OSOceanAcoustics · brandynlucca · Mar 27, 2024 · Mar 27, 2024 · Mar 27, 2024 · Mar 27, 2024
diff --git a/docs/_config.yml b/docs/_config.yml
@@ -12,6 +12,7 @@ execute:
 
 parse:
   myst_enable_extensions:
+  - amsmath
   - colon_fence
   - dollarmath
   - linkify

diff --git a/docs/_toc.yml b/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

diff --git a/docs/images/abundance_to_nasc_conversion.jpg b/docs/images/abundance_to_nasc_conversion.jpg
diff --git a/docs/images/aged_abundance_apportionment.jpg b/docs/images/aged_abundance_apportionment.jpg
diff --git a/docs/images/aged_aged_sex_length_age_number_proportion.jpg b/docs/images/aged_aged_sex_length_age_number_proportion.jpg
diff --git a/docs/images/aged_all_sex_number_proportion.jpg b/docs/images/aged_all_sex_number_proportion.jpg
diff --git a/docs/images/aged_and_unaged_apportioned_sexed_abundance.jpg b/docs/images/aged_and_unaged_apportioned_sexed_abundance.jpg
diff --git a/docs/images/biomass_to_abundance_conversion.jpg b/docs/images/biomass_to_abundance_conversion.jpg
diff --git a/docs/images/initial_biomass_summation.jpg b/docs/images/initial_biomass_summation.jpg
diff --git a/docs/images/total_apportioned_abundance.jpg b/docs/images/total_apportioned_abundance.jpg
diff --git a/docs/images/unaged_abundance_apportionment.jpg b/docs/images/unaged_abundance_apportionment.jpg
diff --git a/docs/images/unaged_all_sex_number_proportion.jpg b/docs/images/unaged_all_sex_number_proportion.jpg
diff --git a/docs/images/unaged_unaged_sex_length_number_proportion.jpg b/docs/images/unaged_unaged_sex_length_number_proportion.jpg
diff --git a/docs/theory/apportion_abundance.md b/docs/theory/apportion_abundance.md
@@ -0,0 +1,229 @@
+(apportion-abundance)=
+# Back-calculating abundances from kriged biomass estimates
+
+```{attention} 
+`Echopop` currently does support kriged abundance back-calculation from biomass estimates. Refer to the <b>[kriged biomass apportionment](apportion_biomass.md)</b> for more information on how `Echopop` incorporates kriging into population estimates.
+```
+
+```{note}
+It is worth noting that all calculations are done for each stratum, $i$. Refer to the <b>[stratification](stratification.md)</b> documentation for more information.
+```
+
+Biomass estimates for each $s$ ($\textrm{M}$ and $\textrm{F}$) along transect interval $k$ are summed across $\ell$ and $\alpha$ via:
+
+$$ 
+B_{\textrm{M}}^{k} = \sum_{\textrm{M}, \ell, \alpha} B_{\textrm{M}, \ell, \alpha}^{k, \textrm{aged}} +  \sum_{\textrm{M}, \ell, \alpha} B_{\textrm{M}, \ell, \alpha}^{k, \textrm{unaged}}
+\label{eq:biomass_M} \tag{1}
+$$
+
+$$
+B_{\textrm{F}}^{k} = \sum_{\textrm{F}, \ell, \alpha} B_{\textrm{F}, \ell, \alpha}^{k, \textrm{aged}} + \sum_{\textrm{F}, \ell, \alpha} B_{\textrm{F}, \ell, \alpha}^{k, \textrm{unaged}}
+\label{eq:biomass_F} \tag{2}
+$$
+
+Similarly, biomass estimates for all fish ($B^{k}$), which is inclusive of both sexed and unsexed fish, are also summed (see: <b>[kriged biomass summation for more details](apportion_biomass.md#total-biomass-apportioned-with-length-and-age)</b>).
