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Kirchoff-ray Mode Backscatter Model

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What is it?

Models and measures of acoustic scattering by zooplankton, fish and their swimbladders are continuously evolving (see table below). Zooplankton have been modeled as fluid-filled spheres and bent, fluid-filled cylinders. Early swimbladder models were based on gas-filled spherical or spheroid bubbles, and gas-filled cylinders. The fish body was first modeled as a gas-filled sphere which ignored the shape of the animal. Arrays of point scatterers have also been used to model the fish body form. More anatomically correct models use fluid-filled cylinders for zooplankton, or a combination of gas- and fluid-filled cylinders for fish. Empirical scattering models of fish and zooplankton are derived using echo measurements and known lengths of caged, tethered, or free ranging organisms. Statistical models of fish backscatter assume a theoretical or tabulate an empirical probability density function (PDF) of scattering amplitudes within an insonified beam.

Organism or Structure
Geometric Form or Measurement
Zooplankton Fluid-filled sphere; Fluid-filled cylinder; Bent fluid-filled cylinder
Fish Body Gas-filled sphere; Array of point scatterers
Fish Swimbladder Gas-filled spherical bubble; Gas-filled spheroid bubble; Gas-filled cylinder
Whole Fish Gas-filled swimbladder; Gas and fluid-filled cylinders
Empirical Models Literature review; Caged; Tethered; In situ; Statistical

Simple geometric shapes (e.g. sphere) regularly used in acoustic modeling efforts do not realistically represent fish body and swimbladder anatomy. To illustrate by example, here is a line drawing and corresponding lateral radiograph of an Atlantic cod (Gadus morhua).

Image - Atlantic Cod Image - Atlantic Cod radiograph
View other species in the Radiograph Gallery

The Kirchhoff-ray mode model represents the culmination of several backscatter modeling efforts. Foote (1985) and Foote and Traynor (1988) used the Helmholtz-Kirchhoff integral to develop an accurate and elaborate method to estimate backscattered sound from fish. This approach was simplified by Clay (1991; 1992) who incorporated Stanton's (1989) finite bent cylinder equation and fluid- or gas-filled cylinders to model fish backscatter. Clay and Horne (1994) combined these approaches to model backscatter by representing the fish body as a contiguous set of fluid-filled cylinders that surround a set of gas-filled cylinders representing the swimbladder.

Image - fishbody tracing Using radiographs like the cod image above, lateral (i.e. side) and dorsal (i.e. back) silhouettes of the fish body and swimbladder are traced, scanned, and digitized. c is the angle of the swimbladder relative to the longitudinal (i.e. sagittal) axis of the fish. Normal resolution of the cylinders is 1 mm.

Backscatter from each cylinder is estimated using a low mode cylinder solution and a Kirchhoff-ray approximation (ka>0.2). Backscattering cross-sections from each finite cylinder are summed over the whole swimbladder or body and then added coherently. The model calculates backscatter as reduced scattering lengths, a non-dimensional linear unit. Reduced scattering length (RSL) is converted to the more familiar target strength (TS) by:

TS = 20 log (RSL) + 20 log (L)

For any digitized fish, we use the KRM model to estimate backscatter as a function of fish length, wavelength (i.e. speed of sound in water/acoustic frequency), and fish tilt. Results from the model can be reported for the swimbladder, body, or the whole fish to show the contribution of the body parts to the total backscatter.

Model results can be combined to summarize backscatter characteristics of a single fish. A backscatter response surface plots reduced scattering length as a function of fish aspect (q) and a ratio of fish length (L) to acoustic wavelength (l). Below is a backscatter response surface for an Atlantic cod.

Image - Backscatter response surface

The dependence of echo amplitude on aspect angle is low at low L/l values. As fish length or acoustic frequency increases, the influence of fish aspect on echo amplitude increases. Since maximum backscatter occurs when the top surface of the swimbladder is parallel to the transducer and corresponding incident wave front, maximum backscatter occurs at 85 degrees with the fish tilted slightly head down. The influence of fish aspect increases as L/l increases. The response surface becomes quasi-symmetrical as q deviates positive or negative from 85o. Along the fish length to acoustic wavelength axis, if fish length is kept constant then higher L/l values correspond to higher acoustic frequencies. Keeping frequency constant illustrates the effect of changes in fish length. The periodic peaks and valleys along the maximum backscatter ridge correspond to constructive and destructive interference between the swimbladder and body.

Component backscatter plots and backscatter response surfaces can be modeled for any species. Tilt angles, lengths, and frequencies are chosen to reflect the species and behavior of interest. You can model your own backscatter component plots and response surface using our web based interactive program KRMCompare.

We have expanded the model to include backscatter calculations as a function of fish roll. This is important as fisheries sonars are insonifying fish aggregations at angles other than 90 degrees incidence (i.e. looking downward). Be sure to see our model visualizations of three-dimensional fish backscatter and model your own.

 

Cited References
Clay, C. S. 1991. Low-resolution acoustic scattering models: fluid-filled
cylinders and fish with swimbladders. The Journal of the Acoustical Society of America 89: 2168-2179.
Clay, C. S. 1992. Composite ray-mode approximations for backscattered
sound from gas-filled cylinders and swimbladders. The Journal of the Acoustical Society of America 92: 2173-2180.
Clay, C. S. and J. K. Horne. 1994. Acoustic models of fish: The Atlantic
cod (Gadus Morhua). The Journal of the Acoustical Society of America 96: 1661-1668.
Foote, K. G. 1985. Rather-high-frequency sound scattering by swimbladdered
fish. The Journal of the Acoustical Society of America 78: 688-700.
Foote, K. G. and J. J. Traynor. 1988. Comparisons of walleye pollock
target strength estimates determined from in situ measurements and calculations based on swimbladder form. The Journal of the Acoustical Society of America 83: 9-17.
Stanton, T.K. 1989. Sound scattering by cylinders of finite length. III.
Deformed cylinders. The Journal of the Acoustical Society of America 86: 691-705.
Relevant Publications
Clay, C.S. and J.K. Horne. 1994. Acoustic models of fish: the Atlantic
cod (Gadus morhua). The Journal of the Acoustical Society of America 96: 1661-1668.
Horne, J.K. and C.S. Clay. 1998. Sonar systems and aquatic organisms:
matching equipment and model parameters. Canadian Journal of Fisheries and Aquatic Sciences 55: 1296-1306.
Horne, J.K. and J.M. Jech. 1999. Multi-frequency estimates of fish
abundance: constraints of rather high frequencies. ICES Journal of marine Science 56: 184-199.
Jech, J.M. and J.K. Horne. Three dimensional visualization of fish
morphometry and acosutic backscatter. ICES FAST working group manuscript.

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