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I n t e r n a t i o n a l C o u n c i l f o r t h e E x p l o r a t i o n o f t h e S e a

C ,M, 1987/B : 2 S e s s , 0

F i s h C a p t u r e Committee

DEPENDENCE OF EQUIVALENT BEAM ANGLE ON SOUND SPEED

Kenneth G . F o o t e

I n s t i t u t e o f Marine R e s e a r c h 5 0 2 4 B e r g e n , Norway

ABSTRACT

The e q u i v a l e n t beam a n g l e i s computed n u m e r i c a i l y , a c c o r d i n g t o i t s r i g o r o u s d e f i n i t i o n , f o r e a c h o f two t r a n s d u c e r s . T h e s e a r e t h o s e u s e d w i t h t h e SIMRAD s p l i t - b e a m e c h o s o u n d e r s a t 38 a n d 70 kKz. The r e s u l t s a r e e x p r e s s e d a s f u n c t i o n s o f t h e medium s o u n d s p e e d .

L ' a n g l e i i q u i v a l e n t du f a i s c e a u e s t c a l c u l i i n m g r i q u e m e n t , s e l o n s a d g f i n i t i o n r i q o u r e u s e , p o u r chacun d e s deux t r a n s d u c t e u r s , u t i l i s g s a v e c l e s s o n d e u r s SIMRAD f a i s c e a u s c i n d g e n 38 e t 70 kHz, Les r g s u l t a t s s o n t exprirniis s o u s La forme de f o n c t - i o n de l a v i t e s s e moyenae du s o n ,

The eqniv,llen"lkjearr, ELIICJLE i s a k e y ( ~ u . a r l C r t ~ in b o t l i the e c h o - c o u n t i n g

- . . . -

and e c h o i n t e c j r z ~ i o r i ifre Lhods .)L ir;c:cic;l*r i.ricj : i s ; > :icn:-,.i L/ (l i:nrri~;l:c-:s r984a) ,

TXi.s i s evident f r o m tile basic: mathernatfr:ai c - x p r e s s i o r ~ s for if3 sh density.

'I'hese n o r m a l i z e a cc!i;it o r echo i.nt:ecjra.i t,o Lhc ob:;ervati.oxi or sampl.ing voliirne , S i n c e ^{t i 7} is volume is d i rectly propori-.iona! to the e q u i v a l e n t . beam a n g l e , the dependence of f i s h density o n i t i s f i r s t - - o r d e r ( F o o t e e t a l , 1.987)

.

^An equivalent beam anqLe v ~ i l l also have ?I f i r s t - o r d e r c C f e c i . c?ri <Fsrl c2k?r,Lsi 1 + T9jb-f :: 3 ",:I-,-. ,.,., "-rjc ' . 'in.7 ^;C i s t r i < j j i - d ] ? e r - e ,

The p a r t i c u l a r aim o f t h i s work i s t o s p e c i f y t h e d e p e n d e n c e o f t h e e q u i v a l e n t beam a n g l e on t h e medium sound s p e e d , This w i l l i n d i c a t e t h e n e e d f o r a d j u s t i n g t h e a n g l e v a l u e t o c h a n g i n g h y d r o g r a - p h i c c o n d i t i o n s .

METHOD

T h e e q u i v a l e n t beam angle i s d e f i n e d as t h e i n t e g r a l o f t h e p r o d u c t o f f a r f i e i t 1 transmit a n d receive beam patterns o v e r all s p a c e . I f t2ae

r e s p e c t i v e beam p a t t e r n s a r e w r i t t e n by ( 8 ,$) and bR ( 8 ,$)

,

where (8 ,$) i s t h e g e n e r a l s p a t i a l d i r e c t i o n , t h e n t h e e q u i v a l e n t beam a n g l e

Y

i s

rick

¹⁹⁷⁵⁾

T h i s d e f i n i t i o n i s q u i t e g e n e r a l , f o r it a l l o w s f o r t h e p o s s i b i l i t y o f u s i n g s e p a r a t e t r a n s m i t t i n g and r e c e i v i n g t r a n s d u c e r s o r i e n t e d i n d i f f e r e n t

d i r e c t i o n s ,

For o r d i n a r y a p p l i c a t i o n s i n f i s h e r i e s a c o u s t i c s , t h e same t r a n s d u c e r performs b o t h f u n c t i o n s , I n t h e absence of i n t e r - e l e m e n t c o u p l i n g , which i s u s u a l l y a Low-frequency e f f e c t o f l i t t l e consequence f o r t y p i c a l s u r v e y f r e q u e n c i e s and t r a n s d u c e r s i z e s , b =b T h i s i s t h e c a s e e n v i s a g e d h e r e ,

