Biological estimates

Biomass estimates

We can obtain an estimate of biomass density (\(\rho_\text{B}\), kg nmi^-2) by multiplying the areal number density (Eq. 1.15) of animals by the average weight \(\left< w \right>\) (kg):

\[\begin{align*} \rho_\text{B} = \rho_\text{A} \left< w \right>, \tag{2.1a} \\[2ex] \left< w \right> = \frac{\sum\limits_{j=1}^N w_j}{N}. \tag{2.1b} \end{align*}\]

Here \(w_j\) is the weight of fish \(j\) and \(N\) is the total number of fish samples. Many fisheries surveys discretize the observed continuous length (\(L\)) distributions, which allows Eq. (2.1b) to instead be expressed as:

\[ \left< w \right> = \tilde{\mathbf{L}}^{\mathsf{T}} \mathbf{w}, \tag{2.2} \]

where \(\textbf{w}\) is the vector of summed \(w\) in length bin \(\ell\) and \(\tilde{\mathbf{L}}\) the vector representing the normalized number frequency \(\tilde{L}_\ell\) of fish samples in \(\ell\):

\[\begin{align*} \tilde{\mathbf{L}} = \left[ \begin{split} \tilde{L}&_1 \\ \tilde{L}&_2 \\ \tilde{L}&_3 \\ \vdots& \end{split} \right]. \tag{2.3} \end{align*}\]

Let \(L_\ell\) denote the nominal frequency (or count) of fish per \(\ell\). The normalized frequency is then:

\[\begin{align*} \begin{split} \tilde{L}_\ell = \frac{L_\ell}{\sum\limits_{\ell} L_\ell}, \end{split} \tag{2.4a} \\[2ex] \begin{split} \sum\limits_\ell \tilde{L}_\ell = 1. \end{split} \tag{2.4b} \end{align*}\]

The weight vector, \(\textbf{w}\), similarly represents the summed \(w\) for each \(\ell\):

\[\begin{align*} \mathbf{w} = \left[ \begin{split} w&_1 \\ w&_2 \\ w&_3 \\ \vdots& \end{split} \right]. \tag{2.5} \end{align*}\]

Values of \(w_\ell\) are estimated by either summing the weights of fish belonging to each \(\ell\), or fitting a log-linear regression to specimen length and weight measurements derived from trawl samples:

\[ \log_{10}(\hat{w}) = \log_{10}(\hat{a}) + \hat{b} \log_{10}(L), \tag{2.6} \]

where \(\hat{w}\), \(\hat{a}\), and \(\hat{b}\) are the weight, \(y\)-intercept, and slope estimates fitted using ordinary least squares (OLS). Let \(L_{\ell}^*\) denote the representative length for bin \(\ell\). Then the weight for bin \(\ell\) is:

\[ \mathcal{W}(\ell) = \left[ 10^{\hat{a}} {(L_{\ell}^{*})}^{\hat{b}} \right] L_\ell. \tag{2.7} \]

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

\[ B = \rho_\text{B} A = \rho_\text{A} \left< w \right> A, \tag{2.8} \]

where \(A\) is the unit area (nmi²) associated with the areal density estimate (Eq. 1.15).