Apportioning back-calculated abundance

Back-calculating abundance from kriged biomass estimates

The biomass estimates for male and female fish (\(s=M\) and \(s=F\), respectively) along transect interval \(k\) across all lengths (\(\ell\)) and all ages (\(\alpha\)) are:

\[ B_{\textrm{i,M}}^{k} = \sum_{\ell} B_{i,\textrm{M}, \ell}^{k, \textrm{unaged}} + \sum_{\ell, \alpha} B_{i,\textrm{M}, \ell, \alpha}^{k, \textrm{aged}} \]

\[ B_{i,\textrm{F}}^{k} = \sum_{\ell} B_{i,\textrm{F}, \ell}^{k, \textrm{unaged}} + \sum_{\ell, \alpha} B_{i,\textrm{F}, \ell, \alpha}^{k, \textrm{aged}} \]

The biomass estimates for all fish including both sexed and unsexed fish in the transect interval \(k\) is then:

\[ B^k_i = B_{i,\textrm{M}}^k + B_{i,\textrm{F}}^k. \]

The estimated abundance \(\hat{N}^k\) can be back-calculated from the kriged biomass estimates using an averaged length-weight relationship \(\overline{\mathcal{W}}(\ell)\) via:

\[ \hat{N}^{k}_i = \frac{B^{k}_i}{\overline{\mathcal{W}}(\ell)_\text{all}}, \]

where \(\overline{W}(\ell)_\text{all}\) is the length-weight regression relationship derived from the catch data.

Similarly, \(\hat{\textit{NASC}_i^{\,\,k}}\) can be back-calculated from the estimated abundance using the averaged differential backscattering cross-section of the \(i^{\text{th}}\) stratum, \(\bar{\sigma}_{bs}^i\), via:

\[ \hat{\textit{NASC}^{\,\,k}}_i = \hat{N}^k_i \times \bar{\sigma}_\textrm{bs}^i, \]

when the transect interval \(k\) falls in stratum \(i\).

Note

In Chu’s Echopro implementation, both \(\hat{N}_{i,s}^{k}\) and \(\hat{N}^{k}_i\) are calculated using a single \(\overline{\mathcal{W}}(\ell)_\text{all}\) fit from all (male, female, and unsexed) fish samples, instead of sex-specific fits.

Apportioning back-calculated abundance

Below, the back-calculated \(\hat{N}^k_i\) Eq. (3) is apportioned similarly to the weight proportions.

Number of fish samples

Unaged fish

The numbers of unaged male and female fish of length \(\ell\) are:

\[\begin{split} \begin{aligned} n_{i,\mathrm{M},\ell}^{\mathrm{unaged}} &= \sum_{j\in J_{i,\mathrm{M},\ell}^{\mathrm{unaged}}} n_j \\ n_{i,\mathrm{F},\ell}^{\mathrm{unaged}} &= \sum_{j\in J_{i,\mathrm{F},\ell}^{\mathrm{unaged}}} n_j \end{aligned} \end{split}\]

Therefore, the total numbers of male and female unaged fish of length \(\ell\) are:

\[\begin{split} \begin{aligned} n_{i,\mathrm{M}}^{\mathrm{unaged}} &= \sum_{\ell} n_{i,\mathrm{M},\ell}^{\mathrm{unaged}} \\ n_{i,\mathrm{F}}^{\mathrm{unaged}} &= \sum_{\ell} n_{i,\mathrm{F},\ell}^{\mathrm{unaged}} \end{aligned} \end{split}\]

Aged fish

The numbers of male and female aged fish of length \(\ell\) and age \(\alpha\) are:

\[\begin{split} \begin{aligned} n_{i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}} &= \sum_{j\in J_{i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}}} n_j \\ n_{i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}} &= \sum_{j\in J_{i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}}} n_j \end{aligned} \end{split}\]

Therefore, the total numbers of male and female aged fish of length \(\ell\) and age \(\alpha\) are:

\[\begin{split} \begin{aligned} n_{i,\mathrm{M}}^{\mathrm{aged}} &= \sum_{\ell,\alpha} n_{i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}} \\ n_{i,\mathrm{F}}^{\mathrm{aged}} &= \sum_{\ell,\alpha} n_{i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}} \end{aligned} \end{split}\]

Number proportions

The sex-specific numbers for unaged Eq. (6) and aged Eq. (8) fish are then summed to calculate the total number of unaged fish (\(n^{\textrm{unaged}}\)), aged (\(n^{\textrm{aged}}\)), and all (\(n\)) fish:

\[\begin{split} \begin{aligned} n_i^{\mathrm{unaged}} &= n_{i,\mathrm{M}}^{\mathrm{unaged}} + n_{i,\mathrm{F}}^{\mathrm{unaged}} \\ n_i^{\mathrm{aged}} &= n_{i,\mathrm{M}}^{\mathrm{aged}} + n_{i,\mathrm{F}}^{\mathrm{aged}} \\ n_i &= n_i^{\mathrm{unaged}} + n_i^{\mathrm{aged}} \end{aligned} \end{split}\]

