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!his paper not to be cited ~ithout orior reference to the authors.

International Council forthe Exploration of the Sea

C.M. 1982/G :34Demersal Fish Committee

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ESTii·..~TlhG POPULA'.l.IOil SIL.t. f ROi'j --

RELA1IVE ABUNDANCE DATA MEASURED ~ITH ERHOR1

J. S. Coll iehoods Hole Oceanographic Institution

~oods Hole, hA 02543

M. P. SissenwineNortheast ~isheries Center

woods Hole Laboratory~oods Hole, ~A 02543

Abstract

A modified DeLury metnod ta estimate fish population sizefrom relative abundance data is developed. In addition toestimating catchability coefficients, the technique accounts forerror in the measurement of relative abundance. A general-purposenonlinear regression subroutine is used to fit tne model. 1he

"technique is demonstrated using Northeast ~isheries Center bottomtrawl survey data as a measure of relative abundance. ritting wasearried out for four fish populations: Georges bank and Southern~ew England yellowtail flounder; Georges bank and NrtiO S.a. 4lhaddock. Catchability coefficients calculated in this manneragree closely with prior estimates. 1n addition, the techniquesmooths the survey data by filtering the measurement error fromtrue fluctuations in population size. Population size estimat~sfar haddock derived by this method are in elose agreement withvirtual population analysis (VPA) es~imates.

lSubmitted far publieation in Can. J. fish. nquat. Sei.

·~lnt~oduction

Li~ect enume~ation of a ~a~ine fish population ove~ a la~ge

a~ea is not gene~ally possible; the~efo~e, indi~ect methods ofestimating population size a~e necessa~y. Standa~dized catch pe~

unit effo~t indices (based on eithe~ fishe~y o~ ~esea~ch vesseldata) p~ovide a measu~e of ~elative abundance; i.e. u:qii whe~e ~

is ~elative abundance, 1I is absolute population s1ze and q 1s apa~amete~ called the catchability coefficient.

Catch pe~ unit effo~t data f~om comme~cial or ~ec~eational

fishe~ies is difficult to standa~dize such that q is constant.Iechnological imp~ovements inc~ease fishing powe~ and bias p~e­

sumably standa~ized indices. Ihe~efo~e, standa~dized ~esea~ch

vessel su~vey data a~e now b~oadly applied in o~de~ to monito~

t~ends in population size (Cla~k 1979, ~issenwine et ale InP~ess).

Absolute abundance is dete~mined by calib~ating u (estimatingq) using othe~ independent info~mation. One app~oach is to esti­mate the efficiency of sampling gea~ (i.e. q) by comparing su~vey

catches with direct enumeration (using subme~sible o~ came~a

sleds fo~ example, Uzman et ale 1917) within a ~elatively smalla~ea. Unfo~tunately, estimates of. q foi any specific a~ea a~e notnecessa~ily ~ep~esentative of a much la~ge~ a~ea inhabited by apopulation of fish.

Relative abundance is usually calib~ated by compa~ison withvi~tual population analysis estimates (VPA; Gulland 1965, Pope1972) of population abundance. Since ~elative abundance indicesbased on resea~ch vessel surveys suffe~ f~om inherent va~iability

(öy~ne et ale 1981), estimates of q a~e u5ually imp~ecise.

Fu~the~mo~e, \PA ~equi~es comme~cial catch-at-age data and isonly useful fo~ histo~ic estimates of population size.

Ihis pape~ introduces an alte~~ate app~oach fo~ estimating .­population size based on ~elative abundance indices and ccmme~-

. cial catch data. T~o alte~nate fo~ms of the model a~e developed.Ihe fi~st can oe applied when only total catch is ~no~n; :hesecond ~equi~es catch-at-age data. (He~eafte~, the ~~o fo~ms of~he model will be ~efe~~ed ~o as 'age-g~ouped' and 'age­st~uctu~ed' ~espectively.) The technique makes use of CeLury's(1947) metnod as modified by Allen (1966). A simila~ method hasbeen used in the estimation of whale population size (e.g. ~i~k­

~ood 1981).

Cur method produces a smoothe~ fit by accounting fo~ measure­ment variability in research vessel data. Statistically, this istne problem of structural relationship between variables (~endall

and Stua~t 1973). Applied to relative abundance data, the tech­nique estimates catchability coefficients and filters the surveydata. using this method, the total error i5 parti:ioned inteobse~vation e~ro~ and variation not accounted for by the dynamicmodel. Ihe filtering theory ~as developed by ~alman (1960) andapplied to ~egression analysis by Luncan ana horn (1972). rarthis application, regression coefficients a~e unknown and the

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~esultant nonlinea~ity in the ~eg~ession equation necessitates anumerical solution.

