NCAM

NCAM: A state-space stock assessment model for Northern cod

General

• An integrated state-space population dynamics model
• Integrated
  • Uses as much information as possible
• State-space
  • Includes process and observation equations + error
  • Population processes are informed by observations from multiple sources
  • Process errors are necessary for realistic stochastic projections
• See Cadigan 2015 [1] and 2016 [2] for details

Data inputs

• Reported landings (1983-2018)
• Catch-at-age information (1983-2018)
• Reported landings by month (1983-2018)
• Offshore research vessel (RV) survey indices (1983-2018)
• Inshore sentinel (SN) gillnet survey indices (1995-2018)
• Inshore acoustic biomass surveys and age composition data from Smith Sound (SS; 1995-2009)
• Tagging data

Biological inputs

Partial catches

Surveys

• Survey estimates are assumed to be log-normally distributed
  • \(E(I_{s,a,y})=q_{s,a}N_{a,y} exp(-t_{s,y}Z_{a,y})\)
  • \(l\left(\left\{I_{s, a, y}\right\} | \boldsymbol{\theta}\right)=\sum_{a} \sum_{y} \log \left(\varphi_{N}\left\{\frac{\log \left(I_{s, a y}\right)-E\left[\log \left(I_{s, a, y}\right)\right]}{\sigma_{s, I}}\right\}\right)\)
• Shift q for the offshore RV survey by estimating the proportion of the stock in Smith Sound \((D_{a,y})\)
  • \(E\left(I_{\mathrm{RV}, a, y}\right)=\left\{\begin{array}{ll}{q_{a} N_{a, y} \exp \left(-t Z_{a, y}\right),} & {y \neq 1995, \ldots, 2009} \\ {q_{a}\left(1-D_{a, y}\right) N_{a, y} \exp \left(-t Z_{a, y}\right),} & {y=1995, \ldots, 2009}\end{array}\right.\)
• Shift q for the inshore sentinel survey by estimating year effects that follow a random walk \((q_{s,y})\)
  • \(log(q_{s,a,y})=log(q_{s,a})+log(q_{s,y})\)

Tagging data

• Age of tagged fish ~ length-at-release | Von Bertalanffy growth model [4]
• Age-based tag returns in each experiment \((x)\) modeled using Baranov catch equation
  • \(R_{x, a, y}=\lambda_{x,y}N_{x, a, y}\left\{1-\exp \left(-Z_{x, a, y}\right)\right\} \frac{F_{x, a, y}}{Z_{x, a, y}}\)
    • note adjustment for reporting rate \((\lambda_{x,y})\)
• Adjust for incomplete mixing in release year: experiment \(~F~\) is assumed to be random deviations from whole stock \(~F~\)
  • \(\log \left(F_{x, a, y}\right) \sim N\left\{\log \left(F_{a, y}\right), \sigma_{f x}^{2}\right\}\)
• Adjust for partial exposure to the fishery \((s_{x,y})\) and other mortality sources\((t_{x,y})\), tag loss \((\gamma_y)\) and tagging mortality \((u)\)
  • \(Z_{x, a, y} = s_{x,y} F_{x, a, y} + t_{x,y} M_{a,y} + \gamma_y + u\)

Population process

• Cohort equation
  • \(N_{a,y}=N_{a-1,y-1}exp(-Z_{a-1,y-1})\)
• Baranov catch equation
  • \(C_{a,y}=N_{a,y}[1-exp(-Z_{a,y})]\frac{F_{a,y}}{Z_{a,y}}\)
• Process error \((\delta)\) is added to baseline \(m\) values
  • \(log(M_{a,y})=log(m_{a,y})+\delta_{a,y}\)
• \(\delta\) and \(F\) modeled using 2D age-year AR1 process
  • \(\operatorname{Cov}\left(\delta_{a, y}, \delta_{a-j, y-k}\right)=\frac{\sigma_{\delta}^{2} \varphi_{\delta, \mathrm{age}} \varphi_{\delta, \mathrm{year}}^{k}}{\left(1-\varphi_{\delta, \mathrm{age}}^{2}\right)\left(1-\varphi_{\delta, \mathrm{year}}^{2}\right)}, \quad \operatorname{Corr}\left(\delta_{a, y}, \delta_{a-j, y-k}\right)=\varphi_{\delta, \mathrm{age}}^{j} \varphi_{\delta, \text { year }}^{k}\)
  • \(\operatorname{Cov}\left[\log \left(F_{a, y}\right), \log \left(F_{a-j, y-k}\right)\right]=\frac{\sigma_{F}^{2} \varphi_{F, \text { age }}^{j} \varphi_{F \text { year }}^{k}}{\left(1-\varphi_{\text {Frage }}^{2}\right)\left(1-\varphi_{F, \text { year }}^{2}\right)}\)

