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2019-10-01

• 2 Stock Reduction Analysis (SRA) for conditioning DLMtool operating models
• 4 Mathematical description of the SRA model

For conditioning an operating model, it is desirable that some objective method be available (as opposed to intuition or a simple guess) to inform key historical parameters. Biological studies can be used to inform life history parameters such as growth and maturity, while other parameters such as depletion and fishing mortality have typically been informed by an assessment model. In) or effort. If the model is conditioned on catch, then the SRA will generate predicted catches that match the observed. If conditioned on effort, the estimated fishing mortality in the model will be proportional to the observed effort. A full time series of the conditioning variable is needed, and length of the historical period OM@nyears will be the length of the conditioned time series.

Ideally, the time series begins at unfished conditions. One could pass the asssumed equilibrium catch or equilibrium effort prior to the first year of data to SRA_scope. The SRA will then attempt to estimate the initial depletion in the first year of the historical period. However, this is generally difficult to estimate in the first place (consider what data are informative to estimate initial depletion, perhaps an age or length sample from that first year that shows the truncation of the composition data relative to unfished conditions).

If catch or effort data are unavailable going back to unfished conditions, then the data could be extrapolated back in time using reconstruction. Examples of catch reconstruction methods for the purposes of a stock asesssment can be found in Porch et al. (2004) and Appendix A of Starr and Haigh (2017).

In addition to the conditioning variable, additional data types can be used: - Indices of abundance (either as surveyed biomass or fishery-dependent catch-per-unit time series) - Age compositions - Length compositions - Mean lengths (this option is generally for very sparse data scenarios when mean length data are available but not the composition data)

Multiple surveys and fleets can be accommodated with SRA_scope. One of these several data types in addition to catch or effort is generally needed to obtain depletion estimates. Availability of these data can be quite sparse over time, yet still informative. For example, a recent age composition sample from a single year that shows a very truncated age structure can sufficient to imply a heavily depleted stock.

Here are the required pre-specified OM parameters needed for SRA scoping:

• Growth (length-at-age) using slots OM@Linf, OM@K, OM@t0 (or alternatively, OM@cpars$Len_age) and OM@LenCV
• Length-weight conversion factors using slots OM@a and OM@b
• Natural mortality using slots OM@M, OM@M2 or OM@cpars$M_ageArray
• Maturity using slots OM@L50, OM@L50_95 or OM@cpars$Mat_age
• Standard deviation of recruitment deviations using slot OM@Perr or OM@cpars$Perr
• Stock-recruit relationship with OM@SRrel
• If no age or length compositions, selectivity is needed with OM@L5, OM@LFS, and OM@Vmaxlen

If growth, natural mortality, or maturity are time-varying in the historical period, then the SRA will implement time-varying life history in the estimation model as well. For example, we’re setting up an operating model where the length of the historical period is 50 years, and we believe that natural mortality has doubled from 0.2 to 0.4 since Year 30 and will remain so into the future. This code can be used to setup this scenario:

The SRA will pick up this change in the model as well.

Note that time-varying life history affects calculation of reference points, in particular unfished depletion. The SRA_scope function will annually calculate the unfished depletion associated with that year’s life history values. The easiest way to turn off time-varying growth and M is to set:

Selectivity is fixed if no age or length compositions are provided. Otherwise, the ascending limb of selectivity is estimated with age or length composition data. If the selectivity is assumed to be dome-shaped, then the descending limb can either be fixed values sampled from slot OM@Vmaxlen or estimated in the SRA.

Information about the slots in the OM object can be viewed through class?OM. If passing custom objects to the operating model that override default inputs (e.g., for time-varying parameters), then DLMtool::validcpars() will be helpful for setting up and indexing the dimensions of the custom objects.

2.2 Updated parameters and SRA model evaluation

Historical OM parameters that are updated by the SRA scoping function include:

• Unfished recruitment OM@R0, only if catch is provided
• Initial depletion OM@cpars$initD
• Depletion OM@D
• Relative effort OM@cpars$Find.
• Annual recruitment deviations OM@cpars$Perr_y.
• If age or length compositions were used, selectivity parameters OM@L5, OM@LFS, and OM@Vmaxlen.

Classe Sra Slot Cars

The SRA model will estimate and return R0 when conditioned on catch. When conditioning on effort, the model is generally scale-independent; there can be information to inform depletion but not the stock size.

The exception occurs when the SRA is conditioned on effort from multiple-fleets, in which case, catch data from all fleets are needed in inform the relative F’s among fleets. In this scenario, R0 is estimated.

