I n t e r n a t i o n a l Council f o r t h e
E x p l o r a t i o n of t h e Sea
,
. C.M.
v
s
1990/B:21
. R
F i s h Capture Committee
EQUIVALENT BEAM ANGLES FOR SEVERAL STANDARD TRANSDUCERS
Kenneth G. F o o t e
I n s t i t u t e o f Marine Research
5024 Bergen, Norway
ABSTRACT
Nominal
transducers.
SIMRAD EK500
120 kHz, and
f r e q u e n c y of
v a l u e s f o r t h e e q u i v a l e n t beam a n g l e a r e g i v e n f o r s e v e r a l
These a r e t h e s t a n d a r d s p l i t - b e a m t r a n s d u c e r s d e s i g n e d f o r t h e
s c i e n t i f i c echo sounder, w i t h o p e r a t i n g f r e q u e n c i e s o f 38 and
t h e new two-beamwidth SIMRAD ES5 t r a n s d u c e r , w i t h o p e r a t i n g
38 kHz.
RESUME: ANGLES EQUIVALENTS DE PLUSIEURS TRANSDUCTEURS
Les v a l e u r s nominales d e s a n g l e s e q u i v a l e n t s s o n t donnees pour p l u s i e u r s
transducteurs.
Ceux-ci s o n t l e s t r a n s d u c t e u r s s t a n d a r d 2 f a i s c e a u s c i n d g
p r e v u s pour l e sondeur s c i e n t i f i q u e SIMRAD EK500, avec l e s f r e q u e n c e s d e
t r a v a i l 38 e t 120 kHz, e t l e s nouveaux t r a n s d u c t e u r s 2 double f a i s c e a u SIMRAD
ES5, pour l a f r e q u e n c e de 38 kHz.
INTRODUCTION
An i m p o r t a n t p a r a m e t e r i n t h e b a s i c e q u a t i o n s used t o determine f i s h
d e n s i t y a c o u s t i c a l l y i s t h e e q u i v a l e n t beam a n g l e ( F o r b e s and Nakken 1972,
MacLennan 1 9 9 0 ) . T h i s i s c o n v e n t i o n a l l y d e f i n e d p u r e l y a s a p r o p e r t y of t h e
transducer.
I t g e n e r a l l y depends on t h e d e t e c t i o n t h r e s h o l d , t h e n c e
b a c k s c a t t e r i n g c r o s s s e c t i o n s o f t a r g e t organisms t o o (Foote 1 9 8 8 ) . However,
even i n t h e g e n e r a l c a s e w i t h h i g h s i g n a l - t o - n o i s e r a t i o , t h e e f f e c t i v e v a l u e
i s e q u a l t o t h e nominal v a l u e .
There a r e a number of methods f o r d e t e r m i n i n g t h e nominal v a l u e $o of t h e
e q u i v a l e n t beam a n g l e .
The most p r a c t i c a b l e of t h e s e depends on knowledge o f
t h e beam p a t t e r n , from which $o can b e computed.
Such knowledge i s o f t e n
c o n t a i n e d i n measurements of t h e beam p a t t e r n , b u t it may a l s o be d e r i v e d by
theoretical calculation.
Transducer m a n u f a c t u r e r s o f t e n s p e c i f y $, i n d a t a s h e e t s accompanying
the transducer.
However, t h e method o f d e t e r m i n i n g t h e q u a n t i t y i s seldom
s p e c i f i e d . The f r e q u e n t p r a c t i c e o f a t l e a s t one m a n u f a c t u r e r i n r e p r e s e n t i n g
t h e beam p a t t e r n by m i n i a t u r e p o l a r p l o t s i n one o r two p l a n e s c a n n o t i n s p i r e
This a u t h o r ' s repeated experience
c o n f i d e n c e i n t h e accompanying v a l u e of $bo.
i n r e c e i v i n g d a t a s h e e t s stamped "approved" t h a t o b v i o u s l y p e r t a i n t o o t h e r
t r a n s d u c e r s t h a n t h e i n d i c a t e d one i s s i m i l a r l y p e r s u a s i v e o f t h e u n r e l i a b i l i t y
of manufacturer-specified d a t a .
User d e t e r m i n a t i o n o f Q
, hence a l s o beam p a t t e r n , i s t h u s recommended.
T h i s may be done by measurement (Simmonds 1984, 1987, Ona a n d Vestnes 1985,
Reynisson 1985, 1986, 1987, Degnbol 1 9 8 8 ) , which may i n c l u d e t h e e f f e c t of
mounting (Simmonds 1 9 8 4 ) , o r by computation b a s e d on knowledge o f t h e
t r a n s d u c e r geometry. T h i s i s t h e approach a d o p t e d h e r e .
The o b j e c t o f t h i s i n v e s t i g a t i o n i s t h e o r e t i c a l s p e c i f i c a t i o n of t h e
nominal v a l u e s of t h e e q u i v a l e n t beam a n g l e f o r t h r e e s p l i t - b e a m t r a n s d u c e r s .
