6.1 Measurement of Volume Back-Scattering Strength
6.2 Mean Volume Back-Scattering Strength (MVBS)
6.3 Measurement of relative density
6.4 Measurement of Absolute Density
6.5 Determination of Mean Fish Density in a given Area
A newcomer to this work, embarking on his first survey cruise aimed at providing an estimate of a given fish stock, may perceive the exercise lying ahead of him as routine. The survey vessel, carrying sophisticated equipment, will sail to a given area, measure a unit fish stock, and then compute the result - mission accomplished! Such a conception is understandable because quantitative acoustic methods are known to represent a powerful tool in present-day fisheries management. But the procedures are more complicated than is generally realised and there are many implications with respect to precision and errors in the estimates.
This section deals with some general concepts and processes related to a biomass estimate by acoustics. Clarification of what comprises an acoustic estimate may be useful at this stage and will facilitate understanding of subsequent discussions. To avoid ambiguity later an assumption is made here that, when referring to a fish stock, we mean a population comprising a single species with a narrow but normal length distribution.
In section 4.5 we considered fish schools as targets in a simple manner. The purpose of Chapter 6 is to examine the theory and to show how the principles of measuring backscattering strength and reverberation level, are put into practise for the quantitative measurement of schools.
A number of simplifying assumptions regarding the distribution and density of fish (scatterers) plus other factors were given in section 4.5 and it is necessary for these assumptions to be fulfilled if accurate results are to be obtained. In practise the actual beam pattern of the transducer is replaced by the equivalent beam y, (section 4.6). The concept that there is unity response inside this beam and zero response outside is now adopted to simplify the explanation of volume backscattering.
In Figure 39 the volume dV has an end face surface area of R2dW, where R is the range from the transducer and dW is the solid angle in steradians subtended by dV at the transducer. The length of dV is sufficiently small that, when insonified by a pulse, all scattering produced from its volume is received by the transducer at the same instant of time. This can be visualised by first considering the front surface of the pulse reaching and acting upon the scatterers at the rear of dV. The energy backscattered from this rear area will arrive at the transducer with the energy backscattered from the front area of dV by the rear surface of the pulse. Thus the length of dV is shown to be ct /2 (a similar principle is illustrated when the minimum separation of two targets in range by ct /2 is necessary to ensure that they are resolved, one from the other, see Figure 18.
Figure 39.
The volume dV is therefore
dV = R2 ct dW (50)
We need to determine the intensity of the acoustic backscattering from this volume and to do this must introduce a term which has some similarity to target strength. This is the volume backscattering coefficient sv. It is defined as the ratio, of backscattered intensity (Ib) produced by unit volume at 1 m from the volume, to the intensity of the incident wave I1, i.e.
sv = Ib/I1 and Sv = 10 log sv (51)
Thus the intensity from dV is
sv·R2·ct ·dW
to include the whole beam we substitute y for dW
sv·R2·ct ·y
It is now possible to determine the reverberation level (RL) at the transducer arising from this expression. If the intensity from the transducer is I, it will be reduced in proportion to R4 i.e. (40 logR) by the journey to and from the volume dV.
RL = IR-4sv·R2 ct y
which in logarithmic terms is RL = SL - 20 logR + 10 log sv + 10 log ct + 10 logy.
Note that we are left with 20 logR the same result as in section 4.5. y can be calculated from either eqn. 34 or 35. Under practical conditions the absorption loss 2a R must also be taken into account so we would have -(20 logR + 2a R), but when an exactly correct TVG is used this term is fully compensated and does not appear in the equation.
In Figure 40 the pressure waves which make up RL act on the transducer to produce a voltage VRT whose amplitude at any instant of time is equal to RL + SRT (SRT is the transducer sensitivity as a receiver expressed in dB/1V/1m Pa). So we can now say
VRT = SL + SRT + Sv + 10 log ct /2 + 10 logy (52)
Figure 40.
This is convenient because SL + SRT is a lumped parameter which can be measured by means of the standard target calibration procedure (see Chapter 7). It can be seen from Figure 40 that the voltage VRT is processed through the echo-sounder first by a fixed gain amplifier G1 and then in the TVG amplifier G2. These gains (in dB) must be taken into account, i.e. included in the eqn. G2 is normally the gain figure at maximum range of the TVG. All voltages from the echo-sounder are squared in the echo-integrator to convert from pressure to intensity proportionality. Adjustment for speed of vessel, selection of depth and depth interval is made.
So far we have only related the Sv to an instantaneous value theoretically produced by one pulse volume. A more meaningful measurement parameter is the MVBS, which is Sv averaged over one or more transmissions and over a certain range interval D R as well as unit distance.
For an integration or depth interval D R = R2
- R1, the volume backscattering strength for one transmission
is a measure of the instantaneous acoustic intensities reflected from each m3
of water added and averaged over D R. Along the
distance sailed, (e.g. one nautical mile) where there are a large number of
successive transmissions, the resulting mean volume backscattering strength
(MVBS) would be the MVBS per transmission, averaged over D R,
and also the relevant unit distance. In logarithmic terms the MVBS denoted by
can be expressed by
(53)
where
= the mean TS of individual fish in the total insonified volume
r v is the mean density (fish/m3)
If is known, the mean density
of fish targets for an integration interval can be estimated provided that
is also known. Two important factors are evident from the above.
(a) the MVBS is a biomass scattering parameter, independent of system parameters(b) when
Figure 41. (a)(b)(c)
Although the MVBS provides a measure of relative biomass density (
not known), like the observed M-values discussed under 6.3, there is an important
difference because the MVBS has a universal meaning, but the M-values represent
arbitrary quantities. Thus, the MVBS obtained from one vessel/system in a given
area is directly comparable with MVBS values collected by another vessel/system
in a different area. Similar comparisons between M-values of two different
survey vessels are not meaningful, because they depend on the scattering parameter
and also on the selected control settings.
Expressing the above more formally we can say that the mean volume backscatter from a single acoustic transmission passing through a range interval D R = R2 - R1, is the sum of the individual pulse volume scattering coefficients divided by the range interval hence
(54)
where
Ci represents the instrument parameters of SL, SRT, y etc
N = D R/(ct /2) is the number of pulse-lengths occurring in D R,
squared output voltage at the nth one of these.
During actual echo surveys, the selected measurement unit could be the MVBS averaged for all acoustic transmissions occurring over a distance of one nautical mile. In this case we have
(55)
where Tx is the total number of transmissions over one nautical mile, and 
is now the doubly averaged pulse volume backscatter.
In summary: The mean volume backscattering strength (MVBS) measured over a
given range interval and per unit distance is the product of an instrument constant
and the average squared output voltage 
of a system using a 20 logR + 2a R TVG function.
The MVBS is exclusively a biomass target parameter which can be compared between
different survey ships and geographic areas, and as such, has a universal meaning.
When a scientific echo-sounder works in conjunction with an echo-integrator acoustically surveying a fish stock, the output is proportional to the observed fish density (Midttun and Nakken, 1977), i.e.
r = C x M
where
r = fish density (tonnes/mile2)C = system calibration constant (tonnes/mile2) per mm, ref. a 1 mile sample with standard control settings
M = observed integrator value (mm)
Imagine a survey vessel with this equipment (without it being calibrated) proceeding along a given survey track in a densely populated area. For each one-mile section sailed, the integrator is automatically re-set and the corresponding readings noted. After some distance there will be a set of values for integrated echo-intensities returned from the layer of the fish population. The observed values M1, M2, M3, etc. represent
(a) a directly measured one-mile echo-abundance index for the insonified (sampled) water volume(b) a relative area density (biomass/mile2) between the individual one-mile sections. It is important to note that the echo-abundance index is not only a function of fish density, but also of the thickness of the actual fish layer occurring within the integrated depth limits. Thus, if a scattering layer of 20 m thickness produces a reading M20 = 100 mm, a layer of equal density but only 10 m thickness would give half the reading, i.e. M10 = 50 mm. To summarise, the use of the integrator as a measuring device results in
(a) a measure of an echo-abundance within a given distance unit
(b) the measurement of relative area density between distance units.
From the simplified relationship for fish density (r v = C x M) the conversion of relative density values in millimetres into 'absolute' fish density (e.g. tonnes per square nautical mile of sea surface area) is achieved through multiplication with the constant 'C'. This 'integrator conversion constant', or scaling factor, is the key to the transformation of relative to absolute fish densities. It can be determined through different methods of calibration as explained in Chapter 7. Often though, the constant has been estimated from experimental data obtained from direct live fish calibration which has particular application to species that either have unknown target strength or simply do not lend themselves individually to acoustic measurements in the field. However, with digital integrators, and an advance in calibration with standard targets (Foote, Knudsen and Vestnes, 1982), coupled with improved knowledge of fish target strength, it has become increasingly attractive to estimate absolute densities through MVBS measurements. This approach can be explained by expanding the mathematical derivations given previously.
(56)
an echo-integrator is a device specifically designed so that its output (M) is
(57)
where Ge is the echo-integrator gain factor, Vo is the output voltage derived from VR, fed directly to the input terminals of the integrator (Figure 42). Combining eqn 56 and 57 we obtain
(58)
Figure 42.
Now, by substituting for sv and rearranging eqn 58 we may write
r v D R = M/CiGe (s /4p) (59)
The left hand side of eqn 59 is interpreted as an area density (r A), i.e. r v = r A D R, commonly expressed in tonnes/mile2. For analog integrators where only one channel is available for integration over the whole depth column of interest, the area density is the most practical unit to work with. Hence, in logarithmic terms the expression for r A becomes
(60)
which, as a working formula, is often written in the antilog form
(61)
demonstrating that, when the relevant instrument constants and the target strength
of fish are known, the observed integrator readings (M) can be readily converted
into absolute fish densities. In the case of digital echo-integrators the situation
and hence the procedures are different. Built-in computing facilities allow
for multi-channel integration with automatic print-out of 
values in each depth interval over the pre-programmed distance unit. Because
the computer memorizes and includes the constants Ge and Ci
(60) in its computation of the MVBS values, the derivation of the equivalent
fish density is now based on the relationship
(62)
where,
D R = the width of the integrator depth interval (m)
= equivalent mean fish density within a depth interval (fish/m3)
= mean target strength of the fish
Written in the antilog form, eqn (62) is
(63)
which, when compared with eqn 61, shows the principal difference between the analog and digital-integrator output expressions.
In practical operations, it may be desirable to determine the MVBS for the whole integrated depth column, in addition to the values of individual depth intervals. Using first linear dimensions, this is achieved by summation of products as follows
(64)
where,
the MVBS for all channels
D Rt = the total width of all integrator channels
the MVBS in the nth channel
D Rn = the width of nth depth channel
By rearranging eqn 64 into a more general form, we have
(65)
where N = number of channels available. By converting into logarithmic form,
and bearing in mind that 10 log 
,
we obtain
(66)
In summary: Measurements of absolute densities with echo-integrators are most commonly achieved by two principal methods. The first is based on experimental calibration using live fish for direct determination of the constant C by the relationship r A = CM. Thus, the product of the constant and the observed integrator reading (M) gives a direct estimate of the absolute density. This method is particularly suitable for analog integrators.
The second method is based on mean volume backscattering strength (MVBS) measurements and is suitable for analog and multi-channel computerized integrators. Provided that the principal instrument constants and target strength are known, the density can be calculated from eqn 63.
In the previous sections we have explained measurement processes that are technical in nature. This section goes a step further and explains how a mean fish density within a given Survey area is determined, and how this is a combined technical and statistical problem. Figure 43 below serves to introduce the problem.
It is desired to determine the mean (average) fish density 
in
the area A using the survey observations M1, M2, M3.....
MN and a calibration constant C. A straightforward procedure would
be to calculate the mean from the formula
(estimated mean) (67)
This has the limitation that it pre-supposes the arithmetic mean is valid, i.e. that the 'M' value, treated as a continuously random variable, exhibits a probability density function (PDF) which can be reasonably well approximated by the normal distribution. If that is true and C is an unbiased calibration constant, then the result would be an unbiased estimate of the mean fish density in the area. However, real-life data have demonstrated clearly that a set of M-data yielded by large-scale acoustic surveys is rarely normally distributed. In most cases it shows a highly positive skewness, or else the observations tend to fit the negative binomial (b), a log-normal (c) or even a geometric distribution (d) where the probabilities of occurrence of successively larger M-values are decreasing geometrically as indicated in Figure 43. Because of this it is important that survey data be submitted to appropriate statistical analyses in order to estimate the most representative mean fish density in the area. A more detailed coverage of the subject is given in section 10.3.
Summary: The determination of mean fish density in a given survey is a combined process of measurement and statistical analysis. The accuracy of the density estimate depends on the accuracy of the calibration constant C, and on the goodness of fit of the probability function on which data averaging is based.
