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An Application of the Level Set Method to Underwater Acoustic Propagation

Published online by Cambridge University Press:  20 August 2015

Sheri L. Martinelli
Sheri L. Martinelli*
Affiliation:
Torpedo Systems Department, Naval Undersea Warfare Center, 1176 Howell Street, Newport, Rhode Island 02841, USA Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA
*
*Corresponding author.Email address:[email protected]
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Abstract

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.

Keywords

65M0665M2265M2543.30.Gv43.30.ZkGeometric acousticslevel set methodhigh-frequency acoustic propagationWENO method
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1359 - 1391
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Osher, S., Cheng, L.-T., Kang, M., Shim, H., and Tsai, Y.-H., Geometric optics in a phase-space-based level set and Eulerian framework, J. Comput. Phys., vol. 179, pp. 622–648, 2002.CrossRefGoogle Scholar
[2]Smith, K., Brown, M., and Tappert, F., Ray chaos in underwater acoustics, J. Acoust. Soc. Amer., vol. 91, pp. 1939–49, 1992.Google Scholar
[3]Collins, M. and Kuperman, W., Overcoming ray chaos, J. Acoust. Soc. Amer., vol. 95, pp.3167–70, 1994.Google Scholar
[4]Godin, O., Restless rays, steady wave fronts, J. Acoust. Soc. Amer., vol. 122, pp. 3353–63, 2007.CrossRefGoogle ScholarPubMed
[5]Osher, S. and Sethian, J., Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., vol. 79, pp. 12–49, 1988.CrossRefGoogle Scholar
[6]Benamou, J., An introduction to Eulerian geometric optics (1992-2002), Jour. Sci. Comp., vol. 19, no. 1–3, pp. 63–93, 2003.Google Scholar
[7]Engquist, B., Runborg, O., and Tornberg, A., High frequency wave propagation by the segment projection method, J. Comput. Phys., vol. 178, pp. 373–390, 2002.Google Scholar
[8]Qian, J., Cheng, L.-T., and Osher, S., A level set-based Eulerian approach for anisotropic wave propagation, Wave Motion, vol. 37, no. 4, pp. 365–379, 2003.Google Scholar
[9]Cheng, L.-T., Osher, S., Kang, M., Shim, H., and Tsai, Y.-H., Reflection in a level set framework for geometric optics, Computer Modeling in Engineering and Sciences, vol. 5, pp. 347–360, 2004.Google Scholar
[10]Cockburn, B., Qian, J., Reitich, F., and Wang, J., An accurate spectral/discontinuous finite-element formulation of a phase-space-based level set approach to geometrical optics, J. Comput. Phys., vol. 208, pp. 175–195, 2005.Google Scholar
[11]Qian, J. and Leung, S., A level set based Eulerian method for paraxial multivalued travel-times, J. Comput. Phys., vol. 197, pp. 711–736, 2004.Google Scholar
[12]Qian, J. and Leung, S., A local level set method for paraxial geometrical optics, SIAM J. Sci. Comput., vol. 28, no. 1, pp. 206–223, 2006.Google Scholar
[13]Engquist, B. and Runborg, O., Computational high frequency wave propagation, Acta Numerica, pp. 181–266, 2003.Google Scholar
[14]Crandall, M. and Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., vol. 277, pp. 1–42, 1983.Google Scholar
[15]Ogilvy, J., Wave scattering from rough surfaces, Rep. Prog. Phys., vol. 50, no. 4, pp. 1553–1608, 1987.CrossRefGoogle Scholar
[16]Jensen, F., Kuperman, W., Porter, M., and Schmidt, H., Computational Ocean Acoustics. AIP Series in Modern Acoustics and Signal Processing, AIP Press, 1994.Google Scholar
[17]Jin, S., Liu, H., Osher, S., and Tsai, Y.-H., Computing multivalued physical observables for the semiclassical limit of the schrdinger equation, J. Comput. Phys., vol. 205, pp. 222–241, 2005.Google Scholar
[18]Jin, S., Liu, H., Osher, S., and Tsai, R., Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems, J. Comput. Phys., vol. 210, pp. 497–518, 2005.CrossRefGoogle Scholar
[19]Liu, H., Osher, S., and Tsai, R., Multi-valued solution and level set methods in computational high frequency wave propagation, Communications in Computational Physics, vol. 1, no. 5, pp. 765–804, 2006.Google Scholar
[20]Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, vol. 153 of Applied Mathematical Sciences. Springer, 2003.Google Scholar
[21]Shu, C., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hy-perbolic conservation laws, ICASE Report 97-65, NASA/CR-97-206253, November 1997.Google Scholar
[22]Gottlieb, S. and Shu, C., Total variation diminishing Runge-Kutta schemes, Mathematics of Computation, vol. 67, no. 221, pp. 73–85, 1998.Google Scholar
[23]Cheng, L.-T., Efficient level set methods for constructing wavefronts in three spatial dimensions, J. Comput. Phys., vol. 226, pp. 2250–2270, 2007.CrossRefGoogle Scholar
[24]Porter, M. and Bucker, H., Gaussian beam tracing for computing ocean acoustic fields, J. Acoust. Soc. Amer., vol. 82, pp. 1349–59, 1987.CrossRefGoogle Scholar
[25]Weinberg, H. and Keenan, R., Gaussian ray bundles for modeling high-frequency propagation loss under shallow-water conditions, J. Acoust. Soc. Amer., vol. 100, pp. 1421–31, 1996.Google Scholar
[26]Kinsler, L., Frey, A., Coppens, A., and Sanders, J., Fundamentals of Acoustics. John Wiley & Sons, third ed., 1982.Google Scholar
[27]Foreman, T., An exact ray theoretical formulation of the helmholtz equation, J. Acoust. Soc. Amer., vol. 86, pp. 234–246, 1989.Google Scholar
[28]Bowlin, J., Spiesberger, J., Duda, T., and Freitag, L., Ocean acoustical ray-tracing software RAY, Tech. Rept. WHOI-93-10, Woods Hole Oceanographic Institute, October 1992.CrossRefGoogle Scholar
[29]Harten, A., Engquist, B., Osher, S., and Chakravarthy, S., Uniformly high-order accurate essentially non-oscillatory schemes III, J. Comput. Phys., vol. 71, pp. 231–303, 1987.CrossRefGoogle Scholar
[30]Shu, C. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys., vol. 77, pp. 439–471, 1988.CrossRefGoogle Scholar
[31]Shu, C. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., vol. 83, pp. 32–78, 1989.Google Scholar
[32]Liu, X.-D., Osher, S., and Chan, T., Weighted essentially nonoscillatory schemes, J. Comput. Phys., vol. 115, pp. 200–212, 1994.Google Scholar
[33]Jiang, G. and Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., vol. 126, pp. 202–228, 1996.CrossRefGoogle Scholar