The statistical ~ethod which we employ has recently beenapplied to the estimation·. of . stock-rec~uitment relationships(Walters and Ludwig 1981, Ludwig and ~alters 1981). La~ge er~ors

in measurement of spawning stock not only obscure the stock­rec~uitment relationship, but failure to account for thesemeasurement errors may also lead to a significant bias in fittinga stock-recruitment curve.

DATA aASE

, 'Since 1963, .the No~theast Fisheries Center (UEFe) has conduc­ted standardized research vessel bottom'trawl surveys during theautumn of each yea~ ~long the northeast coast of the USA' f~omjust south of Hudson Canyon to Nova Scotia and f~om a depth ofabout 27 m to tt~ edge of the continental shelf (Grosslein 1969,Clark 1979). In 1967, the survey area was extended southward toCape Hatteras • A standa~d "36 'tankee" trawl with a 1.25 cmst~etch-mesh cod-end lining has been used throughout the timeseries of autumn surveys. 1he trawl measures 24 m along thefootrope, ave~ages 3 m in height along the headrope, and isequipped with ~ollers to make it suitable for use on ~ough Dot­tom.

Sampling has been based on a stratified random design. Ihe·survey area has been stratified into zones based on depth andlatitude (Figu~e 1). App~oximately 300 stations are sampled dur­ing each su~vey although in recent yea~s this number .hassometimes been nearly doubled. Stations a~e allocated to strataapp~oximately in p~opo~tion to the a~ea of each stratum withspecific assignment within strata on a randorn basis. A 30 minutetow at an ave~age speed of' 3.5 knots is made at each station. Thetotal catch f~om each tow is sorted to species and the number,weight and length f~equency dete~mined and ~ecorded. Samples a~e

collected fo~ sho~eside p~ocessing.

Autumn su~veys a~e more representative of population sizeapproximately two months late~ at the beginning of the next yearof the survey. The~efo~e, the survey index for fish age a-1 fremthe autumn of year t-l is used as a relative abundance index fo~

fish age a in year t. Catch da ta used in this study are fromstatistical bulletins of the No~thwest Atlantic fisheriesOrganization and va~ious unpublished stock assessment dccuments.

ror yellowtail flounder the annual catch was calculated forthe year beginning Octobe~ 1 so as to be in phase with autumnsurveys. io~ haddock, the available compilation of data made itmo~e convenient to use catch on a calenda~ year basis. ~I catchis rather evenly distributed th~oughout the year, then this phaseshift should be of little consequence.

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Relative abundance indices fo~ Geo~ges Lank haddocK and yel­lowtail flounde~ ~e~e based on st~ata 13-25 of the standa~dized

t~awl survey. Ihe sou~he~n ~ew England yellowtail rlounde~ ~ela­

tive abundance index was based on st~ata 1-12 and the 4x haddockindex was based on st~ata 31-34. tollowing b~own and hennemuth(1971), Lux (1969) and Cla~k et al. (1982), natural mo~tality ofboth species was assumed to be 0.2. to~ both species, one yea~

old fish (du~ing autumn su~veys) are conside~ed as p~e-recruits

which ~ecruit to the fishe~y during the following yea~.

AIHl.LYTICAL hE.1HODhge G~ouped MOdel:

rela~es the population size in numbers ror a given yea~, h(t+1),to population size in the preceding year, N(t), minus the catch,C(t), plus recruitment, R(t), all discounted by na~u~al

mortality, M. The model implicity assumes that the fish stock isclosed to migration. llatural mortality is assumed to be constan~

and to occur after the catch. The implications of these assump­tions are considered in the discussion.

The first-order diffe~ence equation,

U(t+1)=(N(t)-C(t)+R(t» exp(-h), (1)

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The survey relative abundance, n(t), is related to theabsolute population size by a catchability coefficient, q, suchthat

n(t)=q N(t)

and assuming that the rec~uits have the same catchability,

r(t)=q R(t).

