Stock Synthesis 2 provides a statistical framework for
calibration of a population dynamics model using a diversity of fishery
and survey data. Such models were first developed in the
1980s (Fournier and Archibald, 1982; Methot, 1989) and began to see
widespread use by the late 1990s.
The original Stock Synthesis model (Methot, 2000)
was developed in 2 versions. One was an age-length structured
model that was developed for assessment of west coast sablefish (Methot
and Hightower, 1988) and the other was an age and geographic area model
developed for Pacific whiting (Hollowed, Methot and Dorn, 1988).
Both versions of synthesis were used for most west coast
groundfish and many Alaska groundfish stock assessments during the
1990s. Stock Synthesis 2 represents a conversion of the
original synthesis from code written in FORTRAN to code written in C++
with ADMB (Otter Research Ltd., 2000). This conversion
provides an opportunity to combine the two previous versions of
synthesis while taking advantage of the advanced features of ADMB and
the many lessons learned over the past 15 years with such models.
Stock Synthesis 2 (SS2) is designed to accommodate both age
and size structure. Thus it is similar to A-SCALA (Maunders
and Watters, 2003); Multifan (Fournier et al, 1990); Multifan-CL
(Fournier, Hampton and Siebert, 1998); Stock Synthesis (Methot 2000)
and CASAL (Bull, et al, 2004) in basic structure and intent.
A general feature of such models is that they tend to cast
the goodness-of-fit to the model in terms of quantities that retain the
characteristics of the raw data. For example, age composition
data that are affected by ageing imprecision are incorporated by
building a sub-model of the ageing imprecision process, rather than to
pre-process the ageing data in an attempt to remove the effect of
ageing imprecision. By building all relevant processes into
the model and estimating goodness-of-fit in terms of the original data,
we are more confident that the final estimates of model precision will
include the potentially relevant sources of variance.
The overall SS2 model is subdivided into three sub-models.
First is the population dynamics sub-model. Here
the basic abundance, mortality and growth functions operate to create a
synthetic representation of the true population. Second is
the observation sub-model. This contains the processes and
filters designed to derive expected values for the various types of
data. For example, survey catchability relates population
abundance to the units in which survey CPUE is measured; an ageing
imprecision matrix transforms the estimated sampled numbers-at-age into
an estimate of the proportions recorded in each otolith ring count.
Third is the statistical sub-model that quantifies the
magnitude of difference between the various types of data and their
expected values and employs an algorithm to search for the set of
parameters that maximizes the goodness-of-fit. An additional
model layer is the estimation of management quantities, such as a
short-term forecast of the catch level that would implement a specified
fishing mortality policy. By integrating this management
layer into the overall model, the variance of the estimated parameters
can be propagated to the management quantities, thus facilitating a
description of the risk of various possible management scenarios.
The complexity of the population sub-model should be
considered relative to the complexity of the data and observation
sub-model. For example, if only biomass-based CPUE data are
available, it is simplest to cast the population sub-model as a simple
biomass-dynamics model such as the delay-difference model (Deriso,
1980). However, with integrated analysis it is possible to
build a more complex, age-structured population sub-model that
collapses to the simple biomass level in the observation sub-model.
If the various mortality, growth and selectivity parameters
necessary in the more complex model are fixed at levels that mimic the
inherent assumptions of the simple biomass dynamics model, then both
models produce identical results. The advantage of the more
complex internal model is that it is primed for a richer array of
sensitivity testing and immediate incorporation of more detailed data
as these data become available.
The SS2 model is primarily designed for a particular, although
not overly restrictive, set of circumstances and data. The
target species are groundfish that are harvested by multiple distinct
fleets and for which there commonly are fishery-independent surveys to
provide a time series of relative abundance. Some age and
length composition data are available from both the fishery and survey,
but they are intermittent, often based on small sample sizes, and the
age data are influenced by a substantial degree of ageing imprecision.
