Accounting for temporal variability in somatic growth improves a state-space assessment model for walleye pollock in the Gulf of Alaska
Giancarlo M. Correa1, Cole Monnahan2, Jane Sullivan3, James T. Thorson2, Andre E. Punt1
1University of Washington, Seattle, WA
2Alaska Fisheries Science Center, NOAA, Seattle, WA
3Alaska Fisheries Science Center, NOAA, Juneau, AK
The Woods Hole Assessment Model (WHAM)
- Fully state-space age-structured model
- Data: catch, indices of abundance, age compositions, empirical weight-at-age, environmental covariates
- Separability of total catch and proportions-at-age, as well as annual F and selectivity
- Random effects in selectivity, M, NAA, Q
- Written in TMB and R (user friendly!) ( see R package ).
Stock and Miller (2021)
The Woods Hole Assessment Model (WHAM)
Limitations:
- Lack of best practices for random effects use
- growth cannot be modeled internally
- selectivity-at-age options but not selectivity-at-length
- single sex, no spatial component, no aging error
WHAM expansion
Data inputs
- (Marginal) length compositions
- Conditional age-at-length
Growth modeling
Growth modeling overview
In the population:
Mean length-at-age (LAA)
Growth equation (von Bertalanffy)
The mean length-at-age at the start of the year (y=1):
˜Ly,a=L∞+(L1−L∞)exp(−k(a−1))
a=1 is first age in WHAM. Then, when y>1:
˜Ly,a={L1,if a=1˜Ly−1,a−1+(˜Ly−1,a−1−L∞)(exp(−k)−1)otherwise
Mean length-at-age (LAA)
Growth equation (von Bertalanffy)
Then, to calculate the mean length-at-age at any fraction of the year:
Ly,a=˜Ly,a+(˜Ly,a−L∞)(exp(−kfy)−1) fy is the year fraction.
Mean length-at-age (LAA)
Growth equation (von Bertalanffy)
Also, Ly,a and variation of length-at-age ( σy,a ) are used to calculate the age-length transition matrix (Stock Synthesis - SS - approach):
φy,l,a={Φ(L′min−Ly,aσy,a)for l=1Φ(L′l+1−Ly,aσy,a)−Φ(L′l−Ly,aσy,a)for 1<l<nL1−Φ(L′max−Ly,aσy,a)for l=nL
Where Φ is standard normal cumulative density function, L′l is the lower limit of length bin l, L′min is the upper limit of the smallest length bin, L′max is the lower limit of the largest length bin, and nL is the largest length bin index.
Mean length-at-age (LAA)
Growth equation (von Bertalanffy)
Random effects on growth parameters can be predicted (notice log scale):
log(L∞t)=μL∞+δ1,t
log(kt)=μk+δ2,t
log(L1t)=μL1+δ3,t
t represents year or cohort effects and can be iid or AR1.
Mean length-at-age (LAA)
LAA random effects
For this case, mean length-at-age ( μ˜La, notice log scale ) are assumed to be parameters and can be estimated. σy,a still needed.
Time variability can be modeled by predicting random effects:
log(˜Ly,a)=μ˜La+δy,a
δy,a can be iid or 2dAR1 (full variance-covariance matrix).
Mean weight-at-age (WAA)
Length-weight relationship
Optional when empirical weight-at-age not provided:
wl=Ω1lΩ2
Random effects on Ω1 and Ω2 can also be predicted.
Then:
ˆwy,a=∑lφy,l,awl
ˆwy,a can also be fitted to wy,a (observed mean weight-at-age)
Mean weight-at-age (WAA)
WAA random effects
Like the LAA random effects. Mean weight-at-age ( μ˜Wa, notice log scale ) are assumed to be parameters and can be estimated.
Time variability can be modeled by predicting random effects:
log(˜Wy,a)=μ˜Wa+δy,a
δy,a can be iid or 2dAR1 (full variance-covariance matrix).
Walleye pollock in the Gulf of Alaska (GOA)
Model overview
Data:
- One fishery (1970 to 2021) and six surveys
- Age compositions for fishery and surveys
- Mean weight-at-age for fishery and surveys
Parameters (penalized ML for time-varying quantities):
- Time-varying fishery age selectivity
- Two Qs vary over time
- Recruitment variability
Starting model
ADMB model vs WHAM model
Growth variability
Observed survey fish lengths:
Growth modeling strategies
Model comparison
Mean SSB estimates:
Model comparison
SSB coefficient of variation:
Model comparison
Growth parameters (only for growth parametric approach):
Model comparison
Predicted mean length-at-age (Jan 1st) vs survey observations (∼ March 1st, not included in the model):
Model comparison
AIC values for models with same input data:
| Model name | (Marginal) AIC | Δ AIC |
|---|---|---|
| LAA random effects (iid) | 827.9 | 0 |
| vB equation (iidy) | 4047.2 | 3219.3 |
| vB equation (iidc) | 4773.2 | 3945.3 |
| WAA random effects (iid) | 1188.5 | 360.6 |
Final thoughts
- New tool to explore growth modeling in state-space assessment models
- Non parametric approach seems to outperform a parametric equation
- Non parametric approach is more flexible and deals with fish shrinkage
Future directions (for this project)
- Include survey length information (e.g. marginal length compositions, conditional age-at-length data)
- Add environmental covariate
- Simulation experiment: compare strategies to account for growth variability using WHAM
- Good practices for modeling growth in state-space models
Future directions (in general)
- Software development: include sex, intraannual variability, tagging data, spatial structure
- Comparison among platforms: e.g. SS vs WHAM?
- Ecological research: meaning of all random effects options
Thanks
Cole Monnahan, Jane Sullivan, Jim Thorson, Andre Punt, Tim Miller, Jim Ianelli, Brian Stock
Contact:
gcorrea@uw.edu
giancarlo.correa@noaa.gov
Find more information:
tinyurl.com/wham-growth
Accounting for temporal variability in somatic growth improves a state-space assessment model for walleye pollock in the Gulf of Alaska Giancarlo M. Correa 1 , Cole Monnahan 2 , Jane Sullivan 3 , James T. Thorson 2 , Andre E. Punt 1 1 University of Washington, Seattle, WA 2 Alaska Fisheries Science Center, NOAA, Seattle, WA 3 Alaska Fisheries Science Center, NOAA, Juneau, AK