(2)

(3)

As noted previously, n(t) is the stratified mean ca~cn per tow of 41recruited age fish f~om the autumn trawl survey of yea~ t-l. Therecruitmenc index is based on su~vey catch of pre-recruits. ~ub­

stituting fo~ rl(t) and flet) in the dynamic equation and multiply-ing by q, we ob~ain the ~eg~ession equation:

n(t+1)=(n(t)-qC(t)+R(t» exp(-h)+.s(C); (t:1,i-1) (4)

whe~e e(t) is the equation e~ror or natural variability notaccounted for by the dynamic model. Stated as such, the problemis a standard linear least-squares estimation of q. Ir ~ is un­known, it can also be estimated, but the estimates of ~ and q maybe hightly correla~ed and tne~efo~e sUbject to conside~able

er~or-.

lhe new twis~ to the p~oblem is that both n(t) and r(t) haveassociated measurement errors. following Clark and Ero~n's (1977)justification fo~ a logarithmic transformation of the surveydata, obse~vation error is assumed to be a lognormally dis­t~ibuted random variable ~ith mean cne. 'ihis makes intuitive

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sense in that ehanges in trawl effieieney ought to have a mul­tiplieative effeet on the eateh and it also insures against nega­tive population estimates. The four populations eonsidered herehave deelined markedly during the years for whieh we have data.Logarithmie transformation tends to equalize the weights given tohigh and low abundanee indices in the fitting proeedure. 1hemeasured relative abundanee, n'et), is related to the 'true'relative abundanee by

n'(t):n(t)exp(n(t»; (t:l,i)

where (t) is a normally distributed randorn variable.for the reeruits,

(5)

Similarly

•r'(t):r(t)exp(ö(t»j (t:1,i-l)

where Q (t) is also a normally distributed random variable .(6)

Errors assoeiated with the eateh measurements are not eon­sidered for two reasons. rirstly, these errors, ~hieh ineludernisreporting and nonreporting of the eommereial eaten and dis­eards, are probably more systematie than random in nature. Inother words, eateh may have been eonsistently underestimated.Seeondly, the eorrelation between the estimators of C(t) and qprecludues reliable estimation of both.

Given a time series of i years of data, there are i expres­sions for Equations 5 and 6 and i-1 expressions for Equation 4.Ihis . gives 31-2 residual errors with 2i parameters to be f1ttedand i-2 degrees of freedom.

Age Struetured ~odel:

The age-struetured model eorresponds to the age-grouped modelexeept that both the survey relative abundanee and the' eommereialeateh are divided by age elass. In lieu of ~quation 1 we write

h ( t+ 1,a+ 1) : (N ( t , a) -C ( t , a) ) ex p (-;-\) • (7)

Assuming the eatehability eoeffieient to be eonstant over all ageelasses,

n(t,a) = q lj(t,a) (8)

Here N(t,a) is the stratified mean eateh per tow of age a-l fishfrom the autumn trawl survey of year t-l.

In praetiee, beeause there are so few survivors in the ter­minal age elasses, it is eonvenient to lump them into one group(i.e. for haddoek our terminal age elass is age 9 and olderfish). Sinee fish ean remain in this terminal age elass for morethan one year, we need two forms of ~quation 4 for the age­struetured model. lf j is the terminal age group (in this ease9+) the dynamie equation for ages 1 co j-l is

n(t+1,a+1) : (n(t,a)-qC(t,a»a:1,j-2)

and fo~ the te~minal age class j

exp(-h) +E(t,a); (t:1,i-1;(9a)

n(t+l,j):[n(t,j)+n(t,j-1)-qC(t,j)-qC(t,j-l)]exp(-M)+(t:l,i-l)

E(t,j);(9b)

again the measu~ed ~elative abundance n'(t,a) is ~elated to the't~ue' ~elative abundance by

n'(t,a}:n(t,a)exP(n(t,a» j (t:l,ija:l,j) (10)

Given data fo~ i yea~s and jage classes we have (i-1)*(j-1)exp~essions fo~ ~quations 9 (a and b) and i*j-l exp~essions fo~

iquation 10. The~e is no exp~ession fo~ n(i,l) because it 1s notincluded in eithe~ of the dynamic Equations 9 a o~ b. ihis givesa total of i*j-1+(i-1}*(j-1) ~esidual e~~o~s with i*j pa~amete~s

to be fitted and (i-1)*{j-1)-1 deg~ees cf freedom.

PARAhE:1E.R ESTL....:ATIOI~:

The catchability coefficient, q, was estimated by fittingEquations (4-6) o~ (9-10) simultaneously using the gene~al­

pu~pose nonlinea~ ~eg~ession sub~outine, ZXSSQ, f~om the lnte~­

national Mathematical and Statistical Lib~a~y on the ~HOI Vax11/780 compute~. Due to the diffe~ence in magnitude between su~­

vey and comme~cial catches, q was scaled by a facto~ of 10 fo~

nume~ical efficiency of the algo~ithm. ZXSSC is a finite­diffe~ence Levenberg-Ma~qua~dt ~outine fo~ solving nonlinea~

least squares p~oblems.

