Open Journal of Marine Science, 2011, 1, 1-17
doi:10.4236/ojms.2011.11001 Published Online April 2011 (http://www.scirp.org/journal/ojms)
Copyright © 2011 SciRes. OJMS
Toward Estimating the Variance in Acoustic Surveys
Based on Sampling Design
Magnar Aksland
Department of Bi olo gy, University of Bergen
E-mail: magnar.aksland@bio.uib.no
Received March 8, 2011; revised March 19, 2011, accepted March 22, 2011
Abstract
This paper develops a sampling method to estimate the integral of a function of the area with a strategy to
cover the area with parallel lines of observation. This sampling strategy is special in that lines very close to
each other are selected much more seldom than under a uniformly random design for the positions of the
parallel lines. It is also special in that the positions of some of the lines are deterministic. Two different vari-
ance estimators are derived and investigated by sampling different man made signal functions. They show
different properties in that the estimator that estimate the biggest variance gives an error interval that, in
some situations, may be more than ten times the error interval computed from the other estimator. It is also
obvious that the second estimator underestimates the variance. The author has not succeeded to derive an
expression for the expectation of this estimator. This work is motivated towards finding the variance of
acoustic abundance estimates.
Keywords: Acoustic Abundance Estimation, Line Surveys, Sampling Design, Sampling in the Area,
Variance Estimation
1. Introduction
This paper is about to find the variance in acoustic sur-
veys to estimate the abundance of fish stocks. Here, usu-
ally a ship with a downward looking echo sounder that
pings sound pulses into the sea and receive sound reflec-
tions from fish, covers an area where the fish stock under
estimation is supposed to be located.
Estimating the variance of acoustic abundance esti-
mates of marine resources is an old problem, but up to
now not much ha s been done based on sampling design.
The author of this paper has worked with acoustic es-
timation of fish populations. Then it is natural that the
methods will be related to the observations of the acous-
tic signal generated by fish echoes, although such meth-
ods will always have the potential to be used in other
applications.
The echo signal received from a modern echo sounder
as a function of the area position of the echo sounder is
an example of a function of which the area integral is
necessary to estimate. This is explained in the next sec-
tion.
The method presented here represents the use of un-
equal probability sampling design, and can be used in
many sampling problems where a resource to be esti-
mated are distributed over an area.
Sampling design in the area or volume is a neglected
field in the sampling literature, but several applications
are demanding appropriated methods. This includes es-
timating resources that are distributed over the area or
volume. Such resources are sometimes observed con-
tinuous along lines, and sometimes at distinct positions.
Reference to such problems is found in Stevens and Ol-
sen (2004).
An important assumption for the present method is
that a quantity associated with each point in an area can
be measured continuously with a movable device. Then,
the quantity may be observed on a system of lines cov-
ering the area of interest.
Here, the unequal probability design has another pur-
pose than is usual. Instead of seeking inclusion prob-
abilities that are positively correlated with the quantity to
be observed, as for the Horvitz-Thompson estimator in
sampling design, we use a sampling design that has a
reduced probability to select object very close to each
other relative to a uniform probability design. The pre-
sent design thus produces samples that are better spatial
balanced than that produced by a uniformly random de-
2
M. AKSLAND
sign.
The present paper is an attempt to give a contribution
to the field of line sampling.
It is a hope that more specialists in sampling theory
will work in this field.
2. Review of Echo Integration in Terms of
Energy
This is a short review of a theory of echo-integration to
estimate the abundance of fish.
The traditional method of Echo Integration for esti-
mating the abundance of sound scatters (MacLennan,
1990) is based on th e integration of the echo-signa l. This
quantity represents sound energy, but the conversion of
integrator values to abundance, or density, of scatters is
based on “back scattering cross-section”, which repre-
sents sound intensity, or power.
In 1982 R. E. Craig proposed to rewrite sonar theory
that is based on power, to a theory based on energy
(Craig, 1983). Before this proposal, however, a method
of echo integration that bases the conversion to scatter
abundance on the energy of single target echoes were in
use (Aksland, 1983), although the theory of the method
was published in 1986 (Aksland, 1986), and published
when the split beam and dual beam system were in regu-
lar use (Aksland, 2005, Aksland, 2006). This alternative
method of echo integration does not use the concept
“back scattering cross-section” or “Target Strength”. The
method is in fact based on much fewer concepts than the
traditional echo integration method. The basis of the
method is given below.
Let
 
21 2
;,
I
zt z tz
t
be the depth integrated 20
Log r TVG (Time Varied Gain) echo signal (echo inten-
sity) between times 1 and 2 after sound transmission.
The parameter r is the distance from the transducer. In
general, the times
t
2,1,
ii
trci
,y2 depend on the
transducer position . Here c is the sou nd speed .
The downward looking transducer is free to be moved
within a horizontal plane with Cartesian coordinates
zx

,
x
y, here called the transducer plan. Let Q be a given
region of the transducer plan.
2()d
Q
EAI z A (1)
where d
A
is the area differential, is defined as the Echo
Abundance within Q subject to the depth chan-
nel.
In applications, the dept channel may
also consist of several disjoint time intervals. This may
be necessary when integrating echoes from a special type
of scatters while excluding others.

11
,tz

tz
Let

2p
I
z be the integrated echo pulse intensity
from a single scatter at 20 log r TVG, as it is when the
echo is resolved.
2d
p
Q
EVCI A (2)
where the integral is over an area where the integrand is
significant is called the Echo Value Constant of the cor-
responding scatter. As for the Echo Abundance, this in-
tegral is also over the different transducer positions in the
transducer plane.
By transforming the Echo Value Constant (2) to polar
coordinates
,
in the object reference system (a
polar reference system with origin in the scatter and
0
pointing vertically upward), and using that
2
dtanddAr

, it may be expressed as
 
2
00
,d tand
c
tr
EVC b
 
 (3)
where
tr
b
is the transmit-receive beam function
(here circular symmetric), c
is an angle where the in-
tegrand outside c
is negligible, and
2
,(,)
ptr
Ib
 
is called the back-scat-
tering energy of the scatter. The back-scattering energy is
the same as the beam compensated integrated 40 log r
TVG echo pulse received from the scatter when the
transducer is in direction
,
in the object reference
system. A necessary condition for this is that the ratio
between the 40 log r TVG and 20 log r TVG functions
satisfies exactly the relation
2
22
40log TVG
20log TVG4
rc
rt
r .
Note that the Echo Value Constant of a scatter de-
pends on both the level of the echo signal and the pulse
length used, as well as on the beam function. For a fixed
acoustic system it is a quantity that varies with the time
since movement and other kinds of behaviour of the
scatters will affect their Echo Value Constants.
If c
in (3) is replaced with a variable angle
, the
corresponding function is called the Echo Value, that is,
 
2
00
,d tand
tr
EV b

 
 (4)
If
increases from zero, it turns out that the Echo
Value increases at first, but flattens out when
ap-
proaches the outer part of the main lobe. When
is
equal to or bigger than the angle 20
where the trans-
mit-receive beam has fallen 20 dB, the Echo Value is
approximately constant due to the strong beam damping.
It is the value of this flat region that is called the Echo
Value Constant, and this property justifies its name.
It can be proved that both the Echo Abundance and the
Echo Value Constants of single scatters are independent
of the distance between the transducer and scatters when
Copyright © 2011 SciRes. OJMS
M. AKSLAND
3
2
I
does not include noise. These properties are due to
the 20 log r TVG.
Let
E
A
EVC be the average, or mean Echo Value Con-
stant over the scatters that contribute to the Echo Abun-
dance (EA). It can be proved (Aksland, 1986) that
1
N
E
A
i
i
EAEVC NEVC

(5)
where N is the number of scatters and i is the Echo
Value Constant of the i-th scatter that contributes to the
Echo Abundance. Relation (5) is true if the depth integral
of overlapping echoes is the same as the sum of the indi-
vidual integrated resolved echo pulses over all scatters. A
necessary condition for this is that the phases of the ech-
oes from each single scatter are uncorrelated, and that
shadow effects are negligible (Zhao and Ona, 2003).
EVC
By writing (5) as
E
A
EA
NEVC
we see that the mean Echo Value Constant correspond-
ing to an Echo Abundance is a constant that converts the
Echo Abundance to the nu mber of scatters.
The mean Echo Value Constant may be estimated
from integrated single target echoes at 40 log r TVG and
corresponding detection angles provided that the echoes
are representative for all echoes that contributes to the
Echo Abundance. The detection angle of an echo is the
angle between the beam axis and the direction between
the transducer and the scatter, and can be detected within
the main lobe with split beam and dual beam echo
sounder systems. See Aksland, (2005), (2006) and (20 10)
for details.
3. Estimating the Echo Abundance
The Echo Abundance is the area integral of the echo in-
tensity as a function of the transducer position over an
area. The alternative echo integrator-method defines and
uses the Echo Abundance, so the sampling methods to be
developed here refer to this method. A review of the the-
ory behind the alternative echo integrator-method is
given in Section 2.
Short reviews of other methods to estimate the Echo
Abundance are given below.
3.1. Review of Estimating the Echo Abundance
A non statistical way of estimating the Echo Abundance
within a given sea area Q is to observe the echo intensi-
ties
 
21 2
;,
I
zt z tz , (see (1)), on a system of lines
covering Q. Next fit some parametric class of surfaces
over Q to the observed values of 2
I
. Then, the volume
under the fitted surface within Q will be an estimator of
the Echo Abundance. An example of th is way to estimate
the Echo Abundance is given in Aksland (1983).
This method does not estimate the precision of the es-
timated Echo Abundance.
The method can, nevertheless be recommended if the
precision of the abundance estimate is not very important,
or if it is believed that the precision of the Echo Abun-
dance is small compared to other factors of uncertainty
affecting the abundance estimate.
To estimate the Echo Abundance with precision is a
challenge. Since acoustic data are observed along lines,
application of probabilistic survey design methods are
not appropriate except for special cases where each strata
are covered with parallel lines of observations. This case
is equivalent with sampling points on a line, where the
integrated echo intensities are projected onto a line or-
thogonal to the parallel lines. The parallel lines are then
projected to points of observation onto the orthogonal
line.
This restriction of probability sampling methods may
be reduced through a generalization of the foundation of
probability sampling theory. Then randomization of
other types of lines covering an area may be used. In
acoustic abundance estimation, coverings with zigzag
lines are common.
Estimation of the Echo Abundance belongs to the
more general problem of estimating an integral
()d
Zfz z
(6)
where Z is a region of the Euclidean space, and
f
z
is observed on a subset of Z with dimension commonly
less than the dimension of Z. In acoustic abundance es-
timation Z is of dimension 2, while the set observed has
dimension 1 (a system of lines).
Foote and Stefánson (1993) have described and dis-
cussed different methods for estimating fish abundance
over an area from line-transects. They recommend
kriging methods, but have only one reference to prob-
ability sampling (Cochran ).
There are mainly two classes of methods that are
available for estimating (6) together with an estimate of
the precision of the estimate. This is methods within
random field models (geostatisics, kriging) and probabil-
ity survey sampling methods, respectively.
3.2. Random Field Models
Here
f
z is considered as a realization of a random
process (field)
,
F
zzZ
.
Methods based on parametric models with stationary
increments are elaborated and known as “Geostatistics”,
(Matheron 1963). Estimators are based on predictors for
Copyright © 2011 SciRes. OJMS
4 M. AKSLAND
the values attained by the process outside the sample
(kriging), (Krige 1951). The sample may be selected
subjectively. The predictors have certain minimum vari-
ance properties for given model parameters (usually
trend and auto covariance functions), and these have to
be estimated in applications.
Objectives against the application of random field
models are that it is difficult to judge whether the models
are well related to the spatial distribution they are sup-
posed to describe. Also the bias caused by spatial distri-
butions that cannot adequately be taken to be a normal
realization of the used random field model is difficult to
judge.
However, the fact that these methods allow subjective
selected samples has lead many scientists to choose these
methods. See the book by Rivoirard et al. (2000) and
references therein.
3.3. Probability Survey Methods
The methods within probability survey sampling are
mainly concerned with the estimation of the mean
1
1N
j
j
y
N
y
or the total Nyof an unknown finite
vector
12
,,,
N
Yyy y The foundation given below
follows Cassel, Särndal and Wretman (1977).
The estimation is based on a sample from the set of
labels
1, 2,,
J
N, where J must be known.
A sampling design, , is a probability measure on
the set of all subsets of J. Selection of a sample is
done in accordance with the selected sampling design.

Ps
The data associated with a sample s is denoted by
,
s
DsY, where
s
Y is the restriction of Y to s. In
general, an estimator is a real function ,

tD s
.
The combination

,tDPs
is called a strategy.
The art of probability sampling is to make use of all
known information about Y to construct a strategy that
will give the most precise and accurate estimates for a
given sampling budget.
An estimator that depends only on the values of Y on
the selected sample and is otherwise independent on the
labels of the sample is denoted label-independent. Most
of the traditional and well-known estimators in probab il-
ity sampling are label-independen t.
A sampling design is non-informative if
Ps is in-
dependent of values of Y. Otherwise the design is infor-
mative.
Many methods within probabilit y sampling may easily
be generalized to sampling from infinite populations.
Sampling from Euclidean space, and from an area in
particular, are good examples, where the infinite set of
points are the sampling objects. If the population from
which samples are selected are a subset of a Euclidean
space, each object has a position, and there is a unique
distance between each pair of points. Unfortunately, such
quantities associated with sample objects are very sel-
dom and poorly treated in the probability sampling the-
ory. In particular, when sampling a subset of the Euclid-
ean space successively to estimate an integral, estimators
that are label-dependent and sampling designs that are
informative are likely to be better than the traditional
estimators in probability sampling.
Fortunately Thompson (1990) and Thompson and Se-
ber (1996) has worked with adaptive sampling methods
that has informative sampling designs that give better
precision than conventional estimates of populations
having aggregation tendencies in their area distribution.
These estimation methods have been used in biological
sampling as well as in acoustic surveys (Harbitz, Ona
and Pennington, 2009), (Conners and Schwager, 2002),
(McQuinn et al., 2005).
When sampling values of some function defined on a
given subset of a Euclidean space, the observed values
give information about the spatial structure of the func-
tion values. Information about this structure should be
used when selecting the rest of the sample, as well as in
constructing a good estimator. When sampling in the
area, estimators that are the volume under some fitted
surface to the observed data are likely to have good
properties relative to other, more traditional, estimators
in the sampling literature. The adapted sampling design
methods mentioned above have not been developed far
enough to be appropriate for all situations when estimat-
ing an integral over Euclidean space.
The case of spatial label sets impose some general
demands on the sampling design.
a) The design should reduce the selection of “spatial
unbalanced” samples (7)
b) For successively selected samples, the design
should be informative
This indicates the need for developing new methods
within the field of probability sampling. Bertil Matern
(1969) has pointed out this and other problems related to
the application of probability sampling methods, but not
much has happened with the foundation of sampling de-
sign since then.
Case a) in (7) above means that samples that do not
cover the space properly, or are too patchy, are “spatial
unbalanced”. A way to avoid th is is to stratify the sample
population into n strata, where 2n is the sample size, and
chose a probability sample of size 2 in each stratum.
When estimating the abundance of animals with aggre-
gation tendencies, wh ich are rath er usual, there is another
advantage of stratifying the sampling population as fine
as possible. The population variance of small strata are
likely to be smaller than the population variance of big-
Copyright © 2011 SciRes. OJMS
M. AKSLAND
Copyright © 2011 SciRes. OJMS
5
ger strata. Then, under the assumption of equal total
sample sizes, the use of many small strata is likely to
give more precise estimates than an estimator based on
fewer and bigger strata. See Des Raj (1968), ch. 4.
lines be l
x
and 1h
x
, respectively, where 1
lh
xx
,
and the values observed are l and h. The strata
width is set equal to 1 since this will not represent any
loss of generality. The function
y y

y
x is then observed
at the values 0, l
x
, h
1
x
and 1 for
x
. In the follow-
ing,
y
x will be called the signal function.
Case b) in (7) holds because when starting to collect a
successive sample, information on the spatial structure of
the population variable is gained. To increase the preci-
sion, this knowledge should be used when selecting the
sampling design for the positio n of the next observation.
We choose the following sampling strategy for the
stratum:
The estimator is given by

01
1111
2lhhll h
Txyxyxyxxyx
4. A Strategy for Parallel Line Survey
 
(8)
Assume that we know approximately the area where a
pelagic resource is located. This resource can be ob-
served with a downward looking transducer that is
moved within the actual area. Cover first the area along
parallel lines that runs completely through the locations
occupied by the resource so that sailing between the dif-
ferent lines are outside th is region. The first covering has
the role of a pilot survey that is selected subjectively.
This is the same as the area under the step function
shown in Figure 2, and defines a label-dependent esti-
mator.
The sampling design is given by the following prob-
ability density:

,120 1
for0, 0,1
lhlhl h
lhlh
f
xxxxx x
xxxx


(9)
Next return and go two parallel legs with random lo-
cations between each neighbour lines of the pilot survey.
The randomized legs should also go completely through
the resource. Fi gu re 1 illustrates this survey.
A plot of this density in terms of x for l
x
and y for
1h
x
is shown in Figure 3.
It is seen that this sampling design reduces the likely-
hood considerable that the lines will be selected close
together, or close to some strata boundary relative to a
uniform sampling design, where the probability density
is constant for all
x
and . y
In this survey, the pilot survey, wh ich is deterministic,
defines strata boundaries. Two random lines are selected
within each stratum, stochastically independent of the
selection within other stratums. To reduce the likelihood
of selecting lines very close to each other, or to close to
one or both lines of the pilot survey, the position s for the
lines are selected with unequal probabilities.
This sampling strategy fulfils to a certain degree de-
mand (7a) for a strategy for estimating an integral over a
subset of the Euclidean space. The estimator (8) repre-
sents the area under a simpl e int erpol ati on t o t he obser ved
In a strata let the positions of the two selected rando m
Figure 1. The pilot survey: thin lines. The probability survey: thick lines. The isolines illustrate the distribution of the echo
intensity that is not known during observation. The whole survey is projected onto the line perpendicular to the parallel
lines.
M. AKSLAND
Copyright © 2011 SciRes. OJMS
6
Figure 2. Graphical illustration of the estimator (area un-
der step function). This is the same as the area under the
straight-line curve through the observed values.
Figure 3. The joint probability density for the positions of
the left (
x
) and right () random lines in a stratum. y
values while the sampling design reduces the likelihood
for selecting lines close together. However, the present
sampling design is not informative, as it is independent
of any observed values from the survey. Unfortunately,
this strategy does not make use of th e observed variation
along the parallel lines.
A generalization of this method where the sampling
design depends on the observations on the strata bounda-
ries, may appear in the future. The following are some
statistical properties of the present strategy, which are
derived in the Appendix I:
The random variables l
x
and h
x
have identical
marginal distributions with probability density

3
20 1, 01xx x
, and the conditional density of
one of the variables, , given the other,
z
x
, is

3
1
6,0
1
zxz z
x
 1
x

,. These distributions are used
when generating random values for l
x
and h
x
. Further





2
2
11
E,E ,E
37
1
1
E,E
28 2
ll lh
l
lh hl
xx xx
x
xx xx
 

2
,
21
(10)


 



1
01 0
1234
01 0
1
E5111
6
or 1
E()()522( )
6
Tyyxxxxyx
Tyy xxxxyxdx
 
 
dx
(11)
In general T is biased, but T is unbiased if
y
x is
linear. Moreover, when the graph of

y
x
1
is a straight
line over the strata, T is equal to for every

0dyx x
selected positions of l
x
and h
x
. This follows from
Figure 2, and implies further that Va for the
class of linear functions r 0T
y
xabx for all real con-
stants and .
ab
It is possible to specify a bigger class of functions
yx for which T is unbiased. However, T is in general
biased, and although most estimators given in the sam-
pling literature are unbiased, the lack of unbiasedness for
T is caused by the fact that T was selected without this in
mind. It is seen from Figure 2 that T has positive bias if
y
x is convex and negative bias if
y
x is concave
on
0,1 , because then
y
x will be below (above) the
straight-line function in Figure 2. However, T is sup-
posed to be summed over several strata, and it is the total
bias over all strata that are important here. This is likely
to be small because the individual strata biases will be
both positive and negative. An in depth theoretical
analysis of the bias of the sum of T over the strata seems
to be difficult. Some considerations are given in the
Discussion.
To be able to find a formula for , the ex-
pected value of is given.

Var T
2
T
 











1
222 234562
0101 0
11
23456 23456
01
00
1223
00
11
E34 351020186d
84 2
510522d 251015102d
601d d
z
Tyyyyxxxxxxyxx
y
xxxxxyxxyx xxxxxyxx
zzyzxxzxx yxxz

 
 


(12)
M. AKSLAND
7
Formula (12) may also be expressed with the polynomials on factorial form.
 






  


 
1
222 2342
0101 0
11
23
23
01
00
0, 0
1
11
E3 41328126d
84 2
151 2d1521d
601111d d
xz
xz
Tyyyyxxxxxxyx
yxxxxyxxy xxxxyxx
xzxzxzyxyzxz




 


x
mean
expression given here is obtained by squaring the last
(11) and subtracting it from the first ex-
2).
With (11) and (12), expressions for the variance and
square error of T may be derived. The variance expression of
pression of (1










1
22234 5 6
0
0101 0
1 1
23456 234562
1
0 0
2
1
Var( )525402066d
126 31
52040306d 351020186d
32
y
Ty
yyy xxxxxxyxx
y

1 1
22
3 234
00 0
60 (1)()()d d2522d
z
x
xxxxxyxxxxxxxxyxx
z

 

(13)
by computing it for te signal function
zz
yzxx zxxyxxxxxxyxx 

An estimator for

Var T may be obtained from (13)

h
y
x
ugh th
y
x
devel , a program that computes this estimator has been
oped.
This estimator cannot take negative values since it is
the variance of T fo r a gi ven sign a l function.
Another variance estimator is:
given by
the piecewise stnction throe observed
values (see Figure 2). However, it is not easy to evaluate
the expectation of this estimator. Since the integrals in
(13) are tractable for piecewise linear functions for
raight-line fu





 
2
1
ˆ
E111
l h
xxyxxy x
22
222 2
01 01
11
Var1311 1
126 4
1
hl lh
lh
Tyyyy xyxxyx
y
 
01
11
11
1
2323
1
11
2
lhhl
hll
yy
xx xyxxyx
xyxx





4
h hl
  
(14)
where 2
ˆ
Eis an estimator of

2
E11 1
hll h
x
yxx yx
not inrm that can be directly applied
. Formula (14) is
because an es-
timator for the last line has to be inserted. The formula is
derived in Appendix I.
Another reason why the proposed variance estimators
cannot be directly used is that it is derived for strata of
with 1. This was done to simplify mathematical deriva-
ut eneralized to the case with
rent strata widths.
a fo
tions, bit may easily be g
diffe
Assume that we have n strata with widths ,1,L,
i
x
in
.
Let (8) be denoted by i
T in strata number i. Then, i
T
is an estimator of 1

dyx x
0, where i
x
xx
, and
x
runs from 0 to i
x
over strata number i. Since the
Echo Abundance in strata number i is equal to

0d
i
xyxx
, we have that ii
x
T is an estimator of the
Echo Abundance in strata number i, while 1i
n
ii
x
T
corresponding estimator r the total Echo Abundance.

2
Var Var
ii ii
is the
fo
Since
x
Tx T
tiplied with 2
i
, (7) or an alternative
mul
x
, is an estimator for
Var ii
x
T
l strata to obtain an esti
, and
these mama-
tor for the var iance of
y be summed over al
1
n
ii
i
x
T
.
Note that in strata number i, lli
x
xx
, and
hhi
x
xx
, where
x
and ih
x
x
zed legs in the
are the absolute
positions of the two randomistrata.
Estimating thence mever, t
t
Many acoustic surveys are carried out in a way where the
Results from sampling a set of artificial functions are
given in Appendix II.
5. Discussion
Echo Abundaay also be useful in
the classic echo integrating method. Howhe pre-
sent methods are independent of what quantity that is
estimated.
5.1. Estimatinghe Echo Abundance
Copyright © 2011 SciRes. OJMS
M. AKSLAND
8
urce is done subjectively using
ns. Amng such decisions there
ca
sa
part of the survey to decide upon
ring. When covering a resource sub-
never selected very close to each other,
ut when using sampling design, this may happen. Many
the sample. Moreover, to be able to estimate
e variance of the used estimator, every pair of objects
ected. This is
estimator
o reduce the prob-
ab
covering of the reso
common sense decisioo
n be found quite wise ideas to generalize the theory of
mpling design. Although most coverings are done in
some systematic ways using the available knowledge of
how the actual resource distributes over the area, the
cruise leaders will always also want to use observation
data from the finished
the remaining cove
jectively, legs are
b
cruise leaders will consider close legs as a waist of
money. But within sampling design from a finite popula-
tion, every object must have a positive probability to be
selected in
th
should have a positive probability to be sel
strict requirement for the Horvitz-Thompsona
based on unequal probabilities given in RAJ (1968). This
is the reason why close legs should have a positive
probability to be selected when using sampling design to
select parallel lines. There are certain problems with un-
equal probability designs. See Tillé (1996), but these do
not affect the present methods.
There is another way to avoid the possible selection of
close legs. This is when the echo intensity as a function
of the area position is modelled stochastically. In Cassel,
Särndal and Wretman (1977) it is shown how super
population models reduce the importance of a sampling
design and may even allow estimation of variances from
subjectively selected samples. However, when variances
are estimated from subjective samples based on some
stochastic model for the area echo intensity, all prob-
abilities comes from this model, and the variance esti-
mates cannot be expected to be more reliable than the
underlying stochastic model.
An advantage with probability sampling methods to
estimate populations with difficult area distribution, is
that the estimators do not depend on some underlying
super population model, but on man made probabilities
expressed by the chosen sampling design. As long as
samples are selected in accordance with the sampling
design, the estimates are objective. Subjective knowl-
edge is used with probability sampling methods in vari-
ous ways, for instance when forming strata, and in gen-
eral when deciding upon the sampling design. Whenever
it is felt that an estimate based on probability sampling
design is biased, or are subject to other kinds of errors,
this feeling comes from knowledge that is not used when
deciding about the sampling strategy. All supplementary
knowledge should be used when deciding the sampling
strategy.
5.2. Miscellaneous
This paper shows that it is possible t
ility to select legs close to each other, but the probabil-
ity should not be zero. An estimator that is biased within
strata in general may bother someone, but since it is the
bias of the estimator summed over all strata that is of
importance, the bias within strata is not serious. The bias
for every signal function

y
x for which (8) can be
integrated can be computed analytically. If
2
yx x
,
which is convex, the bias is + 10.7%. When
2
1yx x
, the bias has the same absolute value, but
is negative. If the likelihood to observe signal functions
with biases that cancel each other is similar, then the bias
of the estimated sum over many strata will tend to zero
when the number of strata increases. However, this is not
likely to occur exactly. An example that may occur is a
function that
alue over
tima
is zero every except fo
a short distance by a fish
y clstrata
tor within the actual stmay ha
where
caused
ose to a
rata
nt fu
r a very high
school. If the
boundary, the
ve a consider-
v
school happens to sta
es
able negative bias. The opposite situation to this, where
the function has a high value everywhere except for a
short distance where it is zero, is very unlikely to occur.
However, to find out more about the bias of the estimator
(8) summed over strata, the best study would be to com-
pute by programming the true value as well as the ex-
pectation for a lot of differenctions

y
x defined
over many strata. The study in Appendix II th
light on this problem, but more signal functions and
rs are needed. However, the figures in Ap-
pendix II show that the bias usually shrinks when the
number of strata is increased from 5 to 10. Otherwise,
the figures in Appendix II indicate that the difference
between (13) and (14) is less when the sampling density
is small and when sampling signal functions with sharp
and big variations.
Some may have noted that the estimator i
T and 1i
T
rows some
strata numbe
corresponding to strata no i and i + 1 both contain the
observed value on the common strata line. But this does
not violate the stochastic independence of i
T and 1i
T
because observed values from the pilot survey (on strata
boundaries) are not stochastic.
5.3. The Estimators of Variance
The two estimators developed in this papy not be
the best. The estimator based on (13) is obviously too
small. This is indicated in the Appendix II. Estimator (14)
is almost unbiased in theory, but Appendix II indicates
that it is too big.
The interval, Estimate

SD T, based on (14), con-
tains the true value in almost all cases in Appendix II, as
well as in other cases. However, the square root of an
unbiased estimator of the variance of an estimator has an
er ma
expectation that is less than the expectation of the Stan-
Copyright © 2011 SciRes. OJMS
M. AKSLAND
9
the estimator. Therefore, if dard Deviation of
Var T
is unbiased, Estimate

Var T, should nt be gra
than the Estimate

SD T in average.
If the estimator based on (13) has much less expecta-
tion than

Var T, it is dangerous to increase it with a
constant factor, because the right factor may vary with
the shape of the signal function
o
, and as well on the sam-
why this estimator
in all strata where
dance generated by a fish population is
th This me
stimator
uce estima
ber of s. It is not sure
th ignal fu
ter
pling density. One possible reason
estimates a too small variance, is that
the observed numbers 0
y, l
y, h
y and 1
y all happen
to have values close to some straight line, the estimated
variance in these strata are close to zero.
A difference between the used man made signal func-
tions in the Appendix II and real signal functions based
on the Echo Abun
at the latter is not static. ans that observed val-
ues are hardly equal if they are observed twice at differ-
ent times. As the man made signal functions used are
continuous with continuous derivative, a function like
this will converge to linear within strata when the strata
widths goes to zero. Therefore, the variance e
based on (13) may prodtes that tend too fast to
zero when the numtrata increases
at the estimator has this property if it used on real
non-static snctions.
Another way to find an estimator for
Var T is to
express it as a multiple polynomial
01
Var( )ijmn
ijmnl h
Tcyyyy
where ijmn
c are constants, or polynomials in l
x
and
h
x
. The problem is to find coefficients ijmn
c so that



EVar VarTT. This is a generalization of the
method given in Des Raj (1968) to find an unbiased
variance estimator when sampling with unequal prob-
abilities. The present author has not yet succeeded to find
an estimator by this method.
5.4. Sampling with Unequal Probabilities
In the literature about sampling design, interesting results
about sampling with unequal probabilities have been
derived (see Des Raj (1968)). It may be tempting to try
the Horvitz-Thompson estimator and variance estimator
here, but in the present case
this estimator is not appro-
prt make
ndaries.
Also, there are different reasons to apply unequal
probabilities in the present case and iHorvitz-
son’s n
spread the sample bet
random sampling stra
The general reason for selecting unequal probabilities
ties can
e chosen that are positive correlated with the variable to
pefully, the
re
is important as a special
riance of an estimated integral based
iate. This is because the estimator does no use of
the observed values on the strata bou
n the
Thomp case. The reason for choosing a not uiform
random sampling strategy when sampling an area is to
ter than is obtained with a uniform
tegy. Living resources’ have social
behaviour, and then it is believed that the echo intensities
at positions close to each other are seldom very different.
This is the reason to spread the sample in the present
case.
in the Horvitz-Thompson’s case is that probabili
b
be observed.
5.5. The Future
The author of this paper does not look upon the present
results as a finite solution. It is more a start of using
sampling design to estimate the variance. Ho
sults of this paper build on some principles that are
new in sampling design. The combination of a subjective
and a randomized covering where the estimator is label
dependent and depends on both deterministic and ran-
domized observations is not common. It is a hope that
generalized methods building on this principle can be
developed. A real challenge is to combine the common
subjective coverings with additional randomized obser-
vations for estimating the variance.
Use of adaptive sampling strategies in sampling design
is difficult. But if sampling designs can be based on the
non-random observations from a deterministic part of the
survey, the same variance reductions may be obtained
with less statistical difficulties.
6. Conclusions
A special sampling strategy is proposed for covering an
area with parallel lines of observation. The strategy con-
sists of a deterministic covering followed by a random-
ized covering between the deterministic covering.
A label dependent estimator is proposed that depends
on both deterministic and randomized observations. The
sampling design is with unequal probabilities with the
purpose to produce better spatial balanced samples.
The theoretical Expectation and Variance of this esti-
mator are derived, and two estimators of the variance
have been found. Further properties of the estimators of
variance are studied by sampling man made functions.
This study showed that one of the estimators is likely to
underestimate the variance.
The two variance estimators may not be the best for
the proposed sampling strategy. However, if the pro-
posed strategy is generalized and based on similar prin-
ciples, the results in this paper
case.
stimating the vaE
on a line sample requires a generalization of the founda-
tion of sampling design. This is a big job.
Copyright © 2011 SciRes. OJMS
M. AKSLAND
Copyright © 2011 SciRes. OJMS
10
d
[3]
na E. Estimation and compensation models
for the shadowing effect in dense fish aggregations. ICES
ournal
theron G. Principles of geostatistics. Economic
.
Osney Mead, Oxford, 2000, 206 pp.
dn. John Wiley and
r, G. A. F. Adaptive Sampling.
., Simard Y., Stroud W. F., Beaulieu J. L.,
ic cod (Gadus morhua) with es-
Y. McGraw-Hill Publish-
nomie et de
7. References
[1] Stevens D. L. and Olsen A. R. Spatially BalanceSam-[13] Rivoirard J. Simmonds K. G., Foote P., Fernandes and N.
Bez. Geostatistics for estimating fish stock abundance
Blackwell Science,
pling of Natural Resources. Journal of the Americal Sta-
tistical Association 99, No. 465, 2004, 262-278.
[2] D. N. MacLennan. Acoustical measurement of fish
abundance. J. Acoust. Soc. Am.; vol. 87(1): 1990, 1-15.
Craig R. E. Re-definition of sonar theory in terms of en-
ergy. In: Selected papers of the ICES/FAO Symposium
on fisheries acoustics. FAO Fish. Rep; (300), 1983, 331
p.
[4] Aksland M. Acoustic abundance estimation of the
spawning component of the local herring stock in Lin-
daaspollene, western Norway. FiskDir. Skr.Ser. HavUn-
ders. 1983, 297-334.
[5] Aksland M. Estimating numbers of pelagic fish by echo
integration. J. Cons. int. Explor. Mer. 43, 1986, 7-25.
[6] Aksland M. An alternative echo-integrating method.
ICES Journal of Marine Science 62, 2005, 226-235.
[7] Aksland M. Applying an alternative method of
echo-integration. ICES Journal of Marine Science 63,
2006, 1438-1452.
[8] Zhao X. and O tima
Journal of Marine Science 60, 2003, 155-163.
[9] Aksland M. Analysing Estimators in the Alternative Echo
Integrator Method. The Open Ocean Engineering J
[20]
3, 2010, 116-128.
[10] Foote K. G., and Stefánson G. Definition of the problem ing C
of estimating fish abundance over an area from acoustic
line-transect measurements of density. ICES Journal of
Marine Science 50, 1993, 369-381.
[11] Ma Geol- Statistique No 44, 1996, 177-189.
ogy 58, 1963, 1246-1266.
[12] Krige D. G. A statistical approach to some mine valua-
tions and allied problems at the Witwatersrand. Master's
thesis of the University of Witwatersrand. 1951
[14] Cassel C. M., Särndal C. E. and Wretman J. H. Founda-
tions of Inference in Survey Sampling. John Wiley &
Sons, 1977.
[15] Thompson, S. K. Sampling, 2nd e
Sons, New York, 2002.
[16] Thompson, S. K., and Sebe
John Wiley and Sons, New York, 1996.
[17] Harbitz A., Ona E. and Pennington M. The use of an
adaptive acoustic-survey design to estimate the abun-
dance of highly skewed fish populations. ICES Journal of
Marine Science 66, 2009, 1-6.
[18] Conners M. E. and Schwager S. J. The use of adaptive
cluster sampling for hydroacoustic surveys. ICES Journal
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[19] McQuinn I. H
and Walsh S. J. An adaptive, integrated “acoustic-trawl”
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ompany LTD, 1968.
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without replacement. Annales D’Éco
M. AKSLAND
11
Appendix I
Derivation of Results
Let us write the probability distribution (9) as

,120 1,0,0,1fxzxzx z xzx z (15)
Where
x
and are variables for l
z
x
and h
x
, re-
spectively. Note that this distribution is symmetric in the
variables
x
and . This means that the marginal di-
stributions of z
l
x
and h
x
are identical, and


EE,0,
and EE
nm mn
lhl h
lh hl
xxxxmn
xx xx

0
The marginal density of l
x
is obtained by integrating
(15) with respect to .
z
 



11
00
12
0
3
,d 1201d
120 1d
20 1
xx
x
f
xz zxzxz z
x
xz zz
xx





Hence, the marginal density is given by
 
3
201, 01
g
xxx x  (16)
Now



13
2
0
12345
0
E201d
203 3d
1E
3
l
h
xxxx
x
xxx
x



x
(17)
and





 

 
11
01,01
11
11
00
11
112
00
23
11
0
13
1
0
E120 1dd
1dd
120 1d
11 1
120d (18)
23
120
E1d
23
nmn m
lh xz z
z
mn
z
mnn
nn
m
n
nm m
lh
xxxzx zxz
zxxzxz
zzxxx
zz z
zz
nn
xxzz z
nn

 







 











By using special cases of (18), the expectation formu-
las except the conditional in (10) are derived.
The conditional distribution of h
x
given l
x
is given
by






3
3
,1201
20 1
1
6,0
11
f
xzxzx z
hzx gx xx
zxz
z
x
x




(19)
The conditional exp ectation follows as:



12
03
61 d
1
2
1
x
hl
zxzz
x
Ex xxx

 
(20)
Below, the relation
 
11
00
d1d
f
xxf xx
 is used
some times. To show the relation, change 1
x
x in
the integral.
By using (8),









01
01
1
EEEE1
2
E1 1
(21)
6
1E1E1 1
2
lh hl
lh
hll h
Tyxyx xyx
xy x
yy
xyxxy x

 
 
Next evaluate the two expectations in the brackets by
using (9 ).
 
 
01
01
11
00
E1
12011d d
12011dd
hl
x
zx
x
xyx
zxzx zyxxz
x
yxzzx zzx






By integrating with respect to
x
, the following expres-
sion is obtained.



1245
0
E1102 2d
hl
x
yxxxxx yxx
(22)
Likewise,


 
 
01
01
11
00
E1 1
120111d d
120111d d
lh
xz
z
z
xy x
x
xzxzyzx y
zyzxxxzx z







By integrating with respect to
x
, and then substitute
1z
with
x
, dd
x
z
, we get.




1345
0
E111023d
lh
x
yx xxxyx 
x (23)
By combining (21), (22) and (23), an expression for
ETis obtained.
Copyright © 2011 SciRes. OJMS
12 M. AKSLAND

 



1
01 0
1234
01 0
1
E( )5111d
6(24)
1522 d
6
Tyy xxxxyxx
yyxx xxyxx
 



Formula (24) has been checked with the functions
for

2
2
,and1abxx x
y
x. These results are
identical with the results obtained by deriving
ETdi-
rectly from (8) and using (10).
Deriving formula (12) is a bit tedious. It follows from
(8) that



2
201
222 22
01
01
01
4 (1)()(1)(1)
()(1)()(1)(1)
2(1)(1)()(25)
2(1)(1)(1)
2(1)(1)()(1
lh hllh
lhh llh
lhhh l
ll hlh
hll h
Tyxyxxyxxyx
y
xyxxyxxy x
yxxyxx yx
yxxyxx yx
xxyxyx
 
 


 )
Then, the expectation of each term is derived by using
(9) and (10) and trying to split double integrals.



222
0101 01
1
E3
21
lh
yx yxyyyy 
4
(26)


 
 

22
22
0, 0
1
11 22
22
00
E1
12011d d
120111d d
hl
xz
xz
x
xyx
zxzxzyxxz
x
yxxz z zz zx






By evaluating the integral with respect to and con-
tracting, the following result are obtained: z




22
12562
0
E1
23553d
hl
xyx
x
xxxyx

x
(27)
Next



 

22
22
0, 0
1
11 22
22
00
E11
120111d d
1201111d d
lh
xz
xz
z
xy x
xxzxzyzxz
zyzzxxxxx z







Integrating with respect to
x
, contracting and chang-
ing the integrator variable with
z1
x
, gives



22
134562
0
E11
21020 13 3d
lh
xy x
x
xxxyx


Continuing

  
0, 0
1
11
22
00
E1
12011d d
120111d d
lhl
xz
xz
x
xxyx
xzxz xzyxxz
x
yxxzz zzzx










Evaluating and contracting gives


12356
0
E1
1022d
lhl
xxyx
x
xxxyx



x
(29)
The next term follows.
 
 

0, 1
11 23
00
E1
12011dd
1201(11dd
hhl
xzo
xz
x
xxyx
zzxzxzyxxz
x
yxx zzzzzx






Evaluating and compressing gives


12456
0
E1
22 510103d
hhl
xxyx
x
xxxxyx

x
(30)
Next term
 
 

2
0, 1
11 23
00
E1
120111d d
12011(11d d
llh
xzo
xz
x
xxyx
xxzxzyzxz
zyzzxxxxx z






Evaluating and compressing


1456
0
E11
2583 d
ll h
xxyx
x
xxyx


x
(31)
The next last term
 
 
0, 0
1
11
22
00
E1
120111d d
1201111d d
hlh
xz
xz
x
xxyx
zxxzxzyzxz
zyzzxx xxzx










x
(28) Integrating and changing with 1z
x
in the inte-
gral with respect to gives
z
Copyright © 2011 SciRes. OJMS
M. AKSLAND
Copyright © 2011 SciRes. OJMS
13




13456
0
E1 1
10 254d
hl h
xxyx
x
xxxyx


this were obtained by the control. The derivation of (11)
and (12) do not involve complicated mathematics, but
rather an extensive algebraic task with risk of making
errors.
x
(32)
And finally the last term


 
0, 0
1
E1 1133
1201111d d
lhl h
xz
xz
xxyxyx
x
zxzxzy x yzxz





This section is concluded with deriving (14). There
may be several ways to obtain an estimator for
Var T.
The version derived here is rather strai g h t f orward.
Since


2
2
Var EETT T
2
T, we try to estimate
both terms. As is an unbiased estimator of
2
ET,
we can use and subtract an estimator for
2
T

2
EETT

2
.
2
T
This formula could be used for the last term in (12).
However, it is possible to derive alternative versions.
One is given here.
 








11
00
1
0
12
0
1223
00
E111
1201111d d
120 11
11 1dd
1201d d
lhl h
z
z
z
xxyxyx
zzyz xx xzyxx
zzyz
zxxxxyx xz
zzyzzxxxx yxxz
 
 


 


is given by (25), but the expectation of the first
term is computable from the data.
Therefore is replaced by its expected
2
01
lh
yx yx
0
l
yx y
2
1
h
x
value


01 01
34
21 yy yy
22
1 in the expression for .
2
T
z
This will reduce the variability of the estimator. This is
confirmed by trying both versions during sampling arti-
ficial signal functions, and the version with the first
terms equal to
22
01 01
34yy yy21 is smallest on
all occasions.
An expression for
2
ET follows from (21).


 

 

2
201
01
2
1
E36
1E11 1
6
1E11 1
4
hll h
hll h
Tyy
yyxyxxy x
xyxxy x


 
Hence





1
0
223
0
E111
120 1
dd
lhl h
z
xxyxyx
zzyz
x
xzxxyxxz
 


(34)
Using the arguments in the expectation operators as
estimators for the expectations, and performing the sub-
traction
22
ˆ
ET
Inserting (26) – (33) or (34) i nt o ( 25 ) gi ves (12).
Formula (12) has been controlled with a linear func-
tion for

y
x. for linear functions, and

Var 0TT
the following estimator is ob-
tained.
ٛ






22
222 2
01 01
01
2
11
Var( )1311
126 4
11
111
2323
11
111E111
24
hl lh
lhhllh
hll hhll h
Tyyyy xyxxyx
yy
xx xyxxyx
xyxxyx xyxxyx



 




  
ٛ
This expression is identical with (14). The last line
cannot be estimated unbiased by using
, as this would yield an
over estimate of

2
111
hll h
xyxxy x


2
E1hl
1 1
l h
x
yx

22
Var EE
x yx
,
This follows from the general result,
X
XX, that holds for any random vari-
able
X
. However,
and it is believed that the variance term is small relative
to the expectation terms in general. Therefore, (14)
with


2
1111
4hl l h
xyxxy x as the last
line was used in Appendix II.



2
2
E11 1
E11 1
Var 111
hll h
hll h
hll h
xyxxy x
xyxxy x
x
yxx yx

 
 
Estimator (14) depends on and even if the
signal function is linear. This means that it takes values
that in general are different from zero even when the
variance is zero, as in the linear case. To try estimator
(14), a program that calculates it has been made. Results
from using (14) in the main paper are given in Appendix
II.
xlxh
14 M. AKSLAND
Estimator (14) depends on l
x
and h
x
even if the sig-
nal function is linear. This means that it takes values that
in general are different from zero even when the variance
is zero, as in the linear case. To try estimator (14), a pro-
gram that calculates it has been made. Results from using
(14) in the main paper are given in Appendix II.
In some difficult cases the variance estimator takes
negative values. This happens on some sampling occa-
sions with 5 strata, but also more seldom with 10 strata
on the signal functions a and b in Appendix II. The stan-
dard deviation is set to zero in such cases.
Appendix II
Sampling man made functions
The two variance formulas based on (13) and given by
(14) have been tried by sampling several artificial signal
functions. An immediate impression from these investi-
gations is that the two variance estimates show different
properties with the variance estimator based on (13) be-
ing the smallest.
The advantage with sampling known artificial signal
functions is that the parameter to be estimated can be
calculated as well as the expected value (11) and varia-
nce (13) of the estimator (8) used. From this, the bias and
deviation between estimated and true value follows. In
the present case, however, the variance (13) was not
computed. From repeated independent samples the vari-
ability of the estimate is shown, and th is indicates some-
thing about the th eoretical variability of the estimator.
Results from 30 repeated independent samples of four
different signal functions are given. Each signal function
is sampled with 5 and 10 strata for comparison.
The signal functions were constructed by means of the
10 first terms in a Fourier series and added some higher
frequencies. To ensure that the signal functions are
non-negative, they were put equal to zero if they were
negative. There are many parameters in such functions,
and it is possible to make any function shape. However,
since real signal functions are integrated echo intensities
over depth and along a fixed direction in the area, they
will seldom show very sharp variations.
The strata widths are constant and equal to one for 5
strata, and half width for 10 strata. Then the integral over
the signal function is the same for 5 and 10 strata.
However, when fish concentrate to schools, real signal
functions may be difficult to sample, in particular when
the area of distribution is small. The signal functions a
and b shown in Figure 4 and 6 are of this kind.
The results of 30 repeated independent samplings are
shown in Figure 5. Note that the y-axis is similar in
these plots.
It is seen from Figure 5 that the error interval based
on (14) seems not to be wider than the error interval
Figure 4. The first signal function (a) sampled with 5 and 10
strata, respectively. On the first sampling occasion, the
function was observed at the locations where there are ver-
tical lines, including the grid.
Figure 5. Estimate, true value, expected value and estimates
of two error bounds for 30 independent samples of signal
function a based on sampling with 5 (upper) and 10 strata
(low er graph).
Copyright © 2011 SciRes. OJMS
M. AKSLAND
15
based on (13). It is also seen that the error interval based
on (14) has zero width several places, especially for 5
strata.
For each signal function that was sampled 30 times, it
was counted how many times the error intervals con-
tained the true value, and how many times they con-
tained the expected value of the estimator. For signal
function a, these numbers are given in Table 1.
The next signal function (b) is shown in Figure 6.
This is also a difficult function to estimate.
The result of 30 independent samples of signal func-
tion b is shown in Figure 7.
Here is seen that the two error interv als seems to be of
the same order for 5 strata, but for 10 strata the error in-
terval based on (14), main paper, is the biggest.
The numbers of times the error intervals contain the
true value and the expected value of the estimator are
given in Table 2. See Table 1 for more information.
The next signal function is shown in Figure 8. This
Table 1: Numbers of times the error interval contained the
true value and expected value, respectively out of 30 inde-
pendent samples of signal function a.
5 strata 10 strata
Estimate ± 2SD based on (13) 13 13 24 21
Estimate ± SD based on (14) 11 7 17 18
Figure 6. Signal function b sampled with 5 and 10 strata,
respectively. The first sampling locations within each stra-
tum are shown as red vertical lines.
Figure 7. Estimate, true value, expected value and two es-
timates of error bound based on 30 independent samples of
signal function b based on sampling with 5 (upper) and 10
strata (lower graph).
Table 2: Numbers of times the error interval contained the
true value and expected value, respectively out of 30 inde-
pendent samples of signal function b.
5 strata 10 strata
Estimate ± 2SD based on (13) 13 13 24 25
Estimate ± SD based on (14) 15 17 30 30
Figure 8. Signal function c sampled with 5 and 10 strata,
respectively. The sampling positions in each stratum are
shown as vertical lines.
Copyright © 2011 SciRes. OJMS
M. AKSLAND
16
values reaches a maximum before
the area under signal func-
tio sed on (14) is the biggest
bo
one
of
gnal function (d) is shown in Figure 10.
Th
he area under signal func-
tio te that the error intervals
st
e error intervals for signal
fu
function start with low
it declines and goes to zero.
The results of estimating
n c is shown in Figure 9.
Here the error interval ba
th for 5 and 10 strata. This case is seldom in that the
bias for 10 strata is bigger that the bias for 5 strata.
The numbers of times the error intervals contain
two parameters are given in Table 3. See Table 1 for
explanation.
The last si
is is a function with small sharp variations, and is an
easy function to estimate.
The results of estimating t
n d are shown in Figure 11.
Figure 11 and others indica
abilize, gets narrower as well as less variable when the
functions are sampled with 10 strata compared with 5
strata. But it also seems that the error interval based on
(13) decreases faster than that based on (14) when the
sampling density increases.
The numbers of times th
nction d contain one of two parameters are given in
Table 4. See Table 1 for more explanation.
Figure 9. Estimate, true value, Expected value and two es
able 3. Numbers of times the error interval contained the
5 strata 10 strata
-
timates of the error bounds for 30 independent samples of
signal function c based on sampling with 5 (upper) and 10
strata (lower graph).
T
true value and expected value, respectively out of 30 inde-
pendent samples of signal function c.
Esti
Figure 10. Signal function d sampled with 5 and 10 strata,
respectively. The function is observed at the x-positions of
the vertical lines, including the grid at the first sampling
occasion.
Figure 11. Estimate, true value, Expected value and tw
es the error interval contained the
trata 10 strata
o
estimates of the error bounds for 30 indepe ndent samples of
signal function d based on sampling with 5 (upper) and 10
strata (lower graph).
able 4. Numbers of timT
true value and expected value, respectively out of 30 inde-
pendent samples of signal function d.
5 s
mate ± 2SD based on (13) 11 14 11 18
Estimate ± SD based on (14) 30 30 30 30 Estimate ± 2SD based on (13)
Estimate ± SD based on (14) 29 30 30 30
22 25 8 26
Copyright © 2011 SciRes. OJMS
M. AKSLAND
Copyright © 2011 SciRes. OJMS
17
signn d sat
t3) shrsider
sa
y biased. It is not
ea
by (14) has not good properties,
as
icult to es-
tim
The results from samplingal functiohows th
he estimator based on (1inks conable from
mpling with 5 strata to sampling with 10 strata. That
the unknown value is within the error interval in only 8
cases for 10 strata is not good, but the expected value is
contained in the error interval 26 of 30 times. Since the
expected value is rather close to the true value here, it
may be concluded that not many estimates are far from
the true value. But this may, nevertheless, indicate an
unfortunate property with th is estimator in that it may be
considerably to small if the sampling density is high
while the signal function is smoo th.
The Figures in Appendix II indicate that the variance
estimator based on (13) is negativel
sy, although not impossible, to derive the expectation
of this estimator. Therefore, the estimator is tried on arti-
ficial signal functions.
The estimator given
the estimates within strata may be negative. When all
within strata estimates are summed over several strata,
the resulting variance has better properties, but based on
the present figures it is hard to be lieve that this estimator
is nearly unbiased. Note that the error interval based on
(14) is calculated as Estimate ± estimated standard de-
viation, while that based on (13) is calculated as Estimate
± two times the estimated standard deviation.
Some of the chosen signal functions are diff
ate, in particular signal function a and b as shown in
the Figures 4 and 6. This may occur in fisheries acous-
tics when fish concentrate to schools. To improve preci-
sion in such cases, the sampling effort has to be in-
creased.