A new technique of acoustic mode filtering in shallow sea
Harish M. Chouhan and G.V.
Anand
Department
ofElectricalCommunication
Engineering,
IndianInstituteofScience,
Bangalore560 012, India
(Received26 May 1989;revised17August1990;accepted23 October1990)
A newtechnique
of normalmodefilteringin an isovelocity
channelusingan equispaced
linear
arrayis presented.
Filteringisachievedby steeringnullsin thedirectionsof arrivalof the
quasiplane
wavepairsassociated
with thenormalmodesthat areto berejected.A theoretical
analysisof theproblemis presented
for bothhorizontalandnonhorizontal
arrays.The
minimumnumberof array elementsand their weightingcoefficients
are determinedfor each
case.The null-steeringtechniqueoffersseveraladvantages
overothertechniques
of mode
filtering.The weightingcoefficients
are independent
of the array depth.A horizontalor
verticalarrayof 2M - 1 elements
with realweightingcoefficients
or a horizontalarrayof M
elements
with complexweightingcoefficients
is sufficient
to selectonemodefroma setof M
modes.There is no constrainton interelementspacing.The filterperformanceis equallygood
underall typesof bottomconditions.If a horizontalarray is deployed,thismethodof mode
filteringcanbe usedin a horizontallystratifiedchannelalso.The null-steeringmethodgives
perfectfilteringif theorientation
of thearrayandthemodalwavenumbers
areknownexactly.
An analysisof the sensitivityof the modefilter to errorsin the assumedvaluesof these
parametersis alsopresented.
PACS numbers:43.30.Wi, 43.30.Bp
k •(z) -- k • + •, (z).
INTRODUCTION
The acousticfield in a water channelcan be represented
by a sumof normalmodesof thechannel.The normalmode
parameters
aredependent
on the acousticproperties
of the
channelandthesourceparameters.Recentlytherehasbeen
considerableinterestin the efficientfilteringof the normal
modes.
•-Iø This is duc to the fact that modefilteringtechniquescanbe usedto determinethe acousticpropertiesof
thechannelfi
'4 On theotherhand,if theacoustic
properties
of the channelare known, thesetechniquescan be usedfor
source
Iocalizationfl
'øModefilteringhasalsofoundapplicationin oceanacoustic
tomography.
m
In a shallowwater channelof constantdepthh, constant
density
p, andsoundspeed
c(z) overlying
a bottomof densityPl, andsoundspeedc,(z), theacoustic
pressure
in the
far field of a point sourceof angularfrequencyo can be
written
as
p(r,z,t)= • pm(r,z)exp
--j or--
(5)
Time resolution
• as well as spatialprocessing
tech-
niques
u'•'• havebeenusedfor modefiltering.Spatialprocessing
techniques
useasteered
horizontal
arrayI• oraverticalarraywitha suitable
modefiltering
processor.•-•'•4'•
The
mode-matching
techniqueof modefilteringutilizestheorthogonalitypropertyof the normalmodeeigenfunctions
as
statedin Eq. (3). In thistechnique,a discreteverticalarray
of Q hydrophones
is used.With theshadingcoefficients
set
proportional
to 0• (z•), wherezq is thedepthof theqth
hydrophone,the outputof the array dueto the nth modeis
givenby
I,,,nr) =B q--I•
Substituting
forp,,(r,z,) fromEq.(2), weget
,
[ 2rr \ •/•
(1)
I,.,(r)=Ba•,,(z,)[k--•,r)
exp(•.r)
(2)
x q--I• •,,(Zq)½,,(zq).
where
pr•(r,z) = (2rr/k,•r)•/'-•,,(z,)½•(z)exp(jk,,r).
In Eq. (2) z, is the sourcedepth,z is the receiverdepth,r is
the range,and ½,,(z) are a set of orthonormaleigenfunctionssatisfyingthe condition
(6)
Comparison
ofEq. ( 3) andEq. (6) indicates
thatI,,, _•0 for
n • m. But thistechnique
hasseveraldisadvantages.
A large
numberof hydrophones
and theirjudiciouspositioningis
requiredto achievesatisfactory
filtering.Furthermorethe
arrayhasto spantheentiredepthwheretheeigenfunctions
nP -•½,(z)½,z)dz+
p•-'½•(z)•,z)dz=•5•,,.
(3)
haveanappreciable
amplitude.
Tindieetal.• haveproposed
(4)
of hydrophones.
But boththesetechniques
are sensitiveto
vertical displacementof the array. A new techniquethat
doesnot sufferfrom thesedrawbacksis presented
in the fol-
Defining
k(z) = o)/c(z),
the horizontalcomponentk,,and the verticalcomponent
•,, of the ruth modewavenumberare relatedas
735
J. Acoust.
Soc.Am.89 (2).February1991
a least-squares
methodthat doesnot requirea largenumber
lowing sections.
