basa_report.tex

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  pdftitle={2023 Bayesian Age-structure Stock Assessment (BASA) Results for Prince William Sound (PWS) herring},
  pdfauthor={Joshua A. Zahner; Trevor A. Branch},
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\title{2023 Bayesian Age-structure Stock Assessment (BASA) Results for
Prince William Sound (PWS) herring}
\author{Joshua A. Zahner \and Trevor A. Branch}
\date{December 14, 2023}

\begin{document}
\maketitle

\hypertarget{executive-summary}{%
\section{Executive Summary}\label{executive-summary}}

The median expected spawning biomass of PWS herring in 2024 was
estimated at approximately 27,050 metric tons, above the minimum
threshold required for the opening of existing herring fisheries (19,958
metric tons). Taking into account uncertainity in this estimate, there
is an approximately 11\% probability that the true biomass of the PWS
herring population is below this lower cutoff. The 2024 biomass estimate
is the highest since 1993, and continues the recent trend of biomass
growth observed since 2018. The estimated age-composition for 2023 shows
a modest proportion of age-7 individuals, continuing to support the
estimate of a strong 2016 cohort, as well as relatively large
proportions of age-3 and age-4 fish, supporting a large 2020 and 2021
year-class. Based on recent trends in both the survey data and model
estimates of biomass, the PWS herring stock appears to be recovering
towards its mid-2000s level. While still a long way from the biomass
levels sustained prior to the 1993 population crash, this is a welcome
sign after nearly a decade of further biomass decline.

\hypertarget{background}{%
\section{Background}\label{background}}

Before 2014, the Alaska Department of Fish and Game (ADF\&G) ran an
Excel-based age structured assessment (ASA) model to forecast PWS
herring biomass for input into harvest control rule. The harvest control
rule has a minimum biomass threshold at 19,958 metric tons, which is
equivalent to 25\% of the unfished biomass under equilibrium determined
from simulations (Funk and Rowell 1995). When forecasted biomass is
between 19,958 and 38,555 metric tons (22,000-42,500 short tons), the
control rules scales the annual harvest rate from 0-20\% (Botz et
al.~2011). These reference points were last revised by the Alaska Board
of Fisheries in 1994.

Since 2014, the ASA has been expanded to include a Bayesian formulation
(BASA) that inherently weights the input data sources based on
statistical probability distributions, and estimates uncertainty through
the sampling of Bayesian posteriors (Muradian et al.~2017). Muradian et
al.~(2017) first demonstrated BASA as a more robust model to the
previous ASA. Since then, BASA has been used in various studies to
evaluate which historical input data were the most informative given the
trade-off between information gain and cost (Muradian et al.~2019) and
which ecological factors most likely regulate herring recruitment and
natural mortality (Trochta and Branch 2021).

Various updates have been made to the equations of BASA since Muradian
et al.~(2017) to reflect changes in the understanding of herring biology
and life history, as well as incorporate new data. These changes
included, but were not limited to:

\begin{itemize}
\tightlist
\item
  Fitting hydroacoustic data to mature biomass instead of total age 3+
  biomass (Trochta and Branch 2021)
\item
  Estimating maturity (proportions mature at ages 3 and 4) over the
  entire modeling time period instead of two time periods split at 1997
\item
  Incorporating aerial age-1 school count data to inform recruitment
  estimates before they enter the spawning population at age 3
\item
  Integrating seroprevalence and disease data to inform natural
  mortality estimates (Trochta et al.~\emph{in review})
\end{itemize}

Furthermore, a more efficient Markov Chain Monte Carlo (MCMC) algorithm
called the No-U-turn Sampler (NUTS) has been introduced to rapidly
sample posteriors of the parameters in BASA (Monnahan et al.~2019).

At present, BASA is primarily used to estimate spawning biomass and
recruitment up to the most recent year with data. Persistent low levels
of biomass and recruitment since the early 1990s continue to preclude
consideration of reopening of fisheries under the current harvest
strategy, and thus forecasts are not conducted. BASA has also been used
as a research tool to investigate hypotheses and evaluate alternative
models. In this report, we present the most recent fits and estimates
from BASA for 2021 and summarize any modifications and alternative
models explored.

\hypertarget{basa-summary}{%
\section{2023 BASA Summary}\label{basa-summary}}

To run the 2023 BASA model, the key software and versions used include:

\begin{itemize}
\tightlist
\item
  AD Model Builder v. 13.1
\item
  R v. 4.3.2
\item
  adnuts v. 1.1.2
\end{itemize}

The no-U-turn sampler (NUTS) was used within ADMB to sample the
posterior distributions of BASA parameters and derived quantities. The
`adnuts' package and its dependencies were used to run NUTS and
diagnostic checking from within R. Four NUTS chains were ran in total
with the default arguments already supplied to `sample\_nuts()'
(e.g.~warmup=700, iter=2000), except for a higher target acceptance rate
(adapt\_delta=0.9) and using the inverse Hessian as the mass matrix
(metric=`mle'). Diagnostics supported convergence in all four chains
(zero divergences and all R-hat convergence values \textless{} 1.05) and
had sufficient sample size (estimated Bulk Effective Sample Size
\textgreater{} 500 from merged chains). The total duration for running
BASA was 1.8 minutes.

