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**Generalization of the Bremmer coupling series.**
*(English)*
Zbl 0879.76088

The one-way “parabolic” approximation to the wave operator arises in the decomposition of the acoustic wave field into constituents with a preferred direction, where two constituents satisfy coupled partial differential equations of a specific type. The direction of preference, the “vertical” direction, arises from the medium’s variations. This paper discusses the mathematical theory underlying the decomposition technique, which is based on the calculus of pseudo-differential operators. An operator formalism is developed to expand the acoustic wave field in a multi-dimensionally smoothly varying medium, generated by a source localized in space and time, into a sum of constituents each of which can be interpreted as a wave that travels up and down a definite number of times with respect to the direction of preference.

This expansion is a generalization of the Bremmer series which yields an expansion of the acoustic wavefield in terms of the spatial derivatives of medium properties. In the proposed expansion, the leading term is a high-frequency (Rytov-like) approximation to the wavefield. The Bremmer coupling series essentially recomposes the solutions of the system of one-way wave equations into a two-way solution, and, as such, it connects the one-way wave formulation of scattering with the Dirichlet-to-Neumann map formulation.

Both the existence and convergence (in the weak sense) of the expansion are discussed. The operator associated with the corresponding generalized vertical slowness generates a full one-way wave operator for the media under consideration. In addition, a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents, are derived.

This expansion is a generalization of the Bremmer series which yields an expansion of the acoustic wavefield in terms of the spatial derivatives of medium properties. In the proposed expansion, the leading term is a high-frequency (Rytov-like) approximation to the wavefield. The Bremmer coupling series essentially recomposes the solutions of the system of one-way wave equations into a two-way solution, and, as such, it connects the one-way wave formulation of scattering with the Dirichlet-to-Neumann map formulation.

Both the existence and convergence (in the weak sense) of the expansion are discussed. The operator associated with the corresponding generalized vertical slowness generates a full one-way wave operator for the media under consideration. In addition, a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents, are derived.

Reviewer: T.Wang (Cherry Hill)

### MSC:

76Q05 | Hydro- and aero-acoustics |

35L45 | Initial value problems for first-order hyperbolic systems |

35S10 | Initial value problems for PDEs with pseudodifferential operators |

### Keywords:

scattering problem; one-way “parabolic” approximation; decomposition of acoustic wave field; high-frequency approximation; wave operator; preferred direction; existence; convergence; interaction operator
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\textit{M. V. de Hoop}, J. Math. Phys. 37, No. 7, 3246--3282 (1996; Zbl 0879.76088)

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### References:

[1] | DOI: 10.1190/1.1440103 |

[2] | DOI: 10.1016/0165-2125(92)90010-Y · Zbl 0786.35085 |

[3] | DOI: 10.1121/1.406845 |

[4] | DOI: 10.1121/1.391561 |

[5] | DOI: 10.1121/1.390272 · Zbl 0545.76083 |

[6] | DOI: 10.1364/JOSA.71.000803 |

[7] | DOI: 10.1007/BF00384672 |

[8] | DOI: 10.1090/pspum/010/0237943 |

[9] | Seeley R. T., Am. J. Math. 91 pp 917– (1969) |

[10] | DOI: 10.1007/BF01405172 · Zbl 0307.35071 |

[11] | DOI: 10.1063/1.526149 · Zbl 0551.76069 |

[12] | DOI: 10.1063/1.526150 · Zbl 0551.76070 |

[13] | DOI: 10.1109/TGRS.1984.6499189 |

[14] | DOI: 10.1111/j.1365-246X.1985.tb05103.x |

[15] | DOI: 10.1121/1.394542 |

[16] | DOI: 10.1063/1.529666 · Zbl 0755.76077 |

[17] | DOI: 10.1029/93RS01632 |

[18] | DOI: 10.1016/0377-0427(87)90139-7 · Zbl 0637.65112 |

[19] | DOI: 10.1016/0165-2125(87)90030-8 · Zbl 0649.35080 |

[20] | DOI: 10.1063/1.527547 · Zbl 0636.35044 |

[21] | DOI: 10.1111/j.1365-246X.1986.tb04531.x |

[22] | DOI: 10.1016/0022-247X(75)90029-3 · Zbl 0313.35020 |

[23] | DOI: 10.1111/j.1365-246X.1990.tb05684.x |

[24] | DOI: 10.1029/RS008i008p00785 |

[25] | DOI: 10.1364/JOSA.71.001224 |

[26] | DOI: 10.1029/93JB02518 |

[27] | DOI: 10.1121/1.390166 · Zbl 0525.73028 |

[28] | DOI: 10.1088/0266-5611/8/6/009 · Zbl 0798.35161 |

[29] | DOI: 10.1190/1.1441601 |

[30] | DOI: 10.1111/j.1365-246X.1986.tb06626.x |

[31] | DOI: 10.1190/1.1441844 |

[32] | DOI: 10.1190/1.1442436 |

[33] | DOI: 10.1016/0370-1573(79)90083-8 |

[34] | DOI: 10.1190/1.1441643 |

[35] | DOI: 10.1190/1.1442850 |

[36] | DOI: 10.2307/2319736 · Zbl 0304.35002 |

[37] | DOI: 10.1121/1.400080 |

[38] | DOI: 10.1016/0022-247X(74)90167-X · Zbl 0296.34006 |

[39] | DOI: 10.1016/0022-247X(60)90001-9 · Zbl 0103.05502 |

[40] | DOI: 10.1016/0165-2125(94)90032-9 · Zbl 0926.74044 |

[41] | DOI: 10.1364/JOSAA.9.000265 |

[42] | DOI: 10.1016/0021-9991(79)90025-1 · Zbl 0416.65077 |

[43] | Papoulis A., Q. Appl. Math. 14 pp 405– (1957) · Zbl 0077.11402 |

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