Equations for internal referencing
TS-length relationship
\[
\textit{TS} = m\log_{10}L+b
\]
See Equation 1.6 for more details.
Relationship between volumetric backscatter and animal number density (animals m-3)
\[
s_\text{v} = \rho_\text{v} \left< \sigma_\text{bs} \right>
\]
See Equation 1.10 for more details.
Vertical integration of volumetric backscatter in the water column
\[
s_\text{a} =
\int\limits_{z_1}^{z_2} s_\text{v} dz =
s_\text{v} H
\]
See Equation 1.11 for more details.
Vertical integration of volumetric backscatter in the water column
\[
\rho_\text{a}(x) = \frac{s_\text{a}(x)}{\left< \sigma_\text{bs} \right>}
\]
See Equation 1.13 for more details.
Areal number density (animals nmi-2)
\[
\rho_\text{A} =
\frac{s_\text{A}}{4 \pi \left< \sigma_\text{bs} \right>} =
\frac{s_\text{A}}{\sigma_\text{sp}}
\]
See Equation 1.15 for more details.
Average animal weights
\[
\left< w \right> = \frac{\sum\limits_{j=1}^N w_j}{N}
\]
See Equation 2.1b for more details.
Discretized length-weight distributions
\[
\left< w \right> = \tilde{\mathbf{L}}^{\mathsf{T}} \mathbf{w}
\]
See Equation 2.2b for more details.
Normalized length frequency distribution
\[\begin{align*}
\tilde{\mathbf{L}} =
\left[
\begin{split}
\tilde{L}&_1 \\ \tilde{L}&_2 \\ \tilde{L}&_3 \\ \vdots&
\end{split}
\right].
\end{align*}\]
See Equation 2.3 for more details.
Weight-length distribution
\[\begin{align*}
\mathbf{w} =
\left[
\begin{split}
w&_1 \\ w&_2 \\ w&_3 \\ \vdots&
\end{split}
\right].
\end{align*}\]
See Equation 2.5 for more details.
Weight-length log-linear relationship
\[
\log_{10}(\hat{w}) =
\log_{10}(\hat{a}) + \hat{b} \log_{10}(L)
\]
See Equation 2.6 for more details.
Weight estimate for length bin \(\ell\)
\[
w_\ell =
\left[
10^{\hat{a}} {(L_{\ell}^{*})}^{\hat{b}}
\right]
L_\ell
\]
See Equation 2.6 for more details.
Areal biomass density (kg nmi-2) for each sex
\[
\rho_{B; s} = \rho^i_{B; s}(x,y) = \rho_A(x,y) (\tilde{\mathbf{L}}^i_s)^\top \mathbf{w}^i_s
\]
See Equation 2.9 for more details.
Sex-specific normalized length frequency distribution
\[\begin{align*}
\tilde{\mathbf{L}}^i_s &=
\left[
\begin{split}
\tilde{L}&^i_{s,1} \\ \tilde{L}&^i_{s,2} \\ \tilde{L}&^i_{s,3} \\ &\vdots
\end{split}
\right],
\\[2ex]
\sum_{s,\ell} \tilde{L}^i_{s,\ell} &= 1.
\end{align*}\]
See Equation 2.10 for more details.
Normalized weight-length distribution for each sex
\[\begin{align*}
\tilde{\mathbf{w}}^i_s =
\left[
\begin{split}
\tilde{w}&^i_{s,1} \\ \tilde{w}&^i_{s,2} \\ \tilde{w}&^i_{s,3} \\ &\vdots
\end{split}
\right].
\end{align*}\]
See Equation 2.11 for more details.
Normalized weight-length distribution bins
\[\begin{align*}
\tilde{w}^i_{s,\ell} &=
\frac{\mathbf{w}^i_s}{\sum\limits_{\ell} w^i_{s,\ell}} =
\frac{w^i_\ell}{\sum\limits_{s,\ell} w^i_{s,\ell}},
\\[2ex]
\sum\limits_{s,\ell} \tilde{w}^i_{s,\ell} &= 1.
\end{align*}\]
See Equation 2.12 for more details.
Transect-specific mean density
\[
\hat{z}^{\,t} =
\frac{\sum\limits_{k \in t} z(x^k)}{d^t}
\]
See Equation 2.15 for more details.
Distance-based transect weights
\[
\tau^t =
\frac{d^t}{\frac{1}{n^i} \sum\limits_{t \in i} d^t}
\]
See Equation 2.16 for more details.
Stratum-specific mean density
\[
\hat{z}^{i} = \frac{1}{n^i} \sum\limits_{t \in i} \tau^t \hat{z}^{t}
\]
See Equation 2.17 for more details.