0001-4966/91/020735-10500.80 (Z01991Acoustical
SocietyofAmerica
735
I. FORMULATION
OF THE PROBLEM
•
sin(y..z•)sin(y
R= •• •Mwqa•,k.,r/
Considera linearhydrophonearray of2Q + 1 identical
omnidirectional
elements
in thefar fieldof a pointsourcein
an isovclocityshallowwater channel.The geometryof the
problemis shownin Fig. 1.
The array is locatedin the xz planemaking an anglea
q=
•
Xexp{j[k,•(r--x•cos/•)+ •/4]}.
M
=
The sourceis locatedat (xs,y,,zs), and
(7)
(13)
sion, we have
.....0.....Q-l,O, withxo=0.
/3'= tan- '(ys/x,)
=1
Interchangingthe orderof summationin the a•ve expres-
with the z axis and the coordinates of the elements are
(Xq,0,zq),q= -Q, -Q+I
-Q
(]4)
where
denotesthe sourcebearingangle.Denoting the horizontal
•
= a• (2•/k•r} •/•sin(•z• )exp[j(k•r+ •/4) ], (15)
rangeof theelement
q fromthesource
byr•, wehave(see
is the complexamplitudeof the ruth modeand
Fig. 1)
'}
2
r• = (r' + x• -- 2rx•cos/•)•/2,
(8)
wherer = roisthehorizontalrangeof thecentralelement.If
r>>(xa -- x_ a ), theprojected
arraylengthin thexy plane,
we get the approximation
r• = r--xq cos/•.
(9)
g,,= •
is the •ray gain factor for the ruth mode.
•e problemofm•e filtering,i.e., selectinga s•ified
normalm•e n w•le rejectingall theothers,canbestatedas
follows:find the sequenceof weightingcoefficients•,
= {m-a,m-o+,,.....too}such
that
If thearrayhasuniforminterelementspacingd, wecanwrite
x• =qdsinct, z•=qdcosa + zo.
(10)
Assumingthe array to be in an isovelocity
waterlayer,
thesoundpressurep•
at theqthelement
canbeexpressed
as
pq -•-
-- 2rr
i a m2[•k,•rq)
\msin
,,=,
(y,,z,)sin(y,•
Zq)
Xexp[j(k,•r• + rr/4) ],
(11)
where a., is the normalization constantfor the rnth mode,
andthe harmonictime dependence
is suppressed.
The array
output, obtainedby weightingthe hydrophoneoutputsand
summingthem,is givenby
a=
•
wqp•,
(12)
wheretheweighting
coefficients
wqmaybecomplex.
Using
a•m•sin(•z•)exp(-jk•x, cos•), (16)
g•=0,
m•n.
(17)
In an isoveloeitywater channel,•eh normal m•e is
fomed by the interference
of two quasiplane
wav• propagatingsymmetheallywith r•et
to the hodzont• plane.
Hens, a m•e can be reject• by steeringnullsof the array
r•n•
in the dir•tions of arrivalof the associated
p•r of
planewaves.Let the sequen• of weightingcoefficients
requir• to steernulls in the directionsof arrival of the plane
wavesass•iat• with the ruth modebe denotedby U•. In
order to filter the nth mode and reject all the othe•, it is
necessa•to designan array wh•e responsehasnullsin the
direc•onsor,rival of all of theunwant• modalplanewave
pairs. If the channelcontainsM modes,the sequenceof
weightingcoefficients
of the nth m•e filteristhereforegiven by the convolution of M--I
elementary sequen•
U•,m•n,
theapproximation
giv,en
byEq.(9) inthephase
termofEq.
w.=
( 11) andtheapproximation
rq_•r in theamplitude
term,we
(zs)
where * denotesconvolution.Expressions
for the elementary sequenceU• undervariousconditionsare derivedin
the followingsections.
can write Eq. (12) as
II. MODE FILTERING
s4'•LY
••,;
WITH A HORIZONTAL
ARRAY
For an equispaced
horizontalarray havingan interelement spacingd and locatedat depthZo,we have
z•=z o, xq=qd.
(19)
The array gain factor for mode m, obtained from Eq. (16),
can now be written as
h
gm=a,, sin(?',,zo)•
wqexp(-jk,,qdcosl•). (20)
The gain factor is dependenton the array depthzo and becomeszero ifz ois the depth at which the rnth modefunction
has a node. We assume this is not the case. Let the directions
FIG. 1. Geometryof the receiverarray and the source.
