SCALIA: Application of an integrated analysis stock ...

SCALIA: Application of an integrated analysis stock assessment model to the 2002 SCTB Methods Working Group

simulated tuna fishery data

Dale Kolody

CSIRO Marine Research Hobart, Australia

SCTB15 Working Paper

MWG−5

SCALIA: Application of an Integrated Analysis Stock Assessment Model to the 2002 SCTB Methods Working Group Simulated Tuna Fishery Data

Dale Kolody

CSIRO Marine Research PO Box 1538, Hobart, Tasmania, 7001, Australia

email: [email protected]

Abstract

This document briefly describes the Statistical Catch-at-Age/Length Integrated Analysis modelling framework (SCALIA) under development by the CSIRO Tropical and Pelagics Ecosystems group (Hobart, Australia) as applied to the assessment of 22 simulated tuna fishery data sets generated for the SCTB Methods Working Group. In this instance, SCALIA is an age-structured, state space model that uses data on total catch, catch length frequencies, effort, and tag release/recapture to make inferences about the fish and fishery dynamics. We have included a brief description of the model dynamics and assumptions, emphasizing differences from similar approaches (MultiFan-CL and A-SCALA) from which several features are derived. Typical results for a reference case are illustrated to show the general character of the model. For several reasons, the resultant maximum likelihood estimates of the fishery dynamics should not be interpreted as our best attempt at 22 stock assessments, but we do consider them to be plausible and sufficient to demonstrate several concerns that we have with SCALIA (and related approaches) that merit further consideration. General comments for future assessment model evaluation projects are included.

Actual results from the 22 simulated data sets are submitted to SCTB-15

MWG in electronic format and not summarized here. The actual MWG operating model state dynamics used to simulate the data remain unknown to the analyst at the time of writing.

Introduction With the increase in cheap computing power and efficient function minimization software, stock assessment modelling methodology is rapidly becoming more complicated, and many are concerned that model evaluation and general understanding has not kept pace with these advances. MultiFan-CL (eg Fournier et al 1998) is one of the most sophisticated single species assessment approaches, originally designed to make inferences about the dynamics of Western and Central Pacific Ocean tuna fisheries. This approach is intuitively appealing because many diverse data are integrated into a unified picture of the stock dynamics with a minimum of intermediate analytical steps. However, it is also recognized that the inferential characteristics of relatively complicated models are not necessarily better than relatively simple models, even though they tend to be more realistic in some sense. This is often described in terms of a trade-off in the bias and precision of the estimators. However, there are additional concerns about the practical implementation of complicated models (eg technical skill and computing power required, general understandability, etc). These problems are not limited to fisheries applications, as there is a general need for more research into methods for identifying the optimal model abstraction required for a given task.

It was proposed that some work should be conducted under the guidance of the SCTB Methods Working Group to evaluate the inferential performance of MultiFan-CL and alternative assessment methods. The initial stages of this evaluation were to be undertaken with 22 (relatively) simple data sets generated by an operating model developed by the Secretariat of the Pacific Community Oceanic Fisheries Programme (SPC-OFP, Marc LaBelle, pers comm.). These simulated fisheries resemble the actual yellowfin tuna fishery, and include complicated fine-scale spatio-temporal dynamics (but these fine-scale data were not available to assessment analysts). Since the true underlying dynamics are known (although ideally not to the assessment analysts beforehand), the inferential performance of the assessment models can be evaluated by comparing the known and estimated dynamics. This document provides a brief overview of the application of the Statistical Catch-at-Age/Length Integrated Analysis modelling framework (SCALIA) under development at CSIRO. Key descriptors of the estimated stock dynamics of plausible assessments are submitted to the SCTB-MWG but not presented here. I have attempted to point out key differences in the SCALIA methodology relative to MultiFan-CL and A-SCALA (Maunder and Watters 2000), and describe some problems encountered (that may be relevant to other approaches). The Simulated Data Two types of fishery operating model scenarios were defined, one with a single aggregated fishery, and another with two separate fisheries (that seemed to have different selectivities and operated in separate areas). Eleven data sets were generated for each of these scenarios (22 total). There were some obvious differences between the data sets (eg recruitment variation), but limited direct prior information about how similar they were (effort patterns were identical among data sets within scenarios). The simulated data consisted of 148 quarterly observations of total catch, catch length frequency distributions and effort by fishery, plus tag releases by area (including release lengths) and recaptures by fishery (and length at release).

Additional biological information was provided on length-at-age, weight-at-length, and maturity-at-age. Total catch and effort fluctuated considerably over the time series (with strong seasonal variations), but strong long term trends were not obvious.

SCALIA SCALIA is really a flexible framework that has been initially developed for Southern Bluefin Tuna stock assessment (Kolody and Polacheck 2001). Most (or all) of the SCALIA features have been implemented and documented by other authors for other species. The acronym is probably most meaningful as an indicator of the group involved with development, rather than any specific features. As applied to the SCTB-MWG data, SCALIA is essentially a less-sophisticated version of MultiFan-CL and A-SCALA (substantive SCALIA features that are distinct from these approaches were not deemed to be useful in this exercise). SCALIA is still under development, and undergoing a series of simulation tests independent of this SCTB-MWG project.