Summary

• \(N\) informed by reported landings and survey indices
  • Challenges: bounds and catchability
• \(F\) and \(M\) informed by catch composition, survey indices and tagging data
  • Challenges: catchability, reporting rate, incomplete mixing
• Different parameters confound population estimates from each data-set
  • Statistical triangulation \(\rightarrow\) underlying population process
  • Same process \(\therefore\) conflicting trends \(\rightarrow\) poor fits

References

1. Cadigan, N. G. A state-space stock assessment model for northern cod, including under-reported catches and variable natural mortality rates. Canadian Journal of Fisheries and Aquatic Sciences 73, 296–308 (2015).

2. Cadigan, N. G. Updates to a northern cod (Gadus morhua) state-space integrated assessment model. DFO Canadian Science Advisory Secretariat Research Document 2016/022, 58 p. (2016).

3. Cadigan, N. G. Weight-at-age growth models and forecasts for northern cod (Gadus morhua). DFO Canadian Science Advisory Secretariat Research Document 2016/016, 19 p. (2016).

4. Cadigan, N. G. & Konrad, C. A cohort time-series von bertalanffy growth model for northern cod (Gadus morhua), and estimation of the age of tagged cod. DFO Canadian Science Advisory Secretariat Research Document 2016/017, 37 p. (2016).

Proportions at age

Proportions at age (frame)

Residuals

based on continuation ratio logits

Proportions at age surface

Fishery selectivity

RV survey index

Y-axes in log scale

Index at age

Y-axes in log scale

Index at age (frame)

Y-axes in log scale

Residuals

Residuals (bubble)

Sentinel gillnet survey index

Y-axes in log scale

Index at age

Y-axes in log scale

Index at age (frame)

Y-axes in log scale

Residuals

Residuals (bubble)

Smith sound acoustic biomass estimates

Numbers at age

Numbers at age (frame)

Residuals

Residuals (bubble)

Proportion of the stock in Smith Sound

Sentinel gillnet survey catchability

Recaptures

Y-axis is in square root scale

Residuals

Observation and prediction axes are in square root scale

Initial F deviations

Thetas

Reporting rates

Stock status

Mohn’s rho

Relative difference = (retro estimate - current estimate) / current estimate; Mohn’s rho = mean(relative difference)

Difference

Difference = retro estimate - current estimate; approximate 95% intervals based on retro standard deviation + current standard deviation

Mortality Rates

Mohn’s rho

Relative difference = (retro estimate - current estimate) / current estimate; Mohn’s rho = mean(relative difference)

Difference

Difference = retro estimate - current estimate; approximate 95% intervals based on retro standard deviation + current standard deviation

Stock status

Abundance at age

SSB at age

Demographics

Totals are lower and upper 95% confidence intervals

Mortality Rates

M at age

F at age

Z, M and F

Z = purples, M = blues, F = reds

Process error

Recruitment

SSB

RPS

SSB x Recruitment

SSB x RPS

log(SSB) x log(RPS)

Catch multiplier projections for SSB_yp / SSB₂₀₁₉

SSB / Blim

Prob(SSB_yp ≥ SSB₂₀₁₉)

Prob(SSB_yp ≥ B_lim)

Landings

Catch at age

RV index

SN index

SS index

SS age composition

Weight at age

Maturity at age

Tag summary

Catch bounds

Baseline M