+
+
+These kriged biomass estimates are then converted to sexed ($\hat{N}_{s}^{k}$) and total ($\hat{N}^{k}$) abundance by using averaged length-weight regression output ($\overline{W}(\ell)$). Consequently, $\overline{W}(\ell)$ can be defined either by using the average length-weight relationship produced or parameterizing $\overline{W}(\ell)$ with the mean length ($\bar{\ell}$). It is important to note, however, that both $\hat{N}_{s}^{k}$ and $\hat{N}^{k}$ are calculated using $\overline{W}(\ell)$ fit from <b>all</b> individuals (i.e. male, female, and unsexed).
+
+$$
+\hat{N}^{k} = \frac{B^{k}}{\overline{W}(\ell)}
+\label{eq:abundance} \tag{3}
+$$
+
+```{note} 
+With $\hat{N}_{\textrm{All}}^{k}$ calculated, $\hat{\textit{NASC}^{k}}$ can then be back-calculated by using the averaged $i^{\text{th}}$ differential backscattering cross-section ($\bar{\sigma}_{\textrm{bs}}$):
+$
+\hat{\textit{NASC}^{k}} = \hat{N}^{k} \bar{\sigma}_{\textrm{bs}}
+$
+```
+
+## Apportioning the back-calculated abundance estimates
+
+### Summing fish counts
+
+The back-calculated $\hat{N}^{k}$ $\eqref{eq:abundance}$ is subsequently apportioned similarly to the [<b>weight proportions</b>](apportion_biomass.md#unaged-biomass-apportioned-with-sex-length-and-age) across sex, length, and age. 
+
+#### Unaged fish
+
+This process is first done across $\ell$ for unaged fish, and both $\ell$ *and* $\alpha$ for aged fish. First, the total number counts for unaged fish ($n_{s,\ell}^{\textrm{unaged}}$):
+
+$$
+\begin{equation}
+\begin{aligned}
+    n_{\textrm{M},\ell}^{\textrm{unaged}} &= \sum_{j \in J_{\textrm{M},\ell}^{\textrm{unaged}}}n_j \nonumber \\
+    n_{\textrm{F},\ell}^{\textrm{unaged}} &= \sum_{j \in J_{\textrm{F},\ell}^{\textrm{unaged}}}n_j \nonumber
+\end{aligned}
+\label{eq:total_unaged_sex_length} \tag{4}
+\end{equation}
+$$
+
+These are then summed across all fish of $s$:
+
+$$
+\begin{equation}
+\begin{aligned}
+    n_{\textrm{M}}^{\textrm{unaged}} &= \sum_{\textrm{M},\ell}n_{\textrm{M},\ell}^{\textrm{unaged}} \nonumber \\
+    n_{\textrm{F}}^{\textrm{unaged}} &= \sum_{\textrm{F}, \ell}n_{\textrm{F},\ell}^{\textrm{unaged}} \nonumber
+\end{aligned}
+\label{eq:total_unaged_sex} \tag{5}
+\end{equation}
+$$
+
+#### Aged fish
+
+The total counts for aged fish across both $\ell$ and $\alpha$ are similarly calculated via:
+
+$$
+\begin{equation}
+\begin{aligned}
+    n_{\textrm{M},\ell,\alpha}^{\textrm{aged}} &= \sum_{j \in J_{\textrm{M},\ell,\alpha}^{\textrm{aged}}}n_j \nonumber \\
+    n_{\textrm{F},\ell,\alpha}^{\textrm{aged}} &= \sum_{j \in J_{\textrm{F},\ell,\alpha}^{\textrm{aged}}}n_j \nonumber
+\end{aligned}
+\label{eq:total_aged_sex_length_age} \tag{6}
+\end{equation}
+$$
+
+These are then summed across all fish of $s$:
+
+$$
+\begin{equation}
+\begin{aligned}
+    n_{\textrm{M}}^{\textrm{aged}} &= \sum_{\textrm{M},\ell,\alpha}n_{\textrm{M},\ell,\alpha}^{\textrm{aged}} \nonumber \\
+    n_{\textrm{F}}^{\textrm{aged}} &= \sum_{\textrm{F},\ell,\alpha}n_{\textrm{F},\ell,\alpha}^{\textrm{aged}} \nonumber \\    
+\end{aligned}
+\label{eq:total_aged_sex} \tag{7}
+\end{equation}
+$$
+
+### Number proportions
+