In this instance, SCALIA is a state-space model that uses difference equations to describe a spatially-aggregated, age-structured fish population harvested by multiple distinct fleets. Dynamics and general assumptions are documented in Appendix 1. The specific assumptions of the model as applied to the SCTB-MWG data are described in Table 1. Some potentially important features in the current application that might differ from other approaches include: (see Appendix 1 for further elaboration)

• Selectivity is parameterized in a pseudo-length-based manner. ie a purely age-based selectivity vector is used in the dynamics, but it is actually derived from a vector of length-based parameters and the fish length-at-age distributions.

• The CV on effective effort deviations is assumed to be inversely related to the

square of the effort. This was implemented to allow for the fact that extremely low and high catch rates tended to be more evident in quarters with very low effort. This observations are qualitatively consistent with a very patchy fish distribution and imperfect fleet searching. However, the selected exponent of 2 was rather arbitrary.

• Tagged fish were assigned a discrete release age based on ‘cohort-slicing’

(and the implicit assumption that all ages were equally abundant at the time of ageing).

• Tags are assumed to be fully mixed into the general population after two

quarters, and the Poisson likelihood for tag recapture probabilities includes an effective release size co-efficient that is intended to reflect the idea that the theoretical distribution under-estimates the recapture variance. The likelihood was based on tag predictions aggregated across both fisheries (in the two fishery case).

• There was a priori information that the fishing gear had monotonically

increasing selectivity with age. This was ignored in the model because global selectivity is dependent on both local gear selectivity and local abundance.

Thus, if the age distribution is not uniformly distributed in space, then the global realized selectivity will not reflect the local selectivity.

• There was a priori information that technological improvements occur over

time (suggesting a monotonic increase in catchability). This was ignored for the same reason as above (global catchability may differ from local catchability if fish or effort distributions are not spatially uniform over time).

• There was a priori information about the upper and lower bounds of fish size-

at-age distributions at discrete ages, however, these are not a sufficient description of the catch length-at-age distributions that are observed when data are aggregated over a quarter (especially for younger ages due to growth within a quarter). The sizes provided were used as a starting estimate for the age-length relationships, but were re-estimated during model fitting.

• A multinomial likelihood for the catch-at-length distribution was used with an

input effective sample size (1000) generally well below the actual sample size to admit that catch-at-length sampling is probably not truly random (and the length-at-age model is imperfect). MultiFan-CL and A-SCALA use a robustified likelihood for catch-at-length (originally derived in Fournier et al 1990).

The best estimate of the current and historical state of the stock is estimated by the minimum of the (pseudo-) log-likelihood objective function (maximum of the joint Bayesian posterior) that includes terms for observed data (catch in numbers, catch length frequency distributions and tagged fish recoveries), a number of priors applied to process error estimates, and additional constraints to reduce the effect that over-parameterization can have on the minimization process: Objective function terms (see Appendix 1):

• total catch in numbers by fishery (log-normal) • catch length frequency distributions by fishery (multinomial) • tag recovery likelihood (Poisson) • recruitment deviation priors (log-normal with weak lag-1 autoregressive

component) • fishery catchability time series change priors (log-normal) • effective effort deviation priors (log-normal) • third difference curvature penalty on age-based selectivity • mortality-at-age deviations from mean (log-normal) • third difference curvature penalty on natural mortality • several constraints that are applied at intermediate stages in the function

minimization for numerical stability (these not active in the final stage) Parameters estimated (approximate number):

• pseudo-length-based selectivity (6 per fishery) • effort deviations (148 per fishery) • catchability (mean, 6 time series deviations, and 3 seasonal effects per fishery) • recruitment (mean plus 168 quarterly deviations; includes cohorts in initial

population)

• natural mortality (mean plus 20 age-class deviations) • tag reporting rates (1 per fishery) • means of length-at-age distribution (4; growth curve) • standard deviations of length-at-age (2, linear function of mean length-at-age) • total scenario 1: 355; total scenario 2: 520

The objective function minimization was conducted with AD Model Builder software (Otter Research, Victoria, Canada) in several phases. In general, parameter constraints are more relaxed in each successive phase. Some constraints were applied during intermediate phases of the minimization, but were not present in the final phase. All of the above parameters were estimated in the penultimate phase, however, length-at-age parameters were shut off in the final phase. This was done to avoid minimization problems that may occur when tag ages are assigned by cohort-slicing (ie a small change in the length-at-age distribution can cause a discontinuity in the objective function because tag ages are re-assigned to discrete integer values). For the results submitted to the SCTB-MWG 15, we assumed that the minimization was successful if the AD Model Builder software successfully calculated the inverse Hessian matrix (but see problems below). In accordance with our understanding of the MWG objectives at this time, we did not attempt to express any uncertainty in the final results. Results and Discussion Weaselly Caveats

SCALIA results that we are presenting at SCTB 15- MWG were conducted under less than optimal conditions and should not be considered our best attempt at 22 stock assessments. The SBT implementation of SCALIA was substantially modified for this exercise, potentially introducing new coding errors and minimization problems. The assumptions summarized in Table 1 represent a rather unsystematic compilation of features, reflecting some guesswork based on experience with real fisheries, but a lot of these values were derived after iterative cycles of attempting to obtain convergence, attempting to second guess the operating model operator, and modifying input parameters somewhat haphazardly. Additional options with appealing characteristics were explored (eg ad hoc likelihood robustification) but did not result in obvious improvements. Without the benefit of informative feedback (and given the goal of submitting a single estimated state history), it was difficult to justify further extensions with the limited time available.

