© 2004 International Council for the Exploration of the Sea
An alternative echo-integrating method
Department of Biology, University of Bergen PO Box 7800, N 5020 Bergen, Norway
*tel: +47 55584479; fax: +47 55584450. e-mail: magnar.aksland{at}bio.uib.no.
An alternative method of estimating the number of individuals in marine populations by echo integration is presented. The method was published almost two decades ago by the same author and here it is reviewed and developed. The method is based on just two defined concepts, the Echo Abundance and the mean Echo-Value Constant, respectively. The ratio of these quantities is equal to the number of scattering individuals within a covered area. As the Echo Abundance is estimated by interpolating echo-integrator values over selected areas, the mean Echo-Value Constant can be estimated from representative resolved echoes and their detection angles from individuals in the population, as obtained by the split-beam echosounder system. Two types of estimators for the mean Echo-Value Constant are given and discussed theoretically with respect to their properties. The approach is described and discussed in relation to the conventional theory of abundance estimation by echo integration.
Keywords: echo integration, fish-abundance estimation, fisheries acoustics, theory
Received 4 March 2004; accepted 22 July 2004.
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Introduction |
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 Top Introduction Popularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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In Aksland (1986) an alternative theory based on the author's concepts of acoustic-abundance estimation was given. As far as is known, this theory has never been adopted by others. Now, after almost 20 years of further development within fisheries acoustics, it is suggested that the main concepts in Aksland (1986) are the most natural and simple for the method of estimating the acoustic abundance of marine populations.
The conventional theory and concepts of echo integration are based, in large part, on the field of acoustics in the physics and engineering sciences. This has led to adoption of, as well as new definitions of, an inconsistent set of concepts, of which many are unnecessary for the estimation of marine populations by echosounders.
One purpose of this paper is to discuss Aksland's concepts in relation to conventional concepts, as for instance given in MacLennan et al. (2002). The second purpose is to develop Aksland's method in accordance with present knowledge, methodology, and equipment within fisheries acoustics.
In the theories described below it is assumed, unless otherwise stated, that marine populations for acoustic-abundance estimation are unmixed with other marine scatterers in the sense that the acoustic contribution from the target population can be separated sufficiently from the contributions from other scattering sources and noise. Although this assumption is often not fulfilled in applications, theories based on it are nevertheless valuable. Besides, the methods and equipment for separating the acoustic contribution to categories of scatterers, as well as noise reduction, have improved recently (Korneliussen, 2000; Korneliussen and Ona, 2002; Korneliussen and Ona, 2003), and are expected to be improved further in the future.
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Popularized review of the alternative model |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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Aksland (1986) models the received echo signal as a superposition of echo-pulse intensities from scattering objects and noise. This is not very original except that his theory is expressed in terms of relative echo-signal intensities. This is done for simplicity, and because absolute units are not needed to derive acoustic-abundance estimators. The method of echo integration is based on just two basic concepts.
One of these is the Echo Abundance and this is not clearly expressed in the conventional theory. First, the echo-signal intensity (with 20 log R Time Varied Gain, TVG) is, for each ping, integrated over a time, or depth interval, containing the echoes of interest. This quantity is an observable function of the horizontal position of the transducer and the time of the actual ping. It is denoted by I2(t1, t2; x, y) in Aksland (1986), where x, y represent the transducer position in a Cartesian coordinate system on the sea surface and t1, t2 the transmit time or depth window of integration. The time of sound transmission is suppressed.
In the conventional theory, several concepts, including the volume, area, and even length-backscattering strengths (MacLennan et al., 2002), are defined in terms of the echo-signal intensities. No such concepts are used in Aksland's theory. Instead, it is focused on considering the echo-signal intensity from one ping as a function of the transducer position in the area where the transducer is free to be moved. The interpretation of I2 is thus nothing else than a quantity of echo energy received at the transducer position in the area. The quantity is, of course, caused by scattering objects in the sound cone below the transducer, but it is nevertheless considered to be a function of one point at the position of the acoustic axis.
