.-. _. ..

International Council for theExploration of the Sea

CM 1997/00:8Assessment Methods

.~., 'I Il':' ,

EVALUATION OF UNCERTAINTY IN STOCK ASSESSMENT, BIOLOGICALREFERENCE POINTS, AND OUTCOME OF A HARVEST CONTROL LAW

WHERE MODEL STRUCTURE IS UNCERTAIN, USING A BAYESIANMETHOD: NORWEGIAN SPRING·SPAWNING HERRING

K R Patterson

FRS Marine LaboratoryPO Box 101, Victoria Road

Aberdeen, AB11 90BScotland, UK

SUMMARY

A method is suggested for calculating statistical distributions of future catches which conformto 0. specified harvest controllaw while incorporating uncertainty in biological reference points,natural mortality, and some aspects of model structure in addition to the usual stochastic noise.A Markov Chain Monte Carlo approach is used to calculate Bayesian posterior distributions forsome parameters of 0. Norwegian spring spawning herring stock assessmemt using a newassessment model that incorporates catch-at-age, sUrVey and tag release and recaptureobservations. Exceptionally, the approach allows prior uncertainty in model structure (egwhether survey observation errors should be treated as normal, lognormal or gamma variates;whether Ricker or Beverton-Holt torms are used to model recruitment). This modelling approachis proposed as 0. useful tool which allows management advice to be provided which takes intoaccount uncertainty in model structures and in some parameters which, by conventionalmethods, need to be specified as arbitrary "best" choices. In 0. simple extension the method canbe used to quantify uncertainty in some biological reference points, and to calculate 0. probabilitydistribution for 0. future catch with respect to 0. specified harvest control law. This has theadvantage of 0. consistent treatment of uncertainty throughout the process of stock modelJing,reference point estimation and concomitant catch forecasting.

INTROOUCTION

CurTent discussion on the provision of fishery management advice is centred on the comparisonof some measure of stock dynamics (eg percentiles of the distribution of SSB or F) with sometarget or limit measure derived from historical parameters estimated from the stock (eg Fmed' Fmsy'

G,oss etc) (Anon, 19960., 19970., 1997c). A methodis suggested here which allows estimationof uncertainty in both the stock assessment and in the reference points so that advice can begiven to management in the form of expected values and probability statements about the levelsof catch that will conform to 0. given harvest controllaw. It can be used to address problems ofproviding advice in cases where fishing mortality is low and close to currently-assumed, but veryuncertain, levels of natural mortaJity.

The method can be used to address questions ot structural uncertainty, such aswhich error­distribution model best represents the spread of observations, or which stock-recruitment model

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best represents the historical relationship of recruitment and spawning stock size. It is fairlyeasily applicable so long as Iikelihood functions for the stock assessment model can becalculated, prior distributions for the unknown parameters can be specified, and if a formalharvest controllaw has been specified. Norwegian spring-spawning herring is used for examplecalculations.

Uncertainty in Natural Mortality

Most reference points are strongly dependent on estimates of natural mortality. In the currentsituation with most demersal stocks in the North Sea, this has usually been considered fairlyunimportant as fishing mortality dominates the stock dynamies, exceeding assumed naturalmortality by three to five times (Table 1). However, that is not necessarily the case throughoutall ICES stocks. Pelagic fish stocks must be managed at quite low levels of F to avoid knownproblems with stock-recruit collapses and concentration effects. Among the main ICES pelagicstocks, current fishing mortalities range from below M to twice M (with the exception of NorthSea herring, for which emergency conservation measures have been implemented).

VPA methods only partition total mortality into an assumed M and a fishing mortality in anarbitrary way, and the basis in data for values of M used in the assessment and advisoryprocedure is often very weak.

Uncertainty in Survey Error Structure

Most fish stock assessments are calculated on the assumption of lognormality in survey errors,. but this assumption may on occasion appear to be violated and other models (eg Gamma) may

appear to be more appropriate. However, altering error-model assumptions may result in largealterations in perceptions of stock size, and the basis in data for choosing one assumption overanother may not be very strong. This is a source of uncertainty that is rarely considered, butwhich is reasonably tractable under the present approach.

Uncertainty in Stock-Recruit Models

In a similar way, the appropriate form of stock-recruit model may have large consequences forperceptions of stock development in the medium term, but the choice of model may be weaklydetermined by data and strongly determined by personal preferences. Again, this type ofuncertainty may be modelied by the procedure suggested here.

Uncertainty in Biological Reference Points

Most fishery reference points are taken as deterministic values, and the uncertainty attached toreference point estimates is only recently being considered (Anon, 1997c), although anexception is the "G10SS" reference point, where the approach has been to evaluate sustainabilityof the fishery in terms of the probability that a fishery is above or below the G,css sustainabilitycriterion, which is itself stochastic (Anon, 1997a). Reference points will typically depend moststrongly on assumed va(ues of natural mortality, and on an estimated selection pattern.

Most of the reference points Iisted in Anon (1997a) are M-dependent. Also, most reasonablerefereoce points suggest low long-term levels of fishing mortality, and these are often quite doseto, or below, M. In some cases, this is stated explicitly, for example F = 213M or F=M have beenproposed as target reference points (Patterson, 1992; Walters and Maguire, 1996). Thesereference points obviously depend directly on the assumed value of M whilst others dependindirectly.

