An approximate form of three-dimensional Cartesian split-step marching solution for the acoustic parabolic equation is derived in order to obtain the efficient algorithm for sound propagation in the three-dimensional ocean. The operator splitting method is used to split the full exponential operator into three exponential operators for depth, cross-range, and the combination of the two. The first two terms are implemented with the split-step Padé algorithm and the final term is implemented with the Taylor series expansion in depth and cross-range operator. In order to resolve the divergence of Taylor approximation out of the interval of convergence, the rational filter of rectangular type is applied to the depth and cross-range operator. The use of the filter improves the stability of the solution but requires extra numerical burdens. Numerical issues involving the accuracy, efficiency, and stability of the proposed model are discussed and illustrated in an ocean wedge environment.
References
1.
Aki
, K.
, and Richards
, P. G.
(2002
). Quantitative Seismology
, 2nd ed. (University Science Books
, Sausalito, CA
), Chap. 6.1.2.
Brekhovskikh
, L. M.
, and Lysanov
, Y. P.
(1991
). Fundamentals of Ocean Acoustics
, 2nd ed. (Springer
, New York
), Chap. 4.3.
Cederberg
, R. J.
, and Collins
, M. D.
(1997
). “Application of an improved self-starter to geoacoustic inversion
,” IEEE J. Ocean. Eng.
22
, 102
–109
.4.
Cederberg
, R. J.
, Collins
, M. D.
, Schmidt
, H.
, and Seigmann
, W. L.
(1997
). “Rational operators for filtering
,” J. Acoust. Soc. Am.
101
, 2518
–2523
.5.
Collins
, M. D.
(1989
). “A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom
,” J. Acoust. Soc. Am.
86
, 1459
–1464
.6.
Collins
, M. D.
(1991
). “Higher-order Padé approximations for accurate and stable elastic parabolic equations with application to interface wave propagation
,” J. Acoust. Soc. Am.
89
, 1050
–1057
.7.
Collins
, M. D.
(1993
). “A split-step Padé for the parabolic equation method
,” J. Acoust. Soc. Am.
93
, 1736
–1742
.8.
Collins
, M. D.
(1999
). “The stabilized self-starter
,” J. Acoust. Soc. Am.
106
, 1724
–1726
.9.
Deane
, G. B.
, and Buckingham
, M. J.
(1993
). “An analysis of the three-dimensional sound field in a penetrable wedge with a stratified fluid or elastic basement
,” J. Acoust. Soc. Am.
93
, 1319
–1328
.10.
Geiger
, H. D.
, and Daley
, P. F.
(2003
). “Finite difference modelling of the full acoustic wave equation in Matlab
,” CREWES Research Report
15
, 1
–9
.11.
Hsieh
, L.
, Chen
, C.
, Yuan
, M.
, and Lin
, Y.
(2007
). “Azimuthal limitation in 3D PE approximation for underwater acoustic propagation
,” J. Comput. Acoust.
15
, 221
–233
.12.
Jensen
, F. B.
, Kuperman
, W. A.
, Porter
, M. B.
, and Schmidt
, H.
(2011
). Computational Ocean Acoustics
, 2nd ed. (Springer
, New York
), Chap. 6.13.
Kim
, T.
(2003
). “Numerical observation of the statistics of acoustic wave propagation through a shallow water environment
,” M.D. thesis, Seoul National University
.14.
Lee
, K.
, and Seong
, W.
(2006a
). “Full 3D one-way parabolic equation algorithm
,” J. Acoust. Soc. Kr.
25
, 67
–68
(in Korean).15.
Lee
, K.
, and Seong
, W.
(2006b
). “The perfectly matched layer applied to the split-step Padé PE solver in an ocean waveguide
,” J. Acoust. Soc. Kr.
25
, 131
–136
.16.
Lee
, K.
, Seong
, W.
, and Na
, Y.
(2019
). “A split-step Padé solver for the three-dimensional Cartesian acoustic parabolic equation in the stair-step representation of ocean environment
,” J. Acoust. Soc. Am.
146
(3
), 2050
–2057
.17.
Lin
, Y.
, Collis
, J. M.
, and Duda
, T. F.
(2012
). “A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximations
,” J. Acoust. Soc. Am.
132
, EL364
–EL370
.18.
Lin
, Y.
, and Duda
, T. F.
(2012
). “A higher-order split-step Fourier parabolic-equation sound propagation solution scheme
,” J. Acoust. Soc. Am.
132
, EL61
–EL67
.19.
Lin
, Y.
, Duda
, T. F.
, and Newhall
, A. E.
(2013
). “Three-dimensional sound propagation models using the parabolic-equation approximation and the split-step Fourier method
,” J. Comput. Acoust.
21
, 1250018
.20.
Milinazzo
, F. A.
, Zala
, C. A.
, and Brooke
, G. H.
(1997
). “Rational square-root approximations for parabolic equation algorithms
,” J. Acoust. Soc. Am.
101
, 760
–766
.21.
Petrov
, P.
(2018
). “Analytic solution for the penetrable wedge shaped ocean based on the G. Deane and M. Buckingham paper (JASA 1993)
,” available at http://oalib.hlsresearch.com/ThreeD/Petrov/ (Last viewed 9/5/2019).22.
Smith
, K. B.
(2001
). “Convergence, stability, and variability of shallow water acoustic predictions using a split-step Fourier parabolic equation model
,” J. Comput. Acoust.
9
, 243
–285
.23.
Sturm
, F.
(2005
). “Numerical study of broadband sound pulse propagation in three-dimensional oceanic waveguides
,” J. Acoust. Soc. Am.
117
, 1058
–1079
.24.
Sturm
, F.
(2016
). “Leading-order cross term correction of three-dimensional parabolic equation models
,” J. Acoust. Soc. Am.
139
, 263
–270
.25.
Sturm
, F.
, and Fawcett
, J. A.
(2003
). “On the use of higher-order azimuthal schemes in 3-D PE modeling
,” J. Acoust. Soc. Am.
113
, 3134
–3145
.26.
Westwood
, E. K.
(1992
). “Broadband modeling of the three-dimensional penetrable wedge
,” J. Acoust. Soc. Am.
92
, 2212
–2222
.27.
Yevick
, D.
, and Thomson
, D.
(1997
). “A hybrid split-step/finite-difference PE algorithm for variable-density media
,” J. Acoust. Soc. Am.
101
, 1328
–1335
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2019
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