The Echo Abundance is defined simply as the area integral of I2 over a given region of the sea surface, and represents the Echo Abundance for this region subject to the depth window associated with I2. To exclude unwanted echoes, the depth window may vary with the transducer position along the track where I2 is observed, and may consist of more than one interval as well. When I2 is generated by all echoes from one target population only, the Echo Abundance is a parameter associated with that population within the area of integration. The Echo Abundance is not directly observable since it would require observation of I2 in each position (x, y) of the actual area. However, it can be estimated by some area interpolation of the multiple values of I2 obtained along the course lines of the ship cruising over the area of interest.
It appears that the conventional theory never uses a concept that is equivalent to the Echo Abundance. In particular, no central concept is used for the potential-backscattered acoustic energy within a sea area. Instead, the conventional method relates a version of I2 to the fish density per unit area below the transducer. In Aksland's theory, the concept of fish density is not used.
The other basic concept of the alternative theory is associated with individual sound scatterers, such as fish or plankton, and is called the Echo-Value Constant. It is defined in analogy with the Echo Abundance, but with I2 replaced by only the echo contribution from one given single scatterer. That is, the Echo-Value Constant is defined as the area integral of the integrated echo intensity from one given scatterer as a function of the transducer position in a horizontal plane above the scatterer. It is thus equal to the Echo Abundance if all except the given scatterer are removed, and represents this scatterer's contribution to the Echo Abundance. In Aksland (1986), Theorem 1, it is shown from the random-phase hypothesis and some usually fulfilled conditions that the Echo Abundance is equal to the sum of the Echo-Value Constants over all scatterers contributing to the Echo Abundance.
The Echo-Value Constants of individual scatterers are not directly observable, but the mean Echo-Value Constant over subpopulations of scatterers may be estimated from representative resolved echoes from the scatterers.
The idea behind Aksland's method is then to estimate the Echo Abundance of a target population as well as the mean Echo-Value Constant for the individuals in the population. The ratio between these quantities is then an estimate of the number of individuals in the population.
The principles of this method belongs to a general, simple principle for estimating animal abundance that may be formulated as follows:
Find a quantity associated with a population that is estimable, and that is the sum of the contributions from its individuals. If the total quantity as well as the mean contribution per individual can be estimated, an estimate of the number of individuals in the population is obtained.
All methods for estimating spawning stocks by means of egg surveys are based on this principle. Here, the total quantity is the number of eggs spawned by the stock while the mean number spawned per individual, or female, is estimated from the gonads in representative samples.
The mean Echo-Value Constant may be estimated from resolved echoes in the same echo signal used to estimate the Echo Abundance, or, if necessary, from a transducer lowered close enough to the individuals in the population to give representative resolved echoes. Both approaches might be used at the same time.
In Aksland (1986) great efforts were made to derive estimators for the mean Echo-Value Constant based on the use of a single-beam transducer. Today, multi-beam transducer systems, such as the dual beam and, in particular, the split-beam system, are in common use. It turns out that estimators for the mean Echo-Value Constant are readily derived in terms of resolved single scatter echoes and their detection angles. These are observed with multi-beam systems.
An estimator based on the use of the split-beam system is given in Aksland (1986), but this estimator is unfortunately not quite correct. The correct estimator is given in the present paper, as well as another new estimator.
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Properties of the alternative method compared with the conventional method of echo integration |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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As long as the Echo Abundance and the mean Echo-Value Constant are estimated by the same echosounder system, this need not be calibrated to give an echo signal in absolute units. The only requirements are a good signal-to-noise ratio, stable and otherwise proper performance, and an acceptable knowledge of the beam function.
Often, however, it is necessary to obtain representative resolved single echoes from a population by means of another transducer that can be lowered closer to the scatterers than their distance from a hull-mounted transducer. This is, in particular, necessary when the target population is very dense, such as for wintering, schooling herring, or when the target population resides at depth. In order to convert data from one echosounder system to another, i.e. the values the data would have if they were collected by the other system, a "calibration" is necessary.
Application of the conventional method of echo integration includes extensive calibration of the echosounder system in use to express both the output echo signal as well as in situ scattering cross-section observations in terms of absolute units. However, such a calibration is not strictly necessary when in situ scattering cross-sections collected with the same echosounder system are used, since any constant of proportion appear both in the echo signal as well as in the observed cross-sections, and will be cancelled out. However, working with quantities in terms of absolute units is rigorously established in fisheries acoustics, and is necessary when comparing data from different echosounders on different vessels, as well as when models of the mean scattering cross-section as a function of species, size, and so on, are used.
A particular inconsistency with the conventional echo-integration method is that the scattering cross-section is expressed in terms of echo intensity, i.e. the peak value of the received echo. This is in agreement with acoustic theory, but the integrated, resolved echo intensity has then to be computed as echo intensity multiplied by the pulse length. In echo integration, the echo-signal intensity is integrated over time, or depth, and then it would be more natural to associate the scattering properties of objects with the integral over the echo-pulse intensities instead of the peak value. This is the reason why only integrated pulse intensities, and not peak values, are used in Aksland (1986). The advantage of this is mainly logical, as peak echo-pulse intensities multiplied by the transmitted pulse length are usually equal to, or at least proportional to, the integrated pulse intensities. It is, however, not difficult to think of cases where this proportionality may be violated, i.e. when the size distribution of scatterers is wide with the biggest sizes significant relative to the pulse volume. Then the backscattered pulse lengths will be variable.
An essential difference between Aksland (1986) and the conventional theory relates to backscattering strength and cross-section. Backscattering strength in the former is expressed in terms of the integrated pulse intensity, while backscattering cross-section is defined in terms of backscattered sound intensity, and hence is a function of pulse intensity throughout acoustic theory. The advantage with Aksland's relative backscattering strength is that a variable for the pulse length is not needed, and that the method is valid whether backscattered pulse lengths and shapes vary or not relative to the transmitted pulses.
Knowledge, or in situ measurement, of the mean backscattering strength or cross-section of the individuals in the population to be estimated is essential in abundance estimation by echo integration. In the conventional method, models of mean backscattering cross-section as a function of species, size (length), and other factors have been much used in combination with biological samples (Foote, 1987). These models, accepted as incorrect in general, were established when only single-beam transducers were in use, and more-or-less updated models are still used with modern echosounder systems.
With the split-beam echosounder system, in situ backscattering cross-section values are observed for each resolved echo that is received. As far as such measurements can be assumed to be representative of a given part of a detected population, estimates of mean backscattering cross-sections are likely to be more reliable than the method based on mean backscattering cross-section models. However, to base the conventional echo-integration method on in situ measurements in general will require transducers that can be lowered close enough to the target scatterers. This is still not routine practice.
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Considerations about backscattering power |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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The relative backscattering strength concept in Aksland (1986) is essentially different from scattering-strength concepts in the acoustic literature. It is therefore given another name in this paper. Assume that a resolved echo is received with 40 log R TVG.
The integrated intensity of this pulse when compensated for the transmit-receive beam effect is called the relative backscattering power of the scatterer and is denoted by 
(
, 
), where (, ) is the direction from the scatterer to the transducer. Note that the relative backscattering power depends on the intensity and pulse length of the transmitted pulse as well as the receiver gain. It is nevertheless normally proportional to the backscattering cross-section within a given echosounder system at constant settings.
Although these considerations are about backscattering power, they also hold for the conventional backscattering cross-section. The other numerous concepts of backscattering coefficients and strengths in the conventional theory are not considered.
In Aksland (1986), the observed relative backscattering strength of a scatterer is considered to be the outcome of a random variable. This view stems from the fact that free-living individuals are known to have a lot of variable states that affect the backscattering power, of which body orientation is the most important when the "body-size to wavelength" ratio is not too small. The rhythm of variation in such states has different characteristics and scales from rapid changes in orientation, daily changes in swimbladder volume by vertically migrating physostomous (not swimbladder, gas-producing) fish to gradual mean changes in behaviour by season. The backscattering power of a randomly selected individual from a given population depends moreover on the size distribution of the population's individuals. These considerations are the reason for representing the backscattering power stochastically in terms of a random variable associated with a formal probability distribution of the form F(y; , ) y 
0, 0 
< 
/2, 0 < 2, corresponding to the direction family of random backscattering power (, ). The parameters and represent the direction of incident and backscattered sound in a spherical-coordinate system with an origin in the centre of the scatter ( = 0 is vertical up and = 0 is fixed relative to the earth). During an observation of ( ), (, ) is the direction from the scatterer to the transducer, but the backscattering power of individuals does exist for each direction at any time of course.
A deterministic formal representation of the backscattering power of an individual could be by a family of time-series (
, , ) over all directions, where is an ordinary time variable. It is necessary to discuss the stochastic representation relative to different properties of the time-series (, ) over individuals and directions. To be useful, the stochastic representation must also represent backscattering power data generated by resolved single echoes observed by a repetitively pinging transducer system. To consider in situ observed-scattering power data as random samples from a probability distribution, (, ) must be assumed to vary in some stationary way, for each direction and individual. This may be reasonable within local areas and limited time periods, but will definitely not hold in general within areas occupied by large populations and time periods long enough to survey the whole population. The estimation method given in this paper must therefore be used for subpopulations where the backscattering powers of the individuals varies in a stationary way, thus requiring a method to stratify a population with respect to area and time in a way that the backscattering power distributions may be represented uniquely within each strata.
There are many ways in which local backscattering power distributions may change in time and space. To mention a few, consider, first, a population with active- and passive-feeding periods during the day. The individuals are likely to show different distributions of body orientations in these periods, and accordingly they represent different backscattering power distributions. Next, consider fish during vertical migration. The change in pressure will affect the volume and shape of the swimbladders that produces a change in the backscattering cross-sections, and hence the backscattering power distributions during these events. This is true, in particular, for physostomous fish like herring that also lose swimbladder gas when the pressure decreases (Huse and Korneliussen, 2000). Finally, consider a population with individuals of considerably different sizes. Then, if the local individual size distribution varies in the area, so does the local backscattering power distribution of a randomly selected individual.
To be able to specify local population strata where the backscattering power distribution of a randomly selected individual may be represented uniquely requires both the best possible knowledge of the biology and behaviour of the target population, and experience with series of in situ analyses backscattering power data obtained from a moving transducer.
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Properties of the Echo-Value Constant |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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For a near-circular, symmetrical beam function, the Echo-Value Constant of a scatterer may be written as
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', where is the off-acoustic axis angle of the scatter-transducer direction.
This definition represents a fundamental difference relative to the conventional theory of the echo integration method that considers scattering individuals from the position of the transducer. The alternative method considers the possible positions of the transducer from the position of scattering individuals instead. The parameter ' in Equation (1) should be an angle where the main transmit-receive lobe has dropped about 20 dB at least. The expression for 
as a function of ' is called the Echo Value.
Aksland (1986) evaluated the Echo Value with a general TVG function, and it is readily seen that the 20 log R TVG makes the Echo Value independent of the depth of the scatter. In terms of a deterministic, backscattering power function (, ) in a spherical-coordinate system with origin in the scatter, and the substitution dA = R2tg d d, where r is the distance between the scatter and the transducer, the Echo Value is given by
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| (2) |
, ) is the transmit-receive beam function of the transducer.
Expression (2) is equivalent to equation (29) in Aksland (1986), and is obtained by expressing Ip2 in Equation (1) by the sonar equation with 20 log R TVG and the backscattering power function (, ).
The Echo Value increases with ', but levels off, in practice, to a constant value while ' is still in the main lobe of the transducer. The Echo-Value Constant is defined as having the value of this "constant" level, a definition that also explains the name. It should be noted that the Echo Value in theory increases everywhere where the integrand is positive, and in fact diverges to infinity when ' 
/2 unless the integrand is zero at ' = /2. However, because of the low value of btr at the edge of and outside the main lobe of usual transducers, the Echo Value is, in practice, constant up to angles very close to /2.
Aksland (1986; Figure 2), has plotted the Echo Value for a homogeneous sphere and two theoretical beam functions (narrow and wide beam). Similar graphs are given here (Figure 1), and indicate that the "constant" level is reached when ' exceeds a value near the angle where the transmit-receive beam has dropped 20 dB.
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The backscattering power of a homogeneous sphere is direction-independent. It might be argued that within the narrow beams used in acoustic-abundance estimation, the backscattering power of marine individuals will also be very close to direction-independent. However, there is nothing to be gained in the derivation of estimators by assuming direction-independent, backscattering powers, and the method to be described will be valid in any case. It is therefore not necessary to separate the backscattering power function and the beam function in this presentation. Then, for simplicity, the beam-dependent, backscattering power
b(, ) = (, )btr(, ) is used. This is usually highly direction-dependent.
Let c be the smallest angle satisfying (c) 
; that is, the Echo Value at angle c is, in practice, equal to the Echo-Value Constant.
When the backscattering power is considered to be a random variable, the Echo-Value Constant is given by
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| (3) |
However, the important parameter to estimate is the mean Echo-Value Constant 
over the individuals of a population. This may be expressed as
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| (4) |
rb is the beam-dependent, backscattering power of a randomly selected individual in the population.
For transducers with a "far-from circular" symmetrical beam function, the above expressions will still represent the Echo-Value Constant if c is the largest angle for all where the beam has dropped around 20 dB. It may, however, be expressed by a smaller region of integration as
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| (5) |
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Integrated, resolved pulse intensities |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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It is convenient to analyse resolved echo pulses at 40 log R TVG, because then the pulse intensities are depth-independent. The integrated intensity of 40 log R pulses will be denoted Ip4.
It is necessary for the derivation of the estimators in the next section that the ratio between the 40 log R and 20 log R TVG functions is equal to R2 or c2t2/4, where R is the range between transducer and scatter, c is the sound speed and t the time between transmit and echo at the transducer (Formula (41) in Aksland (1986)). In this case, an integrated pulse may be written as in Formula (42) in Aksland (1986), and here as
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, 
are random variables for and .
With a single-beam system, the direction variables are not observable, but they can be modelled by concrete probability distributions as in Aksland (1986). In this paper, the scatter direction is assumed to be observable, as is the case with split-beam systems. Then
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| (6) |
, is the observed scatter direction. This means that Ip4 is an observation of the beam-dependent, backscattering power of the scatterer in the observed direction , . From now on, the scatterer is considered to be a randomly selected individual in some population. Then |
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, . This distribution function is called the mean beam-dependent, backscattering power distribution because it is equal to the arithmetic mean of the individual beam-dependent, backscattering power distributions.
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Estimating the mean Echo-Value Constant |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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Let [(w1,
1,1), (w2,2,2),...(wn,n,n)] be n observations of (Ip4, , ). Although such observations from a repetitively pinging transducer will not be stochastically independent in general, we assume here that this has minor effect on the properties of the following estimators as based on independent observations.
A straightforward estimator for the mean Echo-Value Constant is obtained by using the data to estimate E[rb(, )] by some curvilinear regression to a function Ê[rb(, )], and then use Equation (4). This gives the estimator
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| (7) |
c, that is, out to around –20 dB down the main lobe.
To carry out the curvilinear regression, a parametric function of and that can be fitted to the data is needed. Appropriate functions might at first seem difficult to find, but owing to the large amount of resolved-echo data obtainable from a moving, repetitively pinging transducer, general functions with many parameters might be used.
However, the expected beam-independent, backscattering power function is likely to vary insignificantly with direction when , is within the main lobe of a narrow-beam transducer. This is indicated from Aksland's simulated target-strength distributions (Aksland, 1986; Figure 4). Some of these distributions are reshown here (Figure 2), and all show negligible changes within 5° from vertical and rather small changes within 10°. For the kind of deviations shown in Figure 2 this is also likely to hold for their expectations. This means that expected beam-dependent, backscattering power functions tend to be close to proportional with the transmit-receive beam function, at least when c for narrow beams.
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For wide beams where
c is 20° or more, this proportionality may be violated in special cases. Figure 2 indicates that the target-strength distributions of passive fish (with small aspect-angle variation) show the greatest sensitivity for variations the observational aspect. Nevertheless, it may be concluded from these considerations that parametric functions that can be fitted to the beam function of the main lobe may be used in the regression for obtaining Ê[rb(, )] in Equation (7). Such functions are already used routinely for estimating actual beam functions based on echoes from standard spheres (Ona, 1990; Reynisson, 1999).
Techniques for computing the estimator (Equation (7)) for 
are not given here, but will be treated in another paper where a trial of this method in the field will be described. Questions related to the representability of resolved echo data for the distribution function Frb(y; , ) are, however, discussed here.
First, if the fraction of resolved echoes in the echo signals is small, it is likely that the integrated intensities of these echoes are biased relative to those of the overlapping echoes. Use of this method for dense registrations may therefore give doubtful results.
Furthermore care has to be taken with the need to collect resolved echo data from directions out to around –20 dB of the main lobe. Near this edge, the weakest echoes from the target population may be lost due to the low beam factor and the observed detection angles may be unreliable (Reynisson, 1999). However, if echoes close to c are unrepresentable, it is better to extrapolate Ê[rb(, )] from data closer to the acoustic axis so that it continues to be proportional to the beam function. An advantage with the estimator (Equation (7)) for 
is that it will be robust against biases in the echo data far out in the main lobe since echoes there do not contribute much to the Echo-Value Constant. It will be more serious if unwanted weak echoes from small particles central in the beam are wrongly taken for weak echoes from the target population. This may lead to serious underestimation of 
and hence overestimation of the number of individuals in the population. The proper setting of thresholds is therefore important.
An indication of a major presence of unwanted weak echoes in the resolved echo data would be the occurrence of an abnormally "heavy" left tail in the empirical distribution of values of Ip4 from the central part of the beam.
Now, let us look at the estimator given in Theorem 4 in Aksland (1986). This estimator makes use of only angles in addition to integrated resolved echo intensities, and requires that the beam function is circular symmetric. It may thus also be used with dual-beam echosounder systems. First it is noted that condition (ii) in the theorem is unnecessary when is observable since echoes outside ' may be excluded. The proof of Theorem 4 uses the following reasoning:
"Since P(I4 > T |This is not correct. The probability distribution for
given that it is less than or equal to ' is derived in the Appendix, and is given by Equation (13) as: |
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| (8) |
1), (w2, 2), ..., (wn, n)] are n observations of (Ip4,) given that ', and m is the number of echoes received at angles between 0 and '. This estimator accounts for possible losses of echoes outside 0 in the main lobe, but it is required that all resolved echoes with detection angles less than or equal to 0 are observed.
In the special case that 0 can be set to c = ', i.e. all echoes are observed up to c, Equation (8) takes the form
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| (9) |
Better estimators for those above may be derived that use the deviation between the distribution (Equation (13)) and the empirical distribution of the is, and the fact that it is the weakest echoes that are lost far off the acoustic axis. This will not be given here.
Principles for generalization to the case where the beam is not symmetrical, and both and are observed for resolved echoes, now follow. As an alternative to Equation (4), the mean Echo-Value Constant may be expressed as
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, ) bc is not circular, the random variables , , given to be in this region, are not stochastically independent. Let the joint probability density for , be denoted by g(, ). Then, by substituting 
, where frb (
; , ) is the probability density for rb (, ), and multiplying the integrand with g(, )/g(, ) = 1, the mean Echo-Value Constant may be expressed as |
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rb (; , ) g(, ) is the joint probability density for rb (, ), , , the Echo-Value Constant is seen to fulfil:
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| (10) |
, is in the region btr(, ) bc. The function g is obviously proportional the corresponding probability density (Equation (14)), where ' 
: btr(, ) bc. This gives |
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. Noting that the integral of tg/cos2 is tg2/2, we have 
, where 
is the value of at angle where the beam has dropped to bc. Based on representative data [(w1, 1, 1), (w2, 2, 2), ..., (wn, n, n)] for Ip4, , , this suggests an estimator of the arithmetic mean type
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| (11) |
, )bc are observed. It appears that 
does not contain the i, but the i are needed to determine which echoes are received from the region btr(, )bc. If the beam is circular symmetric, then 
is constant and equal to c. Using this, it is easily seen that Equation (9) is a special case of Equation (11). The problem of obtaining representative resolved echoes from a given (sub)population when this cannot be done with the transducer used to estimate the Echo Abundance is now to be considered. In this case a down-looking transducer should be used that can be lowered close enough to the population's scattering layer. This can be a transducer on a towed body, or simply be a transducer platform lowered by its cable from a station-keeping ship at several positions covering the area of distribution of the population. The echo data should be collected from a distance that is short enough to receive mainly resolved echoes, but far enough to ensure that the platform does not affect the behaviour of the individuals.
The two echosounder systems, here denoted A and V, used to estimate the Echo Abundance and the mean Echo-Value Constant, respectively, should be as equal as possible in performance with respect to frequency and pulse length, but may differ in beam function. In addition, the two systems should be calibrated to give equally integrated echo-pulse intensities from a calibration sphere at = 0. Moreover, the transmit-receive beam functions bAtr(, ) and bVtr(, ) associated with systems A and V, respectively, should be accurately measured. Then an integrated pulse intensity IprV observed by the lowered platform has to be transformed to its corresponding value IprA that would have been observed with the other system with the same transducer position and transmit time, by
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| (12) |
, ) is the observed scatter direction.
In order to collect good single-echo data with scatter directions up to c for system A which is around –20 dB down the main lobe, it will be an advantage if system V has some wider beam than system A. Echoes received at angle c will then be stronger with system V than with system A. For the same reason, using a system V with narrower beam than for system A should be avoided.
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Discussion |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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The backscattering power and cross-section of a single marine individual is highly variable unless the ratio between body size and sound wavelength is small. Consequences of this are not fully addressed in the methods of acoustic-abundance estimation of marine populations. In the conventional theory of echo integration, the mean backscattering cross-section of the individuals within the sound beam is a central quantity. However, the variability of this quantity with time and space is not discussed, nor is how to cope with the variation.
The practice of modelling the mean scattering cross-section of fish and other marine organisms in terms of species, size, and other factors is thus more a necessity than a wish. Although such models are often revised, as in Ona (2003), doubt that these models can be reliable in general.
The alternative to use of such models is to measure scattering cross-sections in situ. This is described in the literature (Ona and Barange, 1999; Gauthier and Rose, 2001), but I know of no routine practice for using in situ measurements. This is due to problems with knowing when in situ measurements are a true representation of the population to be estimated, as well as problems with knowing the effect of incorrect measurements to the acceptance of multiple echoes as single echoes. (see Gauthier and Rose 2001 and articles in Ona 1999). These uncertainties are, of course, also a problem with estimating the Echo-Value Constant as described in this paper. But improved instruments and software are likely to give more reliable resolved-echo data in the future. Even with data without technical errors though, there will always be a problem of the degree of representation achieved. This can only be mastered through experience based on the systematic testing of different series of echo data against each other.
It may seem strange to introduce yet another scattering measure in the rich forest of scattering measures in fisheries acoustics. Within the echo-integration method, however, it is not difficult to realize that the backscattering power is the most natural measure. It may be argued that observations of the integrated intensities of resolved echoes are more difficult and demanding than observations of peak values, in particular for weak echoes with tails drowned in noise. This is true, but it is not an argument against using backscattering power as the main scattering measure in the theory of the echo-integration method, because it is easy to estimate the expected backscattering power of an echo from its peak value, or a set of its high values.
Of the two types of estimator for the Echo-Value Constant, represented by Equations (7) and (8), (9) or (11), prefer Equation (7), since it is likely to be more robust than the other approach against the loss of weak echoes far out in the sound beam. It may even be used without echoes far out in the beam by extrapolating Ê[rb(, )] from echo data closer to the acoustic axis. However, both types of estimators should be tried and compared when it is possible to extract the necessary echo data.
As the beam-dependent, backscattering-power expectation estimate Ê[rb(, )] in Equation (7) is expected to be fairly proportional to the transmit-receive beam function, significant deviations from this will indicate errors in the resolved-echo data. In particular, if the backscattering-power expectation estimate seems to be wider than the beam function, this may be caused by the loss of weak echoes far out in the beam or the presence of weak echoes centrally in the beam from small individuals (usually plankton) not belonging to the target population. It may also be a combination of both factors.
The other type of estimator represented by Equations (8), (9), and (11) is based on a simple average and will also be biased if there is over-representation of either weak or strong echoes. Provided the beam is not very wide, it is, however, expected to be more robust against errors in the detection angles than Equation (7), because it affects only the cos term, which, for each echo, takes on values very close to 1.
The alternative method is explained without the use of a lot of common quantities and definitions from acoustics and echo integration, such as sound frequency, pulse length, transmit power, receiver gain, equivalent beam angle, and volume-as well as area- and length-backscattering strength This does not mean that the properties of the method are independent of the technical parameters associated with the echosounder system. Choices of favourable values for transmit power, frequency, pulse length, and beam angle are important for obtaining the necessary resolution and low signal-to-noise ratio that affect the quality of the required data.
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Appendix Probability distribution for the detection angles of echoes |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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Figure 3 shows that if the echosounder transmits with the transducer's centre at a position inside the circle in the transducer plane limited by the angle
' in the scatterer's reference system, then the echo is received at an angle less than or equal to ' relative to the vertical.
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Assume now that the transducer's position is uniformly distributed within the circle limited by
' given that it is inside this circle. Let be a random variable for the detection angle of an echo. Then |
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This is seen from Figure 3, since the radii of the two circles are equal to z tg and z tg, respectively, where z is the depth of the scatterer relative to the transducer. It follows that
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| (13) |
given that it is less or equal to ' as based on an uniformly distributed horizontal position of the transducer relative to the scatterer on a local scale.
The corresponding probability density function is obtained by differentiation with respect to . It is given by
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| (14) |
Provided that scatterers are unaffected by the presence of the transducer and all resolved echoes that hit the transducer at angles not bigger than ' are observed, sequences of observed resolved echoes can be expected to have associated detection angles that are distributed in accordance with Equations (14) and (15). When collecting resolved echoes to angles far out in the main lobe of the transducer, however, the empirical detection-angle distributions are not likely to follow Equation (14) or (15) because many weak echoes will be lost far from the acoustic axis.
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References |
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TopIntroductionPopularized review of the...Properties of the alternative...Considerations about...Properties of the Echo-Value...Integrated, resolved pulse...Estimating the mean Echo-Value...DiscussionAppendix Probability...References |
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  M. Aksland Applying an alternative method of echo-integration ICES J. Mar. Sci., January 1, 2006; 63(8): 1438 - 1452. [Abstract] [Full Text] [PDF]
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