Last minute addendum: according to the operating model descriptions provided in LaBelle (2002), these simulated assessments had at least two substantial flaws:

- I intentionally rejected results that estimated a tag reporting rate of 1.0 (evidently perfect tag reporting is not as implausible as I thought)

- the input maturity-at-age vector (and hence SSB calculations) seems to be incorrect.

LaBelle, M. 2002. Testing the accuracy of MULTIFAN-CL assessments of the WCPO yellowfin tuna fishery conditions. SCTB 15 MWG-1

Fits to Data

In general, a single assessment model specification tended to yield rather similar estimates of M, selectivity and catchability across all data sets (within each of the two scenarios) and it will be interesting to find out whether this is an artifact of the assessment model (and data collection regime), or whether the simulated state realizations were in fact rather similar (except for the obvious recruitment variation). MWG2-1 (ie scenario 2, data files 1.frq and 1.tag) was arbitrarily selected to illustrate a number of fairly typical results from all data sets (although there were some important differences between the scenarios).

Total catch and effort from MWG2-1 are illustrated in fig. 1 to illustrate the strong seasonal variability, and medium-high effort from the beginning of the time series. Predicted and observed total catches are essentially identical as would be expected from the model assumptions (eg CV 0.01, Table 1).

In general, the predicted and observed catch-at-length frequency distributions

were in good agreement (fig. 2), with clearly defined “modal progressions” corresponding to the sequential growth of larger than average cohorts. The largest discrepancies were always associated with the length modes of the younger age classes; the mean and variance of the youngest length-at-age was poorly fit by the model (fig. 3). Estimation of age 1 length-at-age independently of the other ages largely resolved the problem, but this addition was omitted from the baseline results presented. In some cases, the model seemed to estimate two consecutive recruitments of moderate size where a single large recruitment might have been more appropriate. An approximation of the (post-fit) effective sample size (McAllister and Ianelli, 1997) provides a measure of how well CL predictions and observations agree in each year:

�

�

−

−=

lltlt

lltlt

l

po

ppESS

2,,

,,

)(

)1(,

where p and o are the predicted and observed proportions of catch in numbers in each length class frequency distributions. For MWG2-1 the fishery 1 mean post-fit ESS (over all timesteps) was 1687 (minimum of 96) and fishery 2 mean 2006 (minimum 91; fig. 3). This was typical of all data sets, and seems to be reasonable given the input effective sample sizes of 1000 (increasing this input value to 10000 seemed to result in only minor improvements to this measure, while decreasing it to 100 resulted in convergence problems). There were auto-correlated patterns in the post-fit ESS values over time (again likely related to large recruitment events and the bias in the youngest length-at-age), but long term trends were not obvious.

Qualitatively, it appears that predicted tag returns do not fit the data as well as

the CL distributions. While predicted and observed total tag recovery patterns are in reasonable agreement (fig.4), individual age classes were considerably less well fit. There was evidence of an under-estimation bias for plus group tag recaptures. It seems that the tags from the first scenario generally did not fit as well as the second.

This could simply be due to the smaller sample sizes, or possibly the added complexity in the second scenario allowed more freedom to (over-) fit the tags. Increasing the effective tag release co-efficient to 1.0 did not result in an obvious qualitative improvement to the predicted tag recoveries, but did seem to cause irregular behaviour in the estimated selectivities. In scenario 1, tag reporting rates were generally estimated to be around 0.25, and scenario 2: 0.8 for Fishery 1 and 0.5 for fishery 2.

The length-at-age distribution estimated by SCALIA resembles the upper and

lower bounds of length-at-age that were provided with the simulated data (fig. 5). One can infer that the means are similar, while SCALIA variance seems to be much larger than the age-length relationship provided (the comparison is difficult because exact definitions of the bounds were not provided). Part of the discrepancy arises because the provided data are instantaneous values for a given age, while SCALIA estimates represent an aggregate over the entire quarter and reflect the process of growth within a quarter. This is an important factor for younger ages, but much less important for the older ages.

Estimated State Dynamics

Catchability trends were estimated to increase fairly monotonically by a total

of about 10-30% for fishery 1 (fig. 6). This was also true for scenario 1 (not shown). In contrast, fishery 2 was estimated to have a declining trend in the latter part of the time series in almost all cases (unless highly constrained). However, this description is a bit artificial since catchability is highly confounded with effective effort deviations, which also tended to show some auto-correlated behaviour (fig. 6b). Furthermore there appear to be seasonal fluctuations in the effort deviations that were not fully captured by the seasonal catchability effects.