736
J. Aceoat. See. Am., Vol. 89, No. 2, February 1991
of arrival of the ruth modal planewaveswith respectto the
arrayaxisbedenotedby 0,,• and0,,2. For a horizontalarray
0,,] and 0,,2 are equal,and are givenby
H. M. Chouhanand G. V. Anand:Shallow water mode filtering
736
0,,, =0,, 2 =0,, ----cos-'[(k,,k)cos/3)].
(21)
Hencefilteringof moden canbe achievedby a (2M -- 1)
element
realsymmetric
arraywitha weightsequence
W•m
Hencemodem canberejected
bysteering
a singlenullof the
obtained
by
substituting
U,,
=
U•
n
(m
=
1
.....
n
-l,n
array response
in the directionOm.The arrivaldirections
+
1
.....
M)
in
Eq.
(18).
The
sequence
W•J
•
has
the
advanandhencetheinterarrivalangularseparations
arefunctions
of/3. For a sourcein the endfiredirection,i.e.,/3 = 0, we
have maximumseparationamongstthe differentarrivals.
With increasing/3,
thearrivalsgetmoreandmoreclustered.
tageof eliminatingphaseshifters(sinceall the weighting
coefficients
are real) at the costof increasingthe numberof
elementsin the array from M to 2M -- 1.
When the sourceis in the broadsidedirection, i.e., when
/3 = •r/2, all the arrivalanglesare equalto •r/2 and mode
filteringis not possible.
In orderto determinethe weightsrequiredto rejectthe
ruth mode,we considerthe array responseto modem. From
Eq. (20) weseethatg• = 0 if theweightswqarechosen
as
follows:
III. MODE FILTERING
In order to determinethe responseof a nonhorizontal
array with equispacedelements,we have to substituteEq.
(10) intoF.q.(16). The arraygainfactorfor modem isgiven
by
w•. •/w• --- - exp(jk,,dcos/3),forsomep,
we = 0,
forq•p orp q- 1.
g,. = • a,,,wesin[y. (zo+ qdcosa) ]
q
(22)
Hence,modem canberejectedby employinga two-element
arraywhoseweightsaregivenby thesequence
U,•, = {1, -- exp(jk,,dcos/3)},
(23)
wherethesuperscript
H denotesthat the arrayis horizontal.
The weightsequence
W•n of a horizontalnth modefilteris
obtainedby substituting
U,, = Un• (m = !,...,n -- !,n
+ 1.....M) inF.q.(18). SinceU•u contains
twoelements,
the
sequence
W•n contains
M elements.
The response
of a weightedequispaced
lineararrayto a
planewaveincidentat an angle0 with respectto the endfire
directionis givenby
B(O)= • Wqexp(--jqkdcos0).
WITH A NON-HORIZONTAL
ARRAY
Xexp( --jk,.qd sina cos/]).
In order to renderg,. = 0 for all z., we needat leastthree
weightingcoefficients
to be nonzero.It can be readilyverified that g,. = 0 if we assignnonzeroweightsto three consecutiveelementsp-- l,p,p + I sothat
Wp=l,
%+,
--
• sec(y,.dcosa)exp(jk,.d sina cos/3), (30)
andassign
zeroweightsto all theotherelements.
Hence,for
a slantedarray, theweightsequence
requiredto rejectmode
rn can be written as
US,,= ( -- -•sec(y,,d
cosa)exp(--jk,,d sina cosfi),
(24)
1,-- -•sec(y.,d
cosct)exp(jk,,,d
sinatcos/3)).
q
(31)
Hence the free-fielddirectivity function of the array whose
weighting
coefficients
aregivenby thesequence
U •nof Eq.
(23) is givenby
B•,(O) = I -exp[jd(k,• cos/3- kcos0)].
(25)
(29)
The responseof the array definedby Eq. (31) to a plane
waveincidentat an angle0 canbeobtainedusingEq. (24).
The responseis givenby
B,. (0) = 1 -- sec(y,.dcosa)cos(k,.d sina cos/3
As expected,thefree-fielddirectivityfunctionhasa null
at0•, = cos- •[ (k•,/k)cos/3], which
isthedirection
ofar-
- kdcos 0).
rivalof themodalwavesof modem. The free-fielddirectivity
(32)
functionof thenthmodefilterwithweightsequence
IV• is
The zerosof B,. are givenby
givenby
0,. =cos •[(l/k)(k,. sinacosl3ñy.,cosa)],
D,(0)= l-I B•,(O).
m
I
(26)
It hasbeenshownthat modem can be rejectedby employinga two-elementarraywith a complexweightsequence
(33)
which are preciselythe directionsof arrival (with respectto
the array axis) of the two modalwavesassociatedwith mode
1'/2.
