Test apportion abundance doc fix

docs/_config.yml

@@ -12,6 +12,7 @@ execute:
 
 parse:
   myst_enable_extensions:
+  - amsmath
   - colon_fence
   - dollarmath
   - linkify

docs/_toc.yml

@@ -19,8 +19,9 @@ parts:
     - file: theory/acoustics
     - file: theory/bio_estimates
     - file: theory/stratification
-    # - file: theory/kriging
-    - file: theory/apportion
+    # - file: theory/kriging    
+    - file: theory/apportion_biomass
+    - file: theory/apportion_abundance
     - file: theory/kriging_eq
     - file: theory/semivariogram_eq
     # - file: theory/other

docs/theory/apportion_abundance.md

@@ -0,0 +1,242 @@
+(apportion-abundance)=
+# Back-calculating and apportioning abundance estimates
+
+```{attention} 
+Back-calculating and apportioning abundance from kriged biomass estimates as detailed here has not been implemented in `Echopop`.
+```
+
+
+## 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:
+
+$$ 
+B_{\textrm{M}}^{k} =
+\sum_{\ell} B_{\textrm{M}, \ell}^{k, \textrm{unaged}} +
+\sum_{\ell, \alpha} B_{\textrm{M}, \ell, \alpha}^{k, \textrm{aged}}
+\label{eq:biomass_M} \tag{1}
+$$
+
+$$
+B_{\textrm{F}}^{k} =
+\sum_{\ell} B_{\textrm{F}, \ell}^{k, \textrm{unaged}} +
+\sum_{\ell, \alpha} B_{\textrm{F}, \ell, \alpha}^{k, \textrm{aged}}
+\label{eq:biomass_F} \tag{2}
+$$
+
+The biomass estimates for all fish including both sexed and unsexed fish in the transect interval $k$ is then:
+
+$$
+B^k = B_\textrm{M}^k + B_\textrm{F}^k.
+$$
+
+
+These kriged biomass estimates can be converted to sexed ($\hat{N}_{s}^{k}$) and total ($\hat{N}^{k}$) abundance by using an averaged length-weight relationship $\overline{W}(\ell)$ via:
+
+$$
+\hat{N}^{k} = \frac{B^{k}}{\overline{W}(\ell)},
+\label{eq:abundance} \tag{3}
+$$
+
+where $\overline{W}(\ell)$ is the length-weight regression relationship derived from the catch data. 
+
+$\hat{\textit{NASC}^{k}}$ can be back-calculated by using the averaged differential backscattering cross-section for the $i^{\text{th}}$ stratum, $\bar{\sigma}_\textrm{bs}^i$, as
+
+$$
+\hat{\textit{NASC}^k} = \hat{N}^k \times \bar{\sigma}_\textrm{bs}^i,
+$$
+
+when the transect interval $k$ falls in stratum $i$. See [](stratification) for more information.
+
+
+```{note} 
+In Chu's Echopro implementation, both $\hat{N}_{s}^{k}$ and $\hat{N}^{k}$ are calculated using a $\overline{W}(\ell)$ fit from **all** (male, female, and unsexed) fish samples.
+```
+
+
+
+## Apportioning back-calculated abundance
+
+Below, the back-calculated $\hat{N}^k$ $\eqref{eq:abundance}$ is apportioned similarly to the [<b>weight proportions</b>](apportion_biomass.md#unaged-biomass-apportioned-with-sex-length-and-age) across sex, length, and age. 
+
+
+### Number of fish samples
+
+#### Unaged fish
+
+The number of unaged fish of sex $s$ and length $\ell$ is ($n_{s,\ell}^{\textrm{unaged}}$) of length $\ell$ is:
+
+$$
+\begin{equation}
+    n_{s,\ell}^{\textrm{unaged}} = \sum_{j \in J_{s,\ell}^{\textrm{unaged}}}n_j,
+\label{eq:total_unaged_sex_length} \tag{4}
+\end{equation}
+$$
+where $s=M$ and $s=F$ indicate male and female fish, respectively.
+
+Therefore, the total number of fish of sex $s$ is:
+
+$$
+\begin{equation}
+    n_s^{\textrm{unaged}} = \sum_\ell n_{s,\ell}^{\textrm{unaged}}
+\label{eq:total_unaged_sex} \tag{5}
+\end{equation}
+$$
+
+#### Aged fish
