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~(/S 5007
INFORMATIONAL LEAFLET NO.224
PARAMETER ESTIMATION FOR A CLASS OF MODELS DESCRI BING
THE MIGRATORY TIMING OF CHINOOK SALMON
(Qncorhynch us tshawylschal IN THE LOWER YUKON RIVER,ALASKA
By
John E.Clark
STATE OF ALASKA
Bill Sheffield,Governor
DEPARTMENT OF FI SH AND GAME
Don W.Collinsworth,Commissioner
P.O.Box 3-2(XXJ,Juneau 99802
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October 1983
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PARAMETER ESTIMATION FOR A CLASS OF MODELS DESCRIBING
THE MIGRATORY TIMING OF CHINOOK SALMON
(Oncorhynchus tshawytscha)IN THE LOWER YUKON RIVER,AlASKA 1
By
John E.Clark
Alaska Oeparbnent of Fish and Game
Division of Commercial Fisheries
Juneau.Alaska 99802
October 1983
This work was done in partial fulfillment of the degree of Doctor of Philosophy
at Old Dominion University,Norfolk,Virginia.
for cT "results in the following correction:
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Ctt)
(l-qflt))
INFORMATIONAL LEAFLET NO.224
Cit)
l-qf(tJ
n
_(l-B,Elt})1!!.1!!..)
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a {l-qf(tJ}
i,tt)
Ntt)
C tt)
n
L
t=l
n eft)L 11-qf}t=l t
Ctt)is assumed to be proportional to (Ntt)-qf(t)N(t})
[
C(t~J~Cft)-
I1-B2flt }}'"'~ITl_~q~f~ltC7}}
t-l
Equation (41)
Equation (42)
which.upon substituting
or ctt)=a {l -qt(t)JN(t}where a ;s the constant of proportionality.Therefore
Equation 41 and 42 were found to be incorrect.The correct derivation is as
follows:
E-R-R-A-T-A S-H-E-E-T
Since
(equation 39)and
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The revision of equations 41 and 42 will result in small changes in parameter
values of Tables 19 and 20.The differences in parameter values are not large
enough to affect the results or conclusions of the report.
•
E-R-R-A-T-A S-H-E-E-T (Continued)INFORMATIONAL LEAFLET NO.224
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TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
ABSTRACT ..
INTRODUCTION
STUDY AREA AND DATA
INTROOUCTION TO TIME SERIES METHODS
DEVELOPMENT OF THE TIME SERIES MODEL
DISCUSSION OF TIME SERIES RESULTS .
DISCUSSION OF MATHEMATICAL METHODS
METHODS
RESULTS
DISCUSSION AND CONCLUSIONS ON MATHEMA",ICAL MODELS OF MIGRATORY TIMING
ACKNOWLEOGMENTS .
LITERATURE CITED
v
vi i1
3
6
7
33
34
35
46
89
93
98
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liST OF TABLES
Table Page
1.Relationship of mean day of migration for the years 1968 through
1981.and average Apr;1 and May air tempera tures . . . . . . . 8
2.Statistics concerning the time series models and relationships
discussed in the text for Flat Island test fishery catches.
196B-1978 ;'".. . . . . . . . . ....13
3.Parameter values and goodness of fit for two functions proposed as
possible models to d~scribe the time series of the errors,Ct.for
test fishery catches from 1968 throu9h 1978 . . . . . . . . ...18
4.Statistics concerning the time series models and relationships dis-
cussed in the t~xt for Big Eddy test fishery catches.1979·1981 22
S.Parameter values and goodness of fit for two functions proposed as
possible models to describe the time series of the errors,Ct.for
test fishery catches from 1979 through 1981 23
6.Parameter values and goodness of fit for the function proposed as
the model to describe the time series of errors,£t,for 1968-to-
1981 test fishery catches • . . . . . . . . . . . . . . . ....29
7.Estimated values of the mean and variance of the normal curve
equation,F-va1ues from the test of significance for differences
in variances between the first and last quartile and the center
quartile,and the resulting sum of squared deviations from the
fitted model 47
8.Estimated values of the mean and variance of the normal curve
equation,F-values from the test of significance for differences
in variances between the first and last quartile and the center
quartile,and the resulting sum of squared deviations from the
fitted model.Data are the cube roots of reported test fishery
catches for the years 1979 through 1981 ...........•..48
9.Estimated values of the mean and variance of thf normal curve
equation,F-values from the test of significance for d.ifferences
in variances between the first and last quartiles and the center
quartile,and the resulting sum of squared deviations from the
fitted model.Da:a are the reported test fishery catches of Flat
Island for the years 1968 throu9h 1978 .49
10.Estimated values of the mean and variance of the nonnal curve
equation,F-values from the test of significance for differences
in variances between the first and last quartile and the center
quartile,and the resulting sum of squared deviations from the
fitted model.Data are the cube roots of reported Flat Island test
fishery catches for the years 1968 throu9h 1978 . . . . . . .50
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LIST OF TABLES (Continued)
Table Page
11.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopulation.the mean and variance of the later 5ubpop-
u1at10l1,and the proportion assigned to the later subpopulation.
The model assumes a mixture of two subpopulations (model two,equa-
~ion 15)and is fitted by maximization of its likelihood function.
Ulitransfonned data are used from the Big Eddy and Middle Mouth test
fis~ery operations.55
12.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopulation,the mean and variance of the later subpopu-
lation,and the proportion assigned to the later subpopulation.The
model assumes a mixture of two subpopulations (model two,equation
15)and is fitted by minimization of the sum of squared deviations.
Data are cube roots of Bi~Eddy and Middle Mouth test fishery
catches .•56
13.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopulation,the mean and variance of the later subpopu-
lation,and the proportion assigned to the later subpopulation.
The model assumes a mixture of two subpopulations (model two,equa-
tion 15)and is fitted by minimization of the sum of squared devia-
tions.Data are cube roots of Flat Island test fishery
catches ••57
14.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopulation,the mean and variance of the later subpopu-
lation,and the proportion assigned to the later subpopulation.The
model assumes a mixture of two subpopulations (model two.equation
15)and is fitted by maximization of its likelihood function.
Untransformed data are used from Flat Island test fishery opera-
tions •58
15.The est~mate~va1~es of the mean and.variance of a homogeneous
populatl0n mlgrat1ng past the test flsheries,and the values of
the coefficients quantifying the effect of wind speed and commer-
~ial effort on the migration.Model three (equation 27)is fitted
to observed data by minimizing the sum of squared deviations fran
the expected catches.Data are cube roots of Big Eddy and Middle
Mouth test fishery catches.. . . . . . . . . . . . . . . •..68
16.The estimated values of the mean and variance of a homogeneous
population migrating past the test fisheries,and the values of
the coefficients quantifying the effect of wind speed and commer-
cial effort on the migration.Model three (equation 27)is fitted
to observed data by minimizing the sum of squared deviations from
the expected catches.Data are cube roots of Flat Island test
fishery catches 69
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LIST OF TABLES (Continued)
Mean parameter values,confidence limits,and the relation between
mean and variance of model five and average April air tempera-
ture . . . . . . . . . . . . . . . . . . . .....
The estimated values of the mean and variance of a homogeneous pop-
ulation migrating past the test fisheries,and the values of the
coefficients quantifying the effect of wind speed and commercial
effort on the migration.Data are from the Flat Island test fi5h-
ery operations ....•.....................
The estimated values of the mean and variance of a homogeneous pop-
ulation migrating past the test fisheries,and the values of the
coefficients quantifying the effect of wind speed and commercial
effort on the migration.Model five (equation 40)is fitted to
observed data by maximization of the likelihood function.Data are
from the Big Eddy and Middle Mouth test fishery operations
79
75
B4
73
BB
B3
B7
B5
78
fitted to the
predict daily
limits,and the relation between
three and average April air
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the ability of model four and the one population time
to predict daily proportion of total catch and total
parameter values,confidence limits,and the relation between
and variance for model three and average April air ternpera-
Mean parameter values,confidence
the means and variances for model
temperature . . . ....•...
The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to
the earlier 5ubpopulation.the mean and variance of the later sub-
population,and the proportion assigned to the later 5ubpopulation.
The model assumes a mixture of two subpopulations affected by wind
and commercial effort (model four,equation 27)and is fitted by
minimization of the sum of squared deviations.Data are cube roots
of Big Eddy and Middle Mouth test fishery catches .
The estimated values of the mean and variance of the earlier 5ub-
population,the proportion of the total population assigned to the
earlier 5ubpopulation,the mean and variance of the later subpopu-
lation,and the proportion assigned to the later subpopulation.
The model assumes a mixture of two subpopulations affected by wind
and commercial effort (model four,equation 27)and is fitted by
minimization of the sum of squared deviations.Data are cube roots
of Flat Island test fishery catches . . . . . . . . . . . . . . . .
Ccmparison of
density model
annual catch
Comparison of the ability of models three and four.
observed data by minimization of sum of squares,to
proportion of total catch and total annual catch
Mean
mean
ture
20.
,9.
24.
,B.
22.
17.
21.
23.
25.
Table
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LIST OF TABLES (Continued)
Tdble Page
26.Comparison ot the ability of model three and the time series model
to predict daily proportion of total catch and total annual catch ..89
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LIST OF FIGURES
Detail of the Yukon River Delta and fisheries statistical area
Y-l . . . . . . . . . • . . . . . . . . . . . • . .
Average daily proportion of total annual catch in 1968 to 1981
test fishery catches .
5
21
15
28
17
14
16
24
10
12
20
27
25
26
of residuals from
fishery catches.
Autocorrelations of average daily wind speed and daily effort for
all years of test fishery operations ....•.......
Autocorrelations and partial autocorrelations
the autoregressive model for Flat Island test
1968-1978 .
Autocorrelations and partial autocorrelations of transformed
errors of expected proportiof.'i of Flat Island test fishery
catches.1968-1979 .
Correlation of average daily wind speed and daily effort with
residuals fran the autoregressive IIl1Jdel with transfer function
for Flat Island test fishery catches,1968-1978 •.......
Correlations of average daily wind speed and daily effort with
residuals from the autoregressive model for Flat Island test
fis~ery catches,1968-1978 ............•.....
Partial autocorrelations of average daily wind speed and daily
effort for ali years of test fishery operations ....•...
Autocorrelations and partial autocorrelations of residuals from
the fitted autoregressive model of Flat Island test fishery
ca tches.1968-1979 . . . . . . . . . . . . . . . . . . . . . .
Autocorrelations and partial autocorrelations of transformed
errors of expected proportions of Big Eddy test fi~hery catches,
1979-1981 .
Autocorrelations and partial autocorrelations of transformed
errors of expected proportions of combined test fishery cat~hes.
1968-1981 ..
Autocorrelations and partial autocorrelations of residuals fran
the autoregressive model with the transfer function for Big Eddy
test fishery catches.1979-1981 ....•.•..••.....
Correlations of average daily wind speed and daily effort with
residuals from the autoregressive model with transfer function
for 819 Eddy test fj shery catches.1979-1981 .
Correlations of average wind speed and daily effort with residua~s
from the autoregressive model for Big Eddy test fishery catches.
1979-1981 .
7.
2.
1.
5.
3.
4.
6.
9.
8.
1G.
11.
13.
14.
12.
Figure
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Figure
15.
16.
17.
18
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
LIST OF FIGURES (Continued)
Autoconelations and partial autocorre1iJtions of residuals from
the autoregressive model with transfer function for combined
test fishery catches.1968·1981 30
Correlations of average daily winJ speed and daily effort with
residuals of the autoregressive model with transfer function for
combined test fishery catches.196~·198l • •31
Residuals from the autoregressive model with transfer function
for all years (1968-1981)n
Observed da ily propart ions of tota 1 catch and expected prOpal"t.ions
of total catch from model one for 1972 Flat Island catches 51
Observed daily proportions of total catch and expected proportions
of total catch from model one for 1977 Flat Island catches 52
Observed daily proportions of total catth and expected proportions
of total catch from model one for 1981 Flat Island catches 53
Observed daily proportions of total catch and expected proportions
of total catc~from model two for 1972 Flat Island catches 59
ObserveJ daily proportions of tot~l catch and expected proportions
of total catch from model two for 1981 Flat Island catches 61
Observed daily proportions of total catch and expected proportions
of total catch from model two for 1977 Flat Island catches 62
Observed daily proportions of total catch and expected proportions
of total catch from model three for 1972 Flat Island catches 64
Observed daily proportions of total catch and expected proportions
of total catch from model three for 1977 Flat Island catches 65
Observed daily proportions of total catch and expected proportions
of total catch from model three for 1981 Flat Island catches 66
Observed dai ly proportions of tota 1 catch and expected proportions
of total catch frail model four for 1972 Flat Island catches.70
Observed daily proportions of total catch and expected proportions
froo model four for 1977 Flat Island catches "71
Observed daily proportions of total catch and expected proportions
from model four for 1981 Flat Island catches "72
Observed daily proportions of total catch and expected proportions
from model flve for 1972 Flat Island catches •.80
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LIST OF FIGURES (Continued)
Figure Page
31.Observed daily proportions of total catch and expected proportions
from model five for 1977 Flat Island catches.. . . . . . . . . .81
32.Observed daily proportions of total catch and expected proportions
fran model five for 1981 Flat Island catches.. . . . . . . .82
33.Relationship of April mean air temperature to the ll,<!an date of
migration for model three.92
34.Relationship of April mean air temperature to the variance of
migration for model three •94
35.Relationship of April mean air temperature to the influence of
wiild on migratory behavior for model three •95
36.Relationship of year of test fishery catch to the effect of com-
mercial effort on test fishery catch for model three.96
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ABSTRACT
The mlgratory timing of chinook salmon (Oncorhynchus tshawytscha)in the Yukon
River delta is defined as a genetically based.enviroMiental'y mediated pheno-
menon.Fluctuations in abundance are often associated with environmental per-
turbation,renoval by the commercial fleet.and the intrinsic character of the
population comprising the migration.The effects of wind and commercial effort
on test fishery catches are quantified by empirical time series analysis.least-
squares fitting of mechanistic models,and maximum-likelihood estimation of a
derived probability density function.The ability to resolve migratory timing
of Yukon River chinook salmon into a mixture of two populations which are nonmal1y
dist.ributed over til'le and the relative contributions of wind and ccmnercial e-ffort
to daily fluctuations in catch is evaluated.The results are interpreted in tenns
of the ability of each model to predict daily abundance and total abundance.
The fourteen years of available chinook migratory time densities could not con-
sistently be resolved into a mixture of two populations.Abundance forecasts
::np 1oyi ng a two-popu 1ati on model wet'e poorer than predi cti ons fran a one-popula-
tion model.High dai 1y wind speeds were consistently associated with large
catches,while commercial effort depressed the test fishery catch.Daily errors
of proportion in total abundance estimates employing a mechanistic model were
comparable to the errors of an empirical time series model,although the empirical
approach more accurately predicted total abundance.80th models provide a quanti-
tative means of predicting test fishery catch.
A comparison of est ·mated annual parameter values and average April temperdtures
provides insight into the dynamics of the Yukon chinOOk migration.Cold spring
temperaturec:delay the arrival of migrating salmon,shorten the time interval of
migration,and reduce the effect of daily enviroMlental changes on migratory
behavior.Wanns spri.1gs portend an early arrival.a more extended migration,a
greater effect by a covariate of wind speed on migratory behavior.Trends in the
effect of commercial effort suggest an increasing efficiency of the fishery.
KEY I·IORDS:chinook salmon,Y~kon River,time series analysis.migratory timing,
parameter estimatlon of mathematical models,wind speed and migration,
t:mperature and migratory timing,quantitative description of test
flshery ca tches,and ca tchabi 1 i ty
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INTRODUCTION
Abundance is preeminent among the biological attributes of a fish population
which are of concern to fisheries biologists.The primary goal of fishery man-
agement is to control the exploitation of a population or assemblage of popula-
tions so as to obtain the ~~ximum sustainable benefit.Maximum benefit 1S often
synonymous with maximum sustained yield.In order to obtain an estimate of the
optimum harvest of the resource.accurate assessment of total abundance is essen-
tial.Accur.:tely forecasting the magnitude of harvestable surplus depends upon
knowledge of future abundance.
The migratory behavior of adult salmonid populations precludes the use of classic
closed-population estimation methods which ignore the time dimension.Management
benefits little from accurate total abunda.dce estimates once the population has
migrated past the fishery.Predicted time of arrival and distribution of abundance
over time are biological statistics which are also indispensable to salmon managers.
Fortunately,migratory timing of salmon populations is a conservative and predict-
ab 1e phencmenon.
The concept of quantifying mi~ration in terms of its time density distribution has
been well developed by Mundy (1979).In surrrnary,the time density is the relative
abundance of migrating population as a function of time.The time density of the
population is defined with respect to unidirectional movement through a fixed
location by a single life-history stage of a population.Migratory behavior is
measured in units of time,such that the probability of occurrence of any given
interval of time is dependent upon the location of that interval relative to the
center of the migration (mean)and upon the dispersion of ,igration through time
(variance).When the probability assigned to each day of the migration is the
proportion of the total population arriving on a given day,the mean and variance
of the time density distribution can be defined by standard statistical procedures.
The advantage of the time-density approach is that migratory behavior may be
characterized by its time-density distribution and associated statistics fryr any
population.Describing migration in tern,s of its time-density distribution enables
differences in migratory timing b~tween populations (either interspecific or intra~
specific populations)or changes in migratory timing within a population across
generations to be readily quantified.Migratory time densities have been defined
for populations as diverse as sockeye salmon (Oncorhynchus nerka)(Mundy 1979;
Mundy and Mathisen 1981;Hornberger et al.1979;Hornberger and Mathisen 1980i
1981i Brannian 1982),chinook salmon (0.tshawytscha)(Mundy 1982a;1982b;Horn-
berger and Hathisen 1981),pink salmon (0.gorbuscha)(Merritt and Roberson 1981.
Hornberger and Mathisen 1981),chum salmon (0.ketal (Hornberger and Mathisen
1981)and brown shrimp (Penaeus aztecus)(Babcock 1981).
As noted by Mundy (1979),expressing abundance or daily proportion of total abun-
dance as a function of time (calendar date)is conceptually misleading.Migratory
behavior is directly dependent upon the individual physiological state,which in
turn is mediated by ambient physical factors.Time is merely a convenient covar-
iant of ice break-up.wind direction and speed.water temperature.river discharge.
and photoperiod (Favorite et al.1976;Neibauer 1980.and Neibauer 1979;Ingraham
et a 1.1976;Dodimead et a 1.1963),all of whi ch may affect the character of the
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time-density distribution (Alabaster 1970.Barber 1979.Burgner 1978.Ingraham
1979;lorz and Northcote 1966;and other.also see the review by Banks 1969.
Leggett 1977).Mundy (1979)suggests that the next logical advance in modeling
migratory behavior is to define,in lieu of time densities,'temperature density'
or 'photoperiod density'or a multivariate probability density which might be
tenned a 'r'ligratory density'.
lntraannual variability associated with daily fluctuations in abundance is also
a resultant of the stock (defined as a Mendelian population)composition of the
migration.In a study of the migratory timing of Yukon River chinook salmon,
r1undy (1982a)noted that variability in observed daily carrnercia1 catches limited
useful forecasts to a function describing the clJllulative migratory time density.