Mortality-at-age was generally estimated to be between 0.1-0.15 (per quarter),

with smooth trends but never monotonic over all ages (fig. 7). Fishery 1 always showed a selectivity targeting younger ages than fishery 2

(fig. 8). Fishery 2 usually showed a monotonic increase with age, while fishery 1 usually did not (unless constrained to do so).

The estimated stock dynamics showed some fairly consistent features in all

cases. Recruitment was highly variable (log CV around 0.75 for MWG-1), without much autocorrelation (0.07 for MWG-1) this contrasts with the input CV of 0.55 and autocorrelation of 0.4 (the output values did not appear to change substantially unless the input values were drastically altered). Total and spawning stock sizes were estimated to drop to about 25% of unfished levels within the first few quarters, and then fluctuated around ~15-40 % (occasionally higher) due to recruitment variation, for the remainder of the time series (fig. 9).

Perceived Problems with this application of SCALIA There are a couple of major concerns about this application of SCALIA that are presumably relevant to most models of this type:

• I am not confident that global minima were always identified in fitting the

model to these data sets. In some of the preliminary fittings, I found substantially different minima by changing initial values, the order of the parameter phasing and/or the number of minimizer iterations in the intermediate phases. Inspection of the quality-of-fit diagnostics would not always have been sufficient to indicate that a lower minimum ought to exist. The extent of this problem was not thoroughly investigated (it was less obvious in the later stages of the development process, in part because I stopped looking for it), but hopefully this type of exercise will help to identify how important of an issue this really is.

• SCALIA results often appeared to be sensitive to the externally-specified (and

poorly quantified) variance-related parameters (to some extent, this might be an extension of the global/local minimization problem above). I could have made more effort to ensure that input and output variances were more closely aligned, but given the general problem of estimating multiple variances in these types of models, this is not really a solution. Fig. 10 illustrates a range of different Total Stock Biomass (TSB) histories that arise when input variances (related to effort deviations, catachability, CL effective sample sizes and/or mortality constraints) are perturbed from the reference case by a factor of up to 2. The early part of the histories are very consistent in both relative and absolute terms, while TSB over the last 10 quarters are rather divergent (as in most stock assessment models the recent years are most poorly estimated). It is not obvious to me what diagnostic of quality-of-fit or model behaviour could be used to objectively distinguish which trajectory is better. In this case, the trajectory with the highest terminal TSB had a much lower negative log-likelihood (largely due to greater input variances) and similar or better fit to the CL and tagging data (in terms of variance-independent measures: post-fit ESS for catch-at-length and RMSE of predicted tags by time and age). However, this specification was considered less desirable than the baseline case, because tag reporting rates converged on the (presumably implausible) upper bound of 1.0. Clearly, there is a need to test for and express these sensitivities and uncertainties.

Additionally, there are some implementation concerns that are more specific to SCALIA: • The length-at-age estimation could have been more effective through the use

of priors on variances of older ages, and independent estimation of age 1 mean and standard deviation.

• Tagging data might be more informative if dis-aggregation by fishery, and

possibly release event, was maintained, rather than aggregating into a single mixed-tag population. Furthermore, alternative forms of the recapture likelihood should be investigated (eg negative binomial is becoming popular for tagging data). Variance inversely related to release time might be appropriate.

• Robustified likelihoods are worth further consideration, particularly for effort deviations, recruitment deviations, and catch-at-length prediction.

• The confounded nature of temporally changing catchability, seasonal

catchability, and effective effort process error merits further consideration.

• Spatial structure will likely be implemented at some point, to help identify when it is appropriate.

Concluding Comments about Fur ther MWG Simulation-Estimation Evaluation Exper iments

Feedback provided by these types of simulation studies are probably the best way to figure out which system features are important and estimable for a given modelling task. As such, further investigations should be undertaken. I would suggest the following points:

• There will probably need to be several simulated data sets with various levels

of process and observation error. In defining these operating models, we need to be careful not to condition too closely on actual assessments to avoid the problem of the “self-fulfilling prophesy” . Until a range of trials are undertaken, we will not know how well the results generalize across systems.

• These trials are more convincing with greater degrees of blindness in the

assessment analysts. eg In all but the simplest cases, is it possible for the person who coded and parameterized the operating model to make unbiased decisions about how to specify the assessment model? However, there needs to be an appropriate amount of information to prevent the results of the assessment from being determined by circumstances that are irrelevant to the evaluation process. eg In the examples illustrated in fig. 10, I rejected results that estimated a tag reporting rate of 1.0, but the operating model specifier might have been interested in knowing what the effect of perfect reporting rates would be.