Filtering of mode n Call be achievedby usinga (2M
U•. The samefunctioncanalsobeaccomplished
usinga
three-element
arraywith a realweightsequence
U •uRgiven
-- 1) element
arraywhoseweightsequence
W•.' isobtained
bysubstituting
U,. = U,'s•( m = 1.....n -- 1,n q- I,...M) in
by
Eq. (18).
U,•,•
tlu*tl n'
(27)
whereU • ødenotes
a sequence
whoseelements
arethecomplexconjugates
oftheelements
of U •. Substituting
Eq.(23)
in Eq. (27), we get
U• a = (1, -- 2 cos(k•dcos/3),1).
(28)
The elementary
weightsequence
U• definedby Eq.
(31) and the corresponding
sequence
W.w obtainedfrom
Eq. (18) areHermitian,i,e.,weightsof theelementsequidistant from the central element on either side are complex
conjugates
of oneanother.However,for a verticalarray, the
The array definedby Eq. (28) is real and symmetricand
anglea iszeroandtheweightsin thesequence
US.,become
real.Hence,fora verticalarray,thesequences
U• andW.s
steerszerossimultaneouslyin the directions0,, and •r -- 0•,.
are real and symmetric.Putting at = 0 in Eq. (31 ) and res-
737
J. Acoust.Sec. Am., Vol. 89, No. 2, February 1991
H.M. Chouhanand G. V. Artand:Shallow water mode filtering
737
caling,we canwrite the weightsequence
of the elementary
verticalarray for rejectingthe ruth modeas
U• = {1, -- 2 cos(y,,d),!}.
Convolutionof M-
(34)
I suchsequences
givesthe weightse-
quenceW,vof thenthmodefilter.Comparison
of Eq. (34)
with Eq. (28) indicatesthattheweightsequence
of thevertical array modefilter is similarto that of the horizontalend
fire array mode filter (/•=0), with the horizontal wave
numberk,• replacedby the verticalwavenumber
achievedwith short verticalarraysplacedanywherein the
channelif the null-steeringtechniqueis used.
Anothermajoradvantageof the null-steering
technique
stemsform the fact that the weightingcoefficients
are independentof themeanarraydepth.Hence,theperformance
of
the null-steeringmode filter is not affectedby variationsin
the arraydepth.It is pertinentto notethat it is verydifficult
to avoidvariationsin the arraydepthat seaand thesevariationsadverselyaffecttheperformance
of othertypesof mode
filters.
Thesequence
W•s defined
byEqs.(31) and(18) canbe
The null-steeringmodefilter performsequallywell underall bottomconditions,whereasthe mode-matching
filter
performspoorlyin a Perkeftschannelwherethelower-order
useeithertheweightsequence
W,n obtained
byconvolving modefunctionshave significantpenetrationinto the sedi-
usedfor modefilteringwith a horizontalarray (a -• rr/2)
also. However, for a horizontal array, it would be better to
theelementary
sequences
U• [Eq. (23)] or theweightse- ment.
quenceWn
nR obtainedby convolving
the elementary
seEven thoughthe analysisin Sees.II and III is basedon
quences
U•nR[Eq. (28)]. Thesequence
W,nhastheadvan- the assumptionof isovelocitywater channel,the isovelocity
tage of smaller array length (M elementsinstead of
constraintis requiredonly for a nonhorizontalarray. For a
2M -- 1), whilethesequence
W,nRhastheadvantage
ofreal horizontalarray,the elementary
weighting
sequence
U•
and symmetricweightingcoefficients.
IV. ADVANTAGES
OF THE NULL-STEERING
TECHNIQUE
Let w,q, (n = I,...M and q = 1.....Q) denotethe qth
weightingcoefficientof the nth modefilter, and let g,,• denotethe gain factorof the nth modefilter for the ruth mode
input. FollowingEq. (16), we can write
g,, = q• i a•,w,qsin[y,,(qdcosa
+zo)]
Xexp(--)k,,qdsinctcos/g).
(35)
givenby Eq. (23) can be usedto rejectthe mth modeeven
when the channelis horizontallystratified.The gain factor
g.... of the modefilter in a horizontallystratifiedchannelis
obtainedby replacingthefactora,, sin(y,,o) in Eq. (35) by
the appropriatemodefunction•Pm(Zo).
The weightingcoefficients
of a slantedarray modefilter
dependon theanglest• and/g,thewavenumbers)% and km,
andtheinterelement
spacingd [seeEq. (31 ) ]. However,it is
sufficientto knowthevaluesof theparameters
y,• andd for a
verticalarray [ Eq. (34) ], and the parametersk,, cos//and
d for a horizontalarray [Eq. (23) ]. The issueof sensitivity
of the null-steeringmode filter to errors in the assumedvaluesof theseparameters
isdiscussed
in Sec.VI.