Unaged fish

The number proportions of male and female unaged fish of length \(\ell\) Eq. (5) relative to the sex-specific totals of unaged fish Eq. (8) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged/unaged}} &= \frac{n_{i,\mathrm{M},\ell}^{\mathrm{unaged}}}{n_{i,\mathrm{M}}^{\mathrm{unaged}}} \\ r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged/unaged}} &= \frac{n_{i,\mathrm{F},\ell}^{\mathrm{unaged}}}{n_{i,\mathrm{F}}^{\mathrm{unaged}}} \end{aligned} \end{split}\]

The number proportions of male and female unaged fish of length \(\ell\) relative to the total number of fish Eq. (9) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged/all}} &= \frac{n_{i,\mathrm{M},\ell}^{\mathrm{unaged}}}{n_i} \\ r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged/all}} &= \frac{n_{i,\mathrm{F},\ell}^{\mathrm{unaged}}}{n_i} \end{aligned} \end{split}\]

The within-unaged-group proportion integrated over sex \(s\) in stratum \(i\) is:

\[ r_{n,i,\ell}^{\mathrm{unaged/unaged}} = \sum_{s} r_{N,i,s,\ell}^{\mathrm{unaged/unaged}} \]

The number proportions of male and female unaged fish of length \(\ell\) with respect to the total number of fish (unaged and aged combined) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged/all}} &= r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged/unaged}}\times r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged/all}} \\ r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged/all}} &= r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged/unaged}}\times r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged/all}} \end{aligned} \end{split}\]

Aged fish

Similar to the above, the number of male and female aged fish of length \(\ell\) and age \(\alpha\) Eq. (7) relative to the sex-specific totals of aged fish Eq. (8) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged/aged}} &= \frac{n_{i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}}}{n_{i,\mathrm{M}}^{\mathrm{aged}}} \\ r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged/aged}} &= \frac{n_{i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}}}{n_{i,\mathrm{F}}^{\mathrm{aged}}} \end{aligned} \end{split}\]

The number proportions of male and female aged fish of length \(\ell\) and age \(\alpha\) relative to the total number of fish Eq. (9) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged/all}} &= \frac{n_{i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}}}{n_i} \\ r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged/all}} &= \frac{n_{i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}}}{n_i} \end{aligned} \end{split}\]

The number proportions of male and female unaged fish of length \(\ell\) and age \(\alpha\) with respect to the total number of fish (unaged and aged combined) are:

\[\begin{split} \begin{aligned} r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged/all}} &= r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged/aged}}\times r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged/all}} \\ r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged/all}} &= r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged/aged}}\times r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged/all}} \end{aligned} \end{split}\]

Apportioning abundances

Unaged fish

For each transect interval \(k\), the total estimated abundance of male, female, and all unaged fish of length \(\ell\) are apportioned according to the number proportions in Eq. (10):

\[\begin{split} \begin{aligned} \hat{N}_{i,\mathrm{M},\ell}^{k,\mathrm{unaged}} &= \hat{N}_i^{k}\times r_{N,i,\mathrm{M},\ell}^{\mathrm{unaged}} \\ \hat{N}_{i,\mathrm{F},\ell}^{k,\mathrm{unaged}} &= \hat{N}_i^{k}\times r_{N,i,\mathrm{F},\ell}^{\mathrm{unaged}} \\ \hat{N}_{i,\ell}^{k,\mathrm{unaged}} &= \hat{N}_{i,\mathrm{M},\ell}^{k,\mathrm{unaged}} + \hat{N}_{i,\mathrm{F},\ell}^{k,\mathrm{unaged}} \end{aligned} \end{split}\]

Aged fish

Similarly, for each transect interval \(k\), the total estimated abundance of male, female, and all aged fish of length \(\ell\) and age \(\alpha\) are apportioned according to the number proportions in Eq. (13):

\[\begin{split} \begin{aligned} \hat{N}_{i,\mathrm{M},\ell,\alpha}^{k,\mathrm{aged}} &= \hat{N}_i^{k}\times r_{N,i,\mathrm{M},\ell,\alpha}^{\mathrm{aged}} \\ \hat{N}_{i,\mathrm{F},\ell,\alpha}^{k,\mathrm{aged}} &= \hat{N}_i^{k}\times r_{N,i,\mathrm{F},\ell,\alpha}^{\mathrm{aged}} \\ \hat{N}_{i,\ell,\alpha}^{k,\mathrm{aged}} &= \hat{N}_{i,\mathrm{M},\ell,\alpha}^{k,\mathrm{aged}} + \hat{N}_{i,\mathrm{F},\ell,\alpha}^{k,\mathrm{aged}} \end{aligned} \end{split}\]

Combining unaged and aged estimates

Lastly, the estimated abundance of all fish (including unaged and aged fish) of length \(\ell\) can be obtained by:

\[ \hat{N}_{\ell}^{k,i} = \hat{N}_{\ell}^{k,\mathrm{unaged},i} + \sum_{\alpha} \hat{N}_{\ell,\alpha}^{k,\mathrm{aged},i} \]