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The algo~ithm minimizes the sum of squa~es Sex) aboutthe vecto~ of pa~amete~ estimates, (q,e). The cat­chability coefficient, q, is the functional pa~amete~j

.•• ~ep~esents the incidental pa~amete~s. ror the age-g~ouped ()model e :(n(l), •.• ,n(i);~(l),••• ,~(i-1); fo:'" the age strouctu~ed

model e :(n(t,a) foro t:l,i and a:l,i). (li)

lf X is the Jacobian about ~he least-squaroes estimate, x, ~hen

the varoiance-cova~iance mat~ix of x 1s given by

varo ( x) :S ( X) ,. ( X ' X) 1d • f • (12)

As illustroated in ~irokwood (1981) moroe accuroate confidence~egions fo~ x are obtained f~om

(S(x)-S(x»/S(x):f(vl,v2,a)- - - (13)

~he~e i(v1,v2,a) is the critical value with vl, v2 deg~ees oifreedom in the numerator and denominator roespectively. ~n tnenonlinea~ case, the confidence inte~val is exact bu~ ehe p~o­

bability level is approoximate (Droape~ and Smith, 1966).

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1wo alte~nate methods we~e used to evaluate the te~m sex) inEquation 12. The fi~st method is essentially a linea~ization ofthe system of equations about the least-squa~es estimate, x. Ifthe incidental pa~amete~s, e (t, a) a~e assigned thei~ least.­squares values, Equation 12 'can. be solved explicitly for thecritical values of q (see D~aper and Smith 1966, pp. 282).

sex) can be calculated more rigou~ously by fixing the valueof q and fitting the incidental parameters with the sub~outine

ZXSSQ. 1he calculated r statistic is then compared to the alphacritical level. The 'exact' method is more expensive in that itinvolves a grid search of the parameter space. both methods wereused for each trial.

1he-use· of ZXSS~ fo~ this problem is optimal. in thestatistical sense. However, the numerical solution is inefficientbut since we are only demonstrating the method here, the expenseis of little cancern. ior extensive application more efficientmethods are available. Ludwig emd ~~alters (1981) apply .i~ewton' smethod to a similar problem.

RESULTS

The age grouped model was fit fo~ yellowtail flounder(Georges Bank and Southe~n New England populations) and haddock(Gearges Eank). The age-structured model was fit for haddock(Georges Bank and tiAFO S.A. 4x populations). Eoth versions of themodel were applied to Georges Bank haddock,thus allowing the per­formance of the two forms of the model to be eompared. For yel­lowtail flounder, the model was fit with eatch ~eported both inphase (Oetober-September) and out-of-phase (ealendar year). The~e

was little differenee in pa~amete~ estimates, thus alleviatingeonee~n about using out-of-phase eateh fo~ haddock. 0nly resultsbased on in phase catch are repo~ted for yellowtail flounder .

Estimates of q and assoeiated eonfidenee intervals a~e shownin Table 1. lhe eatchability coefficients agree· quite closelywith previous estimates by Clark and brown (1977) even thoughtheir eoefficients are based on weight rather than numbers. Con­fidence intervals based on linear approximations are eonsistentlynarrower than the 'exact' confidence intervals, except for 4xhaddoek. Very similar estimates of q for Georges Bank haddoek areobtained from the age-grouped and age-structured mOdels. However,the eonfidence intervals are eonsiderably narrowe~ for the age­structured model.

Differenees between catchability coefficients witnin speciesbut between areas probably reflect differences in bottarntopography. rar yellowtail flounder, the eatehability coefficient1'or the rougher bottem of Georges Bank (q=O.107 x .10-6 ) ·..Jouldbe expected·· to be lower than' fo~ southern i.e',.; c.ngland( q =0 • 313 x 10-6 . ).

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1he diffe~ence in catcha~ility between Geo~ges cank yellow­tail and haddock could ~eflect thei~ diffe~ent life habits. Onemight speculate that the yellowtail's habit of lying on the bot­tom makes it less p~one to being caught by the otte~ e~awl. Infact, the UE.FC otee~ t~awl su::vey is an out.g~owth of ea~lie~

Geo~ges Bank haddock su~veys. Thus, the su~vey would be expected~o be pa~ticula~ly efficient fo~ this population.

Yellowtail flounde~ is an excellent candidate fo~ applicationof ou~ app~oach because although total catch has been ~epo~teQ,

age composition suitable fo~ VPA is not now available. Ihe~efo~e,

a new method, such as the one applied he~e, is necessa~y.