The output of the nth modefilter is givenby
V. NUMERICAL
M
R, = •'• g,,A,,.
m--
I
(36)
Usingthenull-steering
technique,
to,qcanbechosen
insuch
a way thatg,,, = 0 for m =fin.The outputof the null-steering
modefilter is thereforegivenby
Rn = g,•,•`4n'
(37)
The outputcanbe madeequalto the modeamplitude.4, if
the weighting coefficientsare normalized so as to render
The foregoingdiscussionhighlightsthe fact that the
null-steeringtechniqueprovides,for the first time, a method
of achievingperfectmodefiltering (zero leakagefrom the
unwantedmodes) by employingeither a verticalor a horizontal (2M--1)-element equispaced array with real
weighting coefficients.Furthermore, this method docs not
imposeany constrainteither on the meanarray depthzo or
the interelementspacingd, exceptthat a horizontal array
filterfor thenth modeshouldnot belocatedat a depthclose
to a nodeof thenth mode.On theotherhand,someleakageis
invariablypresentwhenthe mode-matchingmethodusinga
verticalarrayor theleast-squares
methodusingeithera vertical or a horizontalarray are employed.To minimize the
leakage,the mode matchingand least squarestechniques
require the deploymentof a vertical array that spansthe
entire depth of the channel,whereasgood filtering can be
738
J. Acoust.Soc. Am., Vol. 89, No. 2, February1991
SIMULATION
The performanceof the null-steeringmode filter has
beentestedby computersimulationfor isovelocity
channels
with threetypesof bottomconditions,viz., a rigidbottom,a
pressure-release
bottom,and a fluid bottom (Pekerischannel), and it has been verifiedthat perfect filtering can be
achieved in each case. Details of the simulation
results for a
Pekerischannelare presentedbelow.
Numerical values assignedto the parametersin the
computationare as follows:water depth h = 50 m, source
depthZ, = 25 m, sourcefrequency
f= 60 Hz, soundspeedin
water c = 1500 m/s, sound speed in sediment c• = 2000
m/s,density
of waterp = 1000kg/m-•,density
of sediment
p, = 1100kg/m•. Themodaleigenvalues
y,, aretherootsof
theequation
•6
P,Y,nCOt(Y,nh)
+ p([4-•[1 -- (cica)'] -- 7a.,]
"• = O (38)
and the normalizationconstantsa,,, are givenby
Xsin2(y,,h)tan(y,,h)} •
(39)
For the assigned
parametervaluesthereare threepropagating modesin the channel,and the anglesbetweenthe directionsof propagationof the modalplanewavesand the horizontal plane are given by +_ 12.7375ø, _+26.0107ø, and
H.M. Chouhanand G. V. Artand:Shallowwater modefiltering
738
1.00
TABLE I. Verticalarrayweightingcoefficients.
ModeNo.
Weighting
coefficients
n
to,• = to, s
to,,., = tom
!
1.0000
-- 0.8157
2
1.0000
-- 1.6131
1.8510
3
1.0000
-- 2.6040
3.5363
1.9209
0.65
•
0.30
i
i
1.00
MO0•2
4- 39.9738ø. The field at the individual array elementsis
computedusingEq. ( 11).
FIG. 3. Comparisonof modefilter out-
putasa functionof rangeobtainedusing
the null-steeringtechnique(thick line)
and the mode-matching
teehnlque( thin
A. Vertical array
line).
A five-elementequispacedarray with interelement
spacingd = 10 m is setat a rangeof 1 km from the source.
The arraycenterislocatedat a depthzo = 25 m. The modal
wave arrival directionswith respectto the array axis are
given by 0• = 90øq- 12.7375', 02= 90*+ 26.0107',
03 = 90* q- 39.9738*.The corresponding
modefilterweighting coefficients
computedusingEqs. (34) and (18) are giv-
eninTableI. In TableI, w,qdenotes
theqthweighting
coefficientof the nth filter. The free-fielddirectivity functionsof
the modefiltersare shownin Fig. 2, If the filteringis perfect,
one expectsthe mode filter output to be inverselyproportional to the squareroot of ranger. Filter outputsfor each
mode indeedfollow the expectedvariation as shownby the
thick linesin Fig. 3. Also shownin Fig. 3 are the resultsof
filteringwhenthemode-matching
techniqueisusedwith the
same number of elements. In the latter results, one can see
0.30
I
1.00
0.•0
i
lOOO
•
1000
•
9ANOE (m)
B. Horizontal array
We have also considereda horizontal end-fire array,
with an interelementspacingd = 10 m, locatedat a depth
zo = 25 m. A three-element
horizontalarray is adequateto
filter one of the three modes. In this case, the mode filter
weightingcoefficients(Table II) are computedfrom Eqs.