Tagging data are not available for these species and analysis
of tagging data has not been built into the observation sub-model. SS2
allows the stock to be sub-divided into geographic areas such that
surveys and fisheries are specific to particular areas; although the
capability to move fish between areas is incompletely developed at this
time.
The dynamics of fishing mortality and growth have been
incorporated in a way that captures the effect of size-selective
fisheries and surveys on the size and age of fish that are harvested
and sampled, and the effect of size-selective fishery harvest on the
size composition and mean growth characteristics of the fish that
survive the fishery each time period. There are three basic
levels of complexity in modeling of size in age-structured models:
(1) age-selectivity only; (2) size-selectivity influence on
observations; and (3) size-selectivity influence on survivorship.
Many integrated analysis models model the dynamics on an
age-basis only. Some of these allow inclusion of length data,
but only at the level of the observation sub-model (such as Coleraine
and the age-only version of synthesis). In such models, a
fishery that has low selectivity for 3 year old fish is assumed to
capture the same size range of 3 year olds as a fishery that has high
selectivity for 3 year olds. There is no size-selectivity in
such age-structured models even though they can estimate an expected
value for the size composition captured by the fishery. A
more complex approach is to build the size composition into the
population and to allow for size-selectivity in the characteristics of
the fisheries. Now a fishery with delayed size-selectivity
will capture larger 3 year olds and have low overall selectivity for 3
year olds compared to a fishery with higher selectivity for small fish.
Such models, such as MultiFAN, SCALA, Synthesis, model the
effect on size-selectivity on the observed samples, but do not feedback
to influence the size-specific survival. There are
several approaches to capturing the dynamics of size-specific survival.
One is to model the population as simply a size-structured
population and to use a transition matrix to update the
size-composition into the future (reference). Another is to
adjust the moments of the distribution of population size-at-age in
response to the size-selective removals (Parma et al). Here,
a third approach is used.
The stage-one and stage-two models described above treat a
cohort as a collection of homogeneous fish (referred to here as
“morphs”) whose size-at-age is characterized by a
mean and a variance. Thus, in each year the same size-at-age
distribution is recreated, irrespective of the degree of size-specific
fishing mortality. But even these models often partition the
cohort into males and females and, because the genders often have
different growth characteristics, they will experience different
effects of size-specific mortality. The stage-three model
described here extends the computational aspects of genders to multiple
growth morphs within each gender. Each growth morph has
unique growth characteristics and its numbers-at-age are tracked.
Thus growth morphs are differentially affected by
size-selective mortality. Fish within each morph are not
differentially affected by size-selectivity, but the gross effect of
size-selectivity is captured between morphs. Of course, we
have no data to identify fish to morph like we do to gender.
So expected values are summed across morphs within gender in
order to match our data. The operational assumption is that
it is more accurate to model a cohort as a collection of faster and
slower growing morphs than as a single morph.
The structure of SS2 allows for building of stage-one,
stage-two and stage-three age-length models. Selectivity can
be cast as age-specific only to create a stage-one model. A
stage-two model would define just one morph per gender and cast
selectivity as size-specific. Finally, a stage-three model
would cast selectivity as size-specific and subdivide each gender into
multiple, 3-5, morphs to capture the major effect of size-specific
survivorship.
SS2 is able to estimate the variance of parameters and derived
quantities in three ways. First, the model uses a normal
approximation to obtain the variance-covariance matrix of the
parameters and selected output quantities calculated from these
parameters. Second, ADMB based models inherit the capability
to do Monte Carlo Markov Chain (MCMC) investigations of the N-parameter
likelihood surface, thus allows non-parametric description of the
confidence envelope on parameters and derived quantities.
Third, SS2 has the capability to generate N parametric
bootstrap data sets at the completion of a model run.
Re-running SS2 on each of these data sets provides additional
information on the robustness of model convergence and the variability
of the resultant parameter estimates.
