Catch at age
The catch (in numbers) \(C\) at age for fleet \(f\) is \[ C_{y,a,f} = \dfrac{v_{y,a,f} F_{y,f}}{Z_{y,a}} N_{y,a} (1 - \exp[-Z_{y,a}]).\]
If the model is conditioned on catch, then \(F_{y,f}\) can be estimated as parameters or solved iteratively to match the observed catch. If the model is conditioned on effort, then \[ F_{y,f} = q_f E_{y,f},\] where \(E_{y,f}\) is the observed effort and \(q_f\) is the scaling coefficient.
The proportion of the catch-at-age is \[ p_{y,a,f} = \dfrac{C_{y,a,f}}{\sum_a C_{y,a,f}}.\]
Catch at length
The catch at length is calculated assuming a normally distributed length-at-age \(P(\ell,a)\), where \[ C_{y,\ell,f} = \sum_a C_{y,a,f} P(\ell|a) \] and
\[ P(\ell|a) = \begin{cases} \phi(L'_{\ell+1}) & \ell = 1\\ \phi(L'_{\ell+1}) - \phi(L'_\ell) & \ell = 2, \ldots, L - 1,\\ 1 -\phi(L'_\ell) & \ell = L \end{cases} \] with \(L'_{\ell}\) as the length at the lower boundary of length bin \(\ell\) and \(\phi(L'_{\ell})\) as the cumulative distribution function of a normal variable with mean \(\tilde{L}_{y,a}\) (the expected mean length at age \(a\)) and standard deviation \(\tilde{L}_{y,a} \times CV^L\) (\(CV^L\) is the coefficient of variation in mean length at age).
The proportion of the catch-at-length is \[ p_{y,\ell,f} = \dfrac{C_{y,\ell,f}}{\sum_{\ell}C_{y,\ell,f}}.\]