Additionally, if multiple fleets are used for conditioning, then the annual selectivity can change basd on the relative F among fleets. In this case, the annual selectivity is passed the OM output in the OM@cpars$V slot. The default assumption in the projection period of the closed-loop simulation is that the selectivity and relative F among fleets are identical to those in the last historical year. Fleet allocation in management procedures can be exploration in multiMSE, see vignette('multiMSE').

The relative effort provided in the output is the apical F from the SRA. When running the management strategy evaluation with DLMtool::runMSE(), the apical F may be re-scaled to ensure that specified depletion has been reached at the beginning and end of the historical period. For simple operating models, i.e. those with conditions identical to the SRA, the apical F’s in the MSE should be nearly identical to those from the SRA. To confirm that this is the case, one can run the plot function on output returned by SRA_scope:

This function returns a markdown report with:

• Histograms of updated parameters in the OM object
• Fits of the SRA model to the provided data
• Output from the SRA model, e.g. predicted recruitment
• Fits to an additional run of the SRA model to mean life history values among simulations (only when SRA_scope(..., mean_fit = TRUE) is run)
• Comparisons of the historical period of the updated OM to the SRA output

Currently, it is possible to create a more complex operating model than the SRA model itself. For example, discard mortality, movement, and spatial targetting are currently not modeled in the SRA. If these processes are specified in the operating model, then it is possible that the operating model may not necessarily match all output from the SRA.

To be added later.

4.1 Dynamics equations

Selectivity (v) for length bin (ell) and fleet (f) is length-based, where [ v_{ell,f} = left[1 + expleft(-log(19) left(dfrac{L^{50}_f - L_{ell}}{L^{95}_f - L^{50}_f}right)right)right]^{-1} ] if logistic shaped, with (L) is the midpoint of length bin (ell) and (L^{95}_f) and (L^{50}_f) are the lengths at 95 and 50% selectivity, respectively.

If dome shaped, selectivity is [ v_{ell,f} = begin{cases}expleft(-dfrac{(L_{ell} - mu_f)^2}{2(sigma^L_f)^2}right) & L_{ell} < mu_fexpleft(-dfrac{(L_{ell} - mu_f)^2}{2(sigma^R_f)^2}right) & L_{ell} ge mu_fend{cases},] where (mu_f) is the length of full selectivity and (sigma^L_f) and (sigma^R_f) control the shape of the ascending and descending limbs, respectively, of the selectivity function.

Total mortality (Z) in year (y) and for age (a) and length bin is the sum of fishing mortality (F) from all fleets for that length bin and natural mortality (M) for that age,

[ Z_{y,a,ell} = M_{y,a} + Sigma_f v_{ell,f} F_{y,f}.] The catch (in numbers) (C^N) at age and length for fleet (f) is [ C^N_{y,a,ell,f} = dfrac{v_{ell,f} F_{y,f}}{Z_{y,a,ell}} N_{y,a,ell} (1 - exp[-Z_{y,a,ell}]).]

If the model is conditioned on catch, then (F_{y,f}) are estimated. If the model is conditioned on effort, then [ F_{y,f} = q_f E_{y,f},] where (E_{y,f}) is the observed effort and (q^f) is a scaling coefficient.

Classe Sra Sloth

The population abundance (N) at age and length is calculated assuming a normally distributed length-at-age (P(ell,a)), where [ N_{y,a,ell} = N_{y,a} P(ell a) ] and

[ P(ell a) = begin{cases} phi(L'_{ell+1}) & ell = 1 phi(L'_{ell+1}) - phi(L'_ell) & ell = 2, ldots, L - 1, 1 -phi(L'_ell) & ell = Lend{cases}] with (L'_{ell}) as the length at the lower boundary of length bin (ell) and (phi(L'_{ell})) as the cumulative distribution function of a normal variable with mean (tilde{L}_{y,a}) (the expected mean length at age (a)) and standard deviation (tilde{L}_{y,a} times CV^L) ((CV^L) is the coefficient of variation in mean length at age).

After setting the equilibrium population age distribution in the first year of the model, the population abundance (after summing across all length bins) in subsequent years is [ N_{y,a} =begin{cases} R_y & a = 1 Sigma_{ell} N_{y-1,a-1,ell} exp(-Z_{y-1,a-1,ell}) & a = 2, ldots, A - 1, Sigma_{ell} N_{y-1,a-1,ell} exp(-Z_{y-1,a-1,ell}) + Sigma_{ell} N_{y-1,a,ell} exp(-Z_{y-1,a,ell}) & a = Aend{cases}] where (R_y) is the recruitment predicted in year (y).