T R *

The beam p a t t e r n s o f t h e s u b j e c t SZMRAD s p l i t - b e a m t r a n s d u c e r s have been d e f i n e d i n an e a r l i e r r e p o r t t o t h e Council Meeting (Foote 1 9 8 6 ) . Both t h e 38 and 70 kKz t r a n s d u c e r s a r e examples o f p l a n a r a r r a y s o f i d e n t i c a l c i r c u l a r e l e m e n t s . For a l i k e a r r a y of n e l e m e n t s of c o n s t a n t r a d i u s a , t h e beam p a t t e r n i s d e f i n e d f o r 8

E

[0,.ir/2] a s

2 J l ( k a s i n 8 ) l n 2 b ( 8 , 4 ) =

I--- ^-

_n C e x p ( i k - r ) j

,

k a s i n 8 j=L --l

where k i s t h e wavevector and r , i s t h e p o s i t i o n o f t h e c e n t e r of t h e j - t h element. The wavevector s p e c i F l e s t h e f i e l d d i r e c t i o n k=k/k o r p o i n t

( s i n 8 c o s $I, s i n 8 s i n $, c o s 8 ) on t h e u n i t s p h e r e , 1t; magnitude i s k=

2n/h=2.irv/cl where X i s t h e wavelength,

v

i s t h e f r e q u e n c y , and c i s t h e medium sound s p e e d ,

D e t a i l s o f t h e geometry, o r q u a n t i t i e s a a n d

I r . ,

^{j = 1 , 2,}

.

^{, n }}

,

^a-re

g i v e n a s f o l l o w s , "-3

30 kHz t r a n s d u c e r , a=L7,5 ml, n-68, The elements a r e packed as

--

^p----

dense1.y a.s possibl.e '171 a 'nexcaigorl;:l a.rr.ay, T h e n ~ x n h e r o f cr1.emen.t~ per row folLaws t h e sequence: 6 , '7, 8 , 9, 8, 9, 8, 7 , 6 * The r n k d d L e row t h u s Lacks the end e l e m e n t s of the 0 r d - i - n a r y hexagonal s i r z q b e . - k ~ e a ~ n transducer,

70 kHz t r a n s d u c e r ,

---

a=8,5 m m , n=36, The a r r a y geometry i s s q u a r e , The inter-elemerlt d i s t a n c e a?.ong rows a n 2 cr;':r~rnnr; as mcasllred hetw=er) element c e n t e r s is L 8 mm,

In o r d e r eo s t u d y t h e depcnilerzce of ?I' on c:, tile i n t e g r a t i o r : ! i x i equatiori (1) i s performed by rneans of a genera:! mul.ti.di.mensionaL i n t e g r a t i o n r o u t i n e , DOLFCF, i n t h e NAG L i b r a r y (19C4), Corxstraints on t h e computations were t h a t a t l e a s t l000 p o i n t s be used i n each i n t e g r a t i o n and t h a t t h e r e l a t i v e

a c c u r a c y be b e t t e r t h a n 0 . 0 0 1 , The beam p a t t e r n s p e c i f i e d i n e q u a t i o n ( 2 ) was computed f o r t h e p a r t i c u l a r p a r a m e t e r s o f t h e 38 and 70 kKz t r a n s d u c e r s i n a f u n c t i o n s u b r o u t i n e c a l l e d by BOlPCF, The beam p a t t e r n was assumed t o v a n i s h f o r 8 > n / 2 .

RESULTS

These a r e g i v e n i n t h e t a b l e .

E q u i v a l e n t beam a n g l e Y a s a f u n c t i o n o f medium sound s p e e d c f o r two SIMRAD s p l i t - b e a m t r a n s d u c e r s ,

38 kHz t r a n s d u c e r 70 kHz t r a n s d u c e r c (m/s)

Y

10 l o g Y Y 1 0 l o g

Y

1400 0,00859 -20.66 0.01550 - 1 8 - 1 0

1410 0.00871 -20 ^a 60 0,01570 - 1 8 - 0 4

1420 0.00883 -20,54 0,01590 -17 - 9 9

L4 30 0,00896 -20.48 0.01612 -17 - 9 3

1440 0

-

⁰⁰⁹⁰⁸ - 2 0 - 4 2 0,01634 -17.87

1450 0.00921 -20.36 0,01657 - 1 7 , 8 1

1460 0,00934 -20.30 0.01679 -17,75

1470 0,00947 -20.24 0,01702 -17 - 6 9

1480 0.00960 -20.18 0.01726 -17.63

1490 0.00973 -20.12 0.01749 -17,57

1500 0,00986 -20,06 0.01772 -17.52

1510 0.00999 -20.00 0.01797 -17.45

1520 0.01012 -19.95 0.01821 -17.40

1530 0.01026 -19.89 0.01845 -17.34

1540 0.01039 -19.83 0.01869 -17.28

1550 0.01053 -19.78 0.01894 -17.23

1560 0.01066 - 1 9 - 7 2 0 01.918 -17.17

DISCUSSION

The results show a nearly c o n s t a n t 0-06 dB change in the quantity bO loq far each change In c by 10 m/s, Thss can be understood throtaqh small -angle approx?rnat~ ons Lo W , For C I CCUI a f o r r e c t a n g u l ar ?rvays, with 8, and Oh rneasurlnq t h e a n g l e s rn degrees from the a x r s t o t h e - 3 d B levels of one-way bean1 pattern.: sn p e r p e n d l cul au p1 anes or c i ~ t s ,