+The sex-specific abundances for unaged $\eqref{eq:total_unaged_sex}$ and aged $\eqref{eq:total_aged_sex}$ fish are then summed together to calculate the total unaged ($n^{\textrm{unaged}}$), 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 counts of unaged fish across $\ell$ for each $s$ $\eqref{eq:total_unaged_sex_length}$ relative to the sex-specific totals $\eqref{eq:total_unaged_sex}$, $r_{n,s,\ell}^{\textrm{unaged/unaged}}$, are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M},\ell}^{\textrm{unaged/unaged}} &= \frac{n_{\textrm{M},\ell}^{\textrm{unaged}}}{n_{\textrm{M}}^{\textrm{unaged}}} \nonumber \\
+    r_{n,~\textrm{F},\ell}^{\textrm{unaged/unaged}} &= \frac{n_{\textrm{F},\ell}^{\textrm{unaged}}}{n_{\textrm{F}}^{\textrm{unaged}}} \nonumber    
+\end{aligned}
+\label{eq:number_proportions_unaged_sex_length} \tag{9}
+\end{equation}
+$$
+
+In a similar manner, the unaged fish number counts relative to the sum of unaged and aged number counts $\eqref{eq:total_counts}$, $r_{n,s,\ell}^{\textrm{unaged/all}}$, are then calculated via:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M},\ell}^{\textrm{unaged/all}} = \frac{n_{\textrm{M},\ell}^{\textrm{unaged}}}{n} \nonumber \\
+    r_{n,~\textrm{F},\ell}^{\textrm{unaged/all}} = \frac{n_{\textrm{F},\ell}^{\textrm{unaged}}}{n} \nonumber
+\end{aligned}
+\label{eq:number_proportions_unaged_sex} \tag{10}
+\end{equation}
+$$
+
+The number proportions referencing unaged $\eqref{eq:number_proportions_unaged_sex_length}$ and all $\eqref{eq:number_proportions_unaged_sex}$ fish are then combined to calculate the overall sex-specific number proportions:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M}}^{\textrm{unaged}} &= r_{n,~\textrm{M},\ell}^{\textrm{unaged/unaged}} r_{n,~\textrm{M},\ell}^{\textrm{unaged/all}} \nonumber \\
+    r_{n,~\textrm{F}}^{\textrm{unaged}} &= r_{n,~\textrm{F},\ell}^{\textrm{unaged/unaged}} r_{n,~\textrm{F},\ell}^{\textrm{unaged/all}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_unaged} \tag{11}
+\end{equation}
+$$
+
+#### Aged fish
+
+Similar to unaged fish, the number counts of aged fish across $\ell$ and $\alpha$ for each $s$ $\eqref{eq:total_aged_sex_length_age}$ relative to the sex-specific totals $\eqref{eq:total_aged_sex}$, $r_{n,s,\ell,\alpha}^{\textrm{aged/aged}}$, are:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M},\ell,\alpha}^{\textrm{aged/aged}} &= \frac{n_{\textrm{M},\ell,\alpha}^{\textrm{aged}}}{n_{\textrm{M}}^{\textrm{aged}}} \nonumber \\
+    r_{n,~\textrm{F},\ell,\alpha}^{\textrm{aged/aged}} &= \frac{n_{\textrm{F},\ell,\alpha}^{\textrm{aged}}}{n_{\textrm{F}}^{\textrm{aged}}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged_sex_length_age} \tag{12}
+\end{equation}
+$$
+
+In a similar manner, the unaged fish number counts relative to the sum of unaged and aged number counts $\eqref{eq:total_counts}$, $r_{n,s,\ell,\alpha}^{\textrm{aged/all}}$, are then calculated via:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M},\ell,\alpha}^{\textrm{aged/all}} = \frac{n_{\textrm{M},\ell,\alpha}^{\textrm{aged}}}{n} \nonumber \\
+    r_{n,~\textrm{F},\ell,\alpha}^{\textrm{aged/all}} = \frac{n_{\textrm{F},\ell,\alpha}^{\textrm{aged}}}{n} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged_sex} \tag{13}