The catch in weight (tilde{C}) is [ tilde{C}_{y,f} = Sigma_{ell}Sigma_a C^N_{y,a,ell,f} w_{ell},] where (w) is the weight at length bin (ell).

The mean length of the catch (bar{L}_{y,f}) is [ bar{L}_{y,f} = dfrac{Sigma_{ell} L_{ell}Sigma_a C^N_{y,a,ell,f}}{Sigma_aSigma_{ell} C^N_{y,a,ell,f}}. ]

The proportion of the catch-at-age is [ p_{y,a,f} = dfrac{Sigma_{ell}C^N_{y,a,ell,f}}{Sigma_a Sigma_{ell}C^N_{y,a,ell,f}}.]

The proportion of the catch-at-length is [ p_{y,ell,f} = dfrac{Sigma_a C^N_{y,a,ell,f}}{Sigma_a Sigma_{ell}C^N_{y,a,ell,f}}.]

The spawning biomass is (B^S_y) is [B^S_y = Sigma_{ell} w_{ell} Sigma_a m_a N_{y,a,ell},]

where (m_a) is the maturity at age.

4.2 Likelihoods

If the model is conditioned on catch, then the log-likelihood component (Lambda_1) of the catch is [Lambda_1 = Sigma_y Sigma_f left(-log(0.01) - dfrac{[log(tilde{C}^{textrm{obs}}_{y,f}) - log(tilde{C}^{textrm{pred}}_{y,f})]^2}{2 times 0.01^2}right),]

where (textrm{obs}) and (textrm{pred}) indicate observed and predicted quantities, respectively. With a small standard deviation for the catch likelihood relative to the variance in other likelihood components, the predicted catch will generally match the observed catch.

The log-likelihood component (Lambda_2) of survey data is [Lambda_2 = Sigma_y Sigma_s left(-log(sigma_{y,s}) - dfrac{[log(I_{y,s}) - log(q_s B_{y,s})]^2}{2sigma_{y,s}^2}right),] where (I) is the value of the survey with (s) indexing survey, (sigma_{y,s}) is the corresponding standard deviation of the survey, (q) is a scaling coefficient, and (B_{y,s}) is the reference biomass that corresponds to the index, e.g., spawning biomass or vulnerable biomass.

The log-likelihood component (Lambda_3) of catch-at-age data is [Lambda_3 = Sigma_y Sigma_f O^A_{y,f} log(p_{y,a,f})] where (O^A) is the annual sample sizes for the age compositions.

The log-likelihood component (Lambda_4) of catch-at-length data is [Lambda_4 = Sigma_y Sigma_f O^L_{y,f} log(p_{y,ell,f})] where (O^L) is the annual sample sizes for the length compositions.

Classe Sra Slot Car

The log-likelihood component (Lambda_5) of observed mean lengths in the catch is [Lambda_5 = Sigma_y Sigma_f left(-log(omega_f) - dfrac{[tilde{L}^{textrm{obs}}_{y,f} - tilde{L}^{textrm{pred}}_{y,f}]^2}{2 omega^2_f}right),] where (omega_f) is the standard deviation of mean lengths.

The log-likelihood component (Lambda_6) of annual estimated recruitment deviates (delta_y) in log space is [Lambda_6 = Sigma_yleft(-log(tau) - dfrac{delta_y^2}{2 tau^2}right),] where (tau) is the standard deviation of recruitment deviates.

Kimura, D.K. and Tagart, J.V. 1982. Stock Reduction Analysis, Another Solution to the Catch Equations. Can. J. Fish. Aquat. Sci. 39: 1467-1472.

Porch, C.E., Turner, S.C., and Schirripa, M.J. 2004. The commercial landings of red snapper in the Gulf of Mexico from 1872 to 1962. SEDAR7-AW-22. SEDAR, North Charleston, South Carolina. Available at: http://sedarweb.org/docs/wpapers/SEDAR7-AW-22.pdf (Retrieved July 9, 2019)

Starr, P.J. and Haigh, R. 2017. Stock assessment of the coastwide population of Shortspine Thornyhead (Sebastolobus alascanus) in 2015 off the British Columbia coast. DFO Can. Sci. Advis. Sec. Res. Doc. 2017/015. ix + 174 p. Available at: http://www.dfo-mpo.gc.ca/csas-sccs/Publications/ResDocs-DocRech/2017/2017_015-eng.html (Retrieved July 9, 2019)

Walters, C.J., Martell, S.J.D., and Korman, J. 2004. A stochastic approach to stock reduction analysis. Can. J. Fish. Aquat. Sci. 63: 212-223.