Acoustics basics

Backscattering cross-section and target strength

For a given scatterer, the differential backscattering cross section (\(\sigma_{bs}\), units: m²) and backscattering cross section (\(\sigma_{b}\), units: m²) are related by

\[ \sigma_{bs} = \frac{\sigma_b}{ 4 \pi} \]

The target strength (\(TS\), units: dB re 1 m²) is defined as

\[ TS = 10 \log_{10} \sigma_{bs} \]

For a group of \(N\) animals, the mean differential backscattering cross-section is

\[ \left< \sigma_{bs} \right> = \frac{\sum_{j=1}^N \sigma_{bs,j} }{ N }, \]

where \(\sigma_{bs,j}\) is the differential backscattering cross-section of animal \(j\), which often varies as a function of its length \(L_j\):

\[ \sigma_{bs,j} = \sigma_{bs,j}(L_j) \]

TS-length relationship

One common avenute to estimate TS of a scatterer is based on empirical relationshpi between TS and length, which can be expressed by

\[ TS = mL + b, \]

where \(L\) is the total length, \(m\) is the slope, and \(b\) is the y-intercept.

For Pacific hake, the empitical TS-length relationship used in Echopop is

\[ TS = 20 L - 68 \]

where \(L\) is the fish fork length in cm.

Therefore, for Pacific hake

\[ \sigma_{bs} = 10^{TS/10} = 10^{-6.8} L^2 \]