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and Benchmark (Analytic) Solutions

The term 'Three-Dimensional' is used here to refer to or imply the effects of horizontal refraction, e.g., refraction in the lat./lon. plane for a cartesian coordinate system. Although the standard acoustic models can produce full 3D fields by assembling slices in the range-depth plane for different bearing angles, such models do not generally allow for refraction out of each range-depth plane. In spherical coodinates for the globe, we are obviously talking about propagation that deviates from great-circle paths.

When does horizontal refraction matter? Clearly, the stronger the horizontal gradients in the environment, the greater the likelihood that horizontal refraction will matter. Canyons, fjords, and seamounts are examples of bathymetric features with stong gradients. Nonlinear internal waves with particularly short spatial scales (e.g. 1 km) may similarly provide oceangraphic features with strong gradients. The refractive effects tend to accumulate with increasing range so that a continental slope may have no significant effect at a few km in range, but strong effects at tens of kms.

However, an equally important issue is how the signal is observed. One tends to think about transmission loss plots, but those plots represent a jumble of arrivals where differences in the individual components are obscured. In contrast, if the signal is split into parts by observing it through a beamformer, then the horizontal refraction may become a huge effect. Similar comments apply to a signal observed in the time domain where the echoes are distinguished in time (rather than angle). Finally spectrograms, that result when the signial is passed through a sound prism, may also reveal more clearly the 3D effects.

Analytic solution for the penetrable wedge shaped ocean and documentation based on the Modal PE approach of Petrov and Sturm (JASA 2016).
(P. Petrov/V.I.Il'chev Pacific Oceanological Inst. and F. Sturm/École Centrale de Lyon)

Analytic solution for the penetrable wedge shaped ocean based on the G. Deane and M. Buckinham paper (JASA 1993)
(P. Petrov/V.I.Il'chev Pacific Oceanological Inst.)

Analytic solution for the elastic penetrable wedge shaped ocean. This is a mod of the Petrov code above to handle elasticity. (2017)
(唐骏 Jun Tang, College of Underwater Acoustic Engineering, Harbin Engineering University)

BELLHOP3D ray/beam tracing code (part of the Acoustics Toolbox) and
KRAKEN3D adiabatic normal mode code (see FIELD3D in the Acoustics Toolbox)

Source code (in Acoustics Toolbox) (M. Porter)

HWT_3D_mm --- Huygens Wavefront Tracing in 3D for moving media (2013)
(Nikolay Zabotin, Oleg A. Godin, and Liudmila Ye. Zabotina, National Oceanic and Atmospheric Administration)

TRACEO3D: Beam tracing code for 3D (Orlando Camargo Rodríguez and Rogério de Moraes Calazan, SiPLAB/LARSyS, Univ. of Algarve, Portugal) (2018)

USML: Under Sea Modeling Library with WaveQ3D beam tracing code for 3D with eigenray capability(S. Reilly, Univ. Rhode Island and AEgis Technologies Group) (2015)