• The scope of the project should be expanded to include evaluation of

uncertainty quantification. Finally, we note that part of the SCALIA development process (within CSIRO) involves a Simulation-Estimation Stock Assessment Model Evaluation (SESAME) project, with particular emphasis on data and dynamics resembling Southern Bluefin Tuna. We see this project as complementary to the SCTB-MWG project. For compatability, we are adopting the MultiFan-CL data file format conventions to a large extent, and will welcome and encourage other analysts to participate with the SESAME data sets as they become available.

References Butterworth D. S., J. N. Ianelli, and R. Hilborn. 2002. A statistical model for stock

assessment of southern bluefin tuna with temporal changes in selectivity. CCSBT-MP/0203/4

Fournier,D.A., J. Hampton, and J.R. Sibert. 1998. MULTIFAN-CL: a length-based,

age-structured model for fisheries stock assesment, with application to South Pacific albacore, Thunnus alalunga. Can. J. Fish. Aquat. Sci. 55: 2105-2116.

Fournier,D.A., J.R. Sibert, J. Majkowski, and J. Hampton. 1990. MULTIFAN a

likelihood-based method for estimating growth parameters and age composition from multiple length frequency data sets illustrated using data for southern bluefin tuna (Thunnus macoyii). Can. J. Fish. Aquat. Sci. 47: 301-317.

Kolody, D. and T.Polacheck. 2001. Application of a statistical catch-at-age and –

length integrated analysis model for the assessment of southern bluefin tuna stock dynamics 1951-2000. CCSBT-SC/0108/13.

Laslett,G.M., J.P.Eveson and T. Polacheck. (in press) A flexible maximum likelihood

approach for fitting growth curves to tag-recpature data. Can. J. Fish. Aquat. Sci.

Maunder, M. and G. Watters. 2000. A-SCALA: an age-structured statistical catch-at-

length analysis for assessing tuna stocks in the eastern Pacific Ocean. Draft manuscript presented at the scientific working group of the Iner-American Tropical Tuna Commission.

McAllister, M.K. and Ianelli, J.N. 1997. Bayesian stock assessment using catch-age

data and the sampling-importance resampling algorithm. Can. J. Fish. Aquat. Sci. 54: 284-300

Table 1. Summary of externally-specified SCAL IA parameters and constraining assumptions used for all of the simulated SCTB-M WG data sets (scenarios 1 and 2).

Symbol Description Value σC total catch observation error app. CV 0.01 σSR stock-recruitment relationship app. CV (t < -5; -4 <= t <= 148 ) 0.01, 0.55 ρ stock-recruitment auto-correlation co-efficient 0.4 η catch-at-length effective sample size 1000 θ effective tag release co-efficient 0.05 σEDmax maximum effective effort deviation app. CV 0.1 υ effort deviation prior scaling exponent 2 σqTS temporal change in catchability app. CV 0.02 σHC age-based selectivity curvature penalty 0.5 σMC mortality-at-age curvature penalty 0.1 σM app. CV for priors on mortality-at-age deviations from mean mortality 0.2 h Beverton-Holt Stock Recruitment relationship steepness 0.999 b number of quarters that catchability is assumed constant between changes 24 α growth curve parameter 1 β growth curve parameter 3

0 50 100 150

0

e+00

4

e+07

Time (quarter)

Cat

ch (

Num

bers

) Fishery 1 total Catch

0 50 100 1500.0

e+00

1.5

e+07

Time (quarter)

Cat

ch (

Num

bers

)

Fishery 2 total Catch

0 50 100 150

030

000

Time(quarter)

Num

ber

Fishery 1 Total Effort

0 50 100 150

060

0014

000

Time(quarter)N

umbe

r

Fishery 2 Total Effort

Fig. 1. Total Catch and Effort for the two fisheries from the simulated data set MWG2-1. Total catches are predicted (lines) and observed (circles).

Tim

e

138

140

142

144

146

Length-Class (cm)

50

100

150200

n umb

er

0.00

0.01

0.02

0.03

Fishery 1 Catch-at-L en g th Pro p ortio ns PREDICTED

Tim

e

138

140

142

144

146

Length-Class (cm)

50

100

150200

n umb

er

0.00

0.01

0.02

0.03

0.04

0.05

Fishery 1 Catch-at-L en gth Pro p ortio ns OBSERVED

Tim

e

138

140

142

144

146

Length-Class (cm)

50

100

150200

n umb

er

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Fishery 2 Catch-at-L en g th Pro p ortio ns PREDICTED

Tim

e

138

140

142

144

146

Length-Class (cm)

50

100

150200

n umb

er

0.00

0.01

0.02

0.03

Fishery 2 Catch-at-L en gth Pro p ortio ns OBSERVED

Fig. 2. Examples of predicted and observed catch-at-length frequency distributions from fisheries 1 and 2 from simulated data set MWG2-1 (only the final 10 quarters are shown for each).

50 100 150 200

0.0

00

.02

0.0

40

.06

0.08

0.1

0

Length-class (cm)

Pro

por

tion

Fishery 2 Catch-at-Length t = 98

post-fi t effective sam ple size = 91

Fig. 3. Catch-at-length frequency distribution illustrating the worst fit of all 296 samples (from simulated data set MWG2-1) based on the post-fit effective sample size, see text). SCALIA predictions (line) and observed (circles) indicate the clear bias in the youngest length-at-age.