~urthe~mo~e, although t~awl su~veys t~ack t~ends in abundance,individual su~veys sometimes appea~ anomalous. The statisticalmethod used he~e is amenable to this situation. The ~esults a~e

shown in figu~es 2 and 3. Note in figure" 2 how the fitting pro­cedu~e smooths the su~vey abundance and appea~s to distinguishbetween t~ue fluctuation and measu~ement e~ro~. In 1966, fo~ ex- ~ample, the low abundance of age t~o and olde~ (2+) fish and thehigh level of one-yea~-olds is consistent with the model. On eheother hand, ehe peak in 2+ fish in 1972 is inconsistene with themodel because rec~uitment was falling in the p~eceding yea~s.

Haddock was chosen in o~der to al10w a comparison betweenpopulation estimates gene~ated by this method and tnose ob~ained

by ~PA." Age 2+ population estimates we~e gene~ated by multiplyingthe smoothed su~vey abundances by the app~op~iate catchabilitycoefficients. Let h(t) be the age 2+ population size in yea~ t.fo~ the age-g~ouped model N(t) is estimated by ~·(?(t)+a(t»; fo~~he age st~uctu~ed model, r.(t) is estimated by a~Z ,cl • n(t,a). Ihe~esults a~e shown in figu~es 4 and 5. Not only Q~ ou~ two modelsgive simila~ estimates of q fo~ Geo~ges bank, but it can be seenin figu~e 4 that except fo~ th~ final th~ee yea~s, the populationestimates a~e almost indistinguishable.

In both a~eas (Geo~ges Bank and ~AiO S.A. 4x) t.he~e is closeag~eement between the alte~nate estimates of haddock populationsize. It should be noted that. the VPA estimates fo~ the final fewyea~s a~e sensitive to the fishing mo~tality estimaees used toinitiate the analysis. These most cu~~ent estimates a~e the~efo~e

the most p~one ~o e~~o~. ü'~oyle (1981) used" ewo diffe~ent

app~oaches to initialize his VPA fo~ 4x haddock. The sensitivityof the ~esults fo~ the most ~ecent yea~s is demonst~aeed infoigu~e 5.

Ihe dispa~ity between ou~ method and VPA fo~ Geo~ges bankhaddock is g~eatest fo~ the mid 19bO's and late 1970's. Du~ing

the former, ou~ estimates are lower than VPA estimaces. VPA ~ould

ove~estimate actual population size if ehe catch wasove~estimated. In fact, most of the cacch du~ing the mid 1960'swas by discant wate~ fishing vessels Ce.g. USSR) which only~epo~ted thei~ cacch in weights, not numbe~s. lhe numbe~ caughthad to be estimated by assuming that the length composition wascomparable to the length ccmposition of tne ~esea~ch vessel t~awl

su~vey catch (C13~k et al. 1982). Thus, VFÄ ~esults fo~ the b~ief

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per iod when the distant water fleets dorninated the fishery areparticularly uncertain.

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Similarly, ehe disparity between our meehod anti v?A duringthe late 1970's may reflect:bias in"the VPA. During the late1970's catch quotas were extremely restrictive; therefore an un­known amount of the actual catch was unreported, discarded ormisreported as being from S.A. 4x (Clark, personal communica­tion). Thus, the VPA for Georges Sank haddock probably un­derestimates population size during the late 1970's. rurthermore,if same Georges Sank catch was rnisreported as S.A. 4x catch, thenthe VPA for the latter would overestirnate population size. ~his

may explain the disparity between our method and ~PA for the midand late 1970's (figure 5). Population size indices during thisperiod(VPA estimates) are higher than estimated. based. on ·ourmethod •

The relatively large confidence interval ab out ehe estimatesof q fo~ 4x haddock reflects the high variability of the survey­data for that area. The trawl survey sample size for stratawithin S.A. 4x is relatively small.

DlSCUSSIOH

~e have now described and applied a new method of estimatingpopulation size based ·on rela tive abundance ind ices. In addi tionto estimating catchability coefficients, the technique smoothsthe survey data. Population size estimates for haddock using ourmethod compare favorably with VPA estimates.

The Georges Sank haddock applications show that the age­grouped and age-structured models yield virtually· identicalpopulation size estimates. Ihis supports application of the age­grouped model for estimaeing the size of populations, such asyellowtail flounder, where the age composition of the catch i5unknown. ~here commercial catch at age data are available, thereare t~o advantages to using the age-structured modelover theage-grouped. firstly, the age-structured model provides smoothed

'population estimates by age. Secondly, the confidence intervalsabout the parameters estimated by the age-structured model aresmaller because this model has more degrees of freedom.

Ihe similarity between our estimates of population size andthe VPA estimates is perhaps not surprising since both proceduresare based on the same catch history and both make the same assum­ptions about natural mortality. however; an important differencebetween our age-structured model and VPfi should be pointed out.In order to initialize VPA, assumptions must be made about thefishing mortalities (i-values) acting on the oldest ages of eachyear class; thus the system of equations iso underdetermined (Pope1979). In general, it 1s the estimates far the final few years ofthe analysis which are most sensitive to the starting f-values.

