Biological estimates

Number density of scatterers

To obtain the number density of the animal, we define the volume backscattering coefficient (\(s_V\), units: m^-1):

\[ s_V = \rho_V \left< \sigma_{bs} \right>, \]

and its corresponding logarithmic quantity, the volume backscattering strength (\(S_V\), units: dB re 1 m^-1)

\[ S_V = 10 \log_{10} s_V = 10 \log_{10} \rho_V + 10 \log_{10} \left< \sigma_{bs} \right> \]

where \(\rho_V\) is the number density of scatterers (fish) per unit volume (units: m^-3).

In fisheries acoustics, we are often interested in quantities per unit area. Therefore, we define the areal backscattering coefficient (\(ABC\), or \(s_a\), units: m²m^-2)

\[ s_a = s_V H, \]

where \(H\) is the integration height in meter, and the corresponding nautical areal scattering coefficient (\(NASC\), or \(s_A\), units: m²nmi^-2)

\[ NASC = s_A = 4 \pi \times 1852^2 \times s_a, \]

in which the conversion of 1 nmi = 1852 m is used.

Using the above quantities, we obtain

\[ s_a = s_V H = \rho_V \left< \sigma_{bs} \right> H, \]

Let the areal number density (\(\rho_a\), units: m^-2) be

\[ \rho_a = \rho_V H, \]

then

\[ s_a = \rho_a \left< \sigma_{bs} \right>. \]

Similarly, with the corresponding nautical areal number density (\(\rho_A\), units: nmi^-2) being

\[ \rho_A = 1852^2 \rho_a, \]

then

\[ s_A = NASC = 4 \pi \rho_A \left< \sigma_{bs} \right>. \]

Note that \(NASC\) is the typical output from software packages such as Echoview for biological estimates.

Biomass estimates

We can obtain an estimate of biomass density (\(\rho_B\), units: kg nmi^-2) by multiplying the areal number density of animals by the average weight (\(\left< w \right>\), units: kg)

\[ \rho_B = \rho_A \left< w \right>. \]

The average weight is

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

where \(w_j\) is the weight of fish \(j\), and \(N\) is the total number of fish samples.

In the case when the fish length is binned, which is the case for most fisheries surveys,

\[ \left< w \right> = \mathbf{L}^\top \mathbf{w}. \]

Here, \(\mathbf{L}\) is a vector representing the number frequency \(L_\ell\) of fish samples in length bin \(\ell\)

\[\begin{split} \mathbf{L} = \begin{bmatrix} L_1 \\ L_2 \\ L_3 \\ \vdots \end{bmatrix} \end{split}\]

and \(\mathbf{w}\) is a vector representing the weight of fish at length \(L_\ell\)

\[\begin{split} \mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ \vdots \end{bmatrix}. \end{split}\]

Note that the number frequency of fish length is normalized across all length bins, i.e.,

\[ \sum_\ell L_\ell = 1. \]

The \(w_\ell\) values can be estimated by the regressed length-weight relationship derived from trawl samples.

With the above quantities, the biomass (\(B\), units: kg) can then be estimated by

\[ B = \rho_B A = \rho_A \left< w \right> A, \]

where \(A\) is the unit area associated with the density measure.