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It is not immediately c1ear how uncertainty in values of M introduces uncertainty in both theestimation of current stock size, and in the calculation of reference points, nor how this affectsshort-term advice based on a certain harvest control law.

In cases where the dynamics of a stock are at low levels of fishing mortality, there is a significantproblem in the provision of advice, both in the estimation of current stock size, and in thedetermination of appropriate target reference points. At present, only same pelagic stocks arein such an area of stock dynamies. However, in principle the application of the precautionaryapproach would lead other stocks into this area which is problematic for parameter estimation.

A Methodology

Consider an age-structured data set 0, to which is fitted an age-structured stock assessment"model M (which may include a stock-recruit relationship), with model-specific parameters G.One methodology for calculating uncertainty (as a multivariate posterior distribution) in G is givenin McAllister et al. (1994, 1997), but here a simpler approach (Markov Chain Monte-Carlo byhybrid adaptive rejection sampling) is used (Appendix I). This allows calculation of multivariatedraws 0' from the distribution of P(GI 0). Uncertainty in M can also be allowed.

G, M and 0 tagether contain sufficient information for the calculation of target or limit referencepoints. It is therefore feasible to calculate a probability distribution for any reference point T fromthe 0' and D.

Furtherassume that we propose a harvest controllaw which determines catch C in a given yeary as fallows: Cy =H(D,0,T). It is then feasible to calculate the outcome of the application of Has a probability distribution determined by the distribution of G'. This outcome can be specifiedas a distribution in terms of catch, SSB, etc as required.

An Example: Norwegian Spring-Spawning Herring

This is a case ofa stock fished close to assumed values of natural mortality and for which sometagging information is available that should be informative about this parameter. Additionalproblems in the assessment are that the appropriate form of stock-recruit relationship isundecided, the assessment is sensitive to the choice of assumptions about the appropriateerror-distribution model for the surveys, and an disease-induced fish kill is believed to have hada significant impact on the stock in recent years. These aspects are explained further inAppendix 11. Because of these problematic aspects, this stock suits as achallenging case study.

Assessment Model and Harvest Control LawA stock assessment model M was formulated as described in Appendix 11, which is similar to thatdescribed in Anon (1997b). In summary the main features of this are:

• "ADAPT"-type stock assessment, with inclusion of tag return data (assumption of'Poisson errors).

• Normal, lognormal or gamma error distribution for acoustic survey data allowed• Beverton-Holt or Ricker stock-recruit relationships allowed• Natural mortality assumed unknown between specified bounds• Possibility of a disease-induced fish kill admitted

and the MCMC implementation (Appendix I) was used to generate draws of the key stockassessment parameters 0 (listed in Appendix 11).

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~ ," ,.-, t,· .'

For illustrative purposes, a simple harvest controllaw approximating the default harvest controllaw suggested by Anon (1997c) H(0,D,T) was defined as:

If ( (SSBy_1) > Bpa ) then Harvesty = Catch for F = Fpa

otherwise if SSBy_1 > B1im then Harvesty = Catch for F = Fpa .( SSBY_1 - Blim )/(Bpa - Blim)

otherwise if SSBy_1 < B1im then Harvesty = 0

where three reference points are defined: T= ( Bpa = BMSY, Blim = 0.5 Bpa, Fpa = Fa.1 ).

Estimates of the reference points for the harvest controllaw are dependent on 0 , M and 0, andhence uncertainty in the reference points depends on the P(S , M I 0). Here, Fa., is estimatedconventionally from M and the weights at age and the exploitation pattern. In order to investigatethe sensitivity of the precision of the Fa.1 estimate to uncertainty in recent selection, distributionshave been calculated using either a) the selection pattern in 1996, which is obviously highlyuncertain, or b) the mean selection pattern over the whole period 196-1996. which is much morestable.

For present purposes, BMSY has been estimated as half the maximum observed stock size in thetime series (This depends on making an assumption of Schaefer form of stock production, andrelies on low fishing mortalities in the early part of the time series). Clearly BMSY depends verylargely on M and on D.

RESULTS

The Assessment: Perceptions of Stock Dynamics

Estimated posterior probability distributions for stock assessment parameters S, model choicesM and some dependent variables which are of particular management interest are given inFigure 1. These indicate rather wide confidence intervals for the cohort abundances andcatchabilities, with spawning stock size estimates for 1996 in the region 2 to 4 million tonnes.The distribution of natural mortality was rather collapsed onto the lower bound of the uniformprior range, suggesting M = 0.1 or below. The tagging survival model suggests between 30 to40% survival of tagged fish. The data did not appear to allow much discrimination betweenBeverton-Holt and Ricker stock-recruit models, for which the estimated posterior probabilities(0.45 and 0.55) where close to the prior probabilities (0.5 and 0.5). However, anormal error-distribution appeared more likely than either the log-normal or gamma distributions in this case. •There was little indication of significant disease-induced mortality.

