Standard Sphere
Target Strength
Calculator

Sphere parameters

Material:

Diameter: mm

Density: kg/m³

Longitudinal sound speed: m/s

Transversal sound speed: m/s

Water parameters

	Temperature: °C
	Salinity: PSU
	Pressure: dbar
	Sound speed: m/s
	Density: kg/m³

Signal parameters

	Center frequency: kHz
	Pulse duration: µs
	Bandwidth: to kHz
	Resolution: kHz

Frequency (kHz)

TS (dB re 1 m² @ 1m)

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HELP

- What is Target Strength?
- Using the Target Strength Calculator
- Target Strength Calculation
- Calculation notes
- ka limitation
- References
- Acknowledgements

What is Target Strength?

Target Strength (TS; dB) is a logarithmic measure of the backscattering cross-section, providing a measure of how well an object reflects an incident wave. In typical fisheries surveys, acoustic systems must be calibrated to ensure accurate biomass estimates. For this, TS measurements are typically made of a standard metal sphere with known TS. The TS is dependent on the incident wave's frequency, the diameter of the sphere, and the sound speeds and densities of the water and sphere, all of which are inputs into the Target Strength Calculator. Using these inputs, values of TS can then be calculated and used in calibrating the acoustic system.

Below are some typical examples of what the TS is for 60 mm copper (Cu) and 38.1 mm tungsten carbide (WC) with 6% cobalt binding spheres. In both examples, the sound speed is equal to 1500 m/s.

60 mm Cu sphere TS

31.8 mm WC sphere TS

Using the Target Strength Calculator

The TS Calculator is divided into three sections that allow for user input: Sphere, Water, and Signal parameters, as described below. Once these inputs are entered, the TS can then be calculated by clicking the "Calculate" button above the figure window. The progress bar below the button will then indicate the progress of the calculations using the desired inputs.

Once the calculations are complete, the plot of TS versus frequency will update with the results. A constant horizontal line will also indicate the mean TS over the entire frequency range. To obtain the TS at specific frequencies, hover the mouse pointer over a datapoint and a label will appear with the appropriate values.

If the option for specifying a center frequency and pulse duration was selected, then the plot will also contain a weighted TS average using the sinc function from -1 to 1 as the weights.

Sphere

The two types of spheres typically used for echosounder calibrations are tungsten carbide (WC) with 6% cobalt binding and copper (Cu). The TS characterization of a sphere is dependent upon the sphere diameter (mm), density (kg/m³), and longitudinal and transversal sound speeds (m/s). For WC and Cu spheres, the density and sound speeds are well characterized. However, if the user wishes to enter different values for the sphere parameters, they can select "Custom" in the "Sphere material" pull-down menu.

Water

The TS returned by a sphere is dependent upon the density and sound speed contrast with the surrounding medium, in this case fresh or salt water. Therefore, the user is allowed to input the water density (kg/m³) and sound speed (m/s). Alternatively, there is the option to input the temperature, salinity, and pressure from which the sound speed can then be derived. If this option is selected, then once the calculator is run with the current settings, the derived sound speed will be displayed in the greyed-out "Sound speed" box.

Frequency

The TS is lastly dependent upon the wavelength, and thus frequency, of the incident sound wave in relation to the size of the sphere. The user has the option of inputting either a range of frequencies (kHz) or specifying the center frequency (kHz) and pulse duration (µs) defining a CW square pulse, mimicing the output from a typical echosounder.

If specifying a frequency range, the user can choose the frequency resolution defined in the "Resolution" box. If a single frequency is desired, then set the beginning and ending frequencies the same. For example, to calculate TS at 18 kHz, it should read "Bandwidth: 18 to 18 kHz". In this scenario, the resolution is irrelevant.

Target Strength Calculation

The TS values are calculated using equations in MacLennan (1981). The properties of the WC and Cu spheres are from MacLennan and Dunn (1984) and Foote, et al. (1981), respectively.

For calculations of TS when a CW pulse is selected, the frequency range is calculated by finding the bandwidth (BW) of a square pulse at the center frequency (f_c; Hz):

BW = 1/τ
Frequency range = f_c - BW/2 to f_c + BW/2

where τ is the pulse length (m) of the emitted signal. When plotting the TS results, the value of the weighted mean TS is calculated using the sinc function for the weights (w):

Sinc function for the weights

The sinc function defined here is evaluated from -1 to 1, with its maximum aligned with f_c.

Calculation notes

The equations in MacLennan (1981) require calculating Bessel functions of the first and second kind for half-integer orders. To implement this in JavaScript, the Taylor series expansion of the Bessel function is carried out to enough terms such that successive summations result in a change of less than 10^-30.

ka limitation

For values of ka greater than 30, TS results from the calculator start to become less accurate. This is due to a limitation in JavaScript for calculating large values of the gamma function; a requirement for evaluating the Bessel function. For large ka, the gamma function returns a value of Infinity, which when evaluated in a denominator, results in 0. This causes terms in the Taylor series expansion for the Bessel function to begin summing zeros and preventing any better approximation.

References

MacLennan, D.N. 1981. The theory of solid spheres as sonar calibration targets. Scottish Fisheries Research. Report Number 22.

MacLennan, D.N. and Dunn, J.R. 1984. Estimation of sound velocities from resonance measurements on tungsten carbide calibration spheres. Journal of Sound and Vibration. 97(2), 231-331.

Foote, K.G., Knudsen, H.P., Vestnes, G., Brede, R. and Nielsen, L. 1981. Improved calibration of hydroacoustic equipment with copper spheres. ICES CM 1981, Document 8:20, 18 pp.

Acknowledgements

This Standard Sphere Target Strength Calculator was created by Josiah Renfree and David Demer from the Advanced Survey Technologies (AST) group at the Southwest Fisheries Science Center. If referencing this calculator, please use the following citation:

Renfree, J.S., Demer, D.A. 2014. Standard Sphere Target Strength Calculator. Advanced Survey Technologies, Fisheries Resource Division, Southwest Fisheries Science Center, National Marine Fisheries Service, National Oceanic and Atmospheric Administration.

For suggestions or comments, please email Josiah.Renfree@noaa.gov