+
+The total number of fish of sex $s$, length $\ell$, and age $\alpha$ is similarly:
+
+$$
+\begin{equation}
+    n_{s,\ell,\alpha}^{\textrm{aged}} = \sum_{j \in J_{s,\ell,\alpha}^{\textrm{aged}}} n_j
+\label{eq:total_aged_sex_length_age} \tag{6}
+\end{equation}
+$$
+
+The number of aged fish of sex $s$ is then:
+
+$$
+\begin{equation}
+    n_s^{\textrm{aged}} = \sum_{\ell,\alpha}n_{s,\ell,\alpha}^{\textrm{aged}}
+\label{eq:total_aged_sex} \tag{7}
+\end{equation}
+$$
+
+
+
+### Number proportions
+
+The sex-specific numbers for unaged $\eqref{eq:total_unaged_sex}$ and aged $\eqref{eq:total_aged_sex}$ fish are then summed to calculate the total number of unaged fish($n^{\textrm{unaged}}$), aged ($n^{\textrm{aged}}$), and all ($n$) fish:
+
+$$
+\begin{equation}
+\begin{aligned}
+    n^{\textrm{unaged}} &= n_{\textrm{M}}^{\textrm{unaged}} + n_{\textrm{F}}^{\textrm{unaged}} \nonumber \\
+    n^{\textrm{aged}} &= n_{\textrm{M}}^{\textrm{aged}} + n_{\textrm{F}}^{\textrm{aged}} \nonumber \\
+    n &= n^{\textrm{unaged}} + n^{\textrm{aged}} \nonumber
+\end{aligned}
+\label{eq:total_counts} \tag{8}
+\end{equation}
+$$
+
+#### Unaged fish
+
+The number proportions of male and female unaged fish of length $\ell$ $\eqref{eq:total_unaged_sex_length}$ relative to the sex-specific totals of unaged fish $\eqref{eq:total_unaged_sex}$ are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M},\ell}^{\textrm{unaged/unaged}} &= \frac{n_{\textrm{M},\ell}^{\textrm{unaged}}}{n_{\textrm{M}}^{\textrm{unaged}}} \nonumber \\
+    {r_N}_{\textrm{F},\ell}^{\textrm{unaged/unaged}} &= \frac{n_{\textrm{F},\ell}^{\textrm{unaged}}}{n_{\textrm{F}}^{\textrm{unaged}}} \nonumber    
+\end{aligned}
+\label{eq:number_proportions_unaged_sex_length} \tag{9}
+\end{equation}
+$$
+
+The number proportions of male and female unaged fish of length $\ell$ relative to the total number of fish $\eqref{eq:total_counts}$ are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M},\ell}^{\textrm{unaged/all}} = \frac{n_{\textrm{M},\ell}^{\textrm{unaged}}}{n} \nonumber \\
+    {r_N}_{\textrm{F},\ell}^{\textrm{unaged/all}} = \frac{n_{\textrm{F},\ell}^{\textrm{unaged}}}{n} \nonumber
+\end{aligned}
+\label{eq:number_proportions_unaged_sex} \tag{10}
+\end{equation}
+$$
+
+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:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M}, \ell}^{\textrm{unaged}/\textrm{all}} &= {r_N}_{\textrm{M},\ell}^{\textrm{unaged/unaged}} \times {r_N}_{\textrm{M},\ell}^{\textrm{unaged/all}} \nonumber \\
+    {r_N}_{\textrm{F}, \ell}^{\textrm{unaged}/\textrm{all}} &= {r_N}_{\textrm{F},\ell}^{\textrm{unaged/unaged}} \times {r_N}_{\textrm{F},\ell}^{\textrm{unaged/all}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_unaged} \tag{11}
+\end{equation}
+$$
+
+#### Aged fish
+
+Similar to the above, the number of male and female aged fish of length $\ell$ and age $\alpha$ $\eqref{eq:total_aged_sex_length_age}$ relative to the sex-specific totals of aged fish $\eqref{eq:total_aged_sex}$ are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged/aged}} &= \frac{n_{\textrm{M},\ell,\alpha}^{\textrm{aged}}}{n_{\textrm{M}}^{\textrm{aged}}} \nonumber \\
+    {r_N}_{\textrm{F},\ell,\alpha}^{\textrm{aged/aged}} &= \frac{n_{\textrm{F},\ell,\alpha}^{\textrm{aged}}}{n_{\textrm{F}}^{\textrm{aged}}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged_sex_length_age} \tag{12}