oscillations
abouttheideal1/x/• variationdueto leakage (23) and (18) with/3 = 0, and the corresponding
free-field
from the unwanted modes.Also perfect filtering can be
directivityfunctionsare shownin Fig. 4. Ideal filter perforachievedusingthe null-steeringtechniquefor any choiceof
manceis obtainedin thiscasealso.Note that the weighting
valuesofzoandd. On the otherhand,the performanceof the
coefficients
are complexin thiscase.As statedin Sec.III, an
mode-matching
technique,for a givennumberof elementsin
array with real weightingcoefficientscan be designedby
the array,is dependenton the depthsat whichthe elements addingtwoelementsto thearrayandsteeringtwoadditional
are located.A properchoiceof the elementdepthsis neces- nulls.Therealsequence
W,t,
t• canbeobtained
byconvolving
sary to minimize leakage with the mode-matchingtech-
thecomplex
sequence
IVy,
t withitscomplex
conjugate.
Val-
uesof theserealweightingcoefficients
aregivenin TableIII.
VI. SENSITIVITY
FIG. 2. Free-fielddirectivity functions
of the verticalarray modefilters.The responseis normalizedwith respectto the
amplitudein the directionsof arrival of
ANALYSIS
The weighting
coefficients
of thenull-steering
modefilter dependuponthe interelement
spacingd, the wave
numbers
y,. andkm,andtheangles0,. between
thearray
axisandthemodalplanewavearrivaldirections.
Theangles
0,., in turn,depend
ontheangles
atand/3.Hence,anydiscrepancy
between
theactualandtheassumed
values
of the
parameters
listedabove
candegrade
theperformance
ofthe
modefilterdueto the leakageof unwantedmodesinto the
the filtered (selected) modal waves.
TABLE [l. Horizontalend-firearray weightingcoeflicienls.
Three-element
array with complexweightingcoefficients.
Mode No.
I
2
3
739
J. Acoust_Soc. Am., Vol. 89, No. 2, February 1991
Weightingcoefficients
i.0000
1.0000
1.0000
0.9827 +jl.7101
1.1189+jl.5742
1.4061+jl.4092
-- 0.5035 +20.8640
-- 0.3287 +20.9444
-- 0.0023 +jl.0000
H.M. Chouhanand G. V. Artand:Shallow water mode liltenng
739
FIG. 4. Free-fielddirectivity functions
of thehorizontalarraymodefilters.The
õ
responseis normalizedwith respectto
the amplitudein the directionof arrival
of the filtered (selected) modal wave.
RLT ß (OEO#Ef:S)
modefilter output.The valueof the parameterd may be
FIG. 5. Variationof modeleakagefactorsA,.,, of the verticalarray as a
knownwith sufficientaccuracy,but errorsin the assumed functionof tilt 6 in thexz plane:d = 10m, zo= 25 re,f= 60 Hz.
valuesof a and/3 mayoccurdueto practicaldifficulties
in
maintaininga steadyorientationof the array. Errors may
planeof tilt is vertical,i.e., the planeof tilt is thexz plane
also arise in the assumed or estimated values of the environwithat= •r/2 ñ 6,/3 = 0. In Fig. 8, theplaneof tilt ishorimentalparameters
y,, and k,,. Sensitivityof the null-steer- zontal,i.e., the planeof tilt is thexy planewith at=
ing modefilter to theseerrorsis discussed
below.
/3= 6. Figures5-8 indicatethatthemodefilterisconsiderWe definethe modeleakagefactor/t,,,, as
ablymoresensitive
to tiltsin thexz planeascompared
totilts
= 10log<Ig..
whereg,,• isthegainfactordefinedin Eq. (35). The quantity A,,• denotesthe leakageof modem into the outputof the
mode-n filter. For null-steeringmode filter, A,•, = 0 for
m • n, if theparameters
d, at,/3,y,•, andk,, areknownaccurately.
The effectof tilt of a verticalarray from its assumed
verticalpositionis shownin Figs.5 and 6. In thesefigures,
the leakagefactorsA,,• of the verticalarray consideredin
Sec.V areplottedasfunctionsof the angleof tilt 6. In Fig. 5,
the planeof tilt coincideswith the verticalplanecontaining
thesourceandthe arraycenter,i.e., theplaneof tilt is the
planewith at = 6,/3 = 0. In Fig. 6, theplaneof tilt is perpendicular to the vertical planecontainingthe sourceand the
array center,i.e., the planeof tilt is theyz planewith
/3 = ½r/2.Similarplotsfor the horizontalend-firearray consideredin Sec. V are shownin Figs. 7 and 8. In Fig. 7, the
TABLE IIL Horizontalend-firearrayweightingcoefficients.
Five-element
arraywith realweightingcoefficients.