Considering that the Yukon chinook salmon migration is a composite of many separate
stocks distributed over thousands of river miles.the significant deviations from
average values of the migratory time distribution should be expected,due to behav-
ioral differences and intra and interseasonal changes in relative abundance of
each stock.
The recognition of multiple stocks in an exploited migrating species is crucial.
Ricker (1958.1973)mathenatically denonstrated that the maximum yield obtainable
from a mixture of stocks of varying reproductive potential is realized only if
each stock is harvested separately.Overexploitation of the most productive races
can result in greatly diminished future yields.In North Pacific salmon fisheries,
because of the multinational destination of groups of stocks,al~ocation of the
resource to various user groups has international implications.Many studies
have employed discriminant analysis to classify major spawn~ng stocks of Pacific
salmon,usual1y by scale patterns (e.g.,Cook and Lord 1978;Anas and Murai 1969;
Bilton and Messinger 1975.Major et a1.1975).The genetic foundation of migratory
timing implies that migratory timing may serve as one objective criterion to sep·
arate stocks (Mundy 1979)and as an aid to management in optimally exploiting each
stock.
Accurate knowledge of abundance and timing of commercially exploited fish popula-
tions which migrate into or through the fishery are the first priorities of manage-
ment.Fortunately,when migratory behavior is conserved across generations,
abundance and time are related by the characteristic time density of the popula-
tion.As postulated,the apparent time density is a product of the stock composi·
tion,and environmental influences on that mixture of stocks.Knowledge of the
seasonal distribution of the population enables management to forecast both daily
and total catch from observed clfJIulative catch.Forecasts can be updated through-
out the season,providing management with a dynamic method of in-season estimation.
Daily abundance is relevant to fisheries management only in the context of total
abundance.The convention of describing migration in tenms of proportions of
total abundance over time has several advantages.The factors governing migratory
behavior can more easily be discerned through the vagaries of yearly abundance
since the units of relative abundance are dimensior.less.confering an added flex-
ibility to interannual and intraanual comparisons.rredictions of daily or cumu·
lative proportion of total abundance are estimated by averaging previously
observed proportions on the date of interest (Walters and Buckingham 1965.
Mundy 1979~Hornberger et al.1979),by averaging proportions across 'day or run'
(Hornberger and Mathisen 1980~1981;Brannian 1982).or by fitting functions which
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approximate the shape of the temporal distribution of proportions (Mundy and
Mathisen 1981;Hornberger and Mathisen 1980).Studies of the Nushagak Bay
salmon migrations (Hornberger et a 1.1979;Hornbei ger and Mathi sen 1980;Horn-
berger and Mathisen 1981)and the sockeye migration in Togiak Bay.Alaska
(Branoian 1982)seem to indicate that the use of average proportions over day
of run is an appropriate strategy in the pursuit of more accurate estimates of
proportions.
Standard procedures used in fitting nonlinear equations to observed data minimize
the sum of the square deviations frem the expected value.The assumption that
deviations from the expected values (residuals)are independent and normally dis-
tributed with mean zero and constant variance is usually implicit in the statistical
inferences accompanying these procedures.Even in the absence of any assumptions
about the joint distribution of the random variables under cons~deration.the
~ethod of least squares can still serve as a legitimate means of obtaining point
estimates of the parameter values.although no objective ju~gment of the quality
of the estimates can be made.The minimization of the sum of squared residuals is
still the criterion which determines the values of the parameters of the equation.
The association of probabilities with proportions by Mundy (1979)was an important
conceptual achievement.The presumption that proportions accurately reflect prob-
abilities associated with daily migration enables the migratory behl'vior to be
quantified in terms of its probability distribution.This is a subtle.yet fun-
damental distinction.Probability density functions are studied in r~lation to
the 'strategy'of the population itself.availing new methods in point estimates
of parameters of the function (Freund and Halpole 1980).The method of moments
and the method of maximlJll likelihood are examples.The qualifying conditions in
such approaches are that the function sums to unity and all probability values are
greater than or equal to zero .
The objective of the present study is to quantify the effects of environmental
factors.cOlTlllercia1 fishery removal.and differential timing of upstream and down-
stream stocks on the relative abundance of chinook salmon in the low~r Yukon River
as estimated by test fishery catches.Several methods are proposed as means to
achieve this objective.Data from the lower Yukon River test fishery are analyzed
using the statistical techniques of linear time series analysis (Box and Jenkins
1976),and least-squares fitting of nonlinear functions is employed.The iterative
technique of estimating the parameters of a mixture of normal populations by a
maximum likelihood function (Hasselblad 1966)is regularly utilized in size fre-
quency analysis (e.g .•MacDonald and Pitcher 1979;McNew and Summerfelt 1978).
This technique is evaluated as a means to separate the assumed populations on the
basis of migratory timing.Results of the fitting procedures are compared ~n tenns
of the ability of each function to aa:urately pl"edict the daily proportion of the
total catch.and the total catch itself.Improvement in the predictive ability of
each function over the nonmal distribution function.defined by a s:ngle mean and
variance.or in comparison to an empirically-derived stochastic model serves as a
criterion for the applicability of each model .
STUDY AREA AND DATA
Only a synopsis of the lower Yukon River commercial and test fishery methods will
be presented here.More detailed descriptions are available in Mundy (1982a);in
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Yukon Area Management Reports,and in lower Yukon River Test Fishing Reports.
The test fishery began in 1963 near the seaward boundary of the south mouth of
the Yukon River delta at the location known as Flat Island (Figure 1,site A).
Until 1968,test-net sites were chosen by Alaska Department of Fish and Game
(AOF&G)personnel.In 1968 the practice of renting set-net sites from local
residents was initiated and the success of the test fishery effort increased
(the 1963-to-l967 8 1/2-inch mesh yearly average chinook salmon catch was 446
chinook,compared to a 1968-to-1978 yearly average of 708).For logistic rea-
sons,the test fishery was relocated to 1979 approximately 20 miles upriver at
Big Eddy (Figure 1,site B),and another test fishery operation,Middle Mouth,
began.The Middle Mouth site is located near the confluence of the middle and
north-mouth sloughs (Figure 1,site C).
For 20 years the test fishery has assisted management by providing a measure of
relative daily abundance of chinook salmon,summer and fall chum salmon,and
coho salmon.Recently,Brady (unpublished draft)demonstrated that,as manage-
ment had suspected,a high correlation exists between test fishery and commercial
catches,adjusted by one day to account for average travel time.The relation
between Big Eddy catches and commercial catches was the strongest,with Middle
Mouth being only slightly more variable.The use of test fishery catch data to
study the migratory timing of chinook salmon minimizes problems of estimation
relating to censorship and truncation which are inherent in commercial catch data.
Methods of data collection have been consistent for the last 14 years (1968-1981).
Two 25-fathom gill nets of 8 1/2-inch mesh and one 25 fathom 5 1/2-inch mesh gill
net were fished in locations judged to be productive and representative of the
major river channels near the test fishery.Set nets were chosen to standardize
the effort and avoid dependence on the ability of personnel.Except when circum-
stances prev0nted,each net fished 24 hours a day and was checked twice daily.
In the present st~oy,observed daily catch is defined as the total recorded catch
at the 3 net sites for ~ne 24 hours fished.CJtches were adjusted upward propor-
tionally on days with less than 24 hours of fishing.In 1979,Middle Mouth test
fishing did ~ot begin until 18 June,which was half way through the chinook migra-
tion.Therefore,~he 197~Middle Mouth data are not included in this analysis.
Calendar dates are coded relative to 11 June (coded day 1).
Weather observations recorded in Nome and Emmonak were obtained from the National
Weather Service.When available,the Emmonak wind speed was used.In 1979 a
lapse in weather recording required that Nome wind speeds be substituted from
2 July to the end of the migration.The rate of travel of chinook salmon in the
Yukon River has been estimated at between 25 and 30 river-mi1es a day (Trasky
1973).If upriver migration is considered to commence inmediately offshore of
the Yukon River delta south mouth (Figure 1),fish would arrive at the Big Eddy
test fishery approximately one day later.Chinook salmon beginning at the middle-
mouth exit would also arrive at the Middle Mouth test fishery approximately one
day later.Observed average wind speed is defined here as the arithmetic avprage
of all observations on the given day.
Standard statistical techniques and tests of significance used throughout this
study are described in Sakal and Rohlf (1969).The SPSS statistical package
(Nie et al.1975)was employed for standard Pearson's correlations,t-tests,some
of the linear regression analyses,averages,and confidence limits.Preliminary
-4-
II
••••••••••••••••••
Detail of the Yukon River delta and fisheries statistical area Y-l.
Site A is the location of the Flat Island test fishery,Site B the
Big Eddy test fishery.and Site C the Middle Mouth test fishery.
-5-
""
20'
.,'
N
.....
,...w
AREA Y-1
YUKON RIVER
DELTA
..',..-w
20'1
..'
.,'
N
Figure 1.
•••••••••••••••
II
II
II
II
time series analysis was pp~formed with the BOXJ computer program package (Uni-
versity Computer Center,Ulliversity of Massachusetts at Amherst)and the inde-
pendence of residuals was also tested using the same package.
INTROOUCTION TO TIME SERIES METHODS
The use of conventional ~tock and recruitment models is common in the attempts
of fisheries managers to predict the annual abundance of distinguishable stocks
of Pacific salmon migrating through the fishery.The estimated abundance of the
resource is assumed to be a function of past abundance.measured one or more
years prior to the adult migration.Expectations of abundance and harvest guide-
lines are set prior to the fishing season.Intraseasona1 adjustment of the esti-
mation of total abundance is frequently an intuitive.subjective process,the
accuracy of which depends on the expertise of the local resource manager.Develop-
ing an ability to estimate the magnitude of the migration involves comparing the
magnitude of commercial or test catches by date of catch to the historical per-
formance of the migration,environmental factors believed to affect migratory
behavior,and numerous intangibles witnessed by the managerial staff over each
migration.
Techniques to anticipate daily or total abundance by exploiting the conservative
nature of migratory behavior have been developed for several populations of
Pacific salmon (Walters and Buckingham 1975;Mundy 1979;Hornberger and Mathisen
1981;Brannian 1982)and populations of brown shrimp (Babcock 1981).A common
method involves averaging cumulative proportions recorded on each calendar date
of the migration for every year with reliable historical observations.Total
abundance is estimated by the quotient of total catch to date over expected pro-
portion on the given date.Variations on this strategy include defining the day
of nligration relative to the day a set cumulative proportion of total catch is
realized (Brannian 1982;Hornberger and Mathisen 1981),or fitting a deterministic
equation, usually the inverted exponential function,to the distribution (Mundy
and Mathisen 1981;Maty1ewich 1982;see the Methods Section for a discussion of
other appropriate deterministic functions).
Application of a mi9ratory time density function to estimate the total run size
has been successful in several salmon fisheries (Mundy and Mathisen 1981,for
Bristol Bay sockeye salmon;Brannian 1982,for Togiak Bay salmon;Mundy 1982a,
for Yukon River chinook salmon.and Hornberger and Mathisen 1981,for four species
of Nushagak Bay salmon).Because deviations from expected catch are often asso-
ciated with the physical environment of the salmon and,since physical factors are
generally not independent,the accuracy of estimates made during the season can be
enhanced by incorporating measures of the effect of environmental variables on
migratory behavior,and the dependence of sequential daily observations into the
model.The distribution of adult salmon catches over equally spaced intervals of
time (days)is suited to time series analysis.The time series approach proposed
by Box and Jenkins (1976)appears particu1ar1y appropriate.
Several studies have ~p1oyed Box-Jenkins methods to forecast future abundance of
commercially exploited marine organisms.Univariate models have been employed to
forecast monthly rock lobster catch per unit effort (Saila et al.1981),and yearly
-6-
•••••••••••••••••••
DEVELOPMENT OF THE TIME SERIES MODEL
Atlantic menhaden catch (Jensen 1976).Time series analyses which have incor~or
ated univariate and multivariate transfer functions have been applied to the
skipjack tuna fishery (Mendelssohn 1981)and lobster fishery (Boudreault et a1.
1977)respectively.
Catch data used in the study were obtained from the Big Eddy and Flat Island test
fisheries.Procedures employed in the test fisheries have been outlined prevlousl:·.
The bias introduced by yearly differences in total abundance is minimized by trans-
fonning daily catch into the daily proportion of total annual catch.Daily pro-
portions of total abundance,average daily wind speed recorded by the National
Weather Service for Nome,and the proportion of the calendar day open to commercial
fishing was analyzed for the years 1968 through 1981.
Mean dates of migration and variances were calculated by standard methods (see
Mundy 1982;and others)and correlated with average monthly air temperatures for
April and May,as recorded by the National Weather Service for Nome (Table 1).
The hiqh negative correlation between average April air temperature and mean date
of migration (r :=-.913)suggests that the ability of predictive schemes to esti-
mate the lower Yukon River catch distribution by calendar date can be enhanced by
adjusting the calendar dates of migration using average April air temperature.In
the present study,the relative day of migration Dr'~ich corresponds to a calendar
day of migration Dc,is defined as the difference Detween the estimated mean day of
migration Dm•and coded day corresponding to each calendar day (1 :=1 June)(i .e .•
Dr ~Dm -Dc)·The estimated mean day of migration i3 assumed to be a linear
function of April temperature (see Table 1):
(1 )
-7-
Dm :=38.49 -1.025 (April air temperature,oF)
Excellent summaries of the procedures involved in the Box-Jenkins method of time
series analysis may be found i,.,McMichael anlj Hunter (1972).Poole (1976b).and
Mendelssohn (1981).Assuming that the series of observations is stationary.or
can be made stationary by differencing or employing a suitable transfonmation.a
model is proposed which describes an observation as a function of past observations
(autoregressive terms)or of past errors of estimate (moving average terms).The
exact terms in the model are tentatively idp.ntified by studying the autocorrelation
and the partial autocorrelation function exhibited by the data.The parameters of
the postulated model are estimated and the residuals examined to evaluate the ade-
quacy to the model.The three-step process of model identification,model estima-
tion,and diagnostic checking becOolles an iterative routine designed to estimate
the most parsimonious model.
The notation used in.the present study is consistent with that of Box and Jenkins
(1976).Autoregresslve models of order n (AR n ),moving average models of order m
(~Am)'and m1xed autoregre~sive moving average rnode1s of order nm (ARt-lA nm )are
d1SCUSSed.Ihe symbol ~n 1S reserved for the coefficient associated with the
obs~rvation lagged n time.interv~ls prior to the present observation.The symbol
On 1S the parameter ass~c1ated wlth ~he error of estimation n time intervals prior
to the.present observa~lon.The resldual error of the final model,at.are assumej
to be 1ndependent and 1dentically distributed with mean zero and variance al.
II
••••••••••••
II
II..
II
II
II
Tdble I.Relationship of mean day of migration for the years 1968 througn
1981,and average April and May air temperatures.
Mean April Air May Air Predicted Mean
Year Coded Day!Variance Temperature Temperature Day
(oF)(OF)
1968 17.53 51.56 14.4 29.1 23.74
1969 14.48 54.48 21.8 42.8 16.15
1970 21.15 49.81 15.1 36.5 23.02
1971 29.21 64.65 12.9 29.7 25.27
1972 26.36 86.56 11.9 35.4 26.30
1973 20.57 78.84 18.3 35.2 19.74
197'13.57 64.84 20.9 38.2 17 .07
lY75 27.37 40.25 13.4 33.9 24.76
1976 30.37 25.03 9.7 33.1 28.55
1977 28.20 33.94 9.4 32.9 28.86
1978 12.61 95.15 24.9 42.1 12.96
1979 13.91 110.90 25.5 41.8 12.36
1980 14.88 62.48 23.8 43.2 14.10
1981 16.26 103.56 24.3 42.7 13.57
The Pearson Correlation coefficients are:
Mean day and April temperature:r =-.9127
Mean day and May temperature:r =-.7454
Mean day and Variance of "'igration:r =-.5709
The regression of mean day on April air temperature yields the following
equation:
Mean day =38.49 -1.025 (Mean April air temperatures,°F)
with an R2 value of .833 and F-value of 59.84 (p <.001).
1 Number of days after 31 May.
-8-
I
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I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
-9-
Therefore.the pronounced shift of migratory timing due to previous environmental
conditions can be compensated for by defining the day of migration with reference
to a relevant measure of the earlier envirorvnent.The distribution of proportions
of lower Yukon Ri~er test fishery catch averaged over the relative days for years
1968 to 1981 lS presented in Figure 2.
The derivation of a predictive model wtdch incorporates the conservative nature
of the time distribution over generations (Mundy 1979).the dependent distribution
of sequent ia 1 da ily proportions.and the effect of env;rormenta 1 and corrmerci a1
factors on the test fishery catch is attempted.The model consists of two com~
ponents:an average of observed proportions of total catch across the relative
days of migration.and a stochastic component describing the daily deviations from
the average values (et)as a function of average daily wind speed.surrounding
commercial effort.the errors observed o~prior days (et-i)'and the difference
between the estimated proportion and observed proportion of prior days (at_j).
Mathematically the model is:
Visual inspection of the distribution of errors about their mean of 0 revealed
heteroscedasticity of the errors near the annual mean day of the migrations com-
pared to the errors near the beginning or end of the migrations.The square root
transfonnation (.±square root of the absolute value of the error.the sign of the
transformed error being identical to the sign of the original error)normalized
the variances.as demonstrated by the nonsignificance of the F-values of the resi-
duals of the quartiles.Due to the nature of the distribution in time and space
of commercial effort.and the desire to avoid complicated transfonnations of effort
data to correspond to its effect on the test fishery catch.a value of one was
assigned to days of 0.75 or 1.00 days of commercial fishing.and a value of 0
(2 )
n T
P =~T 1:PDr(i)+ei
i=l
where the observed proportion p.observed on relative day Dr is the average of the
proportions PD (i)over the relative days Drli}(i =1 to nTinT being the number
of years with ~ecorded c'Itch on day Dr)plus an error term lei).The series of
errors arranged sequentially by day of mig~ation and by year.(el 19fj8'02.1968 ••.••
e;;.19FiI!'e1.1969.···.ek.1281)where ei,ljr 1S the error assoclated w1th day i of the
mlgratlon (1 - 1 to k)and year !lr (!lr =1968 to 1981).fonns a succe!osion of data
amenable to time series analysis.The parametric time series models proposed by
Box and Jenkins (1976)are used to derive a predictive function of the e t .
The change in location of test fishery sites in 1979 requires a separate analysis
of the daily errors for the years prior to the 1979 relocation (Flat Island catch.
1968 -1978)and subsequent to the move (B;9 Eddy ca tch.1979 -1981).The closer
proximity of Flat Island to the seaward boundary of the Yukon River (approximately
1 day of travel time for chinook salmon (Trasky 1973).the more intense commercial
fishery surrounding the Big Eddy test net sites.and differences in the physical
regime of the Yukon River at each test fishery site imply that the relationship of
test fishery catch to wind.ccmnercial effort.and prior catches may be of a differ-
ent character for each location.
I
••••••••••••
II
II
II
III.
II
.0
,-o,
o
E
R
R
G
E
p
Ropop.
!
H
302S20
o.oaJ /,I I I I I I I b L-
-20 -t5 -10 -5 lJ 5 10 I~
RELRTIVE DRY OF HIGRRTION
Figure 2.Average daily proportion of total annual catch in 1968 to 1981 test fishery catches.Day 0 ;s the
mean day estimated as a linear function of average April air temperature in Nome .