Some indication of the correlation structure of the assessment model can be obtained from amultivariate scatterplot of some of these variates (Fig. 2). This shows marked positivecorrelations between cohort abundances, tagging survival, spawning stock biomass and statusquo catch estimates, and marked negative correlations between spawning stock biomass andfishing mortality (as one would expect). and also between the three catchability parameters andestimates of cohort abundance and stock size. Other parameters (Natural mortality, surveyvariance. disease-induced mortality) did not appear highly correlated to any other parameters.This rather suggests that improvements in the precision of estimates of stock size, fishingmortality or status quo catch could be made by imposing informative prior distributions on thesurvey catchabilities rather than on any other parameters.

Short Term Forecast : Application of Harvest Control Law

Estimated probability densities for the Fa., and Biom reference points are given in Figure 3. Thedistribution of the Biom reflects its dependence on natural mortality and the constraints imposed

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on that parameter. The distribution of 1997 stock size (subjeet to the Hel) spreads from belowBlim to above Bpa, wi,th a mode at about 1.3 B1im • Two distributions of Fo.1 were calculated, basedeither on the 1996 weights at age and exploitation pattern, or on means from 1960 to 1996. Theformer distribution is both wider than the laUer and is located at higtier values of F. It is widerbecause there is more uncertainty in the 1996 exploitation pattern thari in the historie meanexploitation pattern, and it is located at a higher F because the 1996 exploitation pattern doesriot inch.ide any exploitation of young fish, which did occur in the early part of ttie time-series.

The distribution of fishing mortality in 1996 relative to the Fo.1 estimate (based on the 1996weights at age and exploitation pattern) is centred around three to four times Fo.1 •

The probability distributions for the harvest control law catches in 1997 and 1998 indicate thatrather low catches of the order of 120,000 t have a 50% chance of conforming t6 the harvestcontrollaw in 1997, but this value increases to some 220,000 t in 1998. However, given theasymmetric nature of the probability distribution of the harvest controllaw catch, it may be moreappropriate to base advice on the means of the distributions, =215,000 t for 1997 and 324,000t for 1998 in this example.

Medium Term Forecast

Calculations of present perceptions of future harvest control law catches and associated stocksize can also be made. These calculations rely on the stock-recruit models and model-specificparameter estimates. The procedure is similar to that for the short-term projections, except thata two-stage procedure is used to model uncertainty in future recruitment. The first stage is theestimation of uncertainty in the underlying process, which is modelIed as described above andresults in sampies of recruitment model Mr and a model-specific pair of drawn parameters er,say. Uncertainty due to stochastic noise in recruitment around the underlying process, ismodelied by a non-parametric bootstrap method: for each draw of Mr and specific er , ci set offiUed historie recruitments Ry can be calculated. From the same draw, a set of historicrecruitments No,y can be derived (dependent on drawn M and other parameters), and so aspecific set of recruitment residuals on a log scale can be defined,

,

(1)

..• Future recruitments can then be modelIed from a quasi-deterministic component derived fromMr , er in the conventional way , and a random sampie with replacement from the er

y.

An example of this sort of calculation is given in Figure 5, which iIIustrates the perception ofstock development from the stock modelling and harvest controllaw procedure described above.This example shows an interesting feature in that the distributions of catches and stock sizebeeonie increasingly asymmetrie with time: the median and the arithmetic mean of thedistributions diverge inereasingly with time.

DISCUSSION

The approach used here to evaluate harvest control Jaw catches is rather different fromeonventional methods.. Usually one would consider the H as a quota~setting rule which isapplied bya manager in response to current pereeptions of M and e.. Here however H iseonsidered as a process to whieh the manager is attempting to approximate by setting a quota

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according to a perception of the outcome of the application of H while taking account cfuncertainties in the various inputs to H.

This approach has the advantage of putting together the presently somewhat diverse processesof stock assessment, estimation of reference points, medium-term projections and managementsimulation into a single modelling procedure, which has observations, model structures,definitions of biological reference points and prior perceptions defined clearly as separate inputs.Given these inputs, posterior perceptions of quantities of management interest are calculatedin a logically consistent fashion.

This procedure allows calculation of aprediction of the consequence of the application of aharvest controllaw in probabilistic terms in a consistent fashion, and can include uncertainty innatural mortality and other uncertainties as appropriate. Simple extensions can allow uncertaintyin model structure to be modelIed also (eg type of stock-recruit relation, nature of error­distributions, etc). Conversely, it allows advice to be given on the catch to be taken in the shortterm to conform to a given harvest control law with specified probability, or to calculate anexpected catch for a given harvest control law.

The method could also be used to compare outcomes of proposed harvest controllaws. Using •a conventional management simulation procedure allows evaluation of a harvest controllaw Hgiven beliefs about the stock dynamics M and e. A harvest controllaw evaluation is typically alengthy process of evaluating different combinations of Hand M, while ignoring 0 and e.Arguably a more attractive approach would be to specify beliefs about M and e as priordistributions as described above. This would have the advantage that the beliefs would beupdated using the observation data set D. Re-evaluation of medium-term prospects for stockdevelopment for alternative H when new information is obtained which may alter perceptions ofM and e can then be a rather simple and routine task.

REFERENCES

Anonymous. 1996a. Report of the comprehensive fishery evaluation working group. ICES CM1996/Assess: 20.

