J. Acoust. Soc. Am. 93, 22-35 (1993).
4. F. D. Tappert, "The parabolic approximation method," in Wave Propagation in Underwater Acoustics, edited by J. B. Keller and J. S. Papadakis, Lecture Notes in Physics No 70 (Springer-Verlag, New York, 1977), pp. 224-287.
5. R. R. Greene, "The rational approximation to the acoustic wave equation with bottom interaction," J. Acoust. Soc. Am. 76, 1764-1773 (1984).
6. D. J. Thomson and N. R. Chapman, "A wide-angle split-step algorithm for the parabolic equation," J. Acoust. Soc. Am. 74, 1848-1854 (1983).
7. J. van Roey, J. van der Donk, and P. E. Lagasse, "Beam-propagation method: analysis and assessment," J. Opt. Soc. Am. 71, 803-810 (1981).
8. D. Yevick, "A guide to electric field propagation techniques for guided wave optics," Optic. Quant. Electron. 26, S185-S197 (1994).
9. R. T. Seeley, "Complex powers of an elliptic operator," A. M. S. Proc. Symp. Pure Math. 10, 288-307 (1967).
10. R. T. Seeley, "The resolvent of an elliptic boundary problem," Am. J. Math. 91, 917-919 (1969).
11. L. Hörmander, The Analysis of Linear Partial Differential Operators III (Springer-Verlag, Berlin, 1983).
12. J. J. Duistermaat and V. W. Guilleman, "The spectrum of positive elliptic operators and periodic bicharacteristics," Inv. Math. 29, 39-79 (1975).
13. L. Fishman and J. J. McCoy, "Derivation and application of extended parabolic wave theories. I. The factorized Helmholtz equation," J. Math. Phys. 25, 285-296 (1984).
14. L. Fishman and J. J. McCoy, "Derivation and application of extended parabolic wave theories. II. Path integral representations," J. Math. Phys. 25, 297-308 (1984).
15. L. Fishman and J. J. McCoy, "Factorization, path integral representations, and the construction of direct and inverse wave propagation theories," IEEE Trans. Geosci. Remote Sens. GE-22, 682-692 (1984).
16. L. Fishman and J. J. McCoy, "A new class of propagation models based on a factorization of the Helmholtz equation," Geophys. J. R. Astron. Soc. 80, 439-461 (1985).
17. L. Fishman and J. J. McCoy, "Factorization and path integration of the Helmholtz equation: numerical algorithms," J. Acoust. Soc. Am. 81, 1355-1376 (1987).
18. L. Fishman, "Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics - The quadratic profile," J. Math. Phys. 33, 1887-1914 (1992).
19. L. Fishman, "One-way wave propagation methods in direct and inverse scalar wave propagation," Radio Sci. 28, 865-867 (1993).
20. L. Fishman and S. C. Wales, "Phase space methods and path integration: The analysis and computation of scalar wave equations," J. Comp. Appl. Math. 20, 219-238 (1987).
21. J. J. McCoy and L. N. Frazer, "Pseudodifferential operators, operator orderings, marching algorithms and path integrals for one-way equations," Wave Motion 9, 413-427 (1987).
22. V. H. Weston, "Factorization of the wave equation in higher dimensions," J. Math. Phys. 28, 1061-1068 (1987).
23. J. J. McCoy, L. Fishman, and L. N. Frazer, "Reflection and transmission at an interface separating transversely inhomogeneous acoustic half-spaces," Geophys. J. R. Astron. Soc. 85, 543-562 (1986).
24. J. Corones, "Bremmer series that correct parabolic approximations," J. Math. Anal. Appl. 50, 361-372 (1975).
25. M. D. Verweij and A. T. de Hoop, "Determination of seismic wave fields in arbitrarily continuously layered media using the modified Cagniard method," Geoph. J. Int. 103, 731-754 (1990).
26. R. F. Millar, "The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers," Radio Sci. 8, 785-796 (1973).
27. P. M. van den Berg, "Reflection by a grating: Rayleigh methods," J. Opt. Soc. Am. 71, 1224-1229 (1981).
28. R.-S. Wu, "Wide-angle elastic wave one-way propagation in heterogeneous media and an elastic wave complex-screen method," J. Geophys. Res. 99, 751-766 (1994).
29. A. J. Haines and M. V. de Hoop, "An invariant imbedding analysis of general wave scattering problems," accepted for publication J. Math. Phys. (1995).
30. L. Fishman and A. K. Gautesen "An exact, well-posed, one-way reformulation of the Helmholtz equation with application to direct and inverse wave propagation modeling," in New Perspectives on Problems in Classical and Quantum Physics, edited by A. W. Saenz and P. P. Delsanto (Gordon and Breach, Newark, 1996).
31. J. P. Corones, M. E. Davison, and R. J. Krueger, "Direct and inverse scattering in the time domain via invariant imbedding equations," J. Acoust. Soc. Am. 74, 1535-1541 (1983).
