3. PROBLEMS ARISING IN THE ESTIMATION OF POPULATION TOTALS

3.1 Introduction
3.2 The vertical migratory process (daytime observations)
3.3 Night-time biomass estimates

3.1 Introduction

In this chapter we examine some of the peculiarities of the target population (= population of small pelagic fish) which affect the accuracy of the calculated sample estimates.

As we have discussed(Section 2.1) temporal changes in the packing status characterize fish populations of small pelagic fish. Observations of echo records indicate that during night-time fish are spread widely forming layers of fish. During daytime small pelagic fish are packed in schools of various sizes. Schools are not randomly distributed in space but exhibit aggregation patterns which may intuitively be noted by anyone looking at echo records.

Apart from the temporal variation in the packing status of the target population, small pelagic fish exhibit diurnal (vertical) migration which is related to the light intensity. With this information, and with further knowledge that neither echo-integrator surveys nor sonar surveys can achieve a complete vertical coverage of the survey body of water¹ we can readily ascertain that independent population estimates produced by these two kinds of surveys include traces of bias (underestimation).

¹ In echo-integrator surveys approximately the upper 10 m are not covered by the system (4-5 vessel draft, plus 4 m before the echo-sounder operates normally). Also, sample observations obtained within the second upper layer (10-20 m) are biased (underestimation) because of the avoidance effect.

Sonar surveys usually cover the upper 20 m (effective range) and the system can only be used for the detection of fish schools (daytime observations).

3.2 The vertical migratory process (daytime observations)

3.2.1. The structure of sample data
3.2.2 De-biasing sample estimates

It is of interest to examine the vertical migratory process of the schooling fish population by introducing distinct time intervals within the daytime period (during night-time the observed layers of fish are located in the upper layers of water). For purposes of illustration (Fig. 3.2a) we have first divided the survey body of water into the following three depth domains:

Domain A:	Expresses the upper layer of water, 0-10 m. This layer is not covered by echo-integrator surveys, it is covered by sonar surveys.
Domain B:	Expresses the second upper depth layer of water, 10-20 m. This layer is partially covered by echo-integrator surveys because of the avoidance effect (underestimation); it is covered by sonar surveys.
Domain C:	Expresses the remaining body of water below 20 m. This layer is covered only by echo-integrator surveys.

Secondly, we have introduced five time intervals (t, Fig. 3.2a) indicating possible alternatives in the vertical localization pattern of the target schooling population during daytime. Table 3.2b portrays the established temporal vertical localization pattern of the target population and level of coverage obtained by echo-integrator surveys and sonar surveys respectively.

A critical assessment of Table 3.2b leads to the following conclusions:

a) If echo-integrator surveys are conducted only during the time interval one (t-1) they will produce an unbiased estimate of the population total. If echo-integrator surveys are conducted only during the time interval five (t-5) they will produce a zero value of the population total. If echo-integrator surveys are conducted within the time intervals t-2 to t-4 they will produce a biased estimate (underestimation) of the population total.

Table 3.2b. Temporal vertical localization pattern of the schooling target population and level of coverage obtained

t	Depth domains (m)			Level of Coverage
	0-10	10-20	below 20	Echo-integrator		Sonar
1			Ö	below 20 m	Complete coverage	below 20 m	Complete omission
2		Ö	Ö	below 20 m	Complete coverage	below 20 m	Complete omission
				10-20 m	Biased coverage	10-20 m	Complete coverage
3	Ö	Ö	Ö	below 20 m	Complete coverage	below 20 m	Complete omission
				10-20 m	Biased coverage	10-20 m	Complete coverage
				0-10 m	Complete omission	0-10 m	Complete coverage
4	Ö	Ö		10-20 m	Biased coverage	10-20 m	Complete coverage
				0-10 m	Complete omission	0-10 m	Complete coverage
5	Ö			0-10 m	Complete omission	0-10 m	Complete coverage

If the total sample data (daytime observations) are used for estimation purposes they will produce a biased estimate (underestimation) of the population total.

b) If sonar surveys are conducted during the time interval four or five (t-4 or t-5) they will produce an unbiased estimate of the population total.

If sonar surveys are conducted during the time interval one (t-1) they will produce a zero value of the population total.

If sonar surveys are conducted during the time intervals t-2 and t-3, they will produce a biased estimate (underestimation) of the population total.

If total sample data (daytime observations) are used for estimation purposes they will produce a biased estimate (underestimation) of the population total.

3.2.1. The structure of sample data

An insight into the basic structure of the. empirical total sample observations (daytime) can be obtained if the data are broken down according to the established time intervals (post-stratification). In such a case estimates can be calculated of the overall sample mean () e.g., echo-integrator readings/n.mi, and of the sample means within the time intervals (). Observed statistical differences among the calculated means provide an indication of the effect on the sample observations caused by the vertical migratory process of the schooling target population.

It should be noted that, in the case of large-scale acoustic surveys, in order to assess the structure of the empirical sample data the above kind of analyses should be worked out on a space/depth domain basis.

3.2.2 De-biasing sample estimates

3.2.2.1 Matching echo-integrator and sonar estimates

As we have discussed in previous sections, original sample observations (daytime) of echo-integrator surveys and sonar surveys cannot be used independently in order to calculate unbiased population totals. A number of alternatives have been worked out for de-biasing purposes.

1. If the actual sample in the non-biased (complete population average) time interval (s) is sufficient and representative of the survey population and it. can be properly identified (post-stratification) these data can be used in order to provide an independent and unbiased estimate of the population total (echo-integrator, sonar surveys).

2. Sample observations (daytime) of echo-integrator surveys can be used for the estimation of population totals (unbiased) after a proper adjustment of the original sample data.

The de-biasing process asks for an extensive use of the available echograms. Adjustment of the original sample data is made progressively, i.e., on an elementary sampling distance unit basis and by following up the vertical migratory process of the schooling sample population.

Some kind of subjectivity might be inherent in the process of adjusting the original sample observation by using the available echograms.

3. Matching the results of the concurrent echo-integrator and sonar surveys: In such a case an unbiased estimate of the population total is calculated by adding up the estimated totals which are produced independently by these two kinds of acoustic surveys:

Estimated population total = (Echo-integrator total - below 20 m) + (Sonar total - above 20 m).

The mechanism of this alternative is explained in section 3.2.2.1 below.

3.2.2.1 Matching echo-integrator and sonar estimates

An unbiased estimate of the population total is calculated by adding up the independent estimates produced by the echo-integrator survey (below 20 m) on the one hand, and the concurrent sonar survey (upper 20 m) on the other.

For purposes of illustration (Fig, 3.2.2.1a), the survey area has been divided into two depth domains, i.e., A₁: 0-20 m, A₂: below 20. Further, we have introduced three time intervals by taking as criteria both the temporal vertical localization pattern of the schooling target population and its level of coverage by the individual concurrent acoustic surveys:

T-1:	The target population is located below 20 m. The echo-integrator survey alone ensures a complete coverage of the target population.
T-2:	The target population is located in both depth domains A1 and A2. The echo-integrator survey ensures a partial coverage of the total population and the concurrent sonar survey ensures coverage of the remaining part of the target population.
T-3:	The target population is located in the upper 20 m. A sonar survey alone ensures a complete coverage of the target population.

Figure 3.2.2.1

In Figure 3.2.2.1b the overall size of the pre-selected regular line transect sample² (n.mi) is given by the vertical tracks a, b, c which are equal in size; each sample transect is divided into elementary distance sample units (ESDU’s = 1 n.mi).

² See chapter 5.

Variate -(x) expresses number of sample schools per ESDU in the echo-integrator survey, and variate -(y) expresses number of sample schools (effective range) per ESDU in the concurrent sonar survey.

It is obvious that, in the absence of a vertical coverage problem, variable -(x) will take the same value along the sample ESDU’s. An unbiased estimate of the population mean per ESDU is given by the overall sample mean

(1)

where

: Total number of selected ESDU’s

track -a:	values attained x, (mean x)
track -b:	values attained , (mean ); partial vertical coverage = only half of the target population is covered by the echo-integrator survey
track c:	values attained x = 0, (mean 0); complete omission of the target population by the echo-integrator survey

By using the above sample data we can calculate the overall sample mean which provides a biased estimate of the population mean per ESDU:

(2)

or

(biased)

By applying Midttun’s Method (see chapter 4) the estimated number of schools per n.mi can be raised to n.mi² (). A biased estimate of the total number of schools in the population is given by,

(3)

where

A: total survey area in n.mi²

: Estimated total number of schools (Because of the symmetrical case in coverage of our example, this number equals with only a half of the true number of schools in the population).

(4)

where

= average number of schools per n.mi²

: Estimated total number of schools (Because of the symmetrical case in coverage of our example, this number equals with only a half of the true number of schools in the population).

Estimated population total = (5)

3.3 Night-time biomass estimates

As mentioned earlier, during the night-time small pelagic fish are concentrated in the upper layers of water forming layers of fish, which can only be detected by the echo-sounder (echo-integrator surveys).

The effect of bias on the accuracy of the calculated night-time biomass estimates is a function, on the one hand, of the abundance of the survey stocks and, on the other, of the observed localization pattern of fish. For example, fish very close to the surface will be subjected to all types of coverage problems; sub-surface fish will only be subjected to avoidance effect.

For de-biasing the obtained sample observations, the vertical distribution of fish should be taken into account. Specifically, the following kind of distributions can be identified (Fig. 3.3a, b, c).

Figure 3.3

Orthogonal distribution; In the above case (a) simple ratio estimate can be used for de-biasing the sample observations. In such a case, sample data of the depth domain C(= 20-30 m) can be raised to the total by using depth as the raising factor.

Skew distributions: In the above cases (b), (c) for raising sample data of the depth domain C to the total, the kind of skewness (positive, negative) should be taken into account.

Total biomass estimates are calculated by using the adjusted sample data.