Anonymous. 1996b. Report of the Atlanto-Scandian Herring and capelin working group. ICESCM 1996/Assess: 9.

Anonymous. 1996c. Report of the Northern Pelagic and Blue Whiting fisheries Working Group.ICES CM 1996/Assess: 14.

Anonymous. 1997a. Report of the study group on the precautionary approach to fisheriesmanagement. ICES CM 1997/Assess: 7.

Anonymous. 1997b. Report of the northern pelagic and Blue whiting fisheries working group.ICES CM 1977/Assess: 14.

Anonymous. 1997c. Report of the comprehensive fishery evaluation working group. ICES. Inpress.

Draper, D. 1995. Assessment and propagation of model uncertainty. Journal of the RoyalStatistical Society, Series. B, 57(1),45-97.

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Gilks, W.R., Richardson, S. and Spielgehalter, O.J. 1996. Markov Chain Monte CarIo inpractice. Chapman and Hall, London. 486pp.

Gilks, W.R. 1996. Full conditional distributions. Gilks, W.R., Richardson, S. and Spielgehalter,O.J. (eds), pp75-86. Markov Chain Monte Carlo in practice. Chapman and Hall, London.486pp.

Hilbom, R. and Walters, C.J. 1992. Quantitative fisheries stock assessment: Choice, dynamicsand uncertainty. Chapman and Hall, New York. 570pp.

McAllister, M.K and lanelli, J.N. 1997. Bayesian stock assessment using catch-age data andthe sampling-importance resampling algorithm. Can. J. Fish. Aquat. Sei., 54, 284-300.

McAllister, M.K, Pikitch, E.K, Punt, A.E. and Hilbom, R. 1994. A Bayesian approach to stockassessment and harvest decisions using the samplinglimportance resampling algorithm.Can. J. Fish. Aquat. Sei., 51, 2673-2687.

Patterson, KR. 1992. Fisheries for small pelagic species: an empirical approach to• management targets.' Rev. Fish. Biol. Fisheries, 2, 321-338.

Pope, J.G. 1972. An investigation into the accuracy of virtual population analysis using cohortanalysis. ICNAF Res. BulI., 9, 65-74.

Raftery, A.E. and Lewis, S.M. 1996. Implementing MCMC. Gilks, W.R., Richardson, S. andSpielgehalter, O.J. (eds), pp115-127. Markov Chain Monte Carlo in practice. Chapmanand Hall, London. 486pp.

Serchuk, F., Rivard, 0., Casey, J. and Mayo, R. 1997. Report of the Ad Hoeworking group ofthe NAFO scientific council on the precautionary approach. NAFO SC Working Paper97/30.

Thompson, G.G. and Mace, P.M. 1997. The evolution of precautionary approaches to FisheriesManagement, with focus on the United States. NAFO SCR Ooc. 97/26.

Walters, C.J. and Maguire, J.J. 1996. Lessons for stock assessment from the Northem Codcollapse. Reviews in Fish Biology and Fisheries, 6(2), 125-137.•

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TABLE 1

Values of natural and fishing mortality for some demersal and pelagic ICES stocks

StockAdult natural Fishing mortality in

FO.1mortality 1994

North Sea cod 0.35 (age) to 0.2 0.94 0.15

North Sea haddock 1.65 (age 1) to 0.2 0.91 0.20

North Sea whiting .95 (age 1) to 0.2 0.67 0.23

Norwegian spring-spawning 0.15 0.26 0.15 (target)herring

North Sea herring 1(age 1) to 0.1 0.67 0.2 to 0.3proposed

Western mackerel 0.15 0.30 0.15 to 0.2proposed

Western horse mackerel 0.15 0.18

Herring in Vla(N) 1 (aQe) to 0.1 0.12 0.136

(2)

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APPENDIX 1

ASUMMARY OF THE MCMC APPROACH TO ESTIMATING UNCERTAINTY IN STOCKASSESSMENTS

Conventional Bayesian analysis relies on an evaluation of the posterior probability P(0IX) of acertain set of assumptions 0, given prior belief P(0) about those assumptions, a set of newinformation X, and a Iikelihood function allowing the evaluation of P(XI0). The conditionalprobability can be expressed as

P(GIX) = P(Xl0) P(G)

fP(Xl0) P(G) d0

Conventionally the Gare usually a vector of input parameters to a model M, which is assumedto be correct, and upon which the Iikelihood function is predicated. It is also possible to treat theentire model structure M as being an additional parameter and to integrate over uncertainty bothin the model structure M and in the model parameters. In this case, each 0 has a meaningwhich is specific to each M. This method, suggested by Draper (1995), differs from the usualapproach of using available data to choose a "most appropriate" model, and then proceedingon the assumption that the chosen model is correct. The method also provides a posteriorperception of the likelihoods of each structural model being a true representation of the datastructure.

The evaluation of posterior probabilities proceeds exactly analogously:

P(G,MlX) P(XlG,M) P(GIM) P(M)

JP(XlG,M) P(G,M)P(M) cß dM (3)

•It is obviously possible by this approach to evaluate a posterior probability distribution for anyquantity of management interest (such as a catch forecast or medium term projection) that canbe expressed in terms of 0 and M. Such a distribution can be constructed for any reasonablerange of alternative structural model components in M, assuming that for each model componenta Iikelihood term P(XI0,M) can be calculated.

