Process Errors for Population and Fishery Processes
Process errors to generate time-varying dynamics of population and fishery processes can be included in stock projections using AGEPRO. These process errors are defined as independent multiplicative lognormal error distributions for each life history and fishery process.
The general form for a lognormal multiplicative process error term in year \(t\), denoted by \(\varepsilon_i\), is
\[ \begin{split} & \varepsilon_i \sim \exp(w) \\ & where\ w \sim N(-0.5\sigma^2, \sigma^2)\\ \end{split} \tag{46}\label{eq:46} \]
And where the user specifies the coefficient of variation of the lognormal process error as \(CV = \sqrt{\exp(\sigma^2) - 1}\) which sets the value of \(\sigma\) as \(\sigma = \sqrt{\log(CV^2 + 1)}\).
The five population processes and four fishery processes that can include process error along with the input file keyword to specify the process are (keyword):
- Natural mortality at age (
NATMORT) \(M_a(t)\) - Maturation fraction at age (
MATURITY) \(P_{mature,a}(t)\) - Stock weight on January 1st at age (
STOCK_WEIGHT) \(W_{P,a}(t)\) - Spawning stock weight at age (
SSB_WEIGHT) \(W_{S,a}(t)\) - Midyear mean population weight at age (
MEAN_WEIGHT) \(W_{midyear,a}(t)\) - Fishery selectivity at age by fleet (
FISHERY) \(S_{v,a}(t)\) - Discard fraction at age by fleet (
DISCARD) \(P_{y,D,a}(t)\) - Landed catch weight at age by fleet (
CATCH_WEIGHT) \(W_{y,L,a}(t)\) - Discard weight at age by fleet (
DISC_WEIGHT) \(W_{v,D,a}(t)\)
For detailed documentation of projection results, the user can choose to store individual simulated values of these processes through time in auxiliary data files by setting the value of the DataFlag=1 under the keyword OPTIONS (Table 3). The auxiliary file names are constructed from the AGEPRO input filename with file extensions ranging from .xxx1 to .xxx9 for the nine processes in the bullet list above, noting that not all processes may be used in a given projection, e.g., discarding. For processes that are used, the auxiliary file names are assigned in the order in which the process parameters are set in the AGEPRO input file. For example, if the spawning stock weight at age process parameters appeared fifth in the ordering of keywords to specify these nine processes in the AGEPRO input file, then the time series of simulated spawning stock weights at age would be store in the auxiliary output file name input_filename.xxx5. Each row in this file would be the spawning weights at age for one year, in sequence, for each bootstrap replicate and simulation.
Total Stock Biomass
Total stock biomass \(B_T\) is the sum over the recruitment age (\(r\)) to the plus-group age (\(A\)) of stock biomass at age on January 1st. The computational formula for \(B_T\) in year \(t\) is
\[ B_T(t) = \sum\limits_{a=r}^{A}W_{P,a}(t) \cdot N_a(t) \tag{47}\label{eq:47} \] where \(W_{P,a}(t)\) is the population mean weight of age-a fish on January 1st in year \(t\).
Mean Biomass
Mean stock biomass \(\overline{B}\) is the average biomass of the stock over a given year. In particular, mean stock biomass depends on the total mortality rate experienced by the stock in each year. In the AGEPRO model, the user selects the range of ages to be used for calculating mean biomass. One can choose the full range of ages in the model (age-\(r\) through age-\(A\)) or alternatively select a smaller age range if desired. In this case, the upper age \(A_U\) for mean biomass calculations must be less than or equal to the plus group age \(A.\) Similarly, the lower age \(A_L\) must be greater than or equal to the recruitment age \(r\). If \(W_{midyear,a}(t)\) denotes the mean weight of age-\(a\) fish at the mid-point of year \(t\) then the computational formula for \(\overline{B}\) in year \(t\) is
\[ \overline{B}(t) = \sum\limits_{a=A_L}^{A_U}W_{midyear,a}(t) \cdot N_a(t) \cdot \dfrac{\bigl(1-\exp(-M_a(t)-F_a(t))\bigr)}{\bigl(M_a(t)+F_a(t)\bigr)} \tag{48}\label{eq:48} \]
where \(F_a(t)\) is the total fishing mortality on age-\(a\) fish calculated across all fleets.
Fishing Mortality Weighted by Mean Biomass
Fishing mortality weighted by mean biomass \(F_{\overline{B}}(t)\) in year \(t\) is the mean-biomass weighted sum of fishing mortality at age over the age range of \(A_L\) to \(A_U\) (see Mean Biomass above). This quantity may be useful for equilibrium comparisons with fishing mortality reference points developed from surplus production models. The computational formula for fishing mortality weighted by mean biomass is
\[ \begin{split} & F_{\overline{B}}(t) = \frac{\sum\limits_{a=A_L}^{A_u} \overline{B}_a(t) \cdot F_a(t) }{\overline{B}(t)}\\ & \\ & where\ \overline{B}_a(t) = W_{midyear,a}(t)\cdot N_a(t) \cdot \frac{\bigl(1-\exp(-M_a(t)-F_a(t))\bigr)}{\bigl(M_a(t)+F_a(t)\bigr)}\\ \end{split} \tag{49}\label{eq:49} \]
where \(F_a(t)\) is the total fishing mortality on age-\(a\) fish calculated across all fleets.