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by usin6 the su~vey data as independent measu~es of popula­tion abundance, ou~ method avoids the p~oblem of an unde~dete~­

rnined system. lhe dynamic equations (4, 9 A and 0) link enepopulation abundances f~om yea~ to yea~. lhe abundance estimatesfo~ the fi~st and last yea~s of ou~ analysis a~e const~ained bythe dynamic equations only in one di~ection and would the~efo~e

be expected to have a la~ge~ deg~ee of unce~tainty associatedwith them. Since the"model does not include a stock-~ec~uitment

~elationship, the estimate of age 2 abundance should have asimila~ deg~ee of unce~tainty. Ihis p~oblem is common to allmethods of sequential population analysis.

Doubleday (1981) p~esents a method fo~ efficient estirnationof the te~minal fishing mo~tality ~ate necessa~y to initiatesequential population analyses (e .g. VPA). Resea~ch vesselabundance indices, by age, a~e used iteratively to calib~ate sur­vivo~ estimates. lhe catchability coefficient is allowed to va~y

wit .. 3ge.

Since Coubleday's (1981) method ~equi~es catch-at-age data,its application is more limited than ou~ methode Fu~the~mo~e, themethod aoes not take account of measu~ement e~~o~ and the~efo~e

it appea~s sensitive to va~iability in ~elative abundance data.Doubleday applied his method to ~AfO S.h. 4x haedock using ehesame ~esea~ch vessel bottorn t~awl survey data as we used he~ein.

He concluded that the su~vey abundance index was inadequate fo~

quantitative estimation. Although the confidence intervals fo~ qof 4x haddock a~e ~elatively la~ge, the ~esults ~epo~ted inFigu~e 5 a~e certainly encou~aging.

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1he validity of population size and pa~amete~

derived he~e depends on seve~al assumptions; these a~e

below.

estimatesdiscussed

Although the selectivity of the otte~ t~awl is not known, it •seems clear ehat the~e a~e age-speoific diffe~ences in cat­chability coefficients, especially fo~ one and two-yea~-ola fish .

. " lt is not possible to estimate a sepa~ate q fo~ eacn age classbecause of the way in which the ~eg~ession model is fc~mulated.

One app~oach would be to fit a catchability ccefficient fc~

sequential pai~s of age classes. ~his app~oach, howeve~, wouldlack the continuity p~ovided by the dynamic equaeions and theestimates of q woule the~efo~e be ve~y unce~eain. If ehe gene~al

fo~m of the ~elation of q with age we~e known, it might be pos­sible to fit, say, a two-pa~amete~ cu~ve to desc~ibe it. Clea~ly,

it would be desi~able to be able to dete~mine the selectivity ofthe otte~ t~awl independently.

1he catchability ccefficients calculatea by ou~ method ~ay oeconsidered in a sense as weighted means of age-specific q's. Ifso, the estimates of population by age will be biased but thetotal population estimates should be unbiased.

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rhe estimates of ~ are quite highly sensitive to the value ofM, the natural mortality. If M)O.2 then q has been overestimatedanti vice versa. Uatural mortality is probably higher during thefirst few years of" life. In addition to natural mortality, theseyoung fish are also sUbject to the mortality resulting fro~ un­reported discarding of large numbers of one and two-year-olds(Clark et ale 1982). Due to the high correlation betweenestimates of q and H, age-specific natural mortalities cannot bereliably estimated. If the catchability of young fish is lowerthan for adults, but the natural mortality rate is higher then,at least in terms of estimating q, these two biases should tendto cancel out.

The assumption that natural mortality occurs after the catch~affects estimates of q. If catch is evenly distributed over theyear, then average population size is higher, and more naturaldeaths occur~ Thus, fewer fish need to be caught in order toaccount for the difference between n(t) and n(t.1) and thereforeq 1s overestimated by our method if catch 1s distributedthroughout the year. The bias should not be large for heavily ex­ploited populations like those considered here, but additionalwork is required to quantify it and to develop unbaisedestimators.