110 120 130 140

01

0020

03

0040

050

06

00

Time(quarters)

Num

ber

Al l Ages Mixed Tag Recoveries PRED/OBS (al l releases)

Fig. 4. SCALIA Predicted (lines) and MWG2-1 observed (circles) tag recoveries aggregated over all ages and both fisheries.

Fig. 5. Comparison of length-at-age estimated by SCALIA (broken lines are the mean +/- 2 SD) and the bounds provided with data set MWG2-1 (solid lines). The curves are not directly comparable for reasons described in the text.

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25

Age (quarters)

Len

gth

(cm

)

0 50 100 150

0.0

0.4

0.8

Time

q (

not l

og-s

cale

)

Fishery 1 Catchabi l ity (not including seasonal effects)

0 50 100 150

0.0

0.4

0.8

Time

q (

not l

og-s

cale

)

Fishery 2 Catchabil ity (not including seasonal effects)

0 50 100 150

-10

12

Time

Effo

rt D

evia

tion

(log-

scal

e)

Fishery 1 Effort Deviations

0 50 100 150

-40

4

Time

Effo

rt D

evia

tion

(log-

scal

e)

Fishery 2 Effort Deviations

0 50 100 150

02

4

Time

Sea

sona

l Q (

not l

og-s

cale

) Fishery 1 Seasonal catchabil ity effects

0 50 100 150

02

46

8

Time

Sea

sona

l Q (

not l

og-s

cale

) Fishery 2 Seasonal catchabil ity effects

Fig.6. SCALIA estimates of parameters that affect the relationship between effort and fishing mortality for data set MWG2-1.

5 10 15 20

0.00

0.0

50.

10

0.15

0.20

Age

inst

anta

neou

s M

Mortality-at-Age

Fig. 7. SCALIA estimates of instantaneous natural mortality by age (quarterly) for simulated data set MWG2-1.

Tim

e

50

100

Length-Class (cm)

05

1015

20

Se

le ct ivi ty

0.00.20.4

0.60.8

1.0

Fishery 1 Selectivi ty

Tim

e

50

100

Length-Class (cm)

05

1015

20

Se

le c tiv it y

0.0

0.5

1.0

1.5

Fi shery 2 Selecti vi ty

Fig. 8. SCALIA Estimated selectivities for the two fisheries from simulated data set MWG2-1.

0 1000 2000 3000 4000 5000 6000

020

040

0

SSB

recr

utits

MWG2-1 Spawning Stock and Recruitment RelationshipSR Steepness = 0.999

0 50 100 150

020

040

0

time (quarters)

recr

utits

0 50 100 150

020

0050

00

time (quarters)

SS

B

Fig. 9. Estimated Spawning Stock Biomass and Recruitment for simulated data set MWG2-1.

Fig. 10. SCALIA estimated Total Stock Biomass (TSB) over time for simulated data set MWG2-1. The lower solid line indicates the baseline estimates (that were supplied to the SCTB MWG to be compared with the actual operating model states). The upper solid line is the baseline estimate of TSB that would have been observed in the absence fishing. The dashed lines illustrate alternative estimates that resulted by changing assumptions about variance-related parameters.

0

5000

10000

15000

20000

0 50 100 150

time (quarter)

SS

B (

tho

usa

nd

t)

Appendix 1. SCALIA Population Dynamics and Likelihood Objective Function The Statistical Catch-at-Age/Length Integrated Analysis modelling framework is largely an amalgamation of features from other stock assessment models, notably Butterworth et al (2002) and Fournier et al (1998). Differences from the SCALIA SBT application (Kolody and Polacheck, 2001) include:

• selectivity is constant over time • temporal changes in catchability are modeled separately from effort deviations • seasonal catchability effects are added • mortality-at-age is estimated for all ages • length-at-age (and tag ages) are estimated • tag reporting rates are estimated • maturity-at-age is not estimated • stock-recruitment relationship is not estimated (only mean recruitment) • there is no direct catch-at-age information

Notation is summarized in Appendix Table A1. Subscripts and superscripts may be omitted in the following but should be implicit from the context. Population dynamics are based on the standard (Baranov) catch equations, including the usual dynamic pool accumulator for the plus-group:

1,11,1, −−−−= atatat NsN ; for a < A,

AtAtAtAtAt NsNsN ,1,11,11,1, −−−−−− +=

)exp( ,, aatat MFs −−= .

ie The annual survival of a cohort, s, is a function of age-specific natural mortality, M, and fishing mortality, F. Natural mortality is estimated for each age, subject to some constraints (M for each age is assumed to be a random normal deviate from the mean across all ages and/or a third difference curvature penalty is applied; objective function terms L5 and L7 below). Fishing mortality follows a separable assumption, ie F is composed of a time-step component (quarterly in the SCTB-MWG case) and age component for each fishery:

fat

ft

fat HGF ,, = , and

�=f

fat

totalat FF ,,

where Gt is a time-step-specific fishing mortality term, Ht.a is the age-specific fishery selectivity term (for the SCTB-MWG case, selectivity does not change over time so the t sub-script is redundant). Gt is further partitioned into a number of components that have attractive mechanistic interpretations, but are in practice rather confounded in the estimation process:

),0(~);exp( fE

ft

ft

fseason

ft

ft t

NormalEQqG σµγγ == ,

where, q is the fishery catchability, Q is a seasonal catchability effect (Q consists of 4 parameters per fishery corresponding to the 4 quarters within a year; one is defined as unity to prevent complete confounding with q), E is the observed effort in hooks and γ is an effective effort deviation (a process error that describes a potentially large temporary deviation from the mean relationship between catch, effort and abundance, eg due to interannual variability in fish aggregations). It is assumed that effort deviations tend to be larger when effort is low (eg patchy fish distributions and non-random searching produce highly variable catch rates, and the CV would be lower if averaged over more effort units):

max

)max(EDf

t

ff

ED E

Et

σσν

��

�

�

��

�

�=

ie the CV of the prior distribution for the effort deviation in a given quarter is inversely proportional to the ratio of the maximum observed effort over effort in the given quarter (all raised to the power of υ); see L6 below. Catchability is also assumed to change over time, via a random walk process:

),0(~);exp( fq

ft

ft

fbt Normalqq σµδδ ==+ ,

where b indicates the number of time-steps is which q is assumed to remain constant between changes (b can be as small as 1, but in practice, larger time blocks yield almost the same result with fewer parameters). In contrast to the effort deviations, this process is intended to describe gradual, systematic changes in the system which affect the mean relationship between catch, effort and abundance (eg due to cumulative improvements in fishing technology). The predicted catch-at-length frequency distribution for each fishery is predicted by:

)1( ,,,,

,,, atat

attotalat

fat

ala

flt sN

MF

FPC −

+=� ,

where Pa,l is the proportion of age a fish in length-class l. It is assumed that the length frequency distribution for each age-class is normally distributed with means defined by (Laslett et al in press):

��

���

���

���

+−−−+−−−=

−−

∞

β

αβαβµ

/)(

002

12

)exp(1

))(exp(1)(exp(1

kk

a

aaaakL .

This is a weighted average of two regular von Bertalanffy curves, where the weighting is a logistic function. This function is well suited for describing an overall growth curve in which younger ages and older ages seem to follow substantially different von Bertalanffy curves (with a smooth transition between the two). The

standard deviations of the length-at-age is estimated to be a linear function of the mean length-at-age:

interceptslope aa += µσ .

In the SCTB-MWG application, some length-at-age parameters (Linf, k1, k2, a(0), slope, intercept) were estimated, and the others held constant (α,β). Selectivity is modelled as a purely age-based process, however the vector H is actually derived from a length-based concept:

**, ll

laa PH Λ=� .

ie each age-based selectivity is a weighted sum of all the length-based selectivity parameters where the weighting is equal to the proportion of fish age a in length-class l* (in this case, l* is used to indicate that there are actually far fewer length-based parameters estimated than are used for the catch-at-length frequency distributions). Thus consecutive ages must have similar selectivity, depending on the degree of length overlap. The age-based selectivities are re-scaled to a mean of 1. Note that in practice, rather surprising results can occur, and this approach can actually be quite different from a truly length-based selectivity in a model where age and length are jointly tracked. A weak third-difference curvature penalty (which may be redundant in this case) is also applied to smooth out H across adjacent age-classes (L7 below). A Beverton-Holt stock recruitment relationship is estimated as part of the overall parameter estimation (via L4 below). The SR curve is parameterized in terms of mean unfished recruitment R* , and steepness, h (the ratio of mean recruitment at SSB(unfished)/5 over R* ). Lognormally-distributed deviations from the SR are estimated, including the assumption of a lag(1) autoregressive process (Butterworth et al 2002), which was indicated to exist in the SCTB-MWG data.

unfished

unfished

unfished

SRttt

SRtt

tt

SSB

SSBR

h

hSSB

Normal

SSB

SSBR

)(

51

)1(

),,0(~;1

),exp(

*

21

221

+=

−−

=

=−+=

−+

=

−

βα

β

σµτρϖρττ

στβα

Note that in the SCTB-MWG case, a priori information was given to indicate that stock size has no effect on recruitment, so steepness was held constant at h = 0.999. The initial population age structure was also estimated according to the stock-recruitment relationship (and appropriate cumulative natural mortality), however, recruitment deviations corresponding to cohorts older than 5 quarters were much more highly constrained to the mean than the others (because catch-at-length modes cannot resolve individual ages in older fish). Spawning stock biomass was calculated:

�=a

atatat mNMaturitySSB ,,

where mt,a is the mean mass of an individual of age a in year t, and in this case, maturity was defined a priori. Log-normal errors are assumed for the observed total catch in numbers for each fisheries in each timestep ( L1 below). Total catch numbers are assumed to be known essentially perfectly for all fisheries (CV ~ 0.01). It is assumed that numbers at length are truly random samples from the commercial catches, giving rise to multinomial CL likelihoods (L2 below). The effective sample size of catch-at-length is specified at a level that is usually considerably below the true value, to partially admit the effects of non-random sampling. Tagged fish are implemented in a simple age-based manner. We assume that ‘ cohort-slicing’ the length frequency distribution provides reasonable age estimation, particularly for the younger fish that comprise the majority of released tags. In cohort slicing, a fish is assigned to the age-class with the highest proportion of its length frequency distribution in the same length-class as the known length of the fish. In the simple implementation explored here, there was no consideration of differences in relative abundance of age-classes at the time of tagging.