ModeNo.
Weightingcoefficients
TILT & (OEIIqL•S)
I
2
3
1.0000
1.00043
1.0000
1.9655
2.2379
2.8121
2.8833
3.0728
3.9585
FIG. 6. Variation of modeleakagefactors/l,,,,of the verticalarray as a
functionof tilt • in theyzplane:d = 10m, zo= 25 m,f= 60 Hz.
740
J.Acoust.
Soc.
Am.,
Vol.89,No.2,February
1991
H.M.thouhah
andG.V.Artand:
Shallow
water
mode
filtering
740
eitherin theyzplane(in thecaseof a verticalarray) or in the
xy plane (in the caseof a horizontalarray). This difference
in sensitivityisdueto thefactthat a tilt of thearraythrough
an angle$ in thexz planecauses
theangles•,• betweenthe
array axis and the directionof travel of the modal plane
wavesalsoto changeby the sameamountrS.On the other
hand,thesamedegreeof tilt eitherin theyzplaneor in thexy
plane causesthe anglest9,, to changeby muchsmaller
amounts.
As alreadystated,the weightingcoefficients
of thevertical array modefilter dependon the environmentalparametersy,, only, while thoseof the horizontalarray modefilter
dependon the environmentalparametersk,,only. Leakage
factorsof the vertical and horizontal endfire array consideredaboveareplottedasfunctionsof the percentage
error in
?%andkin,respectively,
in Figs.9 and 10.It appears
thatthe
verticalarray is lesssensitiveto error in the environmental
parametersthan the horizontalarray. But this conclusion
cannotbe generalizedsincethe sensitivityalsodependson
the choiceof the otherparameterssuchasd andzoThe sensitivityof the modefilter to errorsin orientation
or environmentalparameterscan be minimizedby an optimal choiceold andzo.The variationof sensitivitywith d and
zoisillustratedin Figs.11-16.In thesefigures,
)t, isa measureof totalleakageof all unwantedmodesintotheoutputof
•32
-25
•
1
I A3•i
2
mode-n filter, defined as
T•LT & (DEGREES)
FIG. 7. Variationof modeleakagefactorsA,,,. of the horizontalend-fire
arrayasa functionof till 6 in thexz plane:d = 10m, z. = 25 m,f= 60 Hz.
2. = 1Olog
g,,,.l2
(Ig,,,,I) '
(4])
In Eq. (41 ) anincoherentsummationispreferredto a coherent summationsincethe phaseof g,,, would be randomly
distributed.For a verticalarray, the variationof/t, with d
-12
-17
•32
-32
-37,
-5
TILT • (DEGREES)
FIG. 8. Varia6onof modeleakagefactorsA,,,,,of the horizontalend-fire
arrayasa functionof tilt t5in thexy plane:d = l0 m, zo -- 25 re,f= 60 Hz.
741
J. Acoust.Soc.Am.,Vol. 89, No. 2, February1991
-25
0
2.5
PERC•-NT œRRO• IN gm
FIG. 9. Varialionof modeleakagefactorsA,,., of the verticalarrayas a
functionof percentage
errorin y.,: d = 10m, z. - 25 m,f-- 60 Hz.
H.M. thouhah and G. V. Artand:Shallowwatermodefiltering
741
-11
-H
I'
s
I
lO
IHT•rR.ELIrMENTDISTANCE
d (METERS)
FIG. 12. Variationof modeleakagefactors/l,, of the verticalarray as a
functionof interelement
spacingd for 1% error in all y,.'s: z. = 25 m,
-5
-2.5
0
2-5
PERCENT ERROR IN km
FIG. 10. Variationof modeleakagefactors/[,,,,of the horizontalend-fire
array asa functionof percentageerror in k.,: d = 10 m, z. = 25 re,f= 60
Uz.
for a 1øtilt in thexz planeandfor 1% errorin all they,,'s is
shown in Figs. 11 and 12, respectively.In both cases,the
largestof the3.,'s decreases
monotonicallyasd is increased.
Numericalcomputations
indicatethat this resultis true in
generalfor all verticalarrays.Hence,the interelementspacingd of a verticalarrayshouldbemadeaslargeaspossible
to
minimizeleakage.Sincethe numberof elementsin thearray
is 2M-- 1, the maximum possiblevalue of d is equal to
h/2(M -- 1), whereh is the depthof the channel.
The dependence
of/i,, on d andZois considerably
more
f= 60 Hz.
in thexz planeare shownin Figs. 13 and 14.Thesefigures
indicatelow sensitivityto tilt for d•_ 10 or 22 m andZoo_10
m. Extensive numerical computationsindicate that this
choiceofd andZois closeto optimum.Optimumvaluesare
thosevaluesof d andzo for which the largestof the ,t,'s is
minimum. Unlike the caseof a vertical array, there is no
upperlimit on the valueof d due to physicalconstraintsin
the caseof a horizontalarray. But practicalconsiderations
rule out the useof very largevaluesold.