+\end{equation}
+$$
+
+The number proportions referencing unaged $\eqref{eq:number_proportions_aged_sex_length_age}$ and all $\eqref{eq:number_proportions_aged_sex}$ fish are then combined to calculate the overall sex-specific number proportions:
+
+$$
+\begin{equation}
+\begin{aligned}
+    r_{n,~\textrm{M}}^{\textrm{aged}} &= r_{n,~\textrm{M},\ell,\alpha}^{\textrm{aged/aged}} r_{n,~\textrm{M},\ell,\alpha}^{\textrm{aged/all}} \nonumber \\
+    r_{n,~\textrm{F}}^{\textrm{aged}} &= r_{n,~\textrm{F},\ell,\alpha}^{\textrm{aged/aged}} r_{n,~\textrm{F},\ell,\alpha}^{\textrm{aged/all}} \nonumber
+\end{aligned}
+\label{eq:number_proportions_aged} \tag{14}
+\end{equation}
+$$
+
+### Apportioning abundances
+
+#### Unaged fish
+
+Total unaged fish abundance estimates for $k$ $\eqref{eq:abundance}$ are then apportioned for each $s$ across $\ell$, $\hat{N}_{s,\ell}^{k, \textrm{unaged}}$ using the computed number proportions $\eqref{eq:number_proportions_unaged_sex_length}$. The sexed estimates are then summed to compute the total unaged fish abundance estimates, $\hat{N}_{\ell}^{k, \textrm{unaged}}$:
+
+$$
+\begin{equation}
+\begin{aligned}
+    \hat{N}_{\textrm{M},\ell}^{k, \textrm{unaged}} &= \hat{N}^{k} r_{n,~\textrm{M}}^{\textrm{unaged}} \nonumber \\
+    \hat{N}_{\textrm{F},\ell}^{k, \textrm{unaged}} &= \hat{N}^{k} r_{n,~\textrm{F}}^{\textrm{unaged}} \nonumber \\
+    \hat{N}_{\ell}^{k, \textrm{unaged}} &= \hat{N}_{\textrm{M},\ell}^{k, \textrm{unaged}} + \hat{N}_{\textrm{F},\ell}^{k, \textrm{unaged}} \nonumber
+\end{aligned}
+\label{eq:abundance_unaged} \tag{15}
+\end{equation}
+$$
+
+#### Aged fish
+
+Total unaged fish abundance estimates for $k$ $\eqref{eq:abundance}$ are then apportioned for each $s$ across $\ell$ and $\alpha$, $\hat{N}_{s,\ell,\alpha}^{k, \textrm{aged}}$ using the computed number proportions $\eqref{eq:number_proportions_aged_sex_length_age}$. The sexed estimates are then summed to compute the total unaged fish abundance estimates, $\hat{N}_{\ell,\alpha}^{k, \textrm{aged}}$:
+
+$$
+\begin{equation}
+\begin{aligned}
+    \hat{N}_{\textrm{M},\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}^{k} r_{n,~\textrm{M}}^{\textrm{aged}} \nonumber \\
+    \hat{N}_{\textrm{F},\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}^{k} r_{n,~\textrm{F}}^{\textrm{aged}} \nonumber \\
+    \hat{N}_{\ell,\alpha}^{k, \textrm{aged}} &= \hat{N}_{\textrm{M},\ell,\alpha}^{k, \textrm{aged}} + \hat{N}_{\textrm{F},\ell,\alpha}^{k, \textrm{aged}} \nonumber
+\end{aligned}
+\label{eq:abundance_aged} \tag{16}
+\end{equation}
+$$
+
+#### Combining unaged and aged estimates
+
+Lastly, unaged $\eqref{eq:abundance_unaged}$ and aged $\eqref{eq:abundance_aged}$ abundance estimates can be consolidated to apportion the total abundances across $\ell$ irrespective of $\alpha$:
+
+$$
+\hat{N}_{\ell}^{k,i} = \hat{N}_{\ell}^{k, \textrm{unaged}} + \sum_{\alpha} \hat{N}_{\ell,\alpha}^{k, \textrm{aged}}
+\label{eq:abundance_length} \tag{17}
+$$
diff --git a/docs/theory/apportion.md → docs/theory/apportion_biomass.md b/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: