Imaging ocean-basin reverberation
Nicholas
via inversion
C. Makris
Naval ResearchLaboratory, Washington,DC 203 75
(Received5 January 1993;acceptedfor publication7 April 1993)
A techniquefor imaging large areas of the ocean floor by inverting reverberationdata is
presented.Acoustic returns measuredon a horizontally towed line array have fight-left
ambiguityabout the array'saxis and decreasingcross-range
resolutionfor off-broadsidebeams
and moredistantscatteringsites.By optimaluseof data takenwith differingarray locationsand
orientations, fight-left ambiguity is eliminated and resolution is maximized. This is
accomplished
via a globalinversionusingthe array'sknownresolvingproperties.The outputof
the inversion can be optimally resolved reverberation,scatteringcoefficient,or physical
propertiesof the seafloor,dependingupon the formulation.Simulationsare presentedto show
the generalsuperiorityof this approachto data averaging.
PACS numbers: 43.30.Gv, 43.30.Pc
INTRODUCTION
Our goal is to image the oceanbasin using acoustic
reverberationdata. When operatingin bottom-limitedenvironments, active sonar systemsmeasure discrete backscatter returns from the seafloor. The nature of these re-
turns depends upon transmissionloss as well as the
morphologyand geo-acoustic
propertiesof the scattering
site. By two-way travel-time analysis,the range of particular scatteringsitescan be deducedfor respectivereturns.
Since low-frequency active systems typically measure
acousticreturns with a horizontally towed line array, the
azimuth of scatteringsites can be determinedby beamforming. Beamformedline-arraydata, however,has (a) an
inherent right-left ambiguity about the array's axis, (b)
decreasingangularresolutionfor off-broadside
beams,and
(c) decreasingcross-rangeresolutionfor more distantscattering sites.Theseambiguitiesmake it difficultto properly
locatereturnsin complexoceanenvironmentswhere scat-
teringsitesarebroadly
distributed
in rangeandazimuth.
•
For future reference,we define an "observation" as an
activeinsonificationand subsequent
towed-arraymeasurement of reverberationat a singlelocationand orientation.
(We sometimesuse "observation"to refer to the array
location in such a measurement.) We define a "reverbera-
tion map" as the magnitude of acousticreturns from a
single observationcharted to the horizontal location of
their respectivescatteringsites. A "scattering strength
map" is a reverberation map normalized by source
strength,scatteringsite area, and transmissionloss.
Averaging together scattering strength maps has
proven to be successful
in reducingright-left ambiguity,
givena sufficiently
diverse
observation
geometry.
2'3However, this techniquehas only been applied to situations
where isolatedscatteringfeatures,such as seamountsor
continentalmargins,disturbsoundchannelsthat otherwise
have excessdepth. It does not work as well for broadly
distributed backscattering in complex bottom-limited
environments.
4
Right-left angularambiguityin ambientnoisedata has
beenresolved
byaniterative
optimization
procedure.
5This
983
J. Acoust.Soc. Am. 94 (2), Pt. 1, August1993
can be further applied to resolve angular ambiguity in
long-rangemonostaticand bistatic reverberationdata by
collapsingtowed-arrayobservationlocationsto a single
point and performing angular inversionsat independent
rangesteps.
6However,
thisapproach
provides
a resolution
no finer than the maximum separationbetweenobservations. Given spatiallydistributedobservationsin complex
bottom-limited environments,higher resolution is often
necessaryfor comparisonswith seafloormorphology.
We proposea more generalimagingmethodbasedon
a global inversionof spatiallydistributedobservations.In
this initial
formulation
we assume that sound scattered
from a particular region is independentof incident and
observationangle,in both the horizontaland vertical.This
approximationis good for monostaticand bistatic reverberationexperimentsin the deepocean,wherepropagation
angles only vary over a narrow vertical width, and the
azimuth of distant scatterersremains relatively constant
from observation to observation. (It can also be used to
make omnidirectionalestimates,givena broaddistribution
of observations,
asis shown
in a preliminary
article.
7)
By simulation, we demonstrate the method for a
monostaticobservationgeometryand operationalparameters used in bottom reverberationexperimentssponsored
by the Officeof Naval ResearchSpecialResearchProgram
in 1991-1993. We first create a syntheticoceanbasin with
a scatteringcoefficientrepresentation,and generatesynthetic reverberationmapsfor the givenobservationgeometry. We then attempt to estimatethe true scatter coefficientsfrom the reverberationmapsby (a) a linear average,
(b) a dB average,and (c) a global inversion.Finally, we
test the accuracyof eachmethodby a statisticalcomparison of the estimated
and true scatter
coefficients.