Results are shown from the BASA model fits to data up to and including
2023 (Figs. 1-3). The inner 95th percentiles of the posterior predictive
distributions of the ongoing biomass survey data (Mile-days milt and
PWSSC acoustic biomass) from BASA encompass all observations (Fig. 1).
Fits of the discontinued data (egg deposition and ADF\&G acoustic
biomass) also fit well the historical time series.

\begin{center}\includegraphics[width=0.85\linewidth]{figures/survey_fits} \end{center}

\textbf{Fig. 1. Estimated survey biomass from Bayesian age structured
assessment (shading showing 50\% and 95\% posterior predictive intervals
in dark and light gray, respectively) compared to indices of biomass in
the population (points and lines showing observation CV).}

Posterior predictions of the juvenile aerial survey index (age-1
schools) bounded all observations, albeit with large uncertainty. BASA
largely overestimated the 2017 index which was the largest in the
available record, although the relative scale of this cohort (2016
age-0) agrees with the large proportions of age-3s, -4s, -5s, and -6s
observed in 2019, 2020, 2021, and 2022 (Fig. 2). The consistent
overestimation of schools since 2017 may be due to bias from a
subjective standardization used to calculate this index; schools were
numerated by four descriptive categories (small, medium, large, and
extra large) and the largest three categories were converted to and
summed as small school equivalents to calculate the index. Furthermore,
the numbers of medium, large, and extra large schools in 2017 each
represented the historical maxima in their respective categories, while
the number of small schools was the third largest. Further investigation
into the accuracy of this standardization is needed. The large aerial
survey index in 2022 is also anomalous, in that two sequential years
have never been observed to have near equally large numbers of age-1
schools. The exact reasons for this discrepency remain unknown at this
time.

\begin{center}\includegraphics[width=0.85\linewidth]{figures/age_compositions} \end{center}

\textbf{Fig. 2. Estimated age structure from the Bayesian age structured
stock assessment (points = median, lines = 95\% posterior predictive
intervals) compared to the age composition data from catches and surveys
(bars). Each color follows a single cohort as it ages through the
fishery. Data are available only for ages-3 and above. The quantity `J'
in the upper-right corner of each panel is the Shannon-Weiner evenness
index score of the age composition for each year (Shannon and Weaver,
1949).}

Posterior predictive intervals for the age composition data mostly show
good fits, except for the age-3 classes in 1987 and 1998 (Fig. 2) as
well as the age-4 and -7 class in 2023. The model misfit in 2023 could
be due to a reduced sample size from the ASL survey being used.
Estimates of the age-3 and age-4 year classes support strong 2019 and
2020 cohort. The median spawning biomass estimate in 2023 was
approximately 28,700 metric tons which is above ADF\&G's lower cut-off
for fishing (Table 1). Additionally, uncertainty in this estimate
indicates there was a 5\% probability that 2023 spawning biomass was
below this lower cutoff (Table 1).

The spawning biomass estimate for 2024 is approximately 27,050 metric
tons (95\% CI: 16,790 - 46,190), corresponding to a \textasciitilde11\%
probability that spawning biomass is below the lower regulatory cutoff
definde by ADF\&G (Figure 3).

\begin{center}\includegraphics[width=1\linewidth]{figures/management_outputs} \end{center}

\textbf{Fig. 3. Bayesian age structured assessment estimates of numbers
of age-3 recruitment in millions, spawning biomass with 95\% credibility
intervals (light gray shading), total exploitation rate, and poterior
probability density of pre-fishery biomass.}\(\\\)

\newpage

\hypertarget{basa-diagnostics}{%
\section{2023 BASA Diagnostics}\label{basa-diagnostics}}

A retrospective analysis was conducted to evaluate the effect that
adding new data has on prior predictions of stock biomass. The Mohn's
rho statistic, commonly used to assess the magnitude and direction of
retrosepctive biad, was \(\rho=-0.088\) in 2023, indicating a small
negative bias in biomass estimates in previous years. This value of
Mohn's rho is not considered indicative of substantial model
misspecification, as it falls within the accepted interval of (-0.22,
0.30; Hurtado-Ferro et al.~2015).

\begin{center}\includegraphics[width=1\linewidth]{figures/retrospective} \end{center}

\textbf{Fig. 4. 5-year retrospectives for the 2023 BASA model. Peels
show no pervasive retrospective bias.}\(\\\)

\newpage

\textbf{Table 1.} Time series of posterior percentile (PCTL) estimates
of the population from BASA. Pre-fishery spawning biomass (SB) is mature
biomass of age-3+ fish. Catch (metric tons) includes all fisheries.
Exploitation fraction is the catch divided by total age-3+ biomass.
P(SB\textless20K) is the proportion of posterior SB samples that are
less than 20,000 tons.

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\end{longtable}



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\hypertarget{references}{%
\section{References}\label{references}}

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\end{document}