These a r e t h e SIMRAD ES38B, ES5, and ES120. The v a l u e s p r e v i o u s l y p r e s e n t e d
( F o o t e 1989) f o r t h e 38-kHz t r a n s d u c e r s , t h e ES38B and ES5, a r e r e v i s e d ; t h e
v a l u e f o r t h e 120-kHz t r a n s d u c e r , t h e ES120, i s p r e s e n t e d f o r t h e f i r s t time.
I t i s hoped t h a t by p r e s e n t i n g t h e s e v e r a l v a l u e s t o g e t h e r , t h e u s e r w i l l be
h e l p e d i n a s s i g n i n g v a l u e s t o $bo.
DEFINITIONS
The nominal v a l u e $bo o f t h e e q u i v a l e n t beam a n g l e i s d e f i n e d a s t h e
2
i n t e g r a l of t h e p r o d u c t o f f a r f i e l d t r a n s m i t and r e c e i v e beam p a t t e r n s , b ,
o v e r t h e hemisphere w i t h a x i s 8=0 c o i n c i d e n t w i t h t h e a c o u s t i c a x i s :
For an a r r a y of n i d e n t i c a l e l e m e n t s i n t h e same p l a n e w i t h c e n t e r p o s i t i o n s
r a n d r e l a t i v e a m p l i t u d e w e i g h t s W , , t h e one-way f a r f i e l d beam p a t t e r n i s
-1
( F o o t e 1990)
k i s t h e wavevector.
where b ( B , $ ) i s t h e s i n g l e - e l e m e n t beam p a t t e r n and 1
F o r s q u a r e elements w i t h s i d e l e n g t h 2a,
b1(8,$) =
s i n ( k a s i n 8 cos $) s i n ( k a s i n 8 s i n $ )
2
(kasine) sin$cos$
I2
1
.
F o r c i r c u l a r elements w i t h d i a m e t e r 2a,
bl ( B l $ ) =
ka s i n 8
l2
where J1(-) i s t h e B e s s e l f u n c t i o n o f f i r s t k i n d and f i r s t o r d e r (Jahnke and
Emde 1945).
TRANSDUCER GEOMETRIES
The t r a n s d u c e r elements f o r t h e 38-kHz t r a n s d u c e r s a r e s q u a r e w i t h s i d e
l e n g t h of 30 mm.
They a r e arranged on square g r i d s with c e n t e r - t o - c e n t e r
d i s t a n c e s of 32 mm along rows and columns. Amplitude weightings f o r t h e
s e v e r a l t r a n s d u c e r s a r e i n d i c a t e d i n F i g s . 1 and 2 f o r a s i n g l e q u a d r a n t , t h e
o t h e r s being d e r i v e d by symmetry.
Fig. 1. R e l a t i v e amplitude weights of elements i n t h e
upper l e f t quadrant of t h e SIMRAD ES38B t r a n s d u c e r .
F i g . 2. R e l a t i v e amplitude weights of elements i n t h e
upper l e f t q u a d r a n t of t h e SIMRAD ES5 t r a n s d u c e r .
A l l
elements a r e used t o form t h e narrow beam.
The elements
t o t h e lower r i g h t of t h e drawn boundary form t h e wide
beam.
The t r a n s d u c e r elements f o r t h e 120-kHz t r a n s d u c e r a r e c i r c u l a r w i t h
The elements a r e arranged on a square g r i d with c e n t e r - t o diameter of 10 mm.
c e n t e r d i s t a n c e s of 11 mm along rows and columns. Amplitude weights f o r t h e
elements a r e shown i n F i g . 3.
Fig. 3 . R e l a t i v e amplitude weights of elements i n t h e
upper l e f t q u a d r a n t of t h e SIMRAD ESl.20 t r a n s d u c e r .
PERFORMANCE MEASURES
I n a d d i t i o n t o t h e nominal v a l u e of t h e e q u i v a l e n t beam a n g l e $,,
t o g e t h e r w i t h i t s l o g a r i t h m i c e x p r e s s i o n y o , a second b a s i c measure o f
t r a n s d u c e r performance i s computed. T h i s i s t h e d i r e c t i v i t y index f o r
i s o t r o p i c n o i s e ( U r i c k 1983) :
D 1 = 10 log-
where t h e i n t e g r a l i s tantamount t o Eq.
of b .
4lT
j b dn
I
( 1 ) b u t with s i n g l e occurrence
The e q u i v a l e n t beam a n g l e i s , c o r r e s p o n d i n g l y , a measure of d i r e c t i v i t y
f o r volume r e v e r b e r a t i o n .
In p a r t i c u l a r , t h e analog t o Eq. (4) i s t h e
d i r e c t i v i t y index f o r volume r e v e r b e r a t i o n ,
Jv = 10 l o g
4lT
/b2
dn
COMPUTATIONAL METHOD
The i n t e g r a t i o n s i n Eqs. (1) and ( 4 ) have been performed s l a v i s h l y on
a d i g i t a l computer by Riemann sums. By symmetry, t h e i n t e g r a t i o n was performed
o v e r t h e o c t a n t 8 E [ 0 , ~ / 2 ] , 4 E [ 0 , ~ / 4 ] , of t h e u n i t hemisphere. A g r i d o f
p o i n t s was chosen w i t h even s p a c i n g i n 8 and even s p a c i n g i n $.
