Selectivity
For length-based parametric selectivity functions (dome or logistic), then parameters \(x^{FS}_b\) and \(x^{L5}_b\) for block \(b\) are estimated over all real numbers, where
\[ \begin{align} L^{\textrm{FS}}_b &= L_{\infty} \times \textrm{logit}^{-1}(x^{FS}_b)\\ L^5_b &= L^{\textrm{FS}}_b - \exp(x^{L5}_b) \end{align}\]
If a third parameter \(x^{V}_b\)is estimated for dome selectivity, then \[ V^{L_{\infty}}_b = \textrm{logit}^{-1}(x^V_b)\]
If the parametric selectivity function is age-based, then
\[ \begin{align} a^{\textrm{FS}}_b &= A \times \textrm{logit}^{-1}(x^{FS}_b)\\ a^5_b &= a^{\textrm{FS}}_b - \exp(x^{a5}_b)\\ V^A_b &= \textrm{logit}^{-1}(x^V_b) \end{align}\]
where \(A\) is the maximum age.
If selectivity is parameterized as free parameters, then \[ v_{y,a,f} = \textrm{logit}^{-1}(x^v_{y,a,f}).\]
For indices, parameterizations are identical except with indexing for survey \(s\).
Fishing mortality
If \(F_{y,f}\) are estimated parameters (condition = "catch"), then one parameter \(x^F_f\) is the estimated \(F\) in log-space in the middle of the time series is estimated and all others are subsequent deviations, represented as \(x^{F_{dev}}_{y,f}\):
\[ F_{y,f} = \begin{cases} \exp(x^F_f) & y \textrm{ is midpoint of the time series}\\ \exp(x^F_f) \times \exp(x^{F_{dev}}_{y,f}) & \textrm{otherwise}\\ \end{cases} \]
If condition = "effort", then \(q_f\) is estimated in log space, where \[F_{y,f} = q_f E_{y,f} = \exp(x^q_f) \times E_{y,f}\]
Index catchability
To scale stock size to index values, the index catchability \(q_s\) is solved analytically in the model as
\[ q_s = \exp\left(\dfrac{\sum_y \log(I^{\textrm{obs}}_{y,s}) - \sum_y \log(\sum_a v_{y,a,s}N_{y,a,s})}{n_s}\right),\]
or \[ q_s = \exp\left(\dfrac{\sum_y \log(I^{\textrm{obs}}_{y,s}) - \sum_y \log(\sum_a v_{y,a,s}N_{y,a,s}w_{y,a})}{n_s}\right),\] for an abundance or biomass based index, respectively, where \(n_s\) is the number of years with index values and the summation is over those \(n_s\) years.
Other parameters
Unfished recruitment is estimated in log-space, \(R_0 = \dfrac{1}{z}\exp(x^{R_0})\) where \(z\) is an optional rescaler, e.g. mean historical catch, to reduce the magnitude of the \(x^{R_0}\) estimate.
Recruitment deviations \(\delta_y\) are estimated in log space.
The support of the steepness parameter \(x^h\) is over all real numbers and is transformed. With the Beverton-Holt SR function: \[ h = 0.8 \times \dfrac{1}{1 + \exp(-x^h)} + 0.2. \]
With a Ricker SR function: \[ h = \exp(x^h) + 0.2.\]
Steepness is fixed unless a prior is used.
Age- and time-constant natural mortality \(M\) can be estimated with a prior. Otherwise, it is fixed to values specified in the OM object. The parameter \(x^M\) is estimated in log space, i.e.,
\[ M = \exp(x^M)\]