The predictive value of this method is limited by the factthat the recruits for the final year of the analysis do not enterinto the dynamic equation. Therefore, it i5 not possible to takeaccount of measurement error. For example, if i is the last yearof the time series for which there is data, the abundance in thefollowing year, n(i.1), can be predicted by prescribing thecatch, C(i.1). The estimate of n(i.1), however, must be based onthe observed recruitment, r'(i), since r(i) is not included inthe dynamic equation (4). Similarly,· for the age-structuredmodel, the population estimate for year i.1 will include theerror in measuring recruitment in year i •

The statistical analysis of the problem of structural,.relationship between variables is ill-defined. Ihere dces not

appear to be a straight-forward method for determining the good­ness of fit of the model. Simulated data sets could be created bythe addition of error terms of a prescribed distribution to atime series of abundance generated by the dynamic model. The fit­ting technique could then be tested for reconvergence about thesimulated abundance indices.,

. It was implicitly assumed in the fitting procedure that theequation and measurement error terms have equal variance. Usingthe age-grouped model, for example, we assume thatvar ( . e: ) ~var ( n )'~var ( oS). It would be desirable to have. an in­dependent estimate of the ratio of the error variances so tnatthe regression equation could be weighted accordingly. 7he esti­mate cf q appears insensitive over an order of magnituae ofweighting factors; it is only the degree of smoothing thatchanges. rinally it should be noted that the'error terms, n (t)and ö (t), are not strictly independent since they come from the

- 11 -

Allen, t~.R. 1966. Some methods 1'0:" estimating exploitedpopulations. J. Fish. Res. 50a:"d Can. 23: 1553-1574.

b:"own, o.E. and R.C. Hennemuth. 1971. P:"edictionof yellow­tail flounde:" population size f:"om p:"e:"ec:"uit catches. Redbooklnt. Comm. Uo:"thw. Atl. fish. Pa:"t 111: 221-228.

Byrne, C.J., T.R. Aza:"ovitz and h.P. Sissenwine. 1981. fac­tors affecting va:"iability af research vessel t:"awl su:"veys. Can.Spec. Publ. fish. Aquat. Sci. 58: 238-273.

Clark, S.H. 1979. Application 01' bottom trawl survey data tofish stock assessment. FISHERIES 4(3): 9-15.

Clark, S.H. and S.E.fishes and squid f:"om1963-1974, as deterrninedU.S. 75: 1-22.

brown. 1977. Changes in biomass 01' fin­the Gulf 01' ~aine to Cape Hatteras,

from research'vessel data. iish. ~ul.

•

Clark, S.H., W.J. Overholtz and R.C. Hennernuth. 1982. Reviewand assessment 01' the Georges bank and Gulf 01' ~aine haddockfishery. Journal 01' Northwest Atlan~ic fishe:"y Service. In Press.

Delury, D.o. 1947. On the estimation 01' biological popula­tions. Eiometrics, 3: 145-167.

Draper, N.H. and H. Smith. 1966. Applied Regression Analysis.~iley, New York. 407 pp.

Doubleday, ~.G. 1981. A method of estimating the abundance 01'survivors 01' an exploited fish population using commercial catchat age and research vessel abundance indices. In ~.G. Loubledayand D. Rivard (eds.), bot~om Trawl Surveys. Can. Spec. Publ.fish. Aquat. Sci. 58.

Cuncan, D.B. and S.D. Horn. 1972. Linear dynamic recursiveestimation from the viewpoint 01' regression analysis. Journal ofthe Amer ican Sta tistical. Associa tion, 67: 815-ö21.

GrossIein, ~.D. 1969. Groundfish survey prog:"am cf BCf ~oods

Hole. Cemm. fish. Rev. 31(8-9): 22-35.

Gulland, J.A. 1965. Estimation af mortality rates. Annex toRep. Artic Fish. ~o~king Group, Int. Counc. Explar. ~ea c.~,.

1965(3). 9p.

r.alman, R. E. 1960. A new approach to linear fil ter ing andprediction problems. Transaction AShE Journal 01' Basic Enginee:,,­ing, 82: 35-45.

hendall, ~.G. and A.Statistics. Val. 11, 3:"d ed.

Stuart. 1973. Ihe Advanced Iheory ofHafner Pub. Co. liew lO:"K, lil.

- 13 -

h.i:--kwooa, G. P. 1981. istimation of stock size using :--elativeabundance data-a simulation study. Rep. lnt. Nhal. Commn. 31:729-735.

Ludwig, D. and C.J. ~alte:--s. 1981. Heasu:--ement e:--:--o:--s and un­ce:--tainty in pa:--amete:-- estimates fo:-- stock and :--ec:--uitment. Can.J. fish. Aquat. Sci. 38: 711-720.

Lux, r. E. 1969. Landings pe:-- uni t effo:--t, age composi tion,and total mo:--tality of yellowtail flounder, Limanda fe:--:--uginea(Sto:--er), off uew England. lnt. Comm. Jorthwest Atl. fish. Res.Dull. 6: 47-52.

ü'Boyle, R. 1981. An assessment of the 4x haddock stock fo:-­the 1961-1980 pe:--iod. CAFSAC Res. Doc. 81/24.