Population dynamics of fully-mixed tagged fish are assumed to be identical to the general population. Predicted recaptures, T, of age a in year t depend on the tags effectively mixed in the general population (Tags) in the same manner as the catch is related to the total population:

)( ,,,

,, atat

at

atpredat sTags

Z

FrT �

��

����

�⋅= ,

where r is the estimated tag reporting rate (assumed constant over time for the SCTB-MWG). In this exercise we did not dis-aggregate tag recoveries by fishery in the likelihood. We assume that tags are only effectively mixed in the population two quarters after release, and prior to that, fishing mortality on tags is independent of the general population:

{

} )exp()exp(

-

)exp())exp(

(

)(

12112

11 timesteprelease1,1

22122

1 timesteprelease2,2

2,2

1,11,1,

−−

+−−

−−−−

−−

−−−−

−−

−−

−+

=

aaat

aaat

at

atatat

Mr

MT

Mr

MTRelease

sTagsTags

ie the number of fully mixed tags is dependent on the surviving fully mixed tags from the previous time step, plus the number of tags that have just achieved fully-mixed status following two timesteps of natural and fishing mortality. For the unmixed individuals, fishing mortality is applied as a pulse fishery in the middle of the timestep. It is assumed that recapture probabilities are described by the Poisson distribution (giving rise to L3 below), with an effective release co-efficient (analogous to the effective sample size in the multinomial CL likelihood, the effective release co-

efficient is intended to admit that errors in tag dynamics are likely larger than the theoretical distribution predicts).

The objective function consists of the following terms:

�=

=9

1iitotal LL ,

� ��� ��

���

�=t l

predflt

l

obsflte

fC

CCL2

,,

,,21 )/(log

)(2

1

σ = total catch in numbers,

��� −=t l

predflte

obsfltf

f

ppL )(log ,,

,,2 η = catch-at-length composition,

( )�� −=t a

predtae

obsta

predta TTTL )(log ,,,3 θ = tag recaptures

�=t

tSR

L2

24 )(2

1 ωσ

= stock recruitment relationship

�=a

aM

L 225 2

1 δσ

= mortality-at-age,

��=t

ft

f f

L 226 )(

2

1 γσ γ

= effort deviations,

�=tt

ttqTS

L 227 2

1 δσ

= temporal changes in catchability,

[ ]� �

===

−

=

+++ −+−=

),,(

3

12

2

,1,2,3,8

21 2

)(log)(log3)(log3)(log

MHHX

A

a XC

ateateateate

ff

XXXXL

σ

= selectivity and natural mortality third-difference curvature penalties.

A number of additional constraints were added to intermediate stages of the function minimization, including :

( )��

���

≥<−=

+

++� ft

ft

ft

ft

ft

ft

t tq

qqqqL

1

1

2

19

;0

;100

= penalty to encourage catchability monotonically increasing over time

( )��

���

≥<−=

+

++� fa

fa

fa

fa

fa

fa

a HH

HHHHL

1

1

2

110

;0

;100

= penalty to encourage selectivity monotonically increasing with age.

Appendix Table A1. Summary of SCALIA notation

Subscripts and Superscripts: t = time (quarterly increments for SCTB-MWG) b = number of consecutive time-steps H or q constant before changing tt = time (increments of b years), a = age (quarterly increments), 0...20+ A = maximum age (20+) l = length intervals, 10-12,12-14, ..., 208-210 cm f = fishery obs = observed (data) pred = predicted (deterministic function of the model parameters) States, Variables and/or Parameters: C = catch (numbers) N = numbers F = instantaneous (quarterly) fishing mortality M = instantaneous (quarterly) natural mortality Z = F + M = total mortality G = fishing mortality time-step-effect H = fishery selectivity T = tag recaptures (numbers) R = recruitment s = time-step survival m = mass E = effort (hooks) η = effective sample size assumed in P = Proportions of catch-at-age, catch-at-length or length-at-age r = tag recovery reporting rate SSB = spawning stock biomass SR = related to stock recruitment relationship µ = distribution mean σ = distribution standard deviation θ = co-efficient down-weighting number of effective Tag Releases (0-1) ω, γ, ε = random deviate from specified distribution τ = an auto-correlated deviate from the stock recruitment relationship ρ = lag(1) correlation co-efficient for SR deviations Linf,k,α,β = parameters describing length-at-age frequency distributions

Λ = a pseudo-length-based selectivity parameter