Typicalplotsof,i,vs d and•[.vs zofor a horizontalendfirearraywith 1% errorin all the k,,'s areshownin Figs. 15
and 16.It isseenthat thechoiceof d= 10or 22 m andzo_ 41
m offerslow sensitivityto errorsin ko,.
complexin thecaseof a horizontalarray.TypicalplotsofA,
vsd andA. vs% for a horizontalend-finearray with a 1' tilt
-4
o
$
lO
NTER-ELEMENT SPACINI], d(METERS)
FIG. 11. Variation of modeleakagefactors,t,. of the verticalarray as a
functionof interelementspacingd for l* tilt in the xz plane:&j = 25 m,
f= 60 Hz.
742
J. Acoust.
Soc.Am.,Vol.89,No.2, February
1991
MEAN ARRAYDEPTH Zo (METERS)
FIG. 13. Variationof modeleakagefactorsA,, of the end-firehorizontal
arrayasa functionof arraydepthzo for 1' tilt in thexz plane.d = 10 m,
f= 60 Hz.
H.M. Chouhan
andG.V.Anand:
Shallow
watermodefiltering
742
1`1
-1S
I
15
•0
•
s
INTER-ELEHEHTOIST,kI,
KEd (METERS)
Io
1•
•o
INTER-ELEMENTSPACINGd(kt'TERS)
FIG. 16. Variationof modeleakagefactorsAo of the end-firehorizontal
array as a functionof interelementspacingd for 1% error in all k,,'s:
FIG. 14. Variationof modeleakagefactors2.,, of the end-firehorizontal
array as a functionof interelementspacingd for 1ø tilt in the xz plane:
zo = 10 re,f= 60 Hz.
z•,-- 41 m,f-- 60 Hz.
VII. CONCLUSIONS
can be reducedto M if complexweightingcoefficients
are
The null-steeringtechniqueof modefiltering presented
in this paperoffersseveraladvantages.
The foremostis that
the array weightingcoefficients
do not dependon the depth
at whichthearrayisdeployed.Hencetheperformance
oft he
filter is not affectedby changesin the positionof the array, as
long as its orientationis maintained.Secondly,there is no
constrainton theinterelementspacingandit isnot necessary
to usean array spanningthe entire depth of the channel.
Equallygoodperformance
canbeobtainedwith all typesof
bottom.For example,unlikethe mode-matching
technique,
perfectfiltering is achievedevenwhen the mode functions
havea significantpenetrationinto the bottom.
A null-steeringmodefiltercan useeithera verticalor a
horizontalarrayof2M - 1 elementswith realweightingcoefficients.For a horizontalarray, the number of elements
used. The horizontal array can be used in a horizontally
stratifiedchannelalso, whereasthe use of a null-steering
vertical array mode filter is restrictedto isovelocitychannels.When a horizontalarray isused,careshouldbetakento
ensurethat ( 1) the sourceis in or near the end-fire direction,
and (2) thearray isnot deployedat a depthcloseto the node
of the moderequiredto be filtered.
A null-steeringverticalarray requiresthe knowledgeof
theverticalwavenumbersYm,whilea null-steering
horizon-
tal array requiresthe knowledgeof the horizontalwave
nt/mberskr•. In an unknownchannel,the horizontalwave
numbersk,• may be estimatedusing Prony'smethod as
shown
byShang
etal.4Twoothermethods
ofestimating
k,•,
usingshortarrays,are described
in the Ph.D. thesisof the
first author.17
Errors in the assumedarray orientationor the assumed
modal wave numberscan lead to degradationof the filter
performance.
The filterisconsiderably
moresensitive
to tilts
in thexz plane(verticalplanecontainingthesourceandthe
arraycenter)thanto tilts in theyzor xy planes.The sensitivity to errorsin theassumed
valuesof at,/3,y,,,and k,, canbe
minimizedby an optimumchoiceof the interelementspacing d and array depthZo.
Details of the hardware implementationof the mode
filteringprocessor
basedon the null-steeringtechniqueand
its usein modelstudiesin a Laboratorytank canbefoundin
Ref. 17.
ACKNOWLEDGMENT
This work was supportedby the Departmentof Electronics, Government of India.
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FIG. ! 5. Variation of mode leakagefactorsA,, of the end-firehorizontal
array as a functionof array depthz•,for 1% error in all k,,'s: d = 10 m,
f= 60 Hz.
743
J. Acoust.Soc. Am., Vol. 89, No. 2, February 1991
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Shallow
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mode
filtering 744