Pope, J.G. 1972. An investigation of the accu:--acy of vi:--tualpopulation analysis using cono:--t analysis. lnt. Comm. ~o:--thw.

Atlant. fish. Res. bulle 9: 65-74. •Pope, J.G. 1979. Population dynamics and management:

status and futu:--e t:--ends. Investigacion Pesque:--a, 43:cu:--:--ent

199-221.

Sissenwine, h.P., T.R. Aza:--ovitz and J.5. Suomala. In f:--ess.Dete:--mining the abundance of fish. Expe:--imental oiology at Sea.A. hacDonald, ditor. Academic P:--ess, Lonaon.

tialte:--s, C.J. and D. Ludwig. 1981. Effects of measu:--emente:--:--o:--s on the assessment of stock-rec:--uitment :--elationships. Can.J. fish. Aquat. Sci. 38: 704-710.

Uzman, J.R., R.A. Coope~, R.c. The:--oux, R.L. ';viggle. 1977.Synoptic comparison of th:--ee sampling techniques fo:-- estimatingabundance and dist:--ibution of selected megafauna: subme:--sible vs.came:--a sIed vs. otte:-- t:--awl :·tar. Fish. Rev. 39(2): 11-19.

- 14 -

•

Table l. Estimated catcnability coefficients. The confidence intervalswere calculated using Equation 12.

95% confidence intervalCatchabili ty

coeffiecien t linearPOPULATIO~ MODEL (x10-6) approximation I~exact"

Yellowtail age-grouped .313 .279-.347 .284-.465flounder (S.N.E.)

Yellowtailflounder (G.B.) age-grouped .107 .079-.135 .065-.160

haddock (G. B.) age-grouped .257 .246-.268 .164-.384• haddock (G.ß.) age-structured .238 .220-.256 .198-.283

haddock (4x) age-structured .340 .222-.424 .261-.425

.1".

IH1PI~

Cl~

15-30

31-60

61-100

> 100

~.

.....

/

'Ir.

Figurc 1. \. strata.trawl BurNEFC bottom

~08SERVED

SOUTHERN NEW ENGLAND

YELLOWTAIL FLOUNDER

AGE 2+

............

"-'\

\\

ESTIMATED \,-'.....

"- .....

""\.

70

50

60

40

:: 300t-

c:::wa.

20--...u....<tU

10z<tw:E

30

20

10 ESTIMATED

AGE 1

63 65 67 69 71 73 75

YEAR OF AUTUMN SURVEY

77 79

FiBure ~ Observed (solid line) against estimated (dashed) abundancefor southern New England yellowtail flounder.

•79

AGE 1

GEORGES BANK

7775

YELLOWTAIL FLOUNDER

AGE 2+

7371

"\.\.

\\

(\BilMArEO \.

22

20

"18

161\

14 / \/ \

I \12 / \

\.

~\

0 10I-

a:: 8wCl.

:I:6

u~ 4u

z 2<tllJ:E

12

10

8

6

4

2

063 65 67 69

YEAR OF AUTUMN SURVEY

Figure~. Observed (solid line) against estimated (dashed) abundancefor Georges Bank yellowtail flounder.

.. 4: l ,.

GEORGES BANK

VPA

AGE-GROUPED MODEL

AGE - STRUCTURED MODEL

8078

/•..........

7674

HADDOCKAGE 2+

726866"'4o.

50

350

500

250

450

100

300

150

200

400

ICf)

G:IJ..oCf)zo::J.....J

:2:

WNCI)

~uot­Cf)

YEAR OF VIRTUAL POPULATION ANALYSIS ( VPA)

Figure 4: Population size of Georges Bank, age 2+ haddock as estimatedby VPA (solid line), age-grouped model (dashed line) and age­structured model (dotted line). The VPA estimates are fromC1ark et a1. (1982).

.. ( ...

300 NAFO S.A. 4X

HADDOCK250

-ü5u.u. 2000(/)

20:::i....J

~ 150wN

VPAtn ,..,,-----.

\ -- "~ ,-- \.u 100 / \.0 / "t; /

../..

- --- ~50 ESTlMATE WITH

AGE STRUCTURE

o 1--...;, , --'-_"--_, ...' --. 1_"""'"----'

64 66 68 70 72 74 76 78 80

YEAR OF VIRTUAL POPULAT10N ANALYSIS (VFA)

Figure 5: Population size of NAFO S.A. 4x, age 2+ haddock as estimatedby VPA (solid line) and age-structured model (dashed line).The VPA estimates are trom O'Boyle (1981). The labels 'a'and 'b' refer to the tvo different methcds used toinitialize the VPA.