+\end{equation}
+$$
+
+The number proportions of male and female aged fish of length $\ell$ and age $\alpha$ relative to the total number of fish $\eqref{eq:total_counts}$ are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged/all}} = \frac{n_{\textrm{M},\ell,\alpha}^{\textrm{aged}}}{n} \nonumber \\
+    {r_N}_{\textrm{F},\ell,\alpha}^{\textrm{aged/all}} = \frac{n_{\textrm{F},\ell,\alpha}^{\textrm{aged}}}{n} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged_sex} \tag{13}
+\end{equation}
+$$
+
+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:
+
+$$
+\begin{equation}
+\begin{aligned}
+    {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged}/\textrm{all}} &= {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged/aged}} \times {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged/all}} \nonumber \\
+    {r_N}_{\textrm{F},\ell,\alpha}^{\textrm{aged}.\textrm{all}} &= r_{n,~\textrm{F},\ell,\alpha}^{\textrm{aged/aged}} \times {r_N}_{\textrm{F},\ell,\alpha}^{\textrm{aged/all}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged} \tag{14}
+\end{equation}
+$$
+
+
+
+
+### 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 $\eqref{eq:number_proportions_unaged_sex_length}$:
+
+$$
+\begin{equation}
+\begin{aligned}
+    \hat{N}_{\textrm{M},\ell}^{k, \textrm{unaged}} &= \hat{N}^{k} \times {r_N}_{\textrm{M},\ell}^{\textrm{unaged}} \nonumber \\
+    \hat{N}_{\textrm{F},\ell}^{k, \textrm{unaged}} &= \hat{N}^{k} \times {r_N}_{\textrm{F},\ell}^{\textrm{unaged}} \nonumber \\
+    \hat{N}_{\ell}^{k, \textrm{unaged}} &= \hat{N}_{\textrm{M},\ell}^{k, \textrm{unaged}} + \hat{N}_{\textrm{F},\ell}^{k, \textrm{unaged}} \nonumber \\
+\end{aligned}
+\label{eq:abundance_unaged} \tag{15}
+\end{equation}
+$$
+
+#### 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 $\eqref{eq:number_proportions_aged_sex_length_age}$: 
+
+$$
+\begin{equation}
+\begin{aligned}
+    \hat{N}_{\textrm{M},\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}^{k} \times {r_N}_{\textrm{M},\ell,\alpha}^{\textrm{aged}} \nonumber \\
+    \hat{N}_{\textrm{F},\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}^{k} \times {r_N}_{\textrm{F},\ell,\alpha}^{\textrm{aged}} \nonumber \\
+    \hat{N}_{\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}_{\textrm{M},\ell,\alpha}^{k, \textrm{aged}} + \hat{N}_{\textrm{F},\ell,\alpha}^{k, \textrm{aged}} \nonumber \\
+\end{aligned}
+\label{eq:abundance_aged} \tag{16}
+\end{equation}
+$$
+
+
+#### Combining unaged and aged estimates
+
+Lastly, the estimated abundance of all fish (including unaged and aged fish) of length $\ell$ can be obtained by:
+
+$$
+\hat{N}_{\ell}^{k,i} = \hat{N}_{\ell}^{k, \textrm{unaged}} + \sum_{\alpha} \hat{N}_{\ell,\alpha}^{k, \textrm{aged}}
+\label{eq:abundance_length} \tag{17}
+$$

docs/theory/apportion.md → docs/theory/apportion_biomass.md

@@ -1,4 +1,4 @@
-(apportion)=
+(apportion-biomass)=
 # Apportioning kriged biomass density
 
 The challenges associated with apportioning the kriged biomass is to properly combine the various pieces of information to distribute the biomass into different sex, length, and age groups. This is because fish samples obtained from a haul (trawl) are processed at two different stations that report different biometric data: