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Fishery Bulletin 96(2), 1998 
at 400 and 6,000 Hz, respectively, digitized at a 20- 
kHz sampling rate, and stored. Digitally stored sig- 
nals were subsequently filtered into 21 1/6 octave 
bands from 500 to 5,000 Hz and amplitude versus 
time envelopes calculated for each band. There was 
little variation between shots in a sequence. Enve- 
lopes of all shots in a sequence were averaged to ob- 
tain a single set that was representative of scatter- 
ing during that sequence, and this averaged data was 
used to calculate volume scattering strength, S , the 
scattering strength of a unit volume of water, as a 
function of depth. Data were displayed as 2-dimen- 
sional images showing S v in dB on a color scale as a 
function of frequency and depth. Scatterer depths 
were determined from the S v images; integration over 
these depths produced a series of layer scattering 
strength (S L ) versus frequency curves. On the basis 
of the distinct nature of the S L curves, scatterers were 
assumed to be swimbladder-bearing fish. Hence, S L 
curves were used to determine swimbladder size with 
a swimbladder scattering model. Fish swimbladders 
are not spherical, nonetheless their size may be con- 
veniently expressed in terms of equivalent spherical 
radii (ESR) (i.e. the radii of spheres of equivalent vol- 
ume). Thus ESR can be derived independently from 
acoustic data and trawl data and compared to evalu- 
ate each method for assessing population statistics. 
Inverse solution 
The ESR of the scatterers were obtained from 
the S L curves of the acoustic data with the non- 
negative least squares solution of 
S = an, (1) 
( Holliday, 1977b) for n, a column vector contain- 
ing the number of scatterers in each of p size 
classes of swimbladder radii. S is a column vec- 
tor of measured layer backscattering cross sec- 
tions a 6 (S L = 10 log a bSL ) for the q frequencies 
at which measurements were made. The ma- 
trix a is of dimension q x p and contains indi- 
vidual backscattering cross sections o bs . that 
are calculated over each of the q = 21 measure- 
ment frequencies ( 1/6 octaves from 500 to 5,000 
Hz) and p = 18 semilogarithmically spaced ra- 
dii (r= 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, 
0.9, 1.0, 1.2 1.4, 1.6, 1.8, 2.0, 2.25, and 2.5 cm) 
with a model of swimbladder resonance (Eq. 3 
below). In solving Equation 1, the number of 
measurement frequencies varied from station 
to station depending on data quality, with from 
4 to 6 of the lower 1/6 octaves excluded. This 
variation was usually due to lower-frequency 
data that were low level and affected by noise. 
The inverse solution was also restricted to ex- 
clude radii smaller than the radius expected to 
be resonant at 5,000 Hz at the depth of each 
layer. These radii were calculated with the sub- 
sequent Equation 4 for the resonance frequency, 
f (y The quality of the inverse solution of Equa- 
tion 1 for n was examined by comparing the 
original data, S, to an estimate of the layer 
strength obtained from the forward calculation 
5 = an- 
Solution of Equation 1 above assumes that the 
a bSL are the backscattering cross sections of a unit 
horizontal area of a layer of dispersed acoustically 
noninteracting scatterers, such that 