.~.•,.,•.•-•..•••II
-11-
Ct =-.0796 +.00936"'t -.0671ft +.294et _1 +.146d t _J +at (4)
After including the effects of wind and cQI'Mlercial effort in the autoregressive
model,the peaks in ACF and PACF at a 3 day lag were found to be significant and
a lag-3 term was added to the model.rhe final procedure involves making small
adjustments in the values of the pdrameters to minimize the sum of squared errors
(dt).Results are presented in Table 3.The ARl 3 and alternate ARHAL3 models
are considered.The parsimonious and minimum variance model which best describes
the errors fran the average daily proportion,Ct.is the ARHAl,3 model:
Correlograms of the AU and PACF of the observed daily cOO111ercial effort and the
average daily wind speed (Figures 4 and 5)demonstrate the strong dependence of
ccmnercial effort on past values of effort and the presence of an autoregressive
process of lag 1 in the distribution of daily wind speed.If the errors of the
test fishery catch are strongly correlated with either variable,large peaks in
the residual ACF and PACF of the ARj "'odel (Figure 6)may be related to the stat-
istical dependence of environmental factors on past measurements of these factors.
The residuals of the fitted ARl model are compared to the average daily wind speed
and commercial effort of 5 days after the date of recorded catch back to 14 days
before the catch to discern the relationship between the three variables.
A very significant (p<.OO1)correlation w~s found between wind speed and cOl1l11ercial
effort,and the errors on the day of recorded test fishery catch (Figures 7 and
Table 2).letting rt (t =1,2,...N;N being the total number of :>bserved devia-
tions from the ARl model)represent the residuals from the AR model;the transfer
function:
(3 )It·-.0052628 +.001l97("'t)-.016341(ft )
is the fitted regression equation to the observed deviations (Table 2).
assigned to days of 0 or 0.25 days of fishing.Although this transformation of
conmercial effort produced little change in the residual sum of squares or esti-
mated parameter values.it d;d result in a sl ightly better fit of predicted to
observed values and was retained in the following models.
The autocorrel~tions (ACF)and partial autocorrelations (PACF)of lag 1 to lag 20
days are presented graphically in Figure 3 for the years of the Flat Island test
fishery catch.The large peak at lag 1 in both ACF and PACF and the absence of
large peaks at other lags implies that the errors can be described as either an
autoregressive process of lag 1 day (ARl model)or a moving average process of
lag 1 day (MAl model),Fitting the HAl and ARl processes to the observed c·
results in slightly lower residua',sum of squares for the ARj model (Table 2>.
Therefore the ARj model is considered to be the more appropriate interim model for
the errors.Inclusion of secondary peaks observed in the ACF and PACF (for example
the positive peak at lag 3 days and negative pOQk at lag 13 days)into the time
series models is deferred until incorporation of the average wind speed and com-
mercial effort transfer function into the model has been accomplished.
I
••••••••••••••
II
II
II
II
.30 _ •.30
P
.20 ~.20
0 eT.10 .100
C 00
R T
R 0.00 g 0.00,I I if rTTl nil,A 2-T -.10 R
N I f -.10•0
N I -.20
S
-.20
N
5
-.30 I -.30
I I I I I I I I I I
0 5 10 15 20 0 5 10 15 20
LAG ([JAT51 OF ACF LRG lOATSI OF PRCF
Figure 3.Autocorrelations and partial autocorrelations of transfonned errors of expected proportions of
Flat Island test fishery catches,1968-1979 .
•.•• • •,.".,,•.•..•_•..•_•••--11
.1••
--------~._-~
Table 2.Statistics concerning the time series models and relationships dis-
cussed in the text for Flat Island test fishery catches.1968-1978.
-13-
1.The residual sum of squares (sum of all e~is .307897,
4.Statistics regarding the fitted MAl model:
Fitted model::£t ::-.012983 +.30688a t _1 +at
Residual sum of c;quares ::6.8769; degrees of freedom ::418
b.Fitted model:E =b +b1W t +b 2 E e +at,0
where:bO -.0621903
b,•,0087317 (F-vo1ue 23.27)
b,-.0638786 (F-va1ue •26.18)
The residl.!al ~um of sQuares of transfonned ei'where Ci ::sign fe.)!c.I J
(sum of all e 2 )is 7.6337 1 1
Statistics regarding the fitted AR 1 model:
Fitted model::£t ::-.013012 +.30418£t_l +at
Residual sum of squares::6.8612.degrees of freedom ::418
5.Statistics regarding the multiple 1 inear regression of average wind speed
(:"'c)and the commercial effort (E c )on errors.
o.Correlation coefficlents:Effort and errors:r -.2514 (p <.0001)
Wind and errors:r::.2390 (p <.0001)
2,
3.
••~'
~••••••••••I.
•
A,
.&0
.40
A
U
T .200e
0
R
R O.DD
E
L
A
T -.20I
0
N
5
-.40
,I I
I I II I
I I I I I I I I I I
-.&0
o 4 8 12 1&
LAG lOATSI Of NINO Aef
20
, ,
o 4 8 12 1&
LAG fOATSI Of EffORT Aef
20
Figure 4.Autocorrelations of average daily wind speed and daily effort for all years of test fishery
operations .
••••••••••••.•••••c ••••
._--_•••••••••••••••
I
I
I
I
I
.21J
'&OJ
.40
-.20
-.40
P
A
R
1
L
Vo O.DD I I ICiiii 'o Iii'iii
R i iR I
~
!
N
,-'",
-.60
I I I I I I
o 4 8 12 16
LnG 100TS)Of WIIIl PACF
20
I 1 I 1 1 ,
o 4 8 J 2 16 20
LnG 100TSI Of EFFORT PA(F
F1gure 5.Partial autocorrelations of average daily wind speed and daily effort for all year:.of test fishery
operations.
.30 .30
.10
-.10
.10
.20
-.20
P
~
!r
o
!
~0.00
oi -.10
I
II
5
.20
-.20
o
T
~o
R
R 0.00 I I I I I I I IE•,I;I I I I I I I r I
L
A
T
b
~
,
~
'",
-.30 -.30
,I I I ,
o 5 10 15 20
LAG lOOTSI Of ACF
I I I I I
o 5 10 15 20
LAG lOATS)OF PA(F
Figure 6.Autocorrelatlons and partial autocorrelations of residuals from the fitted autoregressive model of
Flat Island test fishery catches,1968-1979 .
••••••••m ••••••••••.
..----•••••••••••••••II
.30
.2il
~.10
R
~0.00
,~-.10 I II ·1··II··r~
~,
5
-.2il
-.30
I ,I I !
-IS -10 -5 0 5
lAG lOAYSI Of WIt4J CllftRElRTlQN
!,!I I
-IS -10 -5 0 5
lAG IDRYSI Of EFFORT CORRClRTIONS
Figure 7.Correlations of average daily wind speed and daily effort with residuals from the autoregressive
model for Flat Island test fishery catches,1968-1978.
Table 3.Pal'omelet"values and goodness of fit for two functions proposed as
possible models to describe the time series of the errors,Ct_
fer test fishery catches from 1968 through 1978.
StJtistics regarding the fitting of the ARl.]and ARM/'I,]models.
Model:fiR!,)C t b O +blw t +b 2 f t +Q 1 <:t-l to ~~JCt_)+'t
where b O -.075117
'J .009005
b 2 =.066458
•J .298321
'J .101325
Sum of squares:5.9100;degrees of freedom =394
Model:i,R.'IA 1 •""0 +bjWt +b 2 f t +~JEt_l +')JCt_J +'t.,t
where flO -.079625
"}=.009357
b 2 .067129
• I
.29418
eJ =.14624
SLIJI of squares =5.8748;degrees of freedom =394
-18-
•••••••••••••••••••
-19-
Correlations between the errors and average wind speed and cOfllllercial effort an~
presented in Figure 11.The average wind speed recorded the day before the test
fishery catch produces the highest correlation,followed by the wind speed recorded
the day of the test fishery catch.As was found for the Flat Island catch,there
is d high negative correlation between the residuals of the ARI model and the com-
mercial effort on the same day (Figure 11).The transfer function
Analysis of the residuals (at)of Equation 4 reveals no significant deviations
from a white-noise model (Figure 8).The randomness of the ACF and PACF are
indicative of a time series model which has sufficiently accounted for the depen-
dence of values on preceding values.Correlation coefficients between the J e •
wind speed,and comercial effort are also nonsignificant for all lags (Figure 9).
It is surmised that the multiple regression -::ransfer function has adequately
ex.pla ined the dependence of catch on wind speed and the surrounding cOllTT\el·ci~'
fishery.
Deviations from expected dally proportions of 1979, 1980.and 1981 test fishery
catches were subjected to the same analytical procedures described above for the
Flat Island data (see Tables 4 and 5).The large peak in the ACF and PACF at a
1 day lag which characterized the Flat Io:.land data also appeared in the Big Eddy
ACF and PACF (Figure 10).A secondary peak at a 6 day lag was also present.
MAl and ARl models were fitted to the error;ci'Although the fitted ,Ii;']model
resulted in a lower value of sum of squared deviations (1.7898)than the ll.R]r.lOcel
(1.7912).the difference was small.In order to be consistent with the Flat Island
analysis,an :ill!model WdS employed as the intedm model.
(\)-.041523 +.006072""1 -.08199f,-,
is derived frall the multiple regre.:10n of average wind speed recorded the ddy
before recorded catch and commercial effort the same day as test fishery catch on
the residuals of the ARl model.
Values of the parameters of the MAl and ARl models with a transfer function were
refined by minimizing the sum of squared deviations (Table 5).The models yield
almost identical results.The slightly lower sum of squares and choice of dll I.R]
tenn for the data of the Flat Island catch suggest the selection of an ltRl process
for the Bi 9 Eddy da ta.Cursory inspect i on of the ACF and PACF of the res idua 1s
:Figure 12)and the correlation coefficients between the residuals and the wind
speed ~nd commercial effort (Figure 13)reveal no obvious deficiency in the model.
Autoregressive and partial autoregressive correlations for all years of the test
fishery (1968-1981)are graphically presented in Figure 14 and Table 6 presents
the statistics associated with fitting an ARl time series model with a transfer
function for wind and commercial effort to the combined Flat Island and Big Eddy
test fishery data.Analysis of the residl.als (Figures 15 and 16)demonstrates the
adequacy of this model.No pattern was observed in the ACF or PACF of the rl'si-
duals,and the correlation coefficients were all small and nonsignificant.The
t'esiduals from the full model are presented in Figure 17.No deviations from the
assumption of homogeneity of variance are observed in the distribution of the
residuals.
II
••••••••••••••
It
It
It
II
I I I L I
-.20
20
I I I
10 IS
IDRT51 Of PR[f
.30
P
R .20R
J
R
L .10
ft
Tg 0.00
0
R
R -.10E
Ln!-.20
N
5
-.30
I I
0 5
LRG
20510IS
LAG looT51 Of R[f
o
-.30
.20
.30
R
Ub .10
[
2R oooj I!I , I I IE .I I I i I I I I I I
~
TI -.10
~
,
No,
Figure 8.Autocorrelations and partial (lutoconelations of residuals from the autoregressive model for Flat
Island test fishery catches,1968-1978 .
•••••••••••••••••••-
--••-•••••••••••.•.••.-.-.-.-
.30
.20
C
0 .10R
R
E 0.00
R,T
N I-0 -.10,
N
5
-.20
-.30
I ,I I I
-15 -10 -5 0 5
LAG lOAT51 Of WII(J CORRELRTlOIl
I I I ,I
-15 -10 -5 0 5
LAG lOAT51 Of EffORT CORRfLRTl0Il5
Figure 9.Correlation of average daily wind speed and daily effort with residuals fran the vutoregressive
model with transfer function for Flat Island test fishery catches,1968-1978.
1.The residual sum of squares (sum of all e~'s)is .058757,
Table 4.Statistics concerning the time series models and relationships dis-
cussed in the tex t for 8i 9 Eddy tes t fi shery ca tches.1979-1981.
5.Statistics from the multiple linear regression of average wind speed (0,,;1)
and cOllll1lercial effort (f )on errors.t-,
a.Correlation coefficients:Effort and errors:r =-.3116 (p <.001)
Wind and errors:r =.1582 (p =.052)
••••••••••••••••II
II
•
4.88)
17.08)
+a ,
of freedom =143
+a ,
(F value
(F vaiue =
-22-
model:
-+.17115c 1,-
where ·0 =-.041523
.J =.006072
·z =.08199
Residual sum of squares =1.7912.degrees of freedom =143
Statistics from the fitted Mn 1 model:
Fitted model =c =-.002587 +.17930.1,,-
Residual sum of squares =1.7898.degrees
The residual sum of squares of transformed
The sum of L Z is 1.84703
Statistics from the fitted AR 1
Fitted model =c
t
=-.002287
b.Fitted model:c t =-b o +b 11o't -+-b]f:;+<It
3.
4.
2.
I
•••••••••••••••I.I.
•
Table 5.Parameter values and goodness of fit for two functions proposed as
possible models to describe the time series of the errors,Ct_for
test fishery catches from 1979 through 1981.
Statistics regarding the fitting of the HAl and AR]models.
Model:MA J 't 3bo+bJwe+b 2Et +~l(t-l +at
where bO -.039500
b,•.005826
b,•-.085630.,•.261993
Sum of squares '1.5229;degrees of freedom •144
Model:AR ,'t :bo+bJlo'e +b 2 [t +¢l Ct_l +d t
where b o -.036719
b,.005732
b,•-.090661
.,•.25574
Sum of squares •1.5225;degrees of freedom:144
-23-
.10
,
N
A,
·30
.20
A
~
R
~0.00
A -.10
N
5
-.20
-.30
I I I __l I
o 5 10 15 20
LAG lDArs)Of Ref
.30
P
A
R .20
1
e .10
0rg 0.00
0
Rf -.10
~A -.20
N
5
-.30
,I I I I
o 5 10 15 20
LAG 100rSI Of PllCf
Figure 10.Autocorrelations and partial autocorrelations of transfonued errors of expected proportions of
Big Eddy test fishery catches.1979-1981 .
•••••••••••••••••••
..••..•............'....
.30
Ii
.10
,
N
~,
.20
~
R
R
h 0.00
!-.10
N
5
-.20
-.30
I I I I I
I I I I I I
,I I I ,
-15 -10 -5 0
LflG 100TS)Of NIl«!CORRELRTlON
5
t I I I _J
-15 -10 -5 0
LRG IOAT51 Of EffORT CORRELRTl0N5
5
Figure 11.Correlations of average wind speed and daily effort with residuals from the autoregressive l1Iodel
for 8ig Eddy test fishery catches,1979-1981.
.30
.10
.20
-.20
-.10
,
r
¥o 0 00 I I I I I I I I( .'i iii I I iii i I
2
!
.30.
I
.20
A
U
T .10~
0
RE 0.00
,A
N Tmb -.10,
N
S
-.20
-.30 -.30
I !I ,I I I I I I
o S 10 15
LAG lllATS)Of Ref
20 o 5 10 15
LAG IORTS)Of PA(f
20
Figure 12.Autocorrelations and partial autocorrelations of residuals from the autoregressive model with the
lransfer function for Big Eddy test fishery catches.1979-1981 .
•••••••••••••••••••
•••••••••••••••••••
.30
.20
C
0 .10
R I III IIR I I II,E 0.00 I
N A"T,
Io -.10
N
S
-.20
-.30
I I I I I
-15 -10 -5 0 5
LAG 100TS)OF WINO CORRELATION
I I I I I
-15 -10 -5 0 5
LAG 100TSI Of EFFORT CORRELATIONS
Figure 13.Correlations of average daily wind speed and daily effort with residuals from the autoregressive
model with transfer function for Big Eddy test fishery catches,1979-1981.
.30 .30
.10
-.10
.10
.20
-.10
-.20
P
~
~
Vo 0.00 I'"C j ,iii I Iii I'iii
~
~
T
I
2
S
-.20
.20
V
6
R
R 0.00 1 I I I I IEIIii i I I i I I iii
L
~
I
2
S
,
~,
-.30 -.30
!I I I I !I ,I I
o 5 10 15
LAG lOATS)Of Ref
20 o 5 10 15
LRG lonT51 Of PRCf
20
Figure 14.Autocorrelations and partial autocorrelations of transformed errors of expected proportions of
combined test fishery catches,1968-1981 .
•••••••••••••••••••
-29-
Statistics from the fitting of the AR 1 model.
Table 6.Parameter values and goodness of fit for the function proposed as the
model to describe the time series of errors,Ct"for 1968·to-1981 test
fishery catches.
Sum of squares::7.5396;degrees of freedom ::569
Sum of squares for all years ::II .319193
Average error of estimated proportion::.0151179
'"}'t zbO+bJ""t +bZf t +~lCt_l +.,
t
where bO =-.062750
b,=.007919
b 2 .074693
.,=.29793
Model:
1.The residual sum of squares (sum of all e~'s)is .369689
"2.The residual sum of squares of transformed e~.where £j =sign feiJlejlJ
(sum of .11 ,')is 9.48072
II
•••••••••••••••I.I.I.
~
.30 .30
.10
-.10
.20
.10
-.20
-.10
P
A
R
Je
vo 0.00 I!I ,I I,I( : ,Iii iii iii
R
E
~-.20
.20
A
V
~
Po 0.00 ~I I I ,I I,IEI I I I I I I I I
~
,
is,
-.30 -.30
I ,I ,,,,,I I
o 5 10 15
LAG tOArSI OF ACF
20 o S 10 IS
LAG lOArSI OF PACF
20
Figure 1:.Autocorrelations dnd partial autocorrelations of residuals from the aUltJregressive model with
t~ans fer function for cOfTlbi ned test fi shel'y ca tches.1968-1981 .
•••••••••••••••••••
•••••••••••••••-....
•30
.20
-.-J,I I ,1,,1,uIii
.10
-.10
li
R
E 0 OD~! ,, ,!!1 .I I I I I i I I Iii
2
S
,
w
-.20
-.30
, , ,,,
-15 -10 -5 0
LRG (DRYS)OF WINO CORRELRTION
5
I I I I I
-15 -10 -5 0 5
LRG (DRYS)OF EFFORT CORRELRTIONS
Figure 16.Correlations of average daily wind spet:d and daily effor~with residuals of the autoregressive model
with transfer function for combined test fishery catches,1968-1981.
'I
II
II
II
II
•
•
•
•
•
•
•
•
•
•
•
•
•
,
'"'"m
,,
•
•.'
,,
,,
.•
•,
,
"
•".
..
.
,"
,
'.
,,..
'"
"
.•...
'.•·
.,
,.
".'.
'.'
:
.,
•...
.'.
l ,,
"
•
,,·.
•·
..
,.'
".
"•
.,
••to'• I •••,.
D
'.'
,·
,
"
.....I',
",,
",,,,,",'
•
--!.
"
.,
•
.....
,"""I,..
",
,,
.:..~
•to ,",,,
J,,:,
'.
'.
••
.
"...
.'
,,
,,
..
,.
r.'
•....,.
•.
,,
•
,,
,,,,,.,
••
" I .",~.,,
'c I •·).~r ','.,.
• \.0 ••1
:'.~.'.•,0 •t':f
:,,
,.
.,.
a::l6JVl-O:::JC%....JVl
-32-
,,
.'
..
,,
••
0°'.
'..
..
,
",.,,.',.':;,,,,."
",,,
".Ii ..~,.,,,,
".,
'0 •'.•I!;rl---~'_':""~':""~di''-~ii!r--d=_r--::~r--::~r---=~
,,
•••••••••••••••
III
III
III
III
DISCUSSION OF TIME SERIES RESULTS
Time series ar,alys;5 can benef;t fisheries management in two ways.The resul tant
empirical model can be utilized as an aid in the prediction of future resource
abundance,either by forecasting the catch or relative abundance of the resource
(Mendelssohn 1981;Boudreault et a1.1977~Jensen 1976).or by estimating the
deviation from average performance of seasonal fisheries (Saila et a1.1980;
present study;also see Bulmer 1976.for similar results concerning Canadian
populations of exploited fur resources).The dynamic nature of the Box-Jenkins
approach allows regular updating of parameter values.increasing the reliability
of resource forecasts.The variability found in most fisheries is well suited
to the stochastic nature of the fitted models.The iJnalysis is linear in nature
but can describe fairly ccmplex behavior.Nonstatiollary and seasonal time series
can easily be adapted to analysis.
The second and potentially greater benefit to fisheries management is insight
into the dynamics of the fishery.Mendelssohn (1981)concluded that variability
in the skipjack fishery was more a function of unexpected changes in commercial
effort than the behavior of the fish.Boudreault et a1.(1977)discussed the
possibility that lobster abundance is not only a function of past population levels,
but the effect of temperature on larval survival as well.By includin9 the environ-
mental variables of rainfall and temperature in a time series model describing
mosquito densities.new insights into the effects of the environment on life his-
tory stages of the mosquito were obtained (Hacker et al.1975).Interaction
between various species of Drosophila have been investigated by similar time ser-
ies methods (Poole 1976a).
The effect of surface winds on the migratory behavior of salmon has been shown to
be significant (Lorz and Northcote.1965).yet the nonunifC'nn nature of the distri-
bution of m~gran~s over time has precluded a quantitative description of the
effects.Some authors (Hornberger and Mathisen 1980)have suggested that wind
speed affects the interaction between corrrnercial gear and fish.Others have
attributed the correlations to the influence of wind on the spreading of home
stream water (Lorz and Northcote 1965).Results of the present study indicate
that wind,or a covariate of wind,promotes upriver migratory behavior in chinook
salmon located near the mouth of the Yukon River.The highest correlation was
obtained between deviations from exoected catch and wind speed the same day as
the catch for Flat Island test fishery observations.Big Eddy analysis found the
highest correlation between the deviation and wind speed one day before the catch.
The approximate one-day travel time of migrating chinook salmon from Flat Island
to Big Eddy are in accord with these conclusions.
The relationship of catch and effort remains perplexing.There is strong evidence
that competition between units of fishing gear is high in the lower Yukon chinook
salmon fishery.The large negative correlation between the test fishery catch of
Flat Island and Big Ec'dy and the cOrTlTlercial effort of the same day implies that a
sizable proportion of the population vulnerable to the cOrTlTlercial fishery is har-
vested before it reaches the test fishery.Therefore daily catch per unit of
effort is not solely a function of the initial daily chinook salmon abundance.but
also a function of fishing mortality experienced by the population.In order to
rely on catch per unit effort as an estimator of the relative abundance of the
population,consideration must also be given to the magnitude and effectiveness
of the commercial fishery.
-33-
The advantage that time series analysis has over the fitting of conceptual equa-
tions is that time-series models are constructed empirically,requiring few !
priori assumptions.If the investigator has sufficient knowledge af the pro:_
cesses driving the phenomenon,conceptual models are much preferred over emplrlcal
models.However,in the absence of critical information,the empirical model can
serve as a first step in the development of a conceptual model.Anderson (1977)
extended the Box-Jenkins analysis to include interpreting the results rationally.
Fort~itous autocorrelations 'for which no reasonable explanation can be found,
should not have this effect reflected by the model'(Anderson 1977).
ReducUon in the sum of squares is a standard measure of the ability of a model
to estimate recorded observations.The objective of the present study is to re-
duce the squared deviations between observed daily proportions of total catch and
corresponding average proportions (Figure 2).Time series models provided only a
modest reduction in thE'sum of squares (14%,Table 6).Saila et al.(1981)found
that monthly averages provided a model which had a lower sum of squares than the
time series model,although the time series model forecast fvture values with
somewhat higher accuracy.The importance of the present study is not an improved
ability to predict future catch though obscure,by the time series model,but the
denonstration of a relationship,between wind speed,surrounding cOl11llercial effort,
and catch.
DISCUSSION OF MATHEMATICAL METHODS
The time distribution of abundance of a migrating population for a given locale
is certainly a I:lultivariate phencmenon.The ability of mathematics to accurately
describe and forecast daily lbundance of adult chinook salmon migrating through
the lower Yukon River drainage depends on an understanding of genetic and environ-
mental contributions to the distribution of abundance,or equivalently the propor-
tion of total population abundance,over time.Unfortunately there i~no concensus
on the distribution function governing daily proportion of total catch.Nonnal,
binomial,or multinomial distributions appear adequate considerin9 the genetic
nature or the migratory behavior.The recurrent,apparently Gaussian,distribution
of daily abundance in diverse populations of migrating birds has been approximated
by a cosine power function (Preston 1966).Frohne (unpubliShed manuscript)has
suggested that the inverse gaussian distribution,which describes the passage of
particles randomly dispersing in a flowing medium,may better describe the mech-
anics of migration.A two parameter inverted exponential function was first
employed by Royce (1965)to describe the cumulative proportion of total abundance
per un't time.Others have continued its use (Rothschild and Balsiger 1971;
Mathisen and Ber<j 196B;Dahlber9 1966;Mundy 1979;Mundy and Mat~isen 19B1;
Matylewich 1982).The derivative of this function with respect to time describes
a bell-shaped curve of proportion as a function of time.Other functions Which
generate the familiar bell-shaped curve and have been fitted to the time distri-
bution of proportions include functions from the general class of beta curves
(Vau9han 1954;Hornber<jer 1980)
Migration has been characterized as a genetically controlled environmentally
mediated event (Mundy 1979).Fluctuations of daily abundance in a given locale
are principally a function of short term environmental changes,the commercial
-34-
II
II
•••••••••••••
II
II
II
•
METHODS
The parameters defining the shape and location of the inverse exponential function.
a and b,are related to the mean and variance of the distribution.
can easily be related to the location and dispersion of migratory abundance over
time.The derivative of the two-parameter logistic inverse exponential function,
referred to subsequently as simply the inverse exponential function:
(7)
(9)
(6)
(8)
(10)
-35-
IJ --a/b
Y'(t)z b{exp(-(a+bt))}/{(l+exp(-(a+bt)))Zj,
Y(t):l/[ll+expl-(a+bt)))}
and
where Y'(t)is the daily proportion of total abundance on day t.is more prevalent
in the literature (see Royce 1965;and others)and it can be integrated to yield
an exact solution which describes the distribution of cumulative proportion (y(t))
over time.
The deriva t i ve of the inverted exponent ia1 functi on and the norma 1 probabil i ty
density function are used to quantify the time density of an assllTled homogeneous
population or subpopulation of chinook salmon.Both distributions describe a
bell-shaped curve.However~there are advantages inherent in each model.The
more conventional parameters.mean llJ)and variance loZ).of the equation des-
cribing the normal distribution of the proportions of total abundance.Olt).
over day of the migration,t:
harvest below the reference area.and random variation.Daily peaks in salmon
counts have been successfully correlated with changes in cloud cover (Ellis 1962),
wind speed and direction (lorz and Northcote 1965),barometric pressure (Ellis
1962).and rainfall and increased stream discharge (Alabaster 1970.Libosvarsky
1976;and others:see Banks'1969 review of factors affecting the upstream migra-
tion of salmon).Results of preliminary comparisons of data from ADF&G test
fishery catches and daily weather observations recorded in Nome suggest that a
covariant of wind speed is the principal daily environmental parameter affecting
the lower river mig,aticn of Yukon chinook salmon.
•••••••••••••••I.I.
••
-36-
'1••••••••••••••..
II
II
I
(13 )
tl2l
(11 )
q (t).:erp(-((t-l>)1/(20 1 ))//(2110 1 _Jl/222 2 T
local fishermen are of the oplO1on that the king salmon migration is composed
of two distinct runs,a 'white nose race'and a 'black nose race'.Each race
is supposed distinguishable according to morphological and migratory timing
characteristics.Although a pilot study by U.S.Fish and Wildlife (Anon.1960)
did not discern any meristic differences in fish categorized as 'white nose'or
'black nose',the segregation of major spawning stocks in the Yukon River trib-
utaries implies distinct races separable according to a yet-undiscovered set of
characteristics.The possibility of temporally distinct races can be incorporated
into the analysis.Assume that the time density distribution is c~nposed of two
groups of migrants (subpopulation 1,51'and subpopulation 2,52),each with a dis-
tinct mean h-I'\.12)and variance (0'1.0'2)(or al.a2'b l •b 2 for the inverted
exponential complement of the normal distribution equation).let 01 define the
proportion of total migratory abundance apportioned to 51'and P2 ..1 -01 the
~roportion of 52'Then the expected proportion of 51 on day t (qI(t),or Y'I(t)}
IS
for a population temporally distributed according to the inverted exponential func-
tion.likewise,the equations predicting the expected proportion of s."on day t is
for a normally distribution subpopulation,and
for the nonnal and inverted exponential function respectively.The expected pro-
portion of total abundance can now bl!expressed as either
The same concept can be extended to k subpopulations.The proportion of subpopu-
lation j,where j=l,2,..•,k,on day tis:
q/t)=erp(-((t-IJi'/(20'j))//(]1I0'j):'/2 (17)
or
and
(21)
(19 )
(21 )
(18 )
(13 )
(20)
(t-C)2 [C~;)]n
y',rt)=-b.{ex?(-(a.+b.t))}/{l+exp(-(a.+b.t)))lj.
J )JJ JJ
5'
-,
for a normally distributed migration,and
or
for an inverted exponentially distributed migration.The proportion of total
abundance on day t ;s a sum of proportions of all subpopulations on day t:
-37-
If the number of subpopulations ;s one (Equation 6),the estimation procedure for
the parameters ~and 0 2 reduces to classical mean and variance computations:
~[Cit)]r Jf:1 Oft}Ut}q/t}
where it is assumed that daily catch ;s proportional to daily average abundance at
the test fishery site,cT is the total catch,Cft)is daily catch.and n is the
last_day of effort.The constants a and b may be estimated by solving 52 =nJ/Jb J )
and t =a/b.If the migration is believed to be composed of more than one popula-
tion,the values of the parameters loll'oll'()I'...Uk_I'02 k _l ,01$-1'\Jk,02 k ,
or similarly,aI'b l ,ql'••.ak_I'b k -I ,qk-l'ak ,b k can be obtalned using maxlmum
likelihood procedures with steepest descent methods.
Hasselblad (1966)applied maximum likelihood techniques to mixtures of normal
curves to compute the means,variances.and proportions of total abundance of each
subpopulation.The equations derived by Hasselb1ad (1966)for a mixture of k
normal subpopulations are:
II
••••••••••••••••••
-38-
n
[q /t)(t-\J /lJL[ffiL]
t'l Oft)
a ~.(24)
]n [q/tJ]L [ffiL]
c=l Q(t)
••••••••••••••••
II
••
(25)
(26)
[ffiL]Oft)
n
L
\':here Wt is the proportion of the population potentially 'present and able'to
begin migration into the river.Iof t _1 1s the average wind speed on day t-l;f t is
the effort.in days fished.on day t.and at is random error.
Migration can now be quantified as a product of the inherent time-density distri-
bution of the pOiJulation (Eq.19 and 20).and the effects of daily wind speed and
commercial effort CEq.26).
The system of equations created has no closed-form solution.However.with the
aid of high speed computers.iterative schemes can be devised to approximate the
maximum likelihood estimates of the parameters.
Although minimization of the sum of squared deviations of the observed values from
expected values is not the criterion used here to obtain optimal parameter values,
the sum of squares is a standard measure of the ability of a function to describe
observed data.Sums of squares are inclUded below with tables of parameter values
for comparative purposes.
The relationship of wind speed to migratory behavior is assumed to be linear.
Attempts to assess the importance of environmental factors on in-season migration
of adult salmonids have been confined to either reporting the coincidence of
observed peaks in migration and weather anomalies.or univariate linear regression
(Ellis 1962;Lorz and Northcote 1965).Functional dependence of more complexity
has not been explored.It should be noted that the linear relationship used in
the follOWing analysis does not imply a direct dependence of migration on wind.
Wind spee~.like calendar day.is a convenient covariate of other factors res-
ponsible for the inhibition or stimulation of migration.The assumed linear
dependence of decreases in catch on amount of fishing effort surrounding the test
fishery site follows from classical catch models (Ricker 1975).The linear equa w
tion is:
or
Standard statistical procedures used to find the values of lJj'0 1 i'Pj'flo,61'and
82 for Eq.27 (j .:1 to k)by minimizing the sum of squares,are to dlfferentiate
the expression
(29)
(27)
(28)
n
[Cltl][-wt Jort)q/tJt]E Ort)IJ,)tq/t)
t.:l
uj ..,(30)
n
E ~Cltl][-w JO(t)q .rt)]Oft)Wtq /t)
t=l t J
n ~Clt)][wtq/t)-wt JO(t)q /t)(t-~j)J]E
t=l Qlt)
oJ._(31 )J n
-WtJO(t)q/t)]E [fili][ort)Wtq/t)t-1
-39-
with respect to each of the parameters to be estimated,equilte the expression to
zero and solve the lk+2 equations in tenns of each parameter.The equations des-
cribing a mixture of normally-distributed subpopulations generate the following
set of impl icit equations which produce converging estimates of the parameters
using an iterative procedure:
for a nonmal1y or inverse exponentially distributed mixture of 5ubpopulations
respectively.Under the assumption of a single homogeneous population,Oft}and
y'(t)are defined by equations 6 and 7,and assuming two or more subpopulations,
Qft)and Y'lt)are defined by equations 19 and 20.
The time density of the population characterizes the proportion of the total
population that is 'present and able'to enter the lower river.The effects
of wind speed and removul of fish down river from the test nets modify the dis-
tribut ion of ca tch rea 1fzed by the tes t fi shery.The propart i on of total Cd tch
(Cft)leT )realized on day t is expressed as:
••••••••••••II
•
II
••••
-40-
where all constants and variables are as defined previously,Oft}is defined by
fq.19.and qj(t}by Eq.17;and w,.by Eq.26.Having obtained estimates for \lj
and c'.aj and bj can be approximated accordinq to relationships previously defined
(equations 9 and 10).It should be noted that this system of equations does not
always converge,due to round off error in the computer or the character of the
equations themselves.
u
u
•••••••••••••
I'
I'I.
•
(35)
(34)
(32)
r.C(tJ]O(tpt
LOft)t
[Clti][ ]
--1<1 l(q.(t)-q (tlF
Oft)t J k
;.[Cft)]Oft)IW
'-Oft}t-1t-j
n ~Clti][Oft)-QftJlfB/"t_l+Bltl]~Oft}(;)
(33)So 0
n [Clti]~Oft},
t-j Oft)
o.-J
Regretably.Eq.27 and Eq.28,which express the proportions of total abundance
as a function of time.wind speed.and fishing effort.do not satisfy the condi-
tions defining a probability density function.The proportions do not necessarily
sum to one,and dependin~on the value of (80 +81'"'t-l +8lf ),the proportions
can assume values greater than one or less than zero.In oraer to meet the requ;re~
ments of a probability density function,the expression (80 +8~wt_l +82ft)must
be limited to values between zero and one.The other modificatlon of equations 27
and
-4;-
(37)
(40)
(36)
(38)
(39)
where Y'(t).assuming a single homogeneous population.is defined by Eq.7.and
As discussed above.it is assumed that the percent of offshore fish that do mig-
rate up river is a 1inear function of wind.Redefining We as a 1 inear function
solely of wind speed
and constraining by the condition that 0 <w <1,R't-l is equivalent to l-wt-l)
of the potential inmigrants on day t-l.Thetexpress;on quantifying the proportion
of total abundance present in the lower Yukon delta on day tis:
where Wt is the percent of the 'offshore pool'that does begin the upriver journey
on day t.R't-l is the number of potential migrants remaining frOOI day t-l.Y'(t]
is the proportion of the t')tal population that becomes able to migrate into the
river on day t.and Nt is the total abundance of the population.The proportional
complement of Eq.36 1S derived on divi~ing Eq.36 by NT:
Under the condition that 0 <We -80 +Iilwe-l <I.and sir.ce the integral of
&0 +81"'~_lilRt_l +Y'(e))equals one over the integral --to +-.Eq 39 does
satisfy the requirement for a probab1l ity density function.
and 28 necessary to equate the area under the distribution to one can be developed
within the context of the dynamics of the migration.
Cursory inspection of the distribution of daily catch (Figures 18-33)subs tan·
tiates the ot"c;ervations of cOlTl11ercial fishennen dnd managers that a large 'immigra-
tion of chinook salmon into the lower Yukon River is a one or two day phenomenon,
usually followed by an exceptionally 1C1101 abundance of migrants.It ic:.hypothesized
that the 'offshore pool'of chinook salmon physiologically ready to migrate up
river is temporarily dep~eted after a peak in migratory activity.In more general
terms,the number of chinook salmon.N(tJ.that migrate into the lower river on any
given day t can be expressed as a percent ('if the sum of the number remaining behind
on day t-l and the daily increase in offshore numJers from offshore arrivals and
physiological maturation:
•••••••••
I
••••••••I
(44)
(45)
(46)
(42)
(43)
I.n l...,Z(1}(W 1 )(y'O)+R a )+Z(2}(w
2
}(Y'(2)+R 1 )+•..+
Zft:)(o..t)O"t:/+R t _1 )+....+Z(n}(wn )(Y'(n)+R n _1 ) •
The test fishe:"y catch Cft)is a function of the mmber of fish migrating up the
Yukon River and the co~nercial effort surrounding the test fishery sites on day t.
Glven that the test fishery catch is directly proportional to daily abundance in
the lower river on days of no ccmnercial effort.and directly proportional to a
fraction of the total abundance of total abundance of total abundance on days of
corrrnercial effort,the fraction being a linear function of effort (ft).the follow-
ing rehtionship can be derived:
the expression for R_l can be substituted into Eq.44.giving
Therefore.if Eq.35 is expanded backwards in time to t ---.it follows by
induction that:
-42-
Since
As was the case for Eq.19.the method of maximum likelihood can be utilized to
obtain estimates for the parameter a,b,80'81,and 82'letting
lIt:)..!C(t)/C //(l-B1 f(t)}.the natural logarithm of the lik.:'!lihood function,.T:..n l...1 s:
where constrlnts and variables have been previously defined.Equating Eq.39 to
EQ.41.
To develop an expression for RO.which is a function of the migratory history of
the population before the test fishery begins,regressing backwards in time:
where n is the total number of observations.
-43-
(;0)
(49)
(''')
(48)
i"{t)
o
R O =l:Y'(t)
t=-'"
RO =l/(l+exp(-(a»)
t=-CI'
o
R =l:o
n
[ZIt)][(a)(b)exp(-(a+bt))
<lR t _l ]L +y'(t)+R t _l (l+exp(-(a+bt»)2 a.
t:l (51)•=
n
L [ZIt)][2(b)exp(-2(a+bt»]
t:l y'(t)+R t _1 (l+exp(-(a+bt»)2
r'(t)car.be integrated exactly over the interval of -CD to 0:
Equation 47 can be simplified by either assuming that all the fish arriving off-
shore during time interval [-~.0]which are physiologically able have ilT'Tli'1rated
the day before nets were set (day 1)(i .e .•at t =O.lilt;:::1,and RO =0).)r no
fish have entered the lower Yukon del ta prior to day 1 l i.e .•at t ::0,til to
(';0 =O.and -""
Therefore,under the second assumption,the number of chinook salmon remairing in
the 'offshore pool'on day 1 is
The difference in parameter values assuming either total offshore depletion or
total offshore retention is minimal.The latter assumption is considered tu be
more realistic in view of the observations of other investigators that many salmon
populations mill offshore for various periods of time before ascending their home
river.the supposed difficulty in entering the Yukon River during ice breakl.p.
and the promptness of test net placement after breakup.Taking the partial deri-
vative of Eq.38 with respect to each of the parameters (a,b,60'and B1)'
setting the resul ting expressions equal to zero,and solving for each of the
parameters.the following set of implicit equations are derived:for a
.1
••••••••••••••
II
II
II
•
(60)
(56)
(52)
(55)
(54 )
(53)
(59)
(57)
(58)
+exp(-(a+bt»(t)(bJexp(-(a+bt»
(l+exp(-(a+bt»)1
b€!xp(-(a+bt))
(l+exp(-(a+bt))1
~]+aa '(l-w ) [ilY'(t),,.
o .
2(t)(b)exp(-2(a+bt»
(1+exp(-(a+bt»)1
2bexp{-2 (a+bt»
(1+exp((a+bt)))7
"'/"0 e (l-w )[aRt_l ] -Y'(t-I)-R
t aBo t-2
-44-
'b
(l+exp(-(a+bt»)1
ilY'(t)
ilY'(t),.
where.
where
ard
and
dRO/ilcl exp(-a)/(l+cxp(-a»l .
For b,
n
~[Z (t)] [2(t)bexp(-2(a+bt))+exp(-(a+bt»+Y'{t)+R t _1 (l+exp(-(a+bt»)l {I+exp(-(a+bt»)l
t><l
b n
[ Z (t)][ t (exp(-(a+b»]~l"(t)+R (Hexp((a+bt»)l
t><l '-1
where
ClRe/ilb (1-)[ilY'(e)+ilRt-1]
wt ilb ab
'0
(63)
(64)
(65)
(62)
(61 )
f ',n
L,-1
n
E
'r1
n
E
t=l
and
-45-
and for 81 :
ana
where
At each iteration.a new 82 was calculated for the new estimates of the parameters.
Parameters a and b were estimated by substituting the normal equation into equa-
tions 51 and 54,solving for the mean and variance,and approximating a and b
according to the relationships a -(-bJ\.l and b -1f t /f30 1 ).Equations 58 and 61
were solved by bisection of the ;th and ;+\th iteration values of 80 and 81-
Deciding which function to use for the intrascason prediction of daily and total
abundance requires that several factors of quantitative importance b evaluated to
assess the relative benefits to management.The following criteria are employed
in the present study.The measure of fit of each model is defined as the sum of
squared deviations.The ability to predict daily abundance for each year is
expressed in terms of the relative daily error of prediction:
The value of 82 is not known but can be approximated by regressing the ratio of
observed catch to expected catch ;900r1"9 the effect of cOlTITIercial effort
((C(C}/cr}/Z(tJ]against effort with a given intercept of one (when effort =O.
CfC)/CT =Zit»~:
n [C(t)/CTE(tt)we feT){Y'(t)+R t _1 )
t=1
RESULTS
-46-
Time of convergence is used to quantify the ability of each model to accurately
predict total abundance from expected cumulative proportions of total abundance,
where forecasted total abundance is the ratio of the cumulative catch on day t
over the expected cumulative proportions on day t (Walters and Buckingham,1965).
The criteria comparing time of convergence is the earl fest day that ~posteriori
predicted total catch remains within either 20%.or 50%of observed total catch.
Five equations expressing the daily proportion of total catch as either a function
of calendar day or a function of daily average wind speed.fishing effort,and
calendar day were fitted to observed 1979, 1980.and 1981 Big Eddy daily test
fishery catch;1980,and 1981 Middle Mouth daily test fishery catch;total 1980
and 1981 daily test fishery catch,and the 1968 to 1978 Flat Island test fishery
catch.Estimates of the parameter values were obtained either by maximizing the
likelihood function or minimizing the slim of squares.For conciseness,the five
e~uations will be designated by number.as model one,Ml'to model five,MS'The
standard normal distribution equation (Eq.6)will be referred to as Hi.The
function describing a mixture of two subpopulations (Eq.15).H2'is fitted to
the data by maximization of the likelihood function.Model three quantifies daily
abundance at the test fishery sites as a product of the effects ~f removal of
chinook salmon by commercial effort and daily wind speed on a single homogeneous
population (Eq.27,Oft)defined by Eq.6).Model four describes the results of
the same environmental perturbations on a mixture of two subpopulations (Eq.27.
Oft)defined by Eq.15).Both H]and M4 are fitted by minimization of the sum of
squares.Model five (Eq.42)is analogous tOM3 and fitted by maximization of the
likelihood function.
•II
••••••••••••
I
I
II
II
I
(66)
nl:(I expected -observed I )
t:l
Relative daily error =-'-n
An F-test comparison of the Hi residuals of the first and fourth quartiles of the
time density distributions with those of the central second and third quartiles
demunstrates that the variance of the residuals near the mean day of the migration
is significantly larger than the residual variance of the tails of the time dis-
tribution (Tabl es 7 and 9).Fourteen of the 18 time densities were characterized
by significantly larger residual variances near the center of the distribution at
the P <.01 level.The cube root transfonnation was found to be satisfactory for
attaining homogeneity of variances.The results of fitting Mi to transfonned data
are presented in Tables 8 and 10.Only the 1969 Flat Island and 1979 Big Eddy
residuals remain heterogeneous.For consistency.the cube root transformation
was retained for all minimizations of sum of squares calculations.
The fi t of H1 to da ily catch is sUlrmari zed in Tables 8 and 10.and depi cted for
the representative years of 1972,1977.and 1981 (Figures 18 to 20).Although
each distribution of observed catch over time can be described as somewhat bell-
shaped.with a larger proportion of the observed catch concentrated near the mean
of the migration (Figure 19).large daily deviations from expected values of catch
occur.Several time densities manifest a bimodal (1968.1969.and 1972 Flat Island
catch (Figure 18).pronounced right skewed (1974 and 1978 Flat Island catch.and
II
••••••••••••••••••
Table 7.Estimated values of the mean and variance of the nonmal curve equation,
F-values from the test of significance for differences in variances
between the first and last quartile and the center quartile,and the
resulting sum of squared deviations from the fitted model.
Year Slt.••••Vari••a.'-.81 ••Pro)a1tUU,.S_of
S,ur••--
n"Bl.Il4dy 13.10'110.'"13.003 P (.01 .0421131
1"0 Bl.Il4dy 14."5 62 .476 3.444 P (.01 .0107782
lUO Middle .oatll 19.366 ".650 6.011 P (.01 .0148250
lUO Total Tnt 17.G5 7'.275 5.4U P (.01 .0112911
Fhll.ry
1911 Bl.Il4dy 14.244 103.541 1.6""8 .0116"8
I'll .i441_Moatll 14.567 11 ....6.0G P (.01 .0115667
1911 Total Tnt 15.437 13.714 3.140 P (.01 .0104684
PhJluy
A••ra••14.036 87 .208 .0162021
-47-
Table 8.Estimated values of the mean and variante of the normal curve
equation,F -values from the test of significance for differences in
variances between the first and last quartile and the center quartile.
and the resulting sum of squared deviations from the fitted model.Data
are the cube roots cf reported test fishery catches for the years 1979
through 1981.
Y.er Sih ....Verteso •F-...la.Pl'o~e~Uity S_of
S,••r ••
•--
un Btl Udy 16.43'162.661 1.701 P •.01 .0'00352
1910 Btl Udy 17 .111 121.066 1.20'N 8 .0151434
1'10 Middle .oatb 21.4"121.n2 2.141 P -.0'.0231644
1910 Total Tnt 20.UIl U~.'11 1.411 N 1 .0110163
Pla••ry
1911 Bl.B4',.11."4 143 .11t l.U,N 1 .0224473
1"1 Middl.Mo.t.1I.IU 13'.140 1.661 N 1 .0227J3.
lUI Tohl Tnt 11.'10 143.15'1.041 N'.0217JU
Plu.ry
""'.1'•••11.724 UI.4t3 .0247703
-48-
•••••••••••••••••••
I
••••••••••••••••••
Table 9.Estimated values of the mean and variance of the nonmal curve eq~ntion.
F-values from the test of significance for differences in var~Jnces
between the first and last quartile and the center quarti Ie,and the
resulting sum of squared deviations from the fitted model.Data are
the reported test fishery catches of Flat Island for the y~ars 1968
throu9h 1978.
r.al'Itu ••••Variaa.a F-.el ••Prob.bUlly S_of
Il,una------.~--
19'.Flat Jab••17 .531'51.561 1.756 N 8 .02134'5
1'"Flat lala'ad 14.412 54.4"10.31'P <.01 .0435148
1970 Flat lala..21.155 49.115 7.'27 P <.01 .02"'540
1971 Pht Ida.2'.213 64."~1.U8 N 8 .0198'"
1972 Fht blad 2'.559 •.~'1 5.040 p..01 .0324197
1975 Plat lala...20.'"71.145 '.021 P <.01 .0250171
1974 Flat !abaci 13."6 64 •.,6 7.879 P <.01 .0154575
1975 Plat lab...27.3"40.246 10.467 P <.01 .0_418
1976 Plu Iala..30.373 25.032 2.'"P -
.04 .02479"
1977 Plat lllaad.2'.20'33.'41 4.905 P <.01 .01"310
1971 Flat lata.4 12.612 95.147 2.756 p..01 .0176815
.....:r.,.23 .295 89.157 .0055124
-49-
Table 10.Estimated values of the mean and variance of the normal curve equation,
F-va1ues from the test of significance for differences in variances
between the fir~t and last quartile and the center quartile.and the
resulting sum of squared deviations from the fitted Inodel.Data are
the cube roots of reported Flat Island test fishery catches for the
years 1968 through 1978.
Y••r Siu ....V.:rhaoe Jf-...al ••P:rob.b I 11 t7 S_of
Squ:r ••---
lKI Plat hI...11.711 74.574 1.154 N I
.OU9347
lK,Plu blaH 16.990 14.'74 4.972 P <.01 .0572191
1970 Plat I.laad 22.122 79.310 1.704 N 8
.0285134
1971 Jflat Isla..28.155 12 .071 1.419 N 8
.0271098
1972 Jflat Ill...,26.913 '1.428 1.402 N 8 .0412256
U73 Plat I.b"22.066 105.nO 1.115 N 8
.C32,.26
1914 Pbt labial n .50 152.211 1.195 N 8 .0280416
1915 Fl.t IIl.ad 27.591 64.343 1.526 N 8 .044'112
1916 Pbt hI'"30.464 48.054 1.799 N 8 .023"94
1917 'lat Id...21.007 0.563 1.550 N 8 .01fOO51
1911 Jfbt Ialaad 16.127 141.476 1.162 N 8 .0279536
".........23.2t5 ".157 .0334424
-50-
III
II
•••••••••••••
II
II
II
•
•••••••••••••••••••
•20
1,15
T
Observed cotch •0
Predlct.d cotch •
'44
.......
4
F
,riC
~-e,
~.as
H . .\..··1·····....I ..;..."'«.../
C.ca
9 14
OAT rF HIGRATlC»l II •..IN:11
Figure 18.Observed daily proportion~of total catch and expected proportions of total catch from model one
for 1972 Flat Island catches.
I
.211
'"
Observed catch •0
Predicted catch·
2
....
,21
...-..-.
~""".
p
~.15~
T
~
•10
,~~
N,
~.05
T
(
H
0.00 1
1
.,.....---.".~2-·····
DAY Of HlGRAT ION II •»IE II
Figure 19.Observed daily proportions of total catch and expected proportions of total catch from ~odel one
for 1977 Flat Island catches .
iii ..=.-• ••• •
••••••••---
•••••••••••••••••••
•20
1.15
OAY Of "IGIlATlllH II •.u£II
ObSRrv.d catch •0
Pr_dlct.d catch •----
Figure 20.Observed daily proportions of totai catch and expected proportions of total catch from model one
for 1981 Flat Island catches.
1981 Big Eddy [Figure 20]and Middle Mouth test fishery catch),or left skewed
character (1971 Flat Island and 1980 Bi9 Ed1y catch).
The fluctuations I')f daily catch for 1972, 1977,and 1981 are portrayed in Fi9-
ures 18-20.A pea~catch of almost four times the expected ~atch WdS observed
on 19 June 1972 (Figure 18).large daily catches,over three times the expected
daily catch,were recorded on 9 June 1969;17 June 1973~22 June 1975~and 7 June
and 11 June 1979.large peaks in chinook salmon abundance were generally a one-
day phenomenon:(5 July 1972 [Fi9ure 18);4 July 1977 [Fi9ure 19);and 5 June
1981 [Fi9ure 20);also 9 June 1969;18 June 1970;17 June 1973;3 July 1976;2
June and 4 June 1978;7 June and 11 June at Big Eddy,1979;21 June at Middle
Mouth,1980;and 5 June at Big Eddy,1981)or two days (18-19 June 1972 [Figure
18);29-30 June 1977 [Fi9ure 19));and 7-8 June,10-11 June,and 5-6 July at Big
Eddy,19B1 (Figure 20);also 22-23 June,1975;30 June - 1 July 1976;14-15 June
at Middle Mouth,1980)followed by unusually poor catches.Smaller than expected
catches near the midpoint of migration were a multi-day feature of several annual
distributions (for example 21-27 June 1972 (Figure 18);also 14-18 June 1968;
12-18 June 1969;18-23 June 1973;24-28 June 1975;1£-18 June 1978;12-16 June
1979;and 12-17 June 1981 Middle Mouth catCh).The conventional bell-shaped
curve generated by HI provides a relatively poor fit to such skewed and variable
cbservations.
The mean and variance of the catch distribution do provide a unifonn and computa-
tionally simple means to define the location and dispersion of the migration in
time.Values are relatively consistent (means range from 13 June to 30 June;
variances range from 25 days2 to 111 days2 (Tables 7 and 9)and comparable to
Nushagak Bay chinook salmon catch data (means ranging from 16 June to 29 June;
variances range from 42 days2 to 147 days2 [Mundy et al.1979])and the Bristol
Bay sockeye salmon migrations (means range from 30 June to 9 July;variances
range from 19 days2 to 42 days2 [Mundy 1979]).The means of the Flat Island
test fishery catch are apparently later and the variances smaller than those
of the Big Eddy and Middle Mouth test fisheries.This is a result of the wanner
-than-average springs preceding the 1979. 1980,and 1981 chinook migrations and
will be discussed later.The mean and variance of the Big Eddy test fishery
catch does not appear to be predictably different from that of Middle Mouth.
Big Eddy's 1980 mean date of catch was approximately 4 days earlier and the
variance of catch 17 days2 smaller than the mean date and variance of Middle
Mouth catch.However,the 1981 results are reversed,with the mean date of
Middle Mouth catch being two days earlier and variance 22 days2 smaller than
the mean date and variance of Big Eddy catch (Table 7).Although,based on only
2 years of observations,no large differences were found between the Big Eddy
and Middle Mouth catch distributions;subtle differences in the time densities
of the South Mouth and Middle Mouth may become apparent after several more years
of data have been collected.
Although H2'which quantifies the migratory timing of a mixture of two subpopu-
lations,poorly predicts unexpectedly large or small daily test fishery cat.-:hes.
it adequately describes multi-day departures from a bell+shaped distribution
(Tables 11-14).Migratory distributions which are ostensibly bimodal in charac-
ter are well approximated by H2 (for example the 1972 Flat Island test fishery
(Figure 21).Skewed distributions are also better described by a two population
model.Right-skewed time densities are best fitted by a model consisting of an
early subpopulation with small variance and a late subpopulatfon of larger vari-
-54-
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
un 8ia ..47 7.770 n."5 26 .052 49.611 .664 .JJI .0574620
lUO 8il ..47 4.619 2.513 16.519 54.217 .125 .1".00U245
lUO Ml.41.15.570 32.651 51.746 22.526 .75<1 .244 .0121546
.oau.
lUO Tnt 14.750 57.801 Sl.'46 21.500 .121 .17'.00"146
Piuu,.
lUI Btl Ed,,.7.U5 14.065 21.507 80 .240 .5".154 .00'1571
1911 11.41.1.712 12.280 20.0n 57.554 .410 ."0 .0012"0
.o.t"
HI1 T.at f.II'12.714 20.101 71.7".577 .113 .0076575,ta ...ry
£....r •••'.050 20.225 24.0n 51.036 .4f'.502 .0135229
The estimated values of the mean and variance of the earlier subpop-
ulatian,the proportion of the total population assigned to the
earlier subpopulation.the mean and variance of the later 5ubpoDula-
tion.and the proportion assigned to the later subpopulation.The
model assumes a mixture of two 5ubpopulations (model two.equation
15)and is fitted by maximization of its likelihood function.Un-
transformed data are used from the Big Eddy and Middle Mouth test
fishery operations.
I
I
I
I
i
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Table 11.
IU.Barlier Earlier
••••VarhaCle
Later Later Preportio••Me..Vari....Iar17 Lata
-55-
s..of
S ...r ..
Table 12.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopulation.the mean and variance of the later subpopula-
tion.and the proportion assigned to the later subpopulation.The
model assumes a mixture of two subpopulations (model two.equation
15)and is fitted by f,linimization of the sum of squared deviations.
Data are cube roots of Big Eddy and Middle Mouth test fishery catches.
*signifies that the variance of the residuals in the centered half
of the distribution is significantly different from the varianc~in
the tail quartiles.
Y.a~Sih aadhr Earlbr Later LaUr Proportloal s.of
••••Vari ••oa llI.all Varia.o •Barly La ..Sqa.ar ••
----
1979 Bi.E4dy 5.440 71.769 17.616 11.100 .539 .461 .0434110
191:0 Bi.Eddy-4.516 3.1:0'16.110 144.167 .071 .919 .0149011
1910 .i441.14.154 74.004 33.667 48.083 .659 .341 .0161843
.oath
1910 Tnt 13.981 91.445 34.376 40.745 .741 .158 .0139148
Pi .hary
191:1 Bi8 Eddy 8.674 64.168 18.841 77.431 .543 .457 .0093344
1911 .U4"4.H1 11.487 11.006 136.901 .116 .774 .0117835
.0.0
1981 Ta.t 5.451 16.6H 11.311 149.944 .139 .761 .0106194
Fhhary
A••ra••8.181 50.910 16.377 97 .014 .431 .569 .0171619
-56-
Table 13.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier 5ubpopulation.the mean and variance of the later subpop~
lation.and the proportion assigned to the later subpopulation.
The model assumes a mixture of two 5ubpopulations (model two,equa-
tion 15)and is fitted by minimization of the sum of ~quared devia-
tions.Data are ,:ube rools of Flat Island test fishery catches .
..signifies that th\~vari;o.!'lce of the residuals in the centered hl1lf
of the distribution iC"19nificantly different from the variance in
the tail quartiles.
Later Later Proporti9Aa
••••Variaaae !arly Late
S1m of
Squrea
.114 .02790"
.7".0179039
.101 .•0154996
.413 .0172290
.234 .0173344
.336 .0232732
.121 .0271074
.",.0311016
.670 .0236612
.521 .025"09
.1".0323146
.463 .040161e
.119
.7"
.330
.401
.472
.u,
6.'22 21.447 73.471 .207
-57-
!arlier Earlier
•••a Vari••oa
13.712 33.413 30.21'62.701
Sit_Year
1961 Flat Ial.1.941
1970 Plat 1.1.19.971 11.165 55.405 6.011
un Flat hI.7.'12 14.7"22.'47 4'.237
-.
1'72 Fl.t 1.1.17.321 20.'27 '3.146 40.043
1'71 Fl.t 1.1.24.326 6'.20'3'.'31 '.216
1'73 Fl.t 1.1.10.677 31.406 27.230 17.63'
1'73 Fl.t 1.1.23.104 33.'"34.110 37.303 .337
1"6 Fl.t 1.1.21.301 3.6'1 31.363 43.334 .107
1911 Pht lal.-3.220 10.3'2 1'.029 16'.317
1917 Plat 1.1.26.516 56.264 29.554 44.035 .116
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Table 14.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earlier subpopu1ation,the mean and variance of the later subpopu-
lation,and the proportion assignel.to the later subpopulation.The
model assumes a mixture of two subpopulations (model two,equation
15)and is fitted by maximization of its likelihood function.Un-
transformed data are used,from Flat Island test fishery operations.
Yu..liu &arlb..&a ..U ...Late..Laur Proportlo••S_of...."adaaoe ....Y.d....Early Lau S,..r ••
0
IN'Plat Isi.'.'27 4.507 21."1 27.'"."0 .650 .OUUI
INtPlat Id.I.U5 '.10'22.155 11.7".510 .420 .02"46
It70 Plat ht.U.175 54.120 54.'47 5.4".n5 .01S .0235"
It'71 Pt.t Jd.25."'7 57.452 U.'22 S.U5 .755 .247 .013,.1
It'71 FlU JI1.17.5U 7.5a 54.1'7 25.162 .470 .550 .0ltfOl
1"1 Plat hI.13.101 22.Ul 27.625 40."5 .510 .4fO .025'77
111.Plat lal.10.0U U.5H 22.455 U.UI .715 .21S .010045
If7S Plat Id.U.OU It.511 57.745 1.724 .1".114 .0J75"
I'"Flat Jd.25.760 0.10 u.on 22.46'.0,..f02 .OU517
It'7'Plat J.l.27.'J7 G."I It.U4 0.'06 .774 .2U .00"51
1'"II'lat Jal.5.516 5.171 17.225 77.554 .5".'U .01ua
A.era••17.004 U.la 21.1"2'.4a .5a .4U .0117n
-58-
-------------------
.20
D aa I 1 ..'-....y ~•'v"...~. 9 \4 \9 4 59 54 59 44
p I Ob.erv.d CGich •0
;.15
Predlct.d co1ch •---
F
T .ID
Y,e'"'"~.05[-....··T·····,
~.'".t ,__......,...
(
H
~r tF HI6RATiON II •..u.t:II
Figure 21.Observed daily proportions of total catch and expected proportions of total catch from model two
for 1972 Flat Island catches.
ance (for example the 1981 Big Eddy test fishery catch [Figure 22]~also 1974
and 1978 Flat Island.and 1981 Middle Mouth test fishery catch)(Tables 11 -14).
Left skewed distributions are approximated by a model consisting of an early sub-
population with a relatively large variance and a later subpopulation with
smaller variance (1971 Flat Island).
The most corrrnon pattern of the t;::Jporal distribution of catch and the resultin9
fit of M2 is illustrated by the 1981 test fishery catch of Big Eddy (Figure 22),
where large fluctuations in daily catch (4 June through 12 June)are followed by
a relatively moderate and stable catch of 5 days,which in turn is succeeded by
a week of higher and more variable catches.Model two defines both 1981 Big Eddy
and Middle Mouth catch distributions as a composite of an early subpopulation
comprising about 40%of the total population,and less dispersed about its mean
date of 7 June~and a later more temporally dispersed population centered around
the mean dates of 21 June to 22 June (Table 11).The two subpopulations corres-
pond very closely to the more variable first two weeks of the migration,and the
more consistent later four weeks of the migri\tion.Similar temporal patterns
were evident in the Bi9 Eddy test fishery catch of 1979.Two large catches on
7 June and 11 June were allocated to the earlier sub~opulation and a less-variable
(30 days2)series of catches from 13 June to 13 July corresponded to a later
subpopulation of larger variance (50 days2).The 9 June peak catch of 1969,18
June and 21 June peak catches of 1979,19 June peak catch of 1972 (Fi9ure 21).
17 June peak catch of 1973,10 June and 13 June peak catch of 1974,and 14, 15.
and 21 June peak Middle Mouth catch of 1980 were allocated to the early and less
dispersed subpopulation of the fitted model (see Tables 11-14).The 1968 and
1976 Flat Island and 1980 Big Eddy catch were also best approximated by a mix-
ture of a less dispersed earlier subpopulation and a later subpopulation with
a large variance although larger catches were apportioned to the later subpop-
u1ations (19,22,and 24 June of the 1968 Flat Island catch;30 June -3 July
of the 1976 r1at Island catch;and 10 June -20 June of the 1980 Big Eddy catch).
The distributions of 1971 and 1975 were character'ized by a late peak in catch
(lO July 1971 and 7 July 1975)which resulted in the best approximation to the
distributlon being a combination of an earlier subpopulation with a large vari-
ance and a later subpopulation with a smaller variance.Of the 14 years of test
fishing catch,the leptokurtic distribution of 1977 (Figure 23)was unique in
the small differences in means (28 June and 29 June).The form of a combined
distribution of two populations with coincident means but different variances
is well-illustrated by the 1977 time distribution of catch (Figure 23).
A comparison of the means,variances.and proportions of M]fitted to the data
by maximization of the likelihood function (Tables 11 and 13)and by minimiza-
tion of the square residuals of transformed data (Tables 12 dnd 14)reveals con-
sistent differences in parameter values.The earlier means of the transformed
data are smaller than corresponding earlier means of untransformed data.Con-
sidering the tendency of the cube root transformation to smooth at large fluc-
tuations in catch,the fact that the variances of both early and late subpopu-
lat10ns of the transformed data are larger than those of the untr~nsformed data
is expected.The differences between late means and proportions of transformed
and untransformed data demonstrated no obvious trend.The relationships between
the fitted parameters of M2 are as yet ambiguous.Tendencies are noted for the
varlances of the later subpopulations to be larger than those of earlier subpop-
ulations.The relative abundance of each subpopulation is not consistently
greater than 50%.although some individual proportions differ decidedly from 50%
(see 1980 Big Eddy and 1980 total test fishery catch;Tables 11 and 13).
-60-
---
.20
-••-••• •••••••••
1.15
I (I).rv"catch •0
Pr.dlcted catch.----
p
.ID
,~~,
·l·BS
d.GG I.e .....·---......-~..-4 II 16 l.L...L..L:L_....
DRT lJ'HI6RIlTlIJOI It •.AIlE II
Figure '-2.Observed daily proportions of total catch and expected proportions of total catch from model two
for 1981 Flat Island catches.
.2Il
A
€I»•.,-v.d catch •0
Predlct.d cotch •----
1\
\.
.....
ORr 1Jf HI6RIITllll II •..lIE II
Figure 23.Observed daily proportlons of total catch and expected proportions of total catch from model two
for 1977 Flat Island catches.
•••'.•
I
•
I
•••
I
I
I
I
I
I
I
I
The residuals of /12 fitted by the likelihood function are consistently smaller
than those fitted to the transfonmed data by minimizing the sum of squares.An
average reduction in sum of squares by H2 of 31%for the 1979-1981 test fishing
catch.and 23%for 1968-1978 test fishing catch was achieved.compared to the
sum of squares from MI'The reduction in sum of squares was greatest for cis-
tribution with a bimodal (1972,28t)or skewPd Bi9 Eddy,58%and 1981 Midjle
Mouth,48%)character.Given accurate estimates of the values of the parameters,
HZ offers few advantages over HI in the ability to predict daily fluctuations in
abundance.although the general forn of the distribution is approximated better
by HZ than Hl'
The vicissitudes of daily catch can to some degree be predicted by H].Major
peaks in test fishery catch are paralleled by peaks in predicted catch,although
the magnitude of the peaks differ (11.14.15. 18.19.25,26.28.29 June and
2 July 1972 [Fi9ure 24)and 5,7,10,13-15,25,and 29 June,1981 8i9 Eddy
[Fi9ure 26);also 9,10, 16,and 19 June 1968;9,11,15,18,and 19 June 1969;
11,15,17,21,24,and 25 June 1979;20, 21, 24,27-30 June,and 4 July 1971;
7,17,and 24 June 1973;15. 17,29,30 June,and 1 July 1974;19,22,28 June,
and 7 July 1975;24, 27, 28,30 June,1 and 3 July 1976;10,13,and 14 June
1978;6,11,13,17 June 1979;11, 18,and 19 June 1980;and 7 and 10 June 1981
Middle Mouth).Reduced catches near the center of the migratory distribution
were also accurately estimated (13.16.20, 23.24, 27.30 June and 1 July 1972.
[Fi9ure 24);1-2 July 1977 [Fi9ure 25);and 6,9,12, 16,19,22,and 23 June
1981 Bi9 Eddy [Fi9ure 26);also 11,14,18,and 2,June 1968;10, 13,14,20,
and 21 June 1969;16,19,20,and 23 June 1970;22 June and 5 July 1971;19 and
26 June 1973;4,11, 15,and 18 June 1974;20, 24,and 30 June 1975;25,29 June
2 and 6 July 1976;9,13,and 20 June 1978;8,15,and 22 June 1979;10 and 13
June 1980;and 6 and 16 June 1981 Middle Mouth).On several days.the predicted
peak or depression in catch preceded the observed peak or depressions by one day
(for example 25 June 1972;3 July 1977;and 17 June 1981 [Fi9ures 24-26];also
8 June 1969;17 June 1970;9 and 12 June 1974;26 and 27 June 1975;7 July 1976;
21 June 1978).The predicted peak or depression followed the observed peak or
depression on other occasions (3 July 1972 and 21 June 1981 [Figures ~4 and 26].
also 21 June 1968;11 June 1973;23 June 1974;15 June 1980 Middle Mouth;and 6,
9,and 10 June 1981 Middle Mouth).
No consistent pattern appears to exist between the errors of estimation near the
mean of the migration and the errors of estimation at the beginning or end of
the migration.No reguiar violation of the assumption of inde~endence of obser-
vations was revealed in the autocorrelations of the residuals.Significant (P <
.05)positive autocorrelations of lag 1 were observed in 1970 and 1980,and a
negative autocorrelation of -.29 was calculated for the 1975 residuals.Aver-
age wind speeds were highest in 1980 and 1970 respectively compared to wind
speeds of other years.The high wind speeds in 1970 and 1980 could be respon-
sible for sequences of positive or negative residuals,resulting in the observed
positive autocorrelations.However.wind speeds in 1975 were also relatively
strong.It is also possible that another factor which affects salmon migratory
behavior and which is highly correlated with the number of salmon migrating
through the lower river and also highly autocorrelated could be responsible for
the anomalously high autocorrelations in 2 of 14 years.
The sum of squared deviations is reduced by an average of 43%over those of HI
for the 1979·1981 test fishery catCh,and by an average of 38%for the 1968-1978
-63-
.20
'44
..
'94
Ob.~y.d cotch •0
Predlct.d cotch •----
··.·.:0..,./.
'Ie...-':\<.
P
1.15
T
~.ID
~~
~.as 1\//.....
Ii I:..t ......'......,'.I !I~/'...../\9\:~.aa 14
9
DAT Of "1_1I1Il II •.uE II
Figure 24.Observed daily proportions of total catch and expected proportions of total catch from model
threee for 1972 Flat Island catches.
-------------------
.20
.ID
,
'"~,
!.15
~
T
i
L
~.as
~
Ob...ved cetl:h •0
Predlct.d cetc,..-_
,.,
'..
D.aD I "..e:::,.""'''11''"•f-V IF V~7 2 7 5,5 '-.....",.,7 ~2 ~7 2 I
DRT Of "I6RATlIl'I II •JlI£II
Figure 25.Observed daily proportions of total catch and expected proportions of total catch from model three
for 1977 Flat Island catches.
.2G
p
,.15
P,T .IG'"'"90
~
~.as
H V··,.........-.
G.aa
-4 "'6•
DAT Of HI_TlON II •.lH II
Observed catch •0
Predlct.d catch.----
Figure 26.Observed daily proportions of total catch and expected proportions of total catch from model three
for 1981 Flat Island catches.
I
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test fishery catch (compare values of Tables 15 and 16 with values presented
in Tables 22 and 24).The largest reductions in sums of squares were obtained
for the 1974 (72%),1981 (61%),1972 (51%>'and 1973 (5a)test fishery catches.
Compared to H2.MJ reduced the sum of squared deviations by an average of 10%
and 20'%for 8ig Eddy and Middle Mouth and for Flat Island test fishery catches
respectively (Tables 15, 16. 12,and i3).The ability of HJ to reflect the
daily variation in observed catch and the reduction in sums of squared devia-
tions commend its use as a forecaster of daily abundance over that of HI or M2-
Note the ~onsistent positive correlation between wind speed and abundance (para-
meter B1.Tables 15 and 16)and the consistent reduction of daily test fishery
catch by cOl1l11ercial effort (pararneter S2)'Large test fishery catches corres-
pond to days of high wind speeds and periods of corrmerdal closure.Exceptlon-
ally small catches occur on days of corrmercial fishing and relatively low average
wind speeds.The high and consistently negative values of parameter 32 indicates
a high exploitation rate by the CQnlllercial fishery.Given an average wind speed
of 10 mph,the Big Eddy and Middle Mouth test fishery catch is reduced by an
average of 36%during a coornercial flshing period.Flat Island catches are
reduced by an average of 25%.The mean dates of migration are invariably earlier
than those calculated for modell,while tl'c variances calculated for M]aloe
larger than corresponding variances calculated for ,"fl.With ar:curate estimate!
of H]coefficients,the reduction in the ~rror~af daily chinook catch p:"::'::~ctions
by MJ relative to HJ suggest that the temporal distribution of abundance can bE
accurately described as a function of time and environmental factors.
Fitting H 4 to observed catch results in good agreement between expected and
observed values of test fishery catches (Figures 27-29;Tables 17 and 18).
Unexpectedly large or small daily catches were accurately predicted by M4 for
a single-day or multi-day catch phenomena.As observed for HJ ,exceptionally
large or small catches were sometimes predicted the day before or day after the
observed catch.Scme exceptions to the goodness of fit should be noted.Some
moderate peaks in abundance late in the m;gratian (5 July 1972 [Figure 27];als)
22 June 1969;5 and 10 July 1971;28 June 1973,22 June 1974;22 and 23 June 19'8;
27 and 28 June 1979)were not well predicted.The magnitude of large test fish··
ery catches early or in mid-season were also poorly estimated by the ~odel (29
and 30 June 1977 [Figure 28];also 29 and 30 June;9 June 1969;18 June 1970;
17 June 1973;22 and 23 June 1975;30 June,1 and 3 July 1976;10 and 11 June
1978;11 June 1979;and 18 June 1980).However,the deviat;ons of observed
catch from expected catch were apparently random throughout the migratory dis-
tr;bution and.except as noted previously.comparatively small.
The goodness of fit is reflected in the reductiun of the sum of squares conpared
to previous models.The sum of squared res;duals iJf Flat Island ciltch were
reduced an average of 54t compared to Ml'40:compared to H2'and 25~canpared
to .'f).Middle Mouth and Big Eddy sums of squares were reduced an average of 69%
over those of HI'55%over H 2 ,lnd 46%over H).Although the values of the para-
meters vary by year and test fishery site for H 4 ,some general qualitative con-
e1us ions can be made.The va I ue of the w;nd speed pa rameter,S~.1i es bebleen
the values of -.43 and .59 and,with the exception of 1968. 1976.and 1977 is
positive (Tables 17 and 18).The value of the effort parameter,82.is predict-
ably negative,and lies between the values of -1.7,and -5.7,denoting a high
exploitation rate.The variances corresponding to earlier melns were,with the
exception of the 1970 and 1977 Flat Island catches,smaller than the variances
·67·
Table 15.The estimated values of the mean and variance of a homogeneous
population migrating past the test fisheries,and the values of
the coefficients quantifying the effect of wind speed and commercial
effort on the migration.Model three (equation 27)is fitted to
observed data by minimizing the sum of squared deviations from the
expected catches.Data are cube roots of 81g Eddy and Middle
Mouth test fishery catches.*signifies that the variance of the
residuals in the centered half of the distribution is significantly
different from the variance in the tail quartiles.
Y••r su.••••Vari ••o.BO 1:1 B1 8_of
S,.ar.a
1979 Bt,Eddy·13.915 524.'01 13.56 .5064 -3.307 .043006
1,.0 Bt,Eddy 14.5"210.157 ,..,.3051 -3.556 .OOf052
1'00 .iddl..oatll 19.IU 171.601 11.46 .1"3 -4.121 .011147
19110 Total T••t·11.179 167 .111 11.21 .3055 -3."4 .001574
Fhll.ry
lUl Bi,84'y 15.123 307.401 12.7".3753 -5.10'.00'141
l'al .i.41..oat.14.151 112 .105 14.527 .1464 -5.5".00""
lUl Total Tut 15.27'Ul.,,,13.711 .2404 -5.216 .001500
Fh...ry
A••ral·17.1"301.056 12.46 .1123 -4.421 .014073
-68-
I
I Table 16.The estimdted values of the mean and variance of a homogeneous pop-
ulation migrating past the test fisheries.and the values of the
I coefficients quantifying the effect of wind speed and cOIllllercial
effort on the migration.Model three (equation 27)is fitled to
observed data by minimizing the sum of squared deviations froM the
I
expected catches.Data are cube roots of Flat Islam!test fi~nery
catches.
I Y••I'Sit.If•••VaEI_lloa 1;0 81 8 1 5..of
Sq••ra.--------I un Plat lala_.17.561 142.110 14.10 -.3147 -2."1 .021341
I U69 Plat lalaad 15.43'17'.449 6.15 .SS41 -1.534 .032005
U70 Plat lala..21.5"130.4'4 '.0'.un -1.157 .0199.0
I 1"1 Plat tabu 1'.591 1".110 10.50 .1101 -1.715 .010614
I 1"2 Plat hlaad 24.036 40 •••13 16.0'.1941 -'.790 .023560
U73 Flat hlaad 10.24'246 .046 '.0'.4'"-1.746 .016141
I 1"4 Flat lala.d 12."5 250.246 5.56 .1225 -1."3 .007"3
1:17'Pl:.t bled 17.361 11.257 5.20 .4741 -4.426 .031073
I 1"6 Plat Ial •••30.176 61.514 10.05 -.OSS7 -3.151 .01"30
I u77 Plat hla.d 2'.671 77.755 11.71 -.U76 -1.40£.011300
197.Plat blaad 13.065 32'.205 12.16 .U17 -<5.534 .01408'
I ".'1".'11.796 U1.7"10.03 .2$66 -,.654 .020694
I
I
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I
-69-
I
p ,W I
I.IS
T
Observed catch •0
Predlct.d catch.----
.....
f·.In ·.·.·.·.·.·.·.·.·.· .·.••:~·.· .· .· ....\. .
....\..
o.nn It······""\:Y·,,"(....ie"•.........."\
9 \.\9 •'19 5.59 44
~
~
~.ns
,
~o,
DAY Of HIGRRTll»l II •..lIhf:II
Figure 27.Observed daily proportions of total catch and expected proportions of total catch frOO!model four
for 1972 Flat Island catches.
-------------------
•2ft
1.15
Ob.-rv.d catch •0
flndl ct.d ce"ch •
0
F
.1D
T
,9~e-,
q .as
~
0.DD
7 12
.r··.·.·.· .
!,'........:
'7
ooT CJ'HIGRRTlOH II •..It£II
"......
~2
Figure 28.Observed dai ly proportions of total catcr.and expected proportions from model four for 1977 Flat
Island catches.
.211
Ob,.r~.d catch •o
~~I Predlct.d cotch •
2.15
~
P .10
,~~
N,I,,
~.as ,:/\',',
~,
I,,
O.Dll
6 '0 '5 10 \5 :ItI :25 :Jll ~S ~O ~S
DAT OF HIGRRlION II •JUNE II
Figure 29.Observed daily proportions of total catch and expected proportions from model four for 1981 Flat
Island catches.
I
I
I
•••••
•
••
••
•,
•,,
Table 17.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
earl ier subpopulatlon.the mean and varian.:e of the later 5ubpopula-
ticn,and the proportion assigned to the later subpopulation.The
model assumes a mixture of two 5ubpopulations affected by wind and
commercial effort (model four,equation 27)and is fitted by mini-
mization of the sum of squared deviations.Data are cube roots of
Big Eddy and Middle Mouth test fishery catches.
Y.ar SUa Badiu Badhr IAhr LaUr ProportioJu
••••v.tt ••••)I•••Varia....lady Late
~-
1979 a'i 1l44y 6.109 0.975 15.412 510 .051 .049 .951
1910 a'i 1l44y 9.619 51.914 25.732 149.399 .4&5 .515
1910 Middle 10.701 32.452 27.545 151.411 .211 .712
.0.t1l.
1910 T..t 13.171 11.170 33.116 47.~2 ."3 .307
Fhll.ary
1911 Bi,Eddy '.317 11.259 19.332 247.927 .117 .113
1911 Middle 5.591 10.402 20.021 173.'19 .17'.124
lIo.t1l.
1911 T..t '.042 17.311 i9."4 217 .112 .131 .IU
Flalany
,i..,.era.e 6.671 32.UO 23.114 22....·,..271 .722
-Continued-
-73-
Table 17.Estimated values of the wind.effort.and constant parameters and
sum of squared de',iations fran model,The model assumes a mixture
of two 5ubpopulations affected by wind and commercial effort (model
four,equation 27)and is fitted by minimization of the sum of
squared deviations,Data are cube roots of Big Eddy and Middle
Mouth test fishery catches (continued),
Y..r Site BO B1 B2 S.of
....r ••--
1979 Btl Udy 10.103 .59495 -4.7460 .0194351
1910 Btl Udy 1.105 .3'172 -3.9471 .0077234
1910 M!ddt.Mo.tla '.929 .59402 -4.3516 .00/i5747
1910 Total T..t 9.5".33200 -4.0361 .0044401
Fhllu7
1911 Btl lWi'y 11.570 .37410 -5.U7'.0065742
1"1 Mi441e Mo_tla 12.591 .15209 -5.4117 .0043516
1911 Total Tut
Fhlaery 12.217 .24734 -5.5491 .0044711
ATera ••10.494 .'5413 -4.1226 .0076529
-74-
I
I Table 18.The estimated values of the mean and variance of the earlier sub-
population,the proportion of the total population assigned to the
I earlier subpopulation.the mean and variance of the later subpopu-
lation,and the proportion assigned to the later subpopulation.The
model assumes a mixture of two 5ubpopulations affected by wind and
I commercial effort (model four,equation 27)and ;s fitted by minimi-
zation of the sum of squared deviations.Data are cube roots of Flat
Island test fishery catches.
I
t ••r Sit.Barlhr Eadhl'Later Later ProportioD.
I ••••Varhao •••••Verhao •BarI7 Lat.
I 1961 Flat !ahlUl 1.719 1.11'21.114 76.444 .205 .795
I 1969 Flat bh1Ul 7.lao 17.790 23.153 44.'44 .396 .604
1970 Fht blaad.19.234 77.744 55.052 1.341 .UI .142
I 1971 Plat Jab"22.750 44 .1413 ".115 41.1".573 .427
1972 Flat hlaad 16.441 15.971 13.'35 '3.599 .331 .669
I 1973 Plat blaDd '.956 22.154 26.451 '4.029 .260 .740
I 1974 Flat hlaad 1.7C 0 44.453 23.231 215.111 .374 .626
1975 Flat Ialall4 21.961 25.91'U.721 32.279 .510 .490
I 1976 Plat lalaad 22.111 3.704 31.306 41.723 .OU .'05
1977 Plat lalaado 21.761 17 .'33 29.'06 4.691 .'22 .071
I 1971 Plat lalall4 3.014 7.962 17.701 226 .657 .103 .an
I A"er •••11.411 32.711 24.675 11.066 .421 .57'
-Continued-
I
•
I
I -75-
I
Table 18.Estimated values of the wind.effort,and constant parameters and
sum of squared deviations from model.The model assume a mixture
of two subpopulations affected by wind and commercial effort (model
four,equation 27)and is fitted by minimization of the sum of
squared deviations.Data are cube roots of Flat Island test fish-
ery catches (continued).
r.al'Ilt.8 0 81 8 1 s.of
S,••r ••...-
1"1 Plat Ialaad.14.'"-.43751 -2.3272 .OUU71
1'"Flat Isla..6.543 .33UI -2.1221 .02243U
U70 Flat lahad 7.513 .27755 -2.3361 .012UU
U71 Flat Id....I.OU .320fO -1.7471 .oulln
U72 Flat Ista_10.'".161"-5.1727 .0150231
U73 Flat bla..6.430 .52774 -2.0435 .0140535
1"4 Plat lalaad 1.251 .43506 -2.71U .0072566
U7S Plat lala...4.n5 .4""-4.7514 .0224506
1"6 Plat Idaa4 10.431 -.11567 -3.'336 .015"60
1"7 Plu hI...13.142 -.33247 -2.2277 .014U21
1"1 Plat bla"11.175 .14111 -4.6325 .010""
."'.1'&'.1.421 .160fO -3.0'17 .0154726
-76-
•
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associated with later means.Earl)and late subpopulation proportions average
close to one talf of total abundance.
The results of fitting MS to the distribution of catch over time are similar to
the results obtained in the fitting of HJ (Tables 19 and 20i Figures 30-32).
large or small observed daily catches generally coincide with large or small
estimated catches.Although exceptionally large or small daily chinook salmon
abundances are generally underestimated or overestimated (18.19 June and 5
July 1972 [Fi9ure 30);29,30 June and 1 July 1977 [Fi9ure 31);and 5 and 9
June 1981 Bi9 Eddy [Fi9ure 32);also 9 June 1969;18 and 23 June 1970;I,2,3
July 1971;17 June 1973;22 and 23 June 1975;30 June,1 and 3 July 1976;2 and
4 June 1978;7 and 11 June 19l9;10 June 1980 Big Eddy),the more moderate peaks
in catch are well represented by the estimates of MS-It is observed that in M5
there is a tendency for the catches on the extremities of the temporal distribu-
tion to be underestimated while catches near the mean day of distribution are
generally overestimated,The skewness of the 1981 Big Eddy,1980 Big Eddy.
and 1974 Flat Island distribution is better approximated by the distribution
function of MS than the skewness of the 1975 and 1978 Flat Island distributions.
which are relatively poorly approximated.
The calculated mean day of catch is consistently earlier than the mean day of
catch calculated by Hl'and the variance is smaller.The wind parameter 81.is
positive for all years and the effort parameter.62.is negative for all catch
distributions.Al though minimizing the sum of squared deviations is not the
criterion used in estimating the parameters of the HS _the values of sum of
squares for each year compare favorably with corresponding values of H].The
1969, 1970, 1971, 1972, 1973,1974, 1975,and 1977 Flat Island,the 1979 and 1980
Middle Mouth and Big Eddy.and the 1980 Big Eddy distributions all resulted in
lower sum of squares for HS compared to H].
The relative ability of each model to accurately describe the distribution of
daily catch can be summarized as follows.Models one and H2 demonstrate the
poorest predictive ability.followed by H]and H5'Model four proved to be the
most accurate estimator of daily catch.It can be concluded that models three,
four,and five all accurately depict the time distribution of catch.Obviously,
accurate estimation of the parameters is necessary to make these methods appli-
cable to actual intraseason forecasts.The consistent positive value of the
wind speed coefficient,Pl.and negative value of the commercial effort coeffi-
cient.82'are significitllt and consideration of these two effects would undoubtedly
improve abundance estimates.
The statistics of the estimated values of the parameters of H].H4 •and HS are
surrmarized in Tables 21, 22.and 23 respectively.The wind parameter.61.and
the commercial effort parameter.82.are significantly different from 0 for all
three models.Average April air temperatures were highly correlated with the
fitted mean of I'fJ.the var~ance of H].the mean of 51 and 52 of H4'the variance
of 52 of H 4 •and the mean of H~.linear regression techniques were applied in
the analysis of the relationshlp of the spring air temperatures and the means
and the variances of H]and H-:.and the means of MS'A logarithmic transfonna-
tion was employed to stabilize the variance (0 7).The statistics of the linear
regression analysis are presented in Tables 21.22.and 23.
-77-
Table 19.The estimated values of the mean and variance of a homogeneous poou-
lation migrating past the test fisheries.and the values of the
coefficients quantifying the effect of wind speed and commercial
effort on the migration.Model five (equation 40)is fitted to
observed data by maximization of the 1ik.el ihood function.Data
are from the Big Eddy and Middle Mouth test fishery operations.
Y..r su.M.aD Va:rhao.BO B1 B2 S.of
8qur••-
1979 8i.Eddy 5.247 20.125 0.0443 .0001 ~."U .0363204
1980 Bil Ilddy 13.115 50.602 0.0137 .02329 ~.44".0067951
1980 Middle Moath 16 .101 62.146 0.0292 .01726 -0.5957 .0101361
1980 Total Tnt 15.4"62.506 0.0261 .01156 -0.5313 .0067193
Phll.:ry
1911 Bil Ilddy 4.550 7.50'0.00".00'"-0.6332 .000554
1911 .iddl ••o.t.13.627 74.'"0.2655 .01025 ~.U23 .0113025
1911 Total Tnt 4.,H 1.129 0.0267 .00741 ~.O35 .005"55
Phh.:ry
A....r •••10.534 41.0"0.0511 .01302 ~.5"4 .0120251
-78-
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Table 20.The ~stimated values of the mean and variance of a homogeneous p.op-
ulation migrating past the test fisheries.and the values of the,coefficients quantifying the effect of wind speed and commercial
effort on the migration.Data are from the Flat Island test fisn-
ery operations.
I
I r ••r SU.••••Vart ••o•Bo Bl B2 Sua of
Squrec----
I 1'61 Plat Iliad 14.704 ".412 0.2434 .00000 -il."24 .0223261
I 1"9 Plat IIhad 7.502 7.1U 0.0735 .00433 -il.2536 .0297612
1970 Plat Ia1a:a4 1t.'S~46 .045 0.5611 .03752 -0.2394 .01-43960
I 1971 Plat blad 27.653 59.217 0.1146 .0~131 -il.1439 .0116809
1972 Plat Ill...15.tsO 4.6"0.0320 .00590 -il.5347 .0213216
I 1973 Plat lllaad 11.115 62.721 0.0000 .02174 -il.3510 .0134301
I 1974 Flat Iliad 7.762 15.164 0.0145 .01242 -il.1526 .0061396
1975 Plat Ialaad 25.703 37 .293 0.0000 .03415 -il.4769 .0300980
I 1976 Plat lalaa4 21.265 19.530 0.2611 .00773 -il.4U7 .0210395
,.,,,Plat lala...27.20<1 32.399 0.4614 .00424 -il.2136 .0175581
I 1971 Plat btad 12.726 16 .259 0.0597 .02354 -il.6942 .017J667
•.".ra.a 11.601 37.245 0.1412 .01726 -il."72 .0194313
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.20
Ob.erv.d catch •0
Predlct.d catch •----
...................t...10'...................~
4 39 ~4
OIlY ~HI6RATION II •.IJNE II
Figure 30.Observed daily proportio.of total catch and expected proportions from model five for 1972 Flat
Is land catches.
-------- - -- -
••-••- -
.20
Observed ~Gtch • 0
Predlct.d catch.----
'.
.1lS
.10
o 00 I...............•.r"".·r·····II --V ..··:!:..:=--..
. 7 h7 ~2 ~7 S2 S7 b I
~
~.IS
~
T
~
~
~
,
<Xl
()flY (F 'HGRA11~fl •..A.N:IJ
Figure 31.Observed dally pt'oportions of total catch and expected proportions from model five for 1977 Flat
Island catches.
.20
Observed cetch :0
P~~dlct.d cotch •----
P
E
~.15
T
~
.10
T
,Y
'"R
N L,
~.lIS
l I 1.".1 I ,
H
o.aa
!..4-,I '6
ORT llf HIGRRTlON II •.lJNE II
Figure 32.Observe~dally proportions of total catch and expected proportions from model five for 1981 Flat
Island catches.
•
I Table 21.Mean parameter values.confidence limits.and the relation between
mean and variance for model three and average April air temperature.•••
Mean
Mean day of migration 20.243
Variance of migration 225.159
Constant 10.471
Wind parameter 0.272
Comnlercial effort parameter -3.772
Lower 95%1im;t
16.526
148.667
8.549
0.099
-4.887
Upper 95%limit
23.960
301.651
12.393
0.445
-2.657
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The Pearson Correlation Coefficients are:
Mean day and April tempera ture:r =-.9154 P <.001
Variance and April temperature:r =.6092 P =.021
Na tura 1 log of variance and
Apr;1 t8TIpera ture:r =.6961 P =.006
Mean day =37.933 -1.0056 (Mean April air temperature,Of)
with an R2 value of .838 and F-value of 62.01
Variance =e(3.9242 +.075163 (Mean April air temperature,OF)
with an R2 value of .485 and F-value of 11.28
Sum of squares for all years =:.432103
Average error of estimated observation =:.0169135
-83-
Table 22.Mean parameter values 9 confidence limits 9 and the relation between
the means and variances for model three and average April air
temperature.
Early mean day of migration
late mean day of migration
Early variance of migration
late variance of migration
Proportion of early
subpopulation
Proportion of late
subpopulation
Constant
'.lind parameter
Corrmercia 1 effort parameter
Mean
13.744
26.851
31 .200
133.508
0.377
0.623
9.502
0.223
-3.453
lower 95%limit
9.173
22.712
15.093
45.978
0.219
0.465
7.899
0.042
-4.241
Upper 95%limit
18.315
30.990
47.307
221 .038
0.535
0.781
11.105
0.404
-2.665
The Pearson correlation coefficients are:
Mean day of 51 and April temperature:
Mean day of 52 and April temperature:
Variance of 52 and April temperature:
Natural log of variance and
April temperature:
r =-.8810 P <.001
r =-.7946 P .001
r =.7106 P .004
r =.7395 P =.003
Mean day of 51 =34.681 -1.1901 (Mean April air temperature,oF)
with an R2 value of .776 and F-va1ue of 41.591
Mean day of 52 =43.952 -0.9720 (Mean April air temperature.OF)
with an R2 value of .631 dnd F-value of 20.551
Variance =e(1.3628 +.165373 (Mean April air temperature.OF)
with an R2 value of .5468 and F-value of 14.482
5um of squares for all years =.511062
Average error of estimated proportion =.01800
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Table 23.Mean ~arameter values,confidence limits,and the relation between
mean and variance of model five and average April air temperature.
Mean day:37.958 -1.2338 (Mean April air temperature.OF)
with an R2 value of .725 and F-value of 31.67
The Pearson correlation coefficients are:
Mean day and April air temperature:
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Mean day of migration
Variance of migration
Constant
Wind parameter
Conmercial effort parameter
Mean
16.251
34.909
0.1210
0.016
-0.384
lower 95%11m;t
11.350
20.923
0.0346
0.01 0
-0.477
r'-.8516
Upper 95%1 im;t
21.152
48.895
0.2074
0.022
-0.291
P <:.001
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•
Comparisons of the ability of /'I]."'4.and MS.and the time series model to accur-
ately estimate both daily and total abundance are summarized in Tables 24,25.
and 26.Given accurate least squares estimates of the parameters of HJ and 114
(Tables 15-18)or maximum likelihood estimation of !'Is (Tables 19 and 20).the
average daily error of estimation of 1968 to 1979 test fishery catches is smaller
for both HJ (averaging 0.01449)and H4 (avera9ing 0.01318)than the average daily
error of tlme series model (averaging 0.01666).The time series model more
accurately predicts the daily proportion of 1980 and 1981 total Big Eddy catch
compared to H].although less accurately than /'14'The ability of HJ and /'14 to
forecast total catch win;n a 50%error interval (predicted total catch equals
the ratio of cumulative catch over estimated cumulative proportions of total
catch)is not consistently better or worse than the ability of the time series
model to predict total catch.ModelS results in an average error of daily pre-
diction of .01365 for the 1968 to 1981 proportions of total annual test fishery
catch (Table 25),which is less than the average error of M](6%less)but greater
than the average error of H4 (3%9reater).
Total catch estimated by H]remained within a 50%error interval previous to the
forecasts of the time series model for years 1968,1971,1974, 1976. 1979, 1980.
and 1981,where as 1970. 1972,1975.and 1977 total catch was more prcmptly esti-
mated by the time series model.The time series model estimated total catch
within a SOX error interval in advance of H4 for 1969, 1970, 1971. 1972. 1973,
and 1977 catch data.The time series model was superior to both M]and M4 in
the ability to predict total annual catch within a 20:error interval.Predicted
total annual catches are greater than observed total catches.Total observed
catch was not within 20%of final predicted catch of H)for the years 1969,1971,
1972,1975. 1979,and of H4 for years 1971, 1975,and 1979.
Annual values of M]'S mean,variance,80'81,and 82 were estimated according
to the linear relationship or averages presented in Table 21 and substituted
into ,'fj'In a similar manner the mean,variance,and proportion of 51'the mean,
variance,and proportion of 52.aD,81,and 82 were estimated by a linear function
of temperature (~1'~2'022)or as an average of the l~estimated annual values of
the parameters (0 2 1 ,Pl'p~,80 ,Bl'and B2)'Estimates of both daily and total
abundance of H4 were more 10 error than estimates of the time series model (Table
25)or MJ ,Estimates of daily abundance by the time series model were.on the
average.39%more accurate and those of H3 were 36%closer to the observed daily
catch than the corresponding estimates of H4'Total abundan~e predictions did
not converge to within 50%of observed abundance for 4 years (1972,1975, 1976,
1977)and within 20%of observed abundance for 7 years (1970,1971, 1972, 1975,
1976,1977,1981).The res idua 1s of H]were tes ted for independence ar.d found to
be significantly correlated at lag 1 day.An autoregressive tenn of ¢l1 =.39 was
added to MJ to satisfy the conditions of independence.Both MJ and the time
series model reC:'llted in differences of 0.01 to 0.02 between observed and fore-
casted daily proportion of total abundance.The time series model estimates
were an average of 4%closer to the observed proportions than H]estimates for
14 years of observations.The time series model does anticipate total abundance
with greater accuracy tRan H]{Table 26).Model three estimates of total catch
failed to converge to within 20%of observed total catch for 8 years of test
fishery catches (1969,1974,and 1976-1981).
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Table 24.Comparison of the ability of models three and four.fitted to the
observed data by minimization of sum of squares.to predict daily
proportion of total catch and total annual catch.Average dally
error is defined by equation 66 in the text.NE signifies that the
20:error interval of predicted total catch did not include th/?
observed total catch the last day of the test fishery.
Pitt"·3 Pitted ••Y••r A....r •••total _tUb •....1'•••total .ithill.'.U,.urol'5~2~daUy anor 5~2~
1961 .01776 1 ...11 1 ...17 .01590 1 ...10 1 ...13
1969 .01576 1 ...3 ME .01316 1 ...5 1 ...25
1970 .01442 1...14 J1Ula 2..011"1 ...14 JUDe 20
1971 .01'"1 ...13 ME .01652 1 ...17 NE
1972 .01741 1 ...14 ME .01431 1 ...12 JUDe 15
H73 •01374 1 ...5 ,..&18 .01272 JUD.7 JDDe 18
1974 .00191 1 ...1 1-..12 .00130 .ay 31 1 ...I
1975 .01651 J'u.25 NE .01599 1 ...20 NE
1976 .01515 lu.a 25 J.l,.•.01315 1 ...20 July •
1977 .012"Jua 20 1.1y 1 .011'5 1 ...20 Ine 26
1971 .01020 May 31 1"7 15 .00992 May 30 1...3
1979 .01401 May 27 ME .01036 May 31 NE
1910 .00970 1 ...3 J ...21 .00171 1 ...2 1 ...I
1911 •00953 1 ...3 I_a 14 .00130 ••y 31 1 ...2
-87-
Table 25.COOlpar;son of the abil ity of model four and the one population time
density model to predict daily proportion of total catch and t tal
annual cat~h.Parameters of model four are estimated by statistics
presented 1n Tables 17 and 18.Parameters of model five are F!sti-
mated by maximization of the likelihood function.Average daily
error is defined by equation 66 in the text.NE signifies that the
50:;or 20%error interval of predicted total catch did not include
the observed total catch the last day of the test fishery,
E.ti••t.d t.o-pop.latJoa aod.l Pitt.d ti••••••Ity .0••1.MS
Y••r A....r••••rror Pndicted total """.r....rror Pr.dicted total
••11y predlctJo••3~2~dally pr.dlctlo••3~2~
1961 •03341 1 ....24 1.1y 30 .01119 1 ....12 1 ....16
1969 .01512 1 ....4 1 ....3 .01636 1 ....3 1 •••12
1970 .02016 1...16 lIE .01470 1 ....,1 ....17
1971 .02912 1.1y 10 lIB .01543 1 ....17 HE
1972 .04153 lIE lIB .01713 1 ....17 1 ....26
1973 •014"1 ...11 1 ....U .0UIf 1 ....11 1.0..20
1974 .00911 1 ....2 1 ....I •00164 1 ....3 1 ....I
1975 .02111 lIB lIB .01613 1 ....11 1 ....21
1976 .04977 lIE lIB .01503 1 ....23 1.1y 4
1977 .03471 lIB lIB .01244 1 ....n 1 ....21
1971 .01471 1 ....10 1 ...19 .01207 1 ....6 HE
1979 .01169 1 ....7 1 ...17 .01413 1 ....2 1 ...13
1910 .013,.1 ....15 .1...II .00102 1 ....4 J1UI..18
1911 .01240 1 ....3 lIB .0011'1 ....3 lIB
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Table 26.Comparison of the abil ity of model three and the time series model
to predict daily proportion of total catch and total annual catch.
Pa rameters of mode 1 three are es t ima ted by s ta tis tics presen led in
Table 15 and 16.Average daily error is defined by e~uation 66 in
the text.NE signifies that tile 20%error interval of predicted
tota 1 Cd tch di d not include the observed t.ota 1 Cd tch the 1as t ddy
of the test fishery.
Tt..Sari ••M04.1 Fitted o••-p(~.l.tlo••04al
Y.ar .b'.ra,••,.,.or P,..Uch4 total A...eral.error ?recUct.d total
daily pradlctio••50f0 10f0 dally prediction.50f0 10f0
1961 .02181 1 ....11 111ly 3 .01179 1 ....15 1 ....30
1%9 .01742 1"".3 1"".14 .01651 1"".10 HE
1970 .01519 1 ...7 1"".11 .01'14 1"".11 luo 21
1971 .01770 1 ...16 1"".11 .01931 1 ...14 July 7
1971 .01881 1 ...11 1"".13 .02123 1"".15 July 8
1973 .01599 1 ....,1 ...13 .01573 1"".,JUIlO 19
1974 .01204 1 ....1 1 ...19 .01113 1 ...11 HE
1975 .01034 1 ...11 1"".13 .01985 1"".11 111,U 30
1976 .01901 1 ...11 1 ....30 .01161 1"".30 HE
1977 .01503 10..18 lu1y 1 .01814 1 ....11 HE
1978 .01103 110 7 31 1 ....7 .01135 1 ....3 HE
1979 .01541 1 ....7 10...8 .01489 ··7 18 HE
1980 .00911 1 ....6 1"".16 .00%8 1 ...,IlIl
1981 .00946 1 ...5 1"".6 .01105 1 ...1 IlIl
-89-
DISCUSSION AND CONCLUSIONS ON MATHEMATICAL MODELS OF MIGRATORY TIMING
Two fundamentally different methods of intraseason estimation of abundance are
evolving;the use of cumulative proportions averaged across defined days of
historic data,and t.he adoption of hypothetical models which assume the general
shape of the distribution of proportions over time.The first method was first
formally quantified by Walters and Buckingham (1975)who suggested that total
run size could be quantitatively anticipated by the r.:!tio of catch plus escape-
ment to date and the average cumulative proportion expected to have returned by
that date.The average cumulative proportion on a given calendar date was
defined as the arithmetic mean across years of the cumulative proportions re-
corded on that date.They concluded that the run size estimates were quite
unreliable until over half of the run had past.Variation in average daily
cumulative proportions can be reduced by standardizing the calendar date to day
of the run,where the first day of the run is defined as the maximum of the
second derivative of the normal equation defined by the mean and variance of
each year (Hornberger and Mathisen 1980;Brannian 1982).Although the ~posteriori
assessment of average cumulative daily proportions is refined,the difficulties
of objectively ascertaining the first day of the run during the season have yet
to be resolved.Another means of synchronizing the calendar dates of migration
is to objectively classify the years as early,average,or late according to mean
day of migration (Mundy 1982a).The expected cumulative proportion is the aver-
age cumulative proportion across a calendar date for a subset of years,determined
by spring temperatures.
The second technique developing as a method for intra season abundance estimation
aSSl,.mes the existence of a probability density function which acurrately reflects
the time distribution of proportions of total abundance for all years.Vaughan
(1954)first recommended that temporal distributions of proportion could be
modeled as probability distribution functions which best fit the data.For
Southeastern Alask.an pink salmon migrations,beta curves were chosen the major-
ity of times frem the family of Pearson probability densities.In a study of
Bristol Bay sockeye migrations.a two-parameter logistic function which asymptot-
ically approaches one was chosen as an appropriate mcjel of cumulative proportions
over time (Mundy and Mathisen 1981).The parameters of the equation were esti-
mated intraseason by least-squares fitting to early daily abundance estimates.
Accurate estimates of total abundance were obtained well before the median day
of migration in one of the two years.In contrast,Hornberger and Mathisen
(1980)compared intraseason prediction performance of a normal curve function
fitted to the average abundance distribution with the curve of the historicai
data average.They concluded that the historical average was the preferred means
of intraseason prediction.
The results of the present study demonstrate that variations in the temporal
distribution of abundance can be approximated by fitted probability density
functions or by fitted analytical models;both of which employ equations used
by previously-cited authors to express the daily proportion of total abundance
as a function of time,environmental factors.and fishing effort.Errors of
daily proportions estimated by empirical time series models are comparable to
errors of daily proportions predicted by analytical models whose parameters are
estimated by a function of average April temperatures or as an average of pre-
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iously determined parameter values.However,empirical methods re:ndin ~uperior
to hypothetical models in their ability to forecast total abundance .
One advantage of the analytical approach is that the underlying mechanics of
migratory behavior can be more easily incorporated into existing models without
radically altering the model or the procedure for estimation of parameter vdlues.
The bell-shaped normal curve function is the nucleus of the models presented in
the present study.The use of the normal curve is justified by virtue of tne
genetic nature of migratory timing (Hundy 1979);and the parameters of the nor-
n~l curve function,mean and variance,are more readily related to the migratory
time distribution than parameters of other bell-shaped curves.The assumption
that catch is linearly dependent on wind and co~ercial effort is not contra-
dicted by the results,but more directed studies may uncover a more complex nor-
linear relationship.The ultimate objective of developing refinements in ~he
quantitative description of migration should be a more complete conceptual modE 1
which explains the variation in migratory behavior as a function of intrinslC
behavior of each major stock and the environmental parameters which directly
affect the physiology of the animal.
A second advantage derives from the reduced number of parameters to be estimdtld
in the analytical model.In the present study of chinook salmon migration,thl
problem of intraseason abundance forecasts is now reduced to the accurate esti·
nlation of the parameter values which characterize the analytical model.Mean {ay
and variance of the Yukon River commercial catch have already been shOHn to be
highly dependent on spring temperatures (Mundy 19B2b).A similar relationship
has been reported for Bristol Bay sockeye timin9 and Hay air temperatures (Burf-
ner 1979).The mean and variance of the Yukon River test fishery catch is sh~n
to be highly dependent on spring temperatures.Accurate forecasts of the c~n
stants which determine the distribution of predicted catches over time are esse~
tial for accurate catch estimates.
The results of the present study demonstrate that significant improvements in
the ability to estimate daily proportions of total catch are achieved with the
incorporation of ~ind speed and commercial effort ~asurements into the models.
High wind speeds are often accompanied by larger-than-expected test fishery
catches,intensive commercial effort reduces the test fishery catch.and the
effects of both factors can be quantified.The time density of the migration
is not readily and consistently divisible into two distinct migrations.Stud-
ies in progress confirm that the migrations of major upriver J"d downriver stock;
of chinook salmon are not temporally separated at the Yukon Riv~~delta based on
scale measurements (McBride and Wilcox.pers.camm .•pers.observ.).The inabil··
ity to morpho10Qica11y separate early and late migrants and the less accurate
daily catch estimates of two population models presently restrict the class of
analytical models describing the migratory timing of Yukon River chinook salmon
to those models assuming only one population.However,fisheries of proven l~ix
ture of two or more populations (e.g .•lynn Canal in Southeastern Alaska)would
be appropriate for such an analysis.length frequencies of a mixture of age
classes are routinely described by Hasselb1ad's (1966)iteration equations
(Mac~nald and Pitcher 1979).
As knowledge of migratory behavior continues to accumulate.quantitative des-
criptions will become more conceptual in nature.Yet interim analysis should
not be restricted to the options of average proportional expectation of fitting
-90-
•
nonlinear function to'observed data.Statistical techniques of multiple regre~
sian analysis and time series analysis are relevant.Multivariate methods are
regularly employed in quantifying bird migrations (e.g.,Beason 1978;Blokpod
1978~and review by Richardson 1978).Time series analysis,utilizing the pop-
ular Box Jenkins Models (Box and Jenkins 1976)hav~been applied to yearly catch
data of Atlantic Menhaden (Je,sen 1976)and lobster landings (Boudreault et a1.
1977)and monthly catch data of the New Zealand Rock Lobster (5a;1a et a1.1978)
and the skipjack tuna fishery (Mendelssohn 1981).However,the danger of extra-
polating false relationships ;s inherent in all statistically·based empirical
models.and conceptual models should take precedence if acceptable accuracy is
attainable.
Claims of the ability to foretell the abundance of a sa1monid resource probably
are as old as exploitation of the resource itself.Native Yukon Delta fishermen
gage the magnitude of daily abundance by the speed and direction of wind at the
river mouth.The track record of some old-timers is impressive.but the causal
relation between wind and migratory behavior remains elusive.even though pre-
sently approximated Lya linear function.Several authors have noted the diffi-
culty of trying to statistically quantify the effects of atmospheric and water
conditions on animal migrations.Banks (1969)comments that although there is
an obvious correlation between stream flow and incoming Atlantic salmon migrants.
it cannot be predicted accurately or reduced to a mathematical formula.Ellis
(1962)reported that although correlations between river entry of sockeye salmon
and six environmental measurements and changes of intensity of these measurements
were not significant.upriver movement could be qualitatively predicted by observ-
ing changes from sunny weather to cloudy weather.Onshore wind speed was found
to be more highly correlated with number of kokanee entering their spawning stream
when the tails of the run were deleted (lorz and Northcote 1965).
The relationship between wind speed and migratory behavior is elusive.lorz and
North~ote (1965)suggested that wind affects the spatial distribution of home
stream water.broadcasting the chemical identity of the stream along a greater
area of the shoreline.Reduction of light p~netration was secondarily implicated.
Barber (1979)sppculated that some species of salmon may navigate bv langmuir
circulations in the open ocean.Hornberger and Mathisen (1981)noted that chin-
ook salmon in Nushagak Bay are more susceptible to catch on windy days.
The present level of understanding limits the c1a5sification of analytical models
developed in the present study and fitted to the lower Yukon test fishery data
to an empirical set of relationships.The bridge from proxy data to the casual
factor is either tentatively established (day or run.temperature and photoperiod.
the negative commercial effort parameter and removal of chinook salmon by commer-
cial fleet)or completely unknown (wind speed).To suggest that the proposed
models represent conceptual or mechanistic models would be misleading (Austin
and Ingham 1978).To some extent they approximate the true model.As noted by
Brannian (1982).an empirical model many times is the predecessor of the concep-
tual or mechanistic model.
Thl mechanistic approach often provides new insights into the underlying dynamics
dlnJ character of the migration.Figures 33-35 depict the relationship between
April temperatures and annual mean.variance.and wind coefficients calculated
by M].Chinook salmon returning to the Yukon River in years of warm springs
arrive at an earlier date (Figure 33).the time interval of migration is prolonged.
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------ -..- -- - --- -- --
32
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0
28 rO 0
H 0
E
~
N
0
~24 I-0TE
0
f 20~0
H
~0 0,
<D R
N ~,I I 00N
1&
0 0
00
0
0 0
12
9 13 17 21 25
APRIL MEAN AIR TEMPERATURE.of
Figure 33.Relationship of April mean air temperature to the mean date of migration for model three.
I
as demonstrated by larger variances (Figure 34).and are more susceptible to
the influence of wind (Figure 35).In contrast.the migration of chinook salmon
whose arrival is retarded by a cool spring is characterized by a smaller time
interval and reduced effect of wind.The physiological changes whi~h accompany
migratory behavior may become more insistent for fish arriving late.The para-
meter quantifying the rffect of commercial effort on relative abundance has
predictably become more negative in recent years (Figure 36).The increase in
~fficiency of the commercial fleet and its impact on chinook management is dis-
cussed by Mundy (19820).
In order to avoid overharvesting or underharvesting the temporal components of
a migratory populations managers ne~d reliable estimates of abunda~ce and time
distribution of the resource.Increasing the accuracy of the yield estimates
can result in substantial profit to the fishing community (Mathews 1967~Mundy
a1d Mathisen 1981).To accomplish this objective,control systems have been
~roposed which incorporate long-range abundance estimation with intraseason mod-
ification (Walters and 9uckingham 1965.Mundy and Mathisen 1981).Initially the
estimation process uses escapement counts and the historical recruit-per-spawner
relationship to forecast long range recruitment.Predicted yield is further
refined at observable life history stages (fry,smolt,marine immature)when
total yield is a function of the estimated abundance at the particular life
history stage.Error i~yield estimates tends to decrease for later life stages
(Mundy and Mathisen 1981).The most accurate prediction of total abundance is
expected to be the intraseason estimate which should deliver increasingly accur-
ate forecasts of abundance as the run develops.Development of intra season abun-
dance estimation techniques should be viewed in the context of a complete abundance
estimation system.
ACKNOWLEDGMENTS
Th~author would like to thank the Arctic-Yukon-Kusko~wim regional staff of the
Alaska Department of Fish and Game,whose insight into the dynamics of the fish-
ery was invaluable.The support and ideas of Ron Regnart,Mike Geiger.Bill
Arvey.and James Brady is particularly appreciated.The suggestions of Doug
McBride,John Wilcock,and Scott Marshall of the Stock Biology Group are also
gratefully acknowledged.A special thanks is due all the test fishery crews,
by whose effort daily catch data were obtained.I am particularly indebted to
Helen Hamner,whose enthusiasm and curiosity helped focus parts of this study.
The author was in ?art supported by a research grant (Contract No.Bl-334)from
the Alaska Department of Fish and Game for two years.
Bob Dawson and company at the Old Dominion University computer center were gen-
erous with advice on the various packaged programs.The night shift was also
particularly helpful restoring what the author's Cllr.ilsiness at the computer
would periodically wreck.
This study ~s an edited version of the author1s dissertation,accepted by Old
Dominion University in 1983 in partial fulfillment of the requirements for a
PhD in oceanography.Or.Ram Dahiya,Or.Chester Grosch,and Or.Phil Mundy.
members of the authors dissertation committees devoted much time and effort to
this study;their criticisms and suggestions weie appreciated.Or.Phil Mundy
-93-
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APRIL MEAN AIR TEMPERATURE.OF
Figure 34.Relationship of April mean air temperature to the variance of migration for model thre~.
1.00
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.75
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APRIL ~EAN AIR TE~PERATURE,of
Figure 35.Relationship of April mean air temperature to the influence of wind on migratory behavior for
model three.
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1967 1969 197\1973 1975 \977 1979 1981
YEAR
Figure 36.Relationship of year of test fishery catch to the effect of cQl1TT1ercial effort on test fishery
catch for model three.
chairman of the committee,was more than helpful in developing the concepts of
this study.The foundations of many of the ideas expressed in this dissertation
were introduced by Dr.Mundy in his 1979 dissertation.I also thank Ivan Frohne
for revising this report and pointing out some of the areas which were not clear.
Completion of the dissertation would have been impossible without the support
of many good friends and family who were always available when things went
wrong.This study is.to a large degree.a product of t~eir patience and help.
This brief ackn~ledgment does an injustice to the worth of their contributions.
-97-
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I
I,
I
I
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I
I
I
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