10 I.og ' 1 ' - 1.0 log ( ? 3 ^{) .-} 31-6 (3a)

a b

as g i v e n by ijrick (15375) and used by, among o-l--hers, Ona and Vestnes (1985) Another a p p r a x i m a t i o n t h a t i s i n t h e s ; m e s p i r i t but is applicaS3Le to more g e n e r a l two-dimensional a r r a y s i s

1 0 l o g Y 20 l o g 0

-

3 1 - 6

,

a v e

where 0 i s t h e a v e r a g e a n g l e at. -3 dB l e v e l , a s measured from t h e a x i s , o v e r a l ? v S i r e c t i o n s .

Evaluation of Y thro11gh equations (3a) and (3b) yields similar results.

The values from equation (3a) are generally 0.02 dB less than the respective values from equation (3b) for the 38 kHz transducer and 0.08 dB less for the 70 kHz transducer. Both approximations yive values for Y which are about 0.1 dB higher than those determined by nunierical evaluation of equation (l).

At the -3 dB arlyles for circular and rectangular arrays,

kb sin

B

= constant

,

^{( 4 )}

where b is a characteristic dimension oE the array. For the generally large values of kb for typical transducers in use on survey vessels,

hence

Thus the change in Y due to a change A c in c is

10 log Y 2

-

l0 log W -' 20 log

-

L2 ^{= ac} l 2 O l o g ( l + - )

.

C

I

^C 1..

For small. Ac,

iEenc@, i f d c " - = I 0 i n / s for t h e croi1.nd speed c=-? 475 m j s ,

T h e observation froiri the i-ab11iatet3 result..; is I - ~ L I S confirmed

It is also to be noted that the Listed values are absolute, These are moreover consistent with the manufacturer" test measurements. Helge

Bodholt, S I M R A D Subsea, has reported on test measurements of Y for the 38 kHz split-beam transducer in a personal communication, For the last 17 transducers measured up to the time of the inquiry, the average value was -20-5 dB. The datum for each transducer was based on measurements of the beam pattern in each of two orthogonal planes and use of the small-angle f 0 ~ m u I a in equation ( 3 % ) - The accuracy of the laboratory measurements of Y is estimated by Bodholt to be 5 0 . 5 dB,

A d d i t i o n a l c o n f i r m a t i o n o f t h e a b s o l u t e v a l u e s i s p r o v i d e d by Hood (1987) and Reynisson (1987) , Hood measured a s p l i t - b e a m t r a n s d u c e r i n i t s towed-body housi-ng i n a l a b o r a t o r y t a n k , Use o f t h e -3 dB beam p a t t e r n a n g l e s i n e q u a t i o n (3a) g i v e s l 0 l o g ?=-l9.7 dB, R e y n i s s o n k measurements o f a hull-mounted s p l i t - b e a m t r a n s d u c e r when d r i v e n s o l e l y by t h e i n t e r n a l t r a n s m i t t e r d e t e r m i n e s l 0 l o g ?=-20-1 dB.

The v a l u e o f ? used by t h e I n s t i t u t e o f Marine Research, Bergen, w i t h i t s s p l i t - b e a m t r a n s d u c e r s i s -19.6 dB, D i f f e r e n c e s between t h i s and l a b o r a t o r y measurements and t h e o r e t i c a l computations might b e a t t r i b u t e d t o t h e e f f e c t s o f mounting and housing (Simrnonds l 9 8 4 b , Ona and Vestnes 1 9 8 5 ) , Can t h e s e be computed?

I t i s i m p o r t a n t t o n o t e t h a t r e p e t i t i o n of t h e p r e s e n t n u m e r i c a l computations o v e r much more r e s t r i c t e d r a n g e s of

B

t h a n t h e p r e s e n t range [ 0 , n / 2 ] g i v e s a l m o s t i n d i s t i n g u i s h a b l e r e s u l t s . I n p a r t i c u l a r , d e f i n i t i o n of non-vanishing b by e q u a t i o n ( 2 ) f o r B S [ 0 , n / l 8 ] g i v e s t h e same r e s u l t s t o w i t h i n 0 - 0 1 dB f o r t h e 3 8 kHz t r a n s d u c e r , I n t h e c a s e of t h e 70 kHz t r a n s d u c e r , t h e r e s u l t s a g r e e t o w i t h i n 0.03 d13 f o r 0 E Eorn/91. Thus whatever t h e e f f e c t s of t h e mounting and housing a r e , t h e s e must be s i g n i f i c a n t w i t h i n t h e c e n t r a l p a r t o f t h e beam, i , e , , i n t h e main l o b e .

