The statistical catch-at-age model uses a time series of total catch (in weight), index, and catch-at-age observations, as well as information on weight, maturity, natural mortality at age.
Dynamics equations
The population abundance Na,t of age a in year t is Na,t={Rta=0Na−1,t−1exp(−va−1Ft−1−Ma−1)a=1,…,A−1Na−1,t−1exp(−va−1Ft−1−Ma−1)+Na,t−1exp(−vaFt−1−Ma)a=A, where Rt is the recruitment (age-1), va is the vulnerability at age a, Ft is the apical fishing mortality rate, Ma is the instantaneous natural mortality rate at age a, and A is the maximum age in the model as a plus-group accumulator age.
Selectivity
Assuming logistic vulnerability, the vulnerability is: va=[1+exp(−log(19)a−a50a95−a50)]−1, where a50 and a95 are the estimated ages of 50% and 95% vulnerability, respectively.
Assuming dome vulnerability, a double Gaussian formulation is used: va={f(a;aasc,σasc)a≤aasc1aasc<a≤adesf(a;ades,σdes)a>ades, where f(a;μ,σ)=exp(−0.5(a−μ)2/σ2) is the normal probability density function scaled to one at μ. Four parameters are estimated: a50 the age of 50% vulnerability (ascending limb), aasc the first age of full vulnerability, ades the last age of full vulnerability, and vA the vulnerability at the maximum age in the model. The μ and σ for both the ascending and descending limbs of the double-normal equation are estimated parameters. From these four parameters, σasc=√(a50−μasc)2/log(4) and σdes=√−0.5(A−μdes)2/log(vA) can be derived. The default option sets aasc=ades.
Biomass
The vulnerable biomass Vt at the beginning of year t is Vt=A∑a=0vawaNa,t, where weight-at-age wa is given by wa=W∞(1−exp[K{a−a0}])b.
The mature spawning biomass Et is given by Et=A∑a=1mawaNa,t, where maturity at age ma is ma=[1+exp(−log(19)a−˜a50˜a95−˜a50)]−1, where ˜a50 and ˜a95 are the ages of 50% and 95% maturity, respectively.
Catch-at-age
The estimated catch-at-age ˆCa,t (CAA) is obtained from the Baranov equation: ˆCa,t=ˆvaˆFtˆvaˆFt+Ma[1−exp(−ˆvaˆFt−Ma)]ˆNa,t. The predicted total catch in weight ˆYt is ˆYt=∑awaˆCa,t.
Catch-at-length (SCA_CAL)
If fishery length composition are fitted to the model, then the predicted catch-at-length (CAL) is obtained from the CAA as ˆCLℓ,t=∑aˆCa,tP(ℓ|a) where the ℓ=1,…,L indexes length bin and P(ℓ|a) is the probability of length given age for length bin ℓ.
Index
Index selectivity ˜va,s is fixed in the model.
The estimated s-th index ˆIs,t is ˆIs,t=^qs∑a˜va,swaˆNa,t and ˆIs,t=^qs∑a˜va,sˆNa,t for a biomass and abundance-based index, respectively.
Pope’s approximation (SCA_Pope)
A variant of the SCA is the SCA_Pope
function, which fixes the predicted catches to the observed catches and uses Pope’s approximation to calculate the annual exploitation rate in the midpoint of the year.
The abundance at age is
Na,t={Rta=1Na−1,t−1(1−va−1ut−1)exp(−Ma−1)a=2,…,A−1Na−1,t−1(1−va−1ut−1)exp(−Ma−1)+Na,t−1(1−vaut−1)exp(−Ma)a=A, where ut is the exploitation rate.
The vulnerable biomass in the midpoint of the year is Vmidt=A∑a=0vawaNa,texp(−0.5Ma).
By conditioning the model on catch in weight Yt, the estimated annual exploitation rate ˆut is ˆut=Yt/^Vmidt.
The predicted catch at age ˆCa,t is ˆCa,t=ˆvaˆutˆNa,texp(−0.5Ma).
Time-varying natural mortality
Random walk in M (SCA_RWM)
Annual Mt (age-invariant) can be configured as a random walk in logit space where
˜xMt+1=˜xMt+xMt.
From ˜xMt, Mt, which is bounded between Mlow and Mhigh, is calculated as
Mt=Mhigh−Mlow1−[Mhigh−MlowMt=1−Mlow−1]exp(−˜xMt)+Mlow where the M in the first year Mt=1 is either fixed or estimated with a prior.
The penalty function for constraining the random walk is
LM=∑t(−log(τM)−0.5[ˆxMtτM]2).
Density dependent M (SCA_DDM)
Annual Mt can be a function of biomass depletion (Bt/B0), Mt=M0+(M1−M0)(1−Bt/B0)
where M0 and M1 are the bounds of natural mortality as depletion approaches one and zero, respectively (Forrest et al. 2018). Density-dependence is compensatory if M0>M1 (M decreases as biomass decreases) and is depensatory if M0<M1 (M increases as biomass decreases).
