calc_F: Newton-Raphson search for fishing mortality
In multiSA: Multi-Stock Assessment

calc_F

R Documentation

Newton-Raphson search for fishing mortality

Description

Performs a root finding routine to find the index of F that minimizes the difference between observed catch and the value predicted by the Baranov equation.

Usage

calc_F(
  Cobs,
  N,
  sel,
  wt,
  M,
  q_fs,
  delta = 1,
  na = dim(N)[1],
  nr = dim(N)[2],
  ns = dim(N)[3],
  nf = length(Cobs),
  Fmax = 2,
  nitF = 5L,
  trans = c("log", "logit")
)

Arguments

`Cobs`	Observed catch. Matrix `⁠[f, r]⁠`
`N`	Stock abundance at the beginning of the time step. Array `⁠[a, r, s]⁠`
`sel`	Selectivity. Array `⁠[a, f, s]⁠`
`wt`	Fishery weight at age. Array `⁠[a, f, s]⁠`
`M`	Instantaneous natural mortality. Units of per year `⁠[a, s]⁠`
`q_fs`	Relative catchability of stock `s` for fleet `f`. Defaults to 1 if missing. Matrix `⁠[f, s]⁠`
`delta`	Numeric, the duration of time in years corresponding to the observed catch, e.g., 0.25 is a quarterly time step.
`na`	Integer, number of age classes
`nr`	Integer, number of regions
`ns`	Integer, number of stocks
`nf`	Integer, number of fleets
`Fmax`	Numeric, the maximum Findex value
`nitF`	Integer, number of iterations for the Newton-Raphson routine
`trans`	Whether to perform the search in log or logit space

Details

The predicted catch for fleet f in region r is

C^{\textrm{pred}}_{f,r} = \sum_s \sum_a v_{a,f,s} q_{f,s} F_{f,r} \dfrac{1 - \exp(-Z_{a,r,s})}{Z_{a,r,s}} N_{a,r,s} w_{a,f,s}

The Newton-Raphson routine minimizes f(x_{f,r}) = C_{f,r}^{\textrm{pred}} - C_{f,r}^{\textrm{obs}}.

If trans = "log", F_{f,r} = \exp(x_{f,r}).

If trans = "logit", F_{f,r} = F_{\textrm{max}}/(1 + \exp(x_{f,r})).

The gradient with respect to \vec{x} is

f'(x_{f,r}) = \sum_s \sum_a v_{a,f,s} q_{f,s} N_{a,r,s} w_{a,f,s} \left(\dfrac{\alpha\gamma}{\beta}\right)'

\left(\dfrac{\alpha\gamma}{\beta}\right)' = \dfrac{(\alpha\gamma' + \alpha'\gamma)\beta - \alpha\gamma\beta'}{\beta^2}

where

\alpha_{f,r} = F_{f,r}

\beta_{a,r,s} = Z_{a,r,s} = M_{a,s} + \sum_f v_{a,f,s} q_{f,s} F_{f,r}

\gamma_{a,r,s} = 1 - \exp(-Z_{a,r,s})

\beta'_{a,f,r,s} = v_{a,f} q_{f,s} \alpha'_{f,r}

\gamma'_{a,f,r,s} = \exp(-Z_{a,r,s})\beta'_{a,f,r,s}

If trans = "log", \alpha'_{f,r} = \alpha_{f,r}.

If trans = "logit", \alpha'_{f,r} = F_{\textrm{max}}\exp(-x_{f,r})/(1 + \exp(-x_{f,r}))^2.

This function solves for \vec{x} by iterating until f(\vec{x}) approaches zero, where the vector arrow indexes over fleet and region. In iteration i+1:

\vec{x}_{i+1} = \vec{x}_i - \dfrac{f(\vec{x}_i)}{f'(\vec{x}_i)}

Value

A list containing:

F_afrs Fishing mortality array
F_ars Fishing mortality array (summed across fleets)
Z_ars Total mortality array
F_index Index of fishing mortality. Matrix ⁠[f, r]⁠
CB_frs Catch (biomass) array
CN_afrs Catch (abundance) array
VB_afrs Vulnerable biomass at the beginning of the time step. Array
penalty Penalty term returned by posfun() when F_index exceeds Fmax
fn Difference between predicted and observed catch at the last iteration. Matrix ⁠[f, r]⁠
gr Gradient of fn with respect to F_index in either log or logit space at the last iteration. Vector by ⁠[f, r]⁠