CONCLUS IONS

The p r e s e n t t h e o r e t i c a l computations s p e c i f y t h e dependence o f t h e e q u i v a l e n t beam a n g l e Y f o r two SIMRAD s p l i t - b e a m t r a n s d u c e r s on t h e medium sound speed c i n m e t e r s p e r second t h r o u g h t h e s e e q u a t i o n s : l 0 Log

Y=

-20.66

+

0*0059(c-1400) f o r t h e 38 kBz t r a n s d u c e r and l 0 l o g Y=-18.10

+

Oe0059(c-14001 f o r t h e 70 kHz t r a n s d u c e r . The c o e f f i c i e n t o f t h e sound s p e e d term, 0,0059, i s e x a c t l y t h a t e x p e c t e d from s i m p l e t h e o r e t i c a l c o n s i d e r a t i o n s .

Since Y increases by 0,06 CB f o r iricreases i r a c by L 0 m/s, ordinary cruise applications w i l " l not require adjustment of Yf from t h e i n i t i a l v a l u e , T h i s s h o u l d , however, reflect the la-rge-scale hydrography of the survey region, e - g , , sea water or fresh water, A r c t i - c or tropical conditions, yiven a reference value of ! 'l for the transducer when c n r l f i y u r e d for u s e , P r e s e n t w o r k indi.cates that Y cae; be de-(leminerI to wi.th-i.n *O,I dB (Sirnmoa~ds 1.964a, Qna and Vestnes 1985, Reynisson 29815, Eiood 19891, and t i h i s shou1.d be the aim of researcklers.

c?tl.xer i n f l u e n c e s on Y , such as t h o s e o f aihienir. n o 3 se or r e v e r b e r a t i o n in masking w e a k e r tarqers, require nitentjon,

E, Ona i s thanked f o r reviewing t.he m a n u s c r i p t ,

REFERENCES

F o o t e , K. G . 1986, D i g i t a l r e p r e s e n t a t i o n of s p l i t - b e a m - t r a n s d u c e r beam p a t t e r n s , ICES C , M , / B : Z , 7 pp. [mimeo]

F o o t e , K . G . , Knudsen, B. P., V e s t n e s , G . , MacLennan, D . N . , and Simmonds, E . J. 1987. C a l i b r a t i o n o f a c o u s t i c i n s t r u m e n t s f o r f i s h d e n s i t y e s t i m a t i o n : a p r a c t i c a l g u i d e . ICES C o o p e r a t i v e R e s , Rep. 1 4 4 , 69 pp, Hood, C. R . 1987. Measurements o f a s p l i t beam t r a n s d u c e r . I n t e r n a t i o n a l Symposium on F i s h e r i e s A c o u s t i c s , S e a k t l e , Washington, 22-26 June 1987, 2 3 p p , Emimeo]

NAG L i b r a r y . 1984, NAG FORTRAN L i b r a r y Manual, Mark 11, Vol. 1, N a t i o n a l Algorithms Group, Oxford,

Ona, E . , and V e s t n e s , G. 1985. D i r e c t measurements o f e q u i v a l e n t beam a n g l e on hull-mounted t r a n s d u c e r s , ICES c . M . / B : 4 3 , 6 pp. [mimeo]

Reynisson, P . 1985. A method f o r measuring t h e e q u i v a l e n t beam a n g l e o f hull-mounted t r a n s d u c e r s , I C E S C .M. / B : 4 , 1 3 pp. [mimeo]

Reynisson, P . 1987, Measurements o f t h e beam p a t t e r n and compensation e r r o r s o f s p l i t - b e a m echo s o u n d e r s . I n t e r n a t i o n a l Symposium on F i s h e r i e s A c o u s t i c s , S e a t t l e , Washington, 22-26 J u n e 1987, l 6 pp.

[mimeo ]

Simmonds, E . J. L984a. A comparison between measured and t h e o r e t i c a l e q u i v a l e n t beam a n g l e s f o r seven s i m i l a r t r a n s d u c e r s . J, Sound Vib.

97: l17-128,

S i m o n d s , E , S. L984b, The e f f e c t of mountinq t h e e q u i v a l e n t beam a n g l e o f acoustic survey t r a n s d u c e r s , TCES C , M , / I 3 : 3 2 , 5 pp. [rnirneo]

Urick, R, 9, 1975, Principles of underwater sound, Second e d i t i o n , McGraw-Hill, New York, 384 pp,