Since
somemethodsmay estimatehigh scattercoefficients
better,
we make the above comparisonseparatelyfor different
magnituderegimes.
I. REVERBERATION
MAPS
To comparereverberationmeasuredwith differingarray positionsand orientations,returns from each observa983
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Depth
(100xm)
-17.6
-33.3
-49.0
S•attering
Strength
FIG. 1. Theimage
chosen
torepresent
theocean
basin.
Increasing
negative
values
indicate
increasing
depthasmeasured
fromtheocean's
uppersurface.
Thedataareextracted
andsubsampled
fromtheSRPGeophysical
Survey
9fora 244X244km2region.
Scattering
strength,
•[rn,n]= 10log(•r[n,
rn]),is
chosento be equalto the numericalvalueof depth, in hundredsof meters,for respectiveeastversusnorth grid point coordinatesm-- 1,2,3.....Nx and
n-- 1,2,3.....Ny, whereNx--Ny= 163.
tion must be mappedto absolutespatial coordinates.For
real data, temporal returns measuredon an array of hydrophonescan be accuratelyconvertedto the range and
azimuth of their respectivescatteringsites.Beamforming
yieldsthe necessaryazimuthaldependence,
but with varying resolutionand fight-left ambiguity. Travel time can
then be linearlyconvertedto range,via meansoundspeed,
with high accuracyfor intermediaterangeslarger than a
few oceandepths.More complexconversions
usingdepthdependentslant range or ray travel time are necessaryfor
The image is composedof discretepixels that form a
Cartesianarray of grid points.We arbitrarily choosescattering strengthto be equalto the depth value,in hectometers,of respectivepixelsin the image.This servestwo purposes.It providesscatteringstrengthvaluesthat fall within
therangetypically
measured
1øandit describes
a scattering
region that has variationsrelated to the bathymetry.
We definethe scatteringcoefficientat each grid point
by a discretefunction•r[rn,n],which samplesthe continuous variable •r(x,y) accordingto
localreturns.
8Forsynthetic
data,however,
it ispossible
to
maintain a spatialrepresentation
throughout.
To generate synthetic reverberationmaps, we first
choosea scatteringstrengthrepresentationof the ocean
basinas definedby a singlearbitraryimage,shownin Fig.
cr[m,n]
= •
m=l
6(x--mAx)6(y--nAy)•r(x,y).
n=l
(1)
1.9 We canusea singleimagebecause
wehaveassumed The scatteringcoefficient•r[m,n] is related to scattering
that scatteringstrengthis independentof incidentand observationangle.
984
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
strengthvia •[rn,n] -- 10 log(•r[n,rn]) for grid pointm -- x/
Ax, n -- y/A y, where (x,y) and [re,n] representrespective
NicholasC. Makris:Imagingreverberationvia inversion
984
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(Y-Ys)
continuousand discrete(east, north) coordinatesoriginating at the southwestcornerof the sampledregion.
An ideal line array's azimuthal resolutionis mostcommonly describedby its 3-dB beamwidth/•(0). For steering
Ar
anglesfrom broadside,0= •r/2, to a transitionangleOtnear
Grid
endfire, 0=0,
A
Array
1
(2)
*** "•/•(o)
Eq.(2) reaches
theendfire
beamwidth,
]1
B(O)=2.16Kx[/%/L.
(3)
We approximateB(O)=B(O) for O<O<Ot.
The beamwidth of the simulateddata is determinedby
the wavelength
;[, the array apertureL, andthe taperfuncm, L= 160 m, and K--1.3 for a Ham-
ming window,this translatesto an azimuthalresolutionat
broadsideof B(rr/2)----2.8 ø.The radial resolutionAr is determinedby the mean speedof the deep soundchannelc
and the pulsedurationfor cw signalsr. For c--1.5 km/s
and r= 2s, Ar=cr/2= 1.5km. The grid incrementsfloe,Ay
are chosento be equalto the radial resolutionof the simulated data, floe=Ay= Ar= 1.5 km.
Reverberationmeasureda distancer from the array, at
an angle0 from its axis,is comprisedof returnsweighted
over azimuth and integratedwithin an annulusof width
Ar. The azimuthal weightingis determinedby the array's
beampatternin the 0 direction.In our inversionand simulations,we ignorethe contributionfrom sidelobesin the
beampattern,and approximatethe azimuthalweightingby
unity within the 3-dB beamwidthB(0) and zero outside.