Stock-recruit function
Beverton-Holt stock-recruit function
Age-0 recruitment Rt in year t is Rt=αEt1+βEtexp(δt−0.5τ2), where δt∼N(0,τ2) are recruitment deviations from the stock-recruit relationship in lognormal space and τ is the standard deviation of the recruitment deviations. Parameters α and β are defined as α=4hR0(1−h)B0β=5h−1(1−h)B0, where B0=R0ϕ0. The biomass per recruit ϕ0 is calculated as ϕ0=∑Aa=0mawala, where la={1a=0la−1exp(−Ma−1)a=1,…,A−1la−1exp(−Ma−1)1−exp(−Ma)a=A.
Ricker stock-recruit function
Recruitment Rt in year t is Rt=αEtexp(−βEt)exp(δt−0.5τ2), where α=(5h)1.25R0B0β=54B0log(5h).
No stock-recruit function in assessment model (SCA2)
Recruitment Rt in year t is Rt=ˉRexp(δt−0.5τ2), where δt∼N(0,τ2) are recruitment deviations in lognormal space from the estimated mean recruitment ˉR and τ is the standard deviation of the recruitment deviations. Typically, τ is set to 1 so that recruitment is estimated almost as free parameters (Cadigan, 2016).
Likelihoods
Age composition
The likelihood of the observed catch at age Ca,t, assuming a multinomial distribution, is LCAA=∑tOAt∑apAa,tlog(ˆpAa,t),
where Ot is the assumed sample size of catch-at-age observations in year t and ˆpa,t=ˆCa,t/∑aˆCa,t is annual predicted proportions of catch-at-age.
If a lognormal distribution for the observed proportions at age is assumed, then the likelihood is LCAA=∑t∑a(−log(0.01/ˆpa,t)−0.5[log(pa,t)−log(ˆpa,t)0.01/ˆpa,t]2).
Length composition
Similarly, the likelihood of the observed catch at length CLℓ,t with a multinomial distribution is LCAL=∑tOLt∑ℓpLℓ,tlog(ˆpLℓ,t),
where Ot is the assumed sample size of catch-at-length observations in year t and ˆpLℓ,t=ˆCLℓ,t/∑ℓˆCLℓ,t is annual predicted proportions of catch-at-length.
If a lognormal distribution for the observed proportions at length is assumed, then the likelihood is LCAL=∑t∑ℓ(−log(0.01/ˆpLa,t)−0.5[log(pLℓ,t)−log(ˆpLℓ,t)0.01/ˆpLℓ,t]2).
Catch
The likelihood of the observed catch Yt, assuming a lognormal distribution, is LY=∑t(−log(ω)−0.5[log(Yt)−log(ˆYt)ω]2).
Index of abundance
The likelihood of the observed index Is,t, assuming a lognormal distribution, is LI=∑sλs∑t(−log(σs,t)−0.5[log(Is,t)−log(ˆIs,t)σs,t]2) with λs as a weighting coefficient for survey s.
Recruitment deviations
The likelihood of the estimated log-recruitment deviations ˆδt (penalized parameters) is Lδ=∑t(−log(τ)−0.5[ˆδtτ]2).
Reference points
Time-invariant M
There are two variants. When the stock-recruit relationship is built into the model, productivity parameters R0 and h can be estimated in the assessment model. MSY reference points are derived from the estimates of R0 and h.
When no stock-recruit relationship is used in the assessment model, i.e., annual recruitment is estimated as deviations from the mean recruitment over the observed time series, similar to Cadigan (2016). After the assessment, a stock-recruit function can be fitted post-hoc to the recruitment and spawning stock biomass estimates from the assessment model to obtain MSY reference points.
Time-varying M
Random walk in M
In this case, ϕ0 is calculated from Mt=1. Annual MSY reference points can be calculated for the corresponding Mt with constant stock-recruit α and β. Thus, all MSY reference points decrease as Mt increases.
Density-dependent M
In this case, ϕ0 is calculated from M=M0. MSY is calculated by searching for the F that maximizes yield. The corresponding biomass and natural mortality, subject to the density-dependent relationship and conditional on FMSY, is solved iteratively such that
MMSY=M0+(M1−M0)(1−BMSY/B0).
Priors
See the sub-article on priors.
References
Cadigan, N.G. 2016. A state-space stock assessment model for northern cod, including under-reported catches and variable natural mortality rates. Canadian Journal of Fisheries and Aquatic Science 72:296-308.
Forrest, R.E., Holt, K.R., and Kronlund, A.R. 2018. Performance of alternative harvest control rules for two Pacific groundfish stocks with uncertain natural mortality: Bias, robustness and trade-offs. Fisheries Research 2016: 259-286.