32. 3: II. The Green function approach to inverse scattering,ℝ Inv. Prob. 8, 919-947 (1992).
33. A. J. Berkhout, "Multi-dimensional linearized inversion and seismic migration," Geophysics 49, 1881-1895 (1984).
34. D. E. Lumley, "Angle-dependent reflectivity estimation," SEG Ann. Mtg. Expanded Abstr. 746-749 (1993).
35. D. V. Widder, The Laplace Transform (Princeton U.P., Princeton, NJ, 1946).
36. C. P. A. Wapenaar and A. J. Berkhout, Elastic Wave Field Extrapolation (Elsevier, Amsterdam, 1989).
37. W. Rudin, Functional Analysis (Tata McGraw-Hill, New Delhi, 1981).
38. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).
39. T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1984).
40. M. E. Taylor, Pseudodifferential Operators (Princeton U. P., Princeton, 1981).
41. J. J. Duistermaat, Fourier Integral Operators (Lecture Notes Courant Institute of Mathematical Sciences, New York, 1973).
42. H. Kumano-go, Pseudo-differential Operators (MIT Press, Cambridge, 1981).
43. B. L. N. Kennett, "Wavenumber and wave type coupling in laterally heterogeneous media," Geophys. J. R. Astron. Soc. 87, 313-331 (1986).
44. D. M. Pai, "A new solution method for wave equations in inhomogeneous media," Geophysics 50, 1541-1547 (1985).
45. D. M. Pai, "Generalized f-k (frequency-wavenumber) migration in arbitrarily varying media," Geophysics 53, 1547-1555 (1988).
46. L. Hörmander, The Analysis of Linear Partial Differential Operators I (Springer-Verlag, Berlin, 1983).
47. P. Braam (University of Oxford, personal communication).
48. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
49. C. De Witt-Morette, A. Maheshwari, and B. Nelson, "Path integration in non-relativistic quantum mechanics," Phys. Rep. 50, 255-372 (1979).
50. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
51. L. Fishman, "Phase space factorization and functional integral methods in direct and inverse scattering-symbol analysis," in Wave Phenomena, edited by L. Lam and H. C. Morris (Springer-Verlag, New York, 1988), pp. 76-85.
52. J. Gazdag and P. Sguazzero, "Migration of seismic data by phase shift plus interpolation," Geophysics 49, 124-131 (1984).
53. P. L. Stoffa, J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, "Split-step Fourier migration," Geophysics 55, 410-421 (1990).
54. J. B. Keller and D. W. McLaughlin, "The Feynman integral," Ann. Math. Mon. 82, 451-465 (1975).
55. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics, translation from the Russian by J. Niederle and J. Tolar (Reidel, Dordrecht, 1981).
56. M. Suzuki, "General theory of higher-order decomposition of exponential operators and symplectic propagators," preprint, Dept. of Physics, University of Tokyo (1992).
57. H. Bremmer, "Geometrisch-optische benadering van de golfvergelijking," Handel. Ned. Nat. en Geneeskd. Congr., 88-91 (1939) (in Dutch).
58. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
59. A. T. de Hoop, "Acoustic radiation from an impulsive point source in a continuously layered fluid - An analysis based on the Cagniard method," J. Acoust. Soc. Am. 88, 2376-2388 (1990).
60. G. M. Wing, "Invariant imbedding and generalizations of the WKB method and the Bremmer series," J. Math. Anal. Appl. 48, 400-422 (1974).
61. F. V. Atkinson, "Wave propagation and Bremmer series," J. Math. Anal. Appl. 1, 255-276 (1960).
62. M. V. de Hoop and A. T. de Hoop, "Elastic wave up/down decomposition in inhomogeneous and anisotropic media: an operator approach and its approximations," Wave Motion 20, 57-82 (1994).
63. M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Springer-Verlag, Berlin, 1987).
64. R. Baets, J. Willems, and R. Haes, "Diversity in beam propagation methods." ECIO, 13.1-13.3 (1993).
65. L. Fishman, "Numerical solutions of the Helmholtz, Weyl composition equation in ocean seismo-acoustics," in 13th International Congress on Acoustics, edited by P. Pravica (Sava Centar, Belgrade, 1989), Vol. 5, pp. 149.
66. R. P. Ratowsky, J. A. Fleck, Jr., and M. D. Feit, "Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction," J. Opt. Soc. Am. A 9, 265-273 (1992).
67. M. J. N. van Stralen, M. V. de Hoop, and H. Blok, "Numerical implementation of the Bremmer coupling series," IPR Expanded Abstracts (1996).
68. B. Davies and B. Martin, "Numerical inversion of the Laplace transform; a survey and comparison of methods," J. Comp. Phys. 33, 1-32 (1979).
69. A. Papoulis, "A new method of inversion of the Laplace Transform," Q. Appl. Math. 14, 405-414 (1957).