The integrationregionis an annularsectorof arearB(O) Ar
centeredabout the point (r,O). For our purposes,this integrationregiondefinesthe scatteringareafor the temporal
return mappedto (r,O). Contributionsfrom a symmetric
annular sector centered at (r,--O)
are also added to ac-
countfor the array'sfight-left ambiguity.This is displayed
schematicallyin Fig. 2.
To generatea syntheticreverberationmap for observations, data at grid pointswithin ambiguousannular sectors are summedtogetherincoherentlyvia
Qs[mo,no]=
-ß.•rmin J0mi n rn=l
n=l
Xg(y--n Ay) Ts(r,O)dOdr,
rr( r,O)
(4)
rr( r,-O)
T,(r,O)= 2 f,(r,O)+•
f,(r,--O), (5)
where Q,[mo,no]is the reverberationmeasuredat grid
pointmo=Xo/fi•x,no=Yo/Ay. (The selectedgrid pointsare
alsoshownin Fig. 2.) Polar and Cartesiancoordinatesare
related by
x=rcos(O+cp,)+x,,
i
y...." <:'.-'"<'--Lf
'
' = (•-•')
where K dependson the taper function or window used.
We defineOtby the valueat whichthe fight-handsideof
tion. With I=6.1
i
'
•(0)=K• sin•'
y=rsin(O+cp,)+y,,
(6)
for observations, array axisorientationq%,and array center location (x,,y,). The integration limits are
985
Point
/.
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
FIG. 2. Syntheticreverberationmeasuredat (r,O) from an array positionedat (Xs,Ys).The heavydottedline indicatesthe array axisat angle
q%withrespect
to theCartesian
grid.Symmetric
annularsectors
aboutthe
array axis encompass
grid pointsto be integrated.
0min=00--j•(0)/2, 0max=00q-j•(0)/2, rmin=ro--Ar/2,
rm• = ro+ Ar/2.
The function f•(r,O) consists of two factors
f•(r,O)=gs(r,O)h•(r,O). The first factor gs(r,O) incorporates two-way propagationbetweensourceand scatterer.
For monostaticsituations,reciprocityis usedon the return
tripsothatgs(r,O)=I,2(r,O)/I•o
where-- 10log(I•(r,O))is
the transmission loss at the ocean bottom for horizontal
position(r,O) and 10 log(I•o) is the sourcelevel. For inversionwith actual data, the insonifyingfield is computed
using a range-dependent
propagationmodel such as the
wide angle parabolicequation.
However, for this paper we have assumedzero transmissionlossand sourcelevel, i.e., g,(r,O)= 1 for all (r,O)
and s. Weightingthe data via a complicatedpropagation
model would be lost on our simulationssincethe weights
would simply be removedin the preprocessing
stagesbefore the actualinversion.(By similar reasoning,we neglect
to convertthe data from rangeto travel time back to range
dependence
by maintaininga rangedependence
throughout.)
Due to the fight-left ambiguityof the line-array data
we expectthat Q,[mo,no]= Q,[m•,n•] whenr0 = r•, 00
= -0• for observations. The correction factor,
hs(r,O)=rB(O)Ar/,4[m,n], guaranteesthat this is so regardlessof the grid size, where m=x/fioc, n=y/Ay are
related to (r,O) by Eqs. (6) for observations. Here the
numberof grid pointsper scatteringarea, or the integration number, is
frma,
fOmax
• n--1
•(x--rnfi•x)
ß/s[rn0,n0]
--.,
rmin
J
0min
m--1
X•(y--n
Ay)dO dr,
(7)
wherethe summationlimits are as previouslydefined.This
factor is necessarybecauseof approximationsmade in
mappinginherentlypolar reverberationdata onto the Cartesiangrid usedto digitally definethe image.As the grid
size decreases
with respectto the radial resolutionof the
array Ar, the correctionfactor h,(r,O) approachesunity
NicholasC. Makris:Imagingreverberationvia inversion
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and the resolution of the inversionincreases.However, it is
not advantageous
to choosea grid sizemuch smallerthan
At. As the grid becomesfiner than this, the information
content remainsconstantbut the inversiontime prohibitively increases.
For a givenobservation,we do not computereverberation for grid pointswhosescatteringareasextendbeyond
boundariescorrespondingto the edgesof the image. We
also do not computereverberationfor grid points whose
integrationnumbersexceeda chosenthreshold•/max'This
insures that redundant information from endfire beams,
which is too poor in resolutionto be useful,doesnot slow
downthe processing.
However