Computational Aspects

Most Bayesian stock assessments in the published literature have relied either of direct-mappingapproaches (eg Hilborn and Walters, 1992)"or sampling/importance resampling (eg McAllisterand Pikitch, 1997). The former approach becomes extremely difficult to implement forcomputational reasons if there are more than a few parameters, and the laUer method requiresthe specification of an importance function which, for efficiency, should approximate reasonablyclosely to the posterior distribution. Choice of such an importance function is not trivial:McAllister and lane"i (1997) describe eight separate steps in the development of an importancefunction, each of which requires considerable expertise.

A simpler approach is the Markov Chain Monte Carlo method to generating sampIes fromposterior probability distributions, described accessibly by Gilks et al. (1996). Essentially, thisrelies on setting up a very long Markov chain of model parameter vector 0 with transitionprobabilities defined as the conditional posterior probability of each element of G, depending onthe other elements. So, tor the transition trom iteration k-1 to iteration kin the chain. element

i of 8 is replaced with a new value drawn from the distribution of P(8j 'k I 8 i, i<>i, k.')' Thedistribution to be sampled has pdf: L(DataI8) Pr(8), ie the product of the Iikelihood function andthe prior. The form of this conditional Iikelihood is generally not known analytically and it iscomputationally expensive to evaluate. However, sampies can be taken from it by constructingupper and (ower envelope functions (constructed on simple geometrie rules) from which it easyto sampie. The adaptive rejection sampling procedure involves:

1. Take a sampie from the upper envelope function (X)2. Evaluate the Iikelihood L(X) at the sampled point3. Test whether to accept the sampled value:

if accepted,proceed to next iterationotherwiseupdate the envelope function with the new information X, L(X)Iterate

In the present case, the upper envelope function used is a simple series of rectangularapproximations between evaluated points and estimate of the maximum of the curve (AppendixFig. 1), which is a modification of that proposed by Gilks (1996). Although computationally less •efficient, this algorithm was robust to structural bound-constraints (such as a restrietion of amaximum 100% tagging survival) which this assessment model should logically contain. AMetropolis step was added to ensure robustness of the method to non-Iog-concave conditionalprobability distributions (ie cases where the conditional probability may emerge above the upperenvelope function).

Clearly the adjacent values of 8 k in the Markov Chain will be very strongly correlated, and thiscorrelation will decrease the further apart in the chain that pairs of values are drawn. For mostpurposes we require to draw virtually uncorrelated sampIes from the chain; a method proposedby Raftery and Lewis allows calculation of the appropriate intervals from which to take to takesampies trom the chain so that they have less than a specified correlation (for the Norwegianspring spawning herring example this was calculated as one sampie each 76 iterations tor 5%accuracy in the posterior distributions). This process is sometimes known as "thinning" thechain.

The chain will also clearly be sensitive to starting conditions, and one will wish to discard acertain number of initial iterations, after which the chain is effectively uninfluenced by the startingvalues for 8. This is sometimes known as the "burn-in" period. From the Raftery and Lewis •estimation method this was estimated as 600 iterations.

In order to avoid the recalculation of the burn-in and thinning parameters for the chain, after afew initial trials a value of 1,000 for the burn-in and 100 for thinning was used. Posteriordistributions are approximated by plotting the frequencies of the parameters in around 7,500sampies (implying a chain length of 751,000 iterations).

Values of 8' can be saved from a single run of the chain in order to re-evaluate the outcome ofvarious the harvest controllaws without incurring the computational burden of re-evaluating theentire chain.

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APPENDIX 11

DATA, PRIOR DISTRIBUTIONS AND MODEL STRUCTURES FOR THE ASSESSMENT OFNORWEGIAN SPRING-SPAWNING HERRING

INTRODUCTION

In the case of the Norwegian spring-spawning herring, uncertainty in model structure has led topoor consistency in advice. For example, Anon (1996a) considered alternative structuralassumptions about the error-distributions of the acoustic surveys used to measure the stocksize. On the basis of residual plots, they noted that an assumption of lognormal errorsconfoimed better to the observations than an assumption of normal errors, and so adopted theassumption of lognormality for assessment purposes. However, this change resulted in achange in the perception of the size of the 1983 year class in the beginning of 1995 from 1.8 to3.5 billion individuals. As there are few observations from the surveys, the basis for making achoice of "best" assessment model by comparing error distributions is not very strong. Themodel uncertainty approach described in Appendix I can afford a way in which managementadvice can be provided yet without forcing a choice of "best" assessment model and withoutlosing a perception of the uncertainty due to lack of knowledge about the true structural model.

Additional problematic features of this stock assessment are that natural mortality is unknown,and that arecent outbreak of a disease-induced mortality is thought to have had a significantimpact on the stock, but this effect has proven hard to quantify.

Problems of appropriate parameterisation and perceptions of risk are addressed by examiningposterior distributions for appropriately structured models. By this means, the feasibility ofestimating an additional disease-induced mortality, the uncertainty introduced by the uncertaintyabout the most appropriate error model to use, and the estimation of natural mortality can beaddressed.

OBSERVATIONS

The observations used here are the conventional stock assessment data taken from Anon(1997). Theycomprise the estimates of annuallandings ly, and (separately by age-group a andby year y) numbers of fish caught (Ca,y), their mean weight in catches (CWa,y) and in the stock(SWa,y), and the proportion of fish spawning 0a,y' These are estimated from sampling ofcommercial catches. Data from 1950 to 1996 and of ages 0 to 16+ were used.

Three series U) of acoustic survey measurements (Ua,y,j) of the stock size exist, taken bysurveying overwintering aggregations in December, in January, or on pre-spawning aggregationsin February and March.

Tagging experiments intended to measure fish mortality have been carried out on an annualbasis since 1975 (Anon, 1996a). The ith "experiment" here is defined as the release of Ki,y fishin year y, and all the subsequent recaptures from that release. In years after the release, anumber (my) of fish were screened for tags in the commercial catches Ca,y' The number oftagged fish of each age recaptured was recorded (Xi,a,y)'

The data are taken from Anon (1997).

THESTRUCTURALMODELS

Axiomatic Structural Model Elements

The set of possible structural models is extremely large. For present purposes I define a set ofrelationships as axiomatic, in that they are assumed to hold thraughout and all conc\usionsdrawn are predicated on the assumption that these hold. Other structural relationships in whichuncertainty is admitted are termed candidate structural models. The model structures heretreated as axiomatic are those which have been used as assumed structures for some years byAnon (1997) and are widely accepted in the stock assessment fjeld.

Underlying Population Structural Model

The population structural model is Pope's (1972) cohort analysis model. This is chosen inpreference to the usual Baranov-equation "VPA" model merely on the graunds of fastercomputation. Selection pattern constraints were imposed as described in Anon (1997).

Acoustic Survey Catchability Model

The expected acoustic survey stock size estimates are assumed to be related to populationabundance bya "catchability" coefficient Q specific to each survey time-series j, as:

•(4)

Tag Returns Model

As recaptures of tagged tish are extremely rare events, a Poisson error distribution has beenassumed, tollowing Hilborn and Walters (1992, [p220]). Assurne that of Ki,y fish tagged, aproportion St will survive the pracess of tagging, release and recapture. This proportion iscommon to all tagging experiments. Further assurne that after the tagged fish have suffered theinitial mortality, the fish then suffer no additional mortality and can then be considered to havethe same dynamics as the untagged fish. Obviously it is not possible to know the age of a fishat the time of tagging since it must be released alive, but in the case of this stock the 1983cohort has been exceptionally strang, and it is possible to make rather unambiguousassignations to age merely from length-frequency information. For present purposes, only •tagging information trom this cohort is considered.

Defining the abundance of tagged fish at large as Ta,y, we have for the first year after release:

(5)

and in subsequent years:

{6}

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As only a minuscule fraction of the stock is tagged in this case I do not model the impact of thetagging experiment on the untagged population N. From the usual catch equation, the expectednumber of tag returns from the ith experiment will be:

A _( my ) Fa,yX/'ay- L T;ay (1 -exp( -Fay-M-Mlay», , C" F +M+MI .

a a,y a,y a,y

Under assumption of Poisson errors, the logarithm of the Iikelihood of observing the tag returnsXi,a,y given the structural models as above is:

Li,a,y X;,a)n(Xi,a) -Xi,a,y-ln(X;,a) (8)

Tag model returns appeared to conform weil to both the structural models and to an assumptionof Poisson errors. A single additional parameter St is estimated, and is assumed equal for alltagging experiments.

Candidate Model Elements

For present purposes, uncertainty in the observation error structure for the acoustic surveys isadmitted, and also uncertainty in the appropriate form of stock-recruit relationship.

Acoustic Survey Error Models

The three candidates tested are that the acoustic surveys estimates are distributed with normal,lognormal, or gamma errors, with expectation given as above. The three forms of log-Iikelihoodfunction corresponding to each are:

Normal:

• LogNormal:

Gamma:

1 ( (U ,-ON )2]--L In(2n)+ln(02)+ a,y./ I a,y2 0 2

1 [ In(U ION )2)__"" , In(2n) +In(U 2 ( 2 ) + a,y, I a.y2 La,y,1 a,y,1 I 2

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(9)

(10)

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The Gamma model was included in addition to the normal and lognormal models usedpreviously by Anon (1996b) on account of a perception that there are marked residual patternsin opposite directions when the two foregoing models are fitted. Such trends are apparent whenstandardised residual plots around maximum-likelihood model fits are compared. The Gammamodel is intermediate between those two models in terms of the change in variance with fittedvalue.

Stock-Recruitment Models

Five different forms of recruitment models were considered by Anon (1996b), and theirimplications for medium-term stock development were considered. Here only two of these formshave been admitted as acceptable alternatives:

and the corresponding log-likelihood function is:

1 ( In(R/N )2)--L . In(2TT)+ln(R20 2)+ O.y2 a,y,t Y 2

o r

1. Beverton-Holt form with lognormal error. The structural model is:

R = A sSbyy B+ssby

(12)

(13)

•2. Ricker form, assuming lognormal error. The structural model is:

and the observation error model remains as above.