Survey mean density
\[
\hat{z} = \frac{\sum\limits_{i} A_i \hat{z}^i}{\sum\limits_i A_i}
\]
See Equation 2.18 for more details.
Within-stratum survey variance
\[
\mathbb{V}(\hat{z}^i) =
\frac{\sum\limits_{t \in i} \tau^t (\hat{z}^t - \hat{z}^i)^2}{n^i(n^i - 1)}
\]
See Equation 2.19 for more details.
Area-weighted survey variance
\[
\mathbb{V}(\hat{z}) =
\frac{\sum\limits_i (A^i)^2\, \mathbb{V}(\hat{z}^i)}{\left( \sum\limits_i A^i \right)^2}
\]
See Equation 2.20 for more details.
Random field (or spatial process)
\[
\{ Z(\mathbf{x}) : \mathbf{x} \in D \subset \mathbb{R}^d\}
\]
See Equation 3.1 for more details.
Semivariance
\[
\gamma(h) = \mathbb{C}(0) - \mathbb{C}(h)
\]
See Equation 3.4 for more details.
Empirical semivariogram
\[
\hat{\gamma}(h) = \frac{1}{2N(h)} \sum_{i<j:\; h_{ij}\approx h} \bigl(z(\mathbf{x}_j)-z(\mathbf{x}_i)\bigr)^2
\]
See Equation 3.7 for more details.
Kriging linear predictor
\[
\mathbf{z}^*(\mathbf{u}) =
\sum_{b=1}^n \lambda_b(\mathbf{u}) z(\mathbf{u}_b) +
\left[
1 - \sum_{b=1}^n \lambda_b(\mathbf{u})
\right] \mathbf{m}
\]
See Equation 3.19 for more details.
Ordinary Kriging linear predictor
\[
\mathbf{z}^*(\mathbf{u}) =
\sum_{b=1}^n \lambda_b(\mathbf{u}) z(\mathbf{u}_b) +
\left[
1 - \sum_{b=1}^n \lambda_b(\mathbf{u})
\right] \mathbf{m}
\]
See Equation 3.19 for more details.
Minimized kriging estimate variance
\[
\min \sigma_E^2(\mathbf{u}) = \min \mathbb{V}[\mathbf{z}^*(\mathbf{u}) - Z(\mathbf{u})]
\]
See Equation 3.22 for more details.
Kriging linear matrix
\[\begin{equation*}
\mathbf{\Gamma}_{n \times n} =
\begin{bmatrix}
\gamma(\mathbf{u}_1 - \mathbf{u}_1) & \gamma(\mathbf{u}_1 - \mathbf{u}_2) & \cdots & \gamma(\mathbf{u}_1 - \mathbf{u}_n) \\
\gamma(\mathbf{u}_2 - \mathbf{u}_1) & \gamma(\mathbf{u}_2 - \mathbf{u}_2) & \cdots & \gamma(\mathbf{u}_2 - \mathbf{u}_n) \\
\vdots & \vdots & \ddots & \vdots \\
\gamma(\mathbf{u}_n - \mathbf{u}_1) & \gamma(\mathbf{u}_n - \mathbf{u}_2) & \cdots & \gamma(\mathbf{u}_n - \mathbf{u}_n)
\end{bmatrix}
\end{equation*}\]
See Equation 3.23 for more details.
Kriging estimation variance
\[
\sigma_E^2(\mathbf{u}) =
- \sum_{i=1}^n \sum_{j=1}^n
\lambda_i \lambda_j
\gamma(\mathbf{u}_i - \mathbf{u}_j) +
2 \sum_{j=1}^n \lambda_i \gamma(\mathbf{u}_i - \mathbf{u})
\]
See Equation 3.24a for more details.
Simplified Lagrangian multiplier
\[
\sum_{j=1}^n \lambda_j \gamma(\mathbf{u}_j - \mathbf{u}_b) + \mu = \gamma(\mathbf{u}_b - \mathbf{u})
\]
See Equation 3.26b for more details.
Kriging optimization constraint
\[
\frac{\partial \mathcal{L}}{\partial \mu} = 2 \left(1 - \sum_{j=1}^n \lambda_j\right) = 0
\]
See Equation 3.26c for more details.
Kriging system of equations required for solving for \(\lambda\)
\[\begin{equation*}
\begin{cases}
\sum\limits_{j=1}^n \lambda_j(\mathbf{u})\gamma(\mathbf{u}_j - \mathbf{u}_i) + \mu = \gamma(\mathbf{u}_i - \mathbf{u}) & \text{for } i = 1, \ldots, n \\
\sum\limits_{j=1}^n \lambda_j(\mathbf{u}) = 1
\end{cases}
\tag{3.27}
\end{equation*}\]
See Equation 3.27 for more details.
