### Catch in weight

The catch in weight \(Y\) is \[ Y_{y,f} = \sum_a C_{y,a,f} w_{y,a}.\]

### Mean size

The mean length of the catch \(\bar{L}_{y,f}\) is \[ \bar{L}_{y,f} = \dfrac{\sum_{\ell} L_{\ell} C_{y,\ell,f}}{\sum_{\ell} C_{y,\ell,f}},\] where \(L_\ell\) is the midpoint of the length bin \(\ell\).

The mean weight of the catch \(\bar{w}_{y,f}\) is \[ \bar{w}_{y,f} = \dfrac{\sum_a C_{y,a,f}w_{y,a}}{\sum_a C_{y,a,f}},\]

### Survey

If the \(s^{\textrm{th}}\) survey is biomass-based, then the survey value \(I_{y,s}\) is calculated as \[ I_{y,s} = q_s \sum_a v_{y,a,s} N_{y,a} w_{y,a}, \] where \(q\) is the scaling coefficient and \(s\) indexes survey.

If the survey is abundance-based, then \[ I_{y,s} = q_s \sum_a v_{y,a,s} N_{y,a} . \]

The proportion-at-age vulnerable to the survey is \[ p_{y,a,s} = \dfrac{v_{a,s}N_{y,a}}{\sum_a v_{a,s}N_{y,a}}.\]

The proportion-at-length vulnerable to the survey is \[ p_{y,\ell,s} = \dfrac{\sum_a v_{a,s} N_{y,a} P(\ell|a)}{\sum_{\ell} \sum_a v_{a,s} N_{y,a} P(\ell|a)}.\]