Scalar Parameters and Priors

(14)

Referring to (2) above, in the present model the parameter vector e contains the followingunknown scalars:

123-5678910-11

Population abundance in 1997, age 6 (Ns,1997)

Population abundance in 1997, age 14 (N 14,1997)

The catchabilities Q for each of the three acoustic surveysthe initial tagging mortality Stnatural mortality Mthe additional mortality MI due to IchthyophonusThe variance of the acoustic surveys 0 2

Parameters of the stock-recruit relationship

•

and the alternative model structures M tested are:

1. Normal, Lognormal or Gamma pdt tor acoustic surveys2. Beverton-Holt or Ricker form of stock-recruit relationship

iv

•

Natural Mortality

In the case of Norwegian spring spawnirig-herring, it is hypothesised that there may have beenan additional mortality from an outbreak of the disease Ichthyophonus hoferi, which has beenreported between the years 1991 and 1994. Consequently, additional mortality due to thedisease outbreak (MI) and mortality from other natural causes (M) are treated as separate modelparameters. The disease outbreak has been observed only to affect fish of certain ages incertain years. Based on these observations, the natural mortality model (as used by Anon,1996b) can be written

Natural Mortality =M, unless y>1990 and y<1995 and (y-a»1987, when M =M+MI

In the ICES assessment M and MI take assumed values of 0.13 and 0.10 respectively.

Nuisance Parameters

The survey catchabilities Qj , the survey variances 0 2 and the stock-recruit refationship variances02r are not necessarily of interest for management purposes. Walters and Ludwig (1993)propose analytic solutions for eliminating these from the calculation of Bayesian posteriordistributions, but their findings are specific to Gaussian distributions. Here numericalintegrations over the variances and catchabilities Q are used, and using the MCMC method thenumerical demands of the calculations become tractable.

Prior Distributions

Uniform prior distributions were chosen for each alternative model structure, on the basis thatno model is known apriori to be more Iikely than any other (eg P(lognormal errors) =0.33; P(Ricker Stock-recruit = 0.50)). Uniform prior distributions were also chosen for populationabundance parameters 1-2 and 9-11 above, with range chosen to be sufficiently wide to defineregions of virtually zero posterior probability.

Restrictive prior distributions were chosen for natural mortality, disease-induced mortality andtagging survival. The prior distribution used for the tagging survival parameter St was uniformfrom 0.2 to 1.0. This parameter must have an upper bound set to 1 for obvious reasons.

A uniform prior distribution for the additional disease-induced mortality was chosen based ontesting the estimate used by Anon (1996b) of MI= 0.1, so that a uniform distribution over therange 0.0 to 0.1 was used. Similarly, the range for the baseline natural mortality M was chosenas the range 0.1 to 0.2, centred on the Working Group assumption of M=0.15

A reciprocal prior distribution for the survey variance was used (prior probabilities proportionalto 1/02), according to Jeffreys (1961), with a range chosen to be unrestrictive.

v

---- -- --- ----------

~:cra.co..a... >0-.g :;;.!! c<Il Gloea."t:l

I Gl

~iiw o 5 10 15 20

~:cra.cea... >0­.2 .~.. <Il.!! c<Il Gloea."t:lGliäE~w o 2 3 4 5

Abundance at age 6 (Billions) Abundance at age 14 (Billions)

•~:cra.co..a... >0­0_

"a:: 'iii.!! c<Il Gloea."t:lGl

oera:;W

0.2 0.4 0.6 0.8

>0-:=:cra.co..a... >0­0_

'a:::: 'e;;.!! c<Il Gloea."t:lGl

oera~W

0.1 0.11 0.12 0.13 0.14 0.15

Tagging Survival Natural Mortality

Survey Variance (x1000)•

~:cra.co..a... >0­0_'a:: 'e;;.!! c:<Il Gloea."t:lGl

oera~W o 500 1000 1500 2000

~:cra.co..a... >0­0_'a:: 'iji.!! c<Il Gloea.

"t:lGl-;E~w o 0.5 1

Fishing Mortality(Unweighted Mean on ages 6-14)

1.5

~:cra.co..a... >0­0_';:: 'e:;;.!! c:<Il Gloea.

"t:lGl-;E~w o 0.02 0.04 0.06 0.08 0.1

Figure 1. Estimated Bayes posterior probabilitydistributions for some key parameters in thestock assessment.

Disease-Induced Mortality

In (Q) February-March

~:c<a,go""0-"">..g ~~ cUI CUoe0-"Clcu'l;j

.5'tiw -8.5 -8 -7.5 -7

~:c<a,g0""0-

"" >.0.;: ~UIcu c'ti cu

0 e0-"Clcu'l;jE:;::Oll

-6.5 w-8.5 -8 -7.5 -7

In (Q) January

-6.5

•>.

=:c<a,goQ:

o~.~ 'Ci)~ cUI cuoe0-"Clcu'l;jE~w -8.5 -8 -7.5

In (Q) December

-7

Stock- Recruitment Models

~ 0.6:c<a,g 0.50Q:

0.4 1]11\\%0.;:0.3 I\Wcu

'ti *~~i_00.20-

I~:;n~w"Cl

~ 0.1 ill~N:'1fi-6.5 E ..::.,~~

~ 0W Beverton .Holt Ricker

Error Models

•

>.::5:c<a,goQ:"" >..g ~~ cUI cuoe0-"Cl~<a

.5'tiw o 246

SSB (Million t)

8

>.~:c<a,g0 0.8""0-

"".2 0.6t-Oll 0.400-"Cl 0.2cu'l;j

.5 0'tiw Normal

-Log­

normalGamma

Figure 1 (contd.). Estimated Bayes posterior probability distributions for additional parameters inthe stock assessment.

•

•

0'

. ,:}.,"'.p' NU

'. \ ~L ,,',\;,.: ...~.~.: ~'" ,~I' ='. . ..

.. #> -. .. .' ... #> • • • •

}::' '/~" ;'~. ~\.'" ..\ ~;.., t,;;.: .i~ .li ~

Figure 2. Multivariate scatterplot of40Qsamples from the Markov Chain ofparameters. N6and N14, Cohort abundanees at ages 6 and 14 on 1 January 1997. St, tagging mortality.QFebMar, QDec and QJan, acoustie survey eatehabiities for the surveys on the spawninggrounds in February and March, for surveys on the wintering grounds in Deeember, andsurveys on the wintering grounds in January. M, natural mortality. MI, additional naturalmortality eaused by a disease outbreak. Var, aeoustie survey variance (pooled for all surveys).SQC, Fstatus quo eatch forecast for 1997. SSB, spavming stock size in 1996. F, fishingmortality in 1996).

,------------ ---

~:EiIII.c1:Q.

... >.0_.~ 'inGI c1ii GIoeQ.

"'0GI

'1;jE~w

2.1 2.3 2.5 0.0 1.0 2.0 3.0 4.0

Estimate of Blim (Million t) Estimate of SSB97/Blim

Estimate of Fpa =FO.t

(Unweighted Mean, ages 6.14)

~~ ref. 1960-"'. 96:: L.-- _

, .

765432

Estimate of F961 Fpa (Ref. 1996)(Unweighted Mean, ages 6-14)

~:EiIII.co..

Q.

.. >..g .~GI C1ii GIoeQ.

"'0

~ilw

0.140.120.1

...' ..

.,, ., ., ,, ,, ., .,

0.08

>.:!::EiIII.coa:... >.0_.~ '(ji~ CIII GIoe

Q.

"'0GI

'1;jE~w

Figure 3. Estimated probability densities of Blim and Fpa =FO.1 reference points, and theassociated probability densities of stock size in 1997 and of fishing mortality in 1996 relative tothose reference points. Two calculations of FO.1 are made, based either on 1996 stock weightsand selection pattern, or else on mean weights and selection in the period 1960-1996.

~

11997

1'

1:.ci >- 0.9

11997

1"'..c ~ 0.80... ..ca. .2: 0.7... >-

~ 0.60_'i: 'e;;

~ 0.5GI c:1ii GI

~ 0.4OQa. "''C 'S 0.3.2! § 0.2"'E o 0.1tlw 0

0 200 400 600 800 1000 0 200 400 600 800 1000

Harvest Control Law Catch Harvest Control Law Catch(Thousand t) (Thousand t)

~ 1:.ci

11998

1 ~0.9

11998

1"'..c 0.80 :.cie ...0.7a. "'... >- ..c

0 0.60_ ...'i: 'üj a. 0.5.2! c: GItI) GI > 0.4~c i'C 'S 0.3<SI E- 0.2"' :::lE 0 0.1~w 0

0 200 400 600 800 1000 0 200 400 600 800 1000

Harvest Control Law Catch Harvest Contral Law Catch(Thousand t) (Thousand t)

Figure 4. Left, estimated probability densities of harvest control law catches for 1997 and1998. Right, cumulative probability that a given catch will exceed the catch corresponding tothe harvest controllaw. Distributions calculated using the estimates of FO.1 using 1996estimates.

..

2000

Year

1999

-----------

19981997

2000 ~ ---.195% I,

1800 ",1600 ",,

I,,;- 1400'1:l

~ 1200Ul;:,~ 1000!=.'1:l 800Gi>= 600

:~--~------~-----~--~

8 -.----------------------, 195% I

20022001

,I,,,,,

II,,-------

2000199919981997

;-7c§ 6

~CIl 5N

iii

g4~~~~(;) 3Clc _-- _

.~ 2-...Co

CI) 1

IYear

Figure 5. Perceptions of future catches and future stock size inthe medium term, resulting from the implementation of theharvest controllaw.Upper panel, Yield. Lower panel, spawningstock size,

..

Adaptive Rejection Sampling

"Coo

J::.'Qj::;;::.JC,o-Iftjc:~'Sc::oo

Value of Parameter

Appendix Figure 1. Construction of upper envelope function foradaptive rejection sampling. Bold line, conditional probability distributionof unknown form. Fine lines, constructed upper envelope function.Dashed line, construction Iines. Arrows show the points at which thefunction has been evaluated. New points are added and the envelopereadjusted until a sampIe has been taken from the conditionaldistribution, which typically takes 5 to 8 function evaluations.