The v a l u e s f o r
c o n s t a n t 8 were a v e r a g e d and weighted by t h e a r e a o v e r t h e u n i t liemisphere
between 8-A8/2 and 8+A8/2, namely
COS (8-A0/2) - c o s (8+A8/2)1
l
where A8 i s t h e s p a c i n g i n 0 , w i t h p r o p e r t r e a t m e n t of t h e e n d p o i n t s a t 8=0
and 8=.rr/2.
S e v e r a l c r i t e r i a were used t o d e t e r m i n e t h e number of p o i n t s on t h e
i n t e g r a t i o n g r i d s p a n n i n g he d e f i n e d o c t a n t .
The f i r s t i s t h a t a p r e v i o u s
r e s u l t o b t a i n e d w i t h u s e of t h e powerful i n t e g r a t i o n r o u t i n e DOlFCF from t h e
NAG L i b r a r y (1984) be d u p l i c a t e d .
T h i s was t h e computation of $o f o r t h e
SIMraAD 70-kHz s p l i t - b e a m t r a n s d u c e r , which r e s u l t was p r e s e n t e d by Foote ( 1 9 8 7 ) R e p e t i t i o n o f t h e computations f o r s u b s t a n t i a l l y f i n e r g r i d s ensured t h e
second c r i t e r i o n o f numerical convergence.
I n b r i e f , s u b d i v i s i o n of t h e r a n g e
of 8 by 180 and t h a t o f $ by 16 gave r e s u l t s t h a t a g r e e d t o w i t h i n 0.02 dB
i n t h e w o r s t c a s e f o r r e s u l t s o b t a i n e d u s i n g s u b d i v i s i o n s o f 5000 and 200 f o r
9 and $, r e s p e c t i v e l y .
The c o m p u t a t i o n s were performed on a d i g i t a l computer with 3 2 - b i t
s i n g l e - p r e c i s i o n f l o a t i n g - p o i n t word s i z e . Computations f o r t h e s u b d i v i s i o n
180-16 were performed i n s i n g l e p r e c i s i o n , w h i l e t h o s e f o r t h e extreme
s u b d i v i s i o n 5000-200 were performed i n d o u b l e p r e c i s i o n .
RESULTS AND DISCUSSION
The r e s u l t s a r e p r e s e n t e d i n t h e t a b l e .
These a p p l y t o a medium sound
They may b e e x t r a p o l a t e d t o o t h e r sound speeds by use of
speed of 1470 m / s .
t h e r e l a t i o n (Foote 1987)
where c =l470 m / s .
0
Table 1. Nominal v a l u e s $o of t h e e q u i v a l e n t beam a n g l e and o t h e r performance
measures of t h r e e SIMFUAD split-beam t r a n s d u c e r s .
Frequency
(kHz)
2
Transducer
Beamwidth
mode
ES 3 8B
Single
38
0.0171
28.67
0.00798
31.97
-20.98
ES5
Narrow
38
0.00852
31.69
0.00374
35.26
-24.27
ES5
Wide
38
0.0169
28.70
0.00760
32.18
-21.19
ES120
Single
120
0.0340
25.67
0.0135
29.69
-18.70
IbdR
(sr) '
(dB)
(sr)
Jv
(dB)
(dB)
D1
.!bdR=y o
yo
1
~t i s emphasized t h a t t h e p r e s e n t work s p e c i f i e s nominal v a l u e s of t h e
s e v e r a l parameters.
I n g e n e r a l , f o r a p p l i c a t i o n t o f i s h and o t h e r marine
organisms, t h e t h r e s h o l d e f f e c t should b e considered. Use of t h e e f f e c t i v e
e q u i v a l e n t beam a n g l e $, however, w i l l n o t u p s e t use o f t h e nominal value $ho
i n c a l i b r a t i n g a c o u s t i c equipment. The p l a c e t o a p p l y $ i s t h e same a s t h a t
where t h e mean b a c k s c a t t e r i n g c r o s s s e c t i o n i s a p p l i e d . T h i s i s a t t h e d a t a
p r o c e s s i n g s t a g e where t h e v a l u e s of a r e a b a c k s c a t t e r i n g c o e f f i c i e n t a r e
converted t o p h y s i c a l q u a n t i t i e s of s c a t t e r e r d e n s i t y .
REFERENCES
Degnbol, P .
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ICES C.M. 1988/B:8.
10 pp. [mimeo]
1987. Dependence of e q u i v a l e n t beam a n g l e on sound speed.
1987/B:2. 6 pp.
[mimeo]
Foote, K. G .
C.M.
ICES
1988. Acoustic sampling volume v e r s u s e q u i v a l e n t beam angle.
Foote, K. G.
ICES C.M. 1988/B:5. 13 pp.
[mimeo]
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1989. Wideband beam p a t t e r n .
ICES C.M.
1989/B:2. 9 pp.
Foote, K. G.
1990. Designing an improved t r a n s d u c e r a r r a y geometry.
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[mimeo]
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