Propagation Modeling

Ocean acoustic propagation models estimate how acoustic energy travels through the ocean after being generated by an underwater sound source. The output of these models can be used to help understand and extract information about ocean acoustic phenomena and to predict potential impacts to animals from noise-generating activities. However, all practical acoustic models make some kind of simplifying assumptions about how sound travels. It is important to note that these assumptions are not necessarily true for all circumstances, and the quality of an acoustic model’s output is only as good as the input data used.

The simplest acoustic models are known as “power law” models, and include spherical, cylindrical, and so-called “practical” spreading models. Despite their simplicity, they still have useful applications, and the reader is encouraged to review the Advanced Topic on Cylindrical vs. Spherical Spreading before proceeding further: Cylindrical vs. Spherical Spreading .

This Advanced Topic will describe more complex propagation models, based on a webinar presented by Dr. Aaron Thode (2022 Webinar Acoustic Propagation Modeling ).

Spherical spreading, the simplest possible propagation model, describes the decrease in sound level whenever a sound wave propagates away from a source in deep water a good distance away (more than ten wavelengths) from the ocean surface and bottom. The model assumes that the speed of sound in water is the same at all locations, with the result that the sound spreads as a spherical surface, with no interactions with the sea surface or seafloor. The sound amplitude decays in proportion to the inverse distance from the source, and the sound intensity decays in proportion to the square of the inverse distance from the source. .

Once sound reaches the sea surface or seafloor, its propagation gradually transitions from spherical to cylindrical spreading, which assumes propagation is restricted at the top and bottom by boundaries (the sea surface and seafloor) and that the ocean floor is flat. As a result, sound energy spreads as a cylinder, with all acoustic energy completely reflecting from the ocean surface and floor. Sound decays less quickly with increasing horizontal range under a cylindrical spreading model than with a spherical spreading model. Models also exist that generate transmission losses midway between cylindrical and spherical spreading models and are dubbed “practical spreading” models. These models attempt to account for energy lost into the seafloor.

Complex models require more details about the propagating sound (specifically its frequency, and the depth of the source and receiver) and the ocean environment (including the water depth, bottom composition, bathymetry, sound speed profile, and surface roughness). Several online databases provide information on the ocean environment. Sediment data that describe the sea floor for the United States coastal regions can be found through the U.S. Geological Survey (e.g., ; ). The General Bathymetric Chart of the Ocean (GEBCO) is a global terrain model, providing elevation in meters on a grid. The World Ocean Atlas is a collection of profile data of temperature, salinity, and other parameters. The HYCOM model is a sophisticated data assimilation model that predicts the sound speed profile at most locations in the ocean at a specific time or date in the past.

The choice of which model to use depends on the wavelength of the sound relative to the water depth. This determines whether propagation is occurring in an acoustically deep ocean, where knowledge of the sound wavelength is not required. Alternatively, in an acoustically shallow ocean, knowledge of the wavelength is crucial for accurate modeling. A general rule of thumb to differentiate acoustically deep from shallow water is to define a “differentiating depth” that is ten acoustic wavelengths deep:


Differentiating Depth = 10λ

Where λ = wavelength (m)

Differentiating Depth = 10 c/f

f=frequency (Hz, cycles/s)


If the differentiating depth is less than the actual water depth, the ocean is considered an acoustically deep-water scenario, and the frequency of the sound has less effect on propagation. However, if the differentiating depth is greater than the water depth, then the frequency of the sound cannot be ignored, and the modeling results strongly depend on frequency. For example, for a 1.5 kHz sound, the water depth needs to be greater than 10 m to be considered an acoustically deep ocean:


Differentiating Depth = 10 c/f

Differentiating Depth = 10 x 1,500 m/s / 1,500 Hz = 10 m


Note it is possible for an ocean to be acoustically deep at a high frequency, but the same ocean can be considered shallow at a lower frequency.

Sophisticated models estimate the effects of reflection, refraction, and scattering. The most accurate estimation requires choosing a model that best fits the modeling scenario. This includes determining whether the ocean is acoustically deep or shallow. The Ocean Acoustics Library provides acoustic modeling software, documentation, and data. The three most common models are ray tracing, normal mode, and parabolic equation models. Each approach uses a simplified version of the full wave equation to increase computational speed, at the cost of requiring certain assumptions about the propagation scenario.

Ray tracing models generate rays that predict travel time and acoustic amplitude. The greater the number of rays created, the more accurate the results. Eigenrays are rays that connect the source and receiver. The key assumption of ray tracing is that the acoustic wavelength is much smaller than the ocean depth or any bathymetric features. Therefore, ray tracing is a good choice for acoustically deep water.

Visualization of a ray tracing model in an environment with a constant sound speed profile. Figure courtesy of Kevin Souhrada, Scripps Institution of Oceanography.


A great advantage of ray tracing is that it is easy to model situations where the water depth, sound speed profile, or bottom composition change with range. A disadvantage is that ray tracing doesn’t capture frequency-dependent effects that are important for lower-frequency sound propagation. The most common ray tracing software is BELLHOP created by Dr. Michael Porter (, which also has some ability to model acoustic amplitude as well as travel time.

“Normal mode models” are appropriate and efficient for acoustically shallow waters, under circumstances where the source and receiver are horizontally separated by distances greater than a few water depths. For example, in an ocean that is 100 m deep, normal mode models are valid for horizontal separations greater than 500 m. Under such circumstances, the only rays propagating over those distances are those that propagate at certain special angles with respect to the horizontal. At these angles, the ray’s up-and-down reflections interfere constructively, or reinforce one another, and the resulting propagating field can be mathematically described as a set of normal modes. The total field radiating from an acoustic source can then be interpreted as the sum of these normal modes, where the contribution of each mode to the acoustic field depends on the source and receiver depths.

Visualization of normal mode modeling. The source, an orange point at c(z), creates rays that can be interpreted as modes propagating horizontally (mode 1, mode 2, mode n). Adapted from Kuperman, W., Roux, P. (2007). Underwater Acoustics. In: Rossing, T. (eds) Springer Handbook of Acoustics. Springer Handbooks. Springer, New York, NY.


The key advantage of normal mode modeling is that it is extremely fast, particularly in acoustically shallow water. Two disadvantages are that the modeling range must be greater than a few water depths away from the source, and incorporating range dependent effects in the oceanography or seafloor is challenging. Furthermore, the execution of these models slows down dramatically at higher frequencies and deeper water (as defined by the differentiating depth). Kraken by Dr. Michael Porter is a commonly-used normal mode software program (

Parabolic equation models, or PE models, use a simplified version of the wave equation, to permit the calculation of the sound field in range-dependent environments. Some PE models can even model 3-D propagation, where the sound field propagates in range, depth, and azimuth.

Visualization of parabolic equation modeling. The color bar shows transmission loss (TL) in terms of decibels (dB). Figure from Lynch, J.F., G.G. Gawarkiewicz, Y.-T. Lin, T.F. Duda, and A.E. Newhall. 2018. Impacts of ocean warming on acoustic propagation over continental shelf and slope regions. Oceanography 31(2):174–181,


The key assumption of PE models is that most of the acoustic energy must be traveling in a relatively horizontal direction, although modern PE models can accurately estimate acoustic propagation along steeper angles. Its biggest advantage is its fast computation time in complex range-dependent environments. However, PE models share similar disadvantages with normal mode models, in that their results are only accurate at ranges beyond a few water depths, and their execution speed slows down at higher frequencies and/or deep waters. In addition, they can have issues with situations where energy propagates at near-vertical angles. RAM-GEO by Dr. Michael Collins is a popular PE software program (

While at least one out of these three models will cover most ocean acoustic propagation scenarios, most standard propagation models will not handle situations where backpropagating or backscattered energy is a significant part of the acoustic field. Such situations include extremely steep bathymetries, such as submarine canyon walls or seamounts.

Propagation modeling is always advancing. More complex models are being developed that refine three-dimensional propagation (Lin et al. 2019) and which specialize in very long-range propagation scenarios. Other models specialize in propagation over seafloors with complex structure or ice-covered ocean surfaces. Two examples of these specialized approaches are wave number integration and finite element modeling. Despite these sophisticated modeling options, simple models such as spherical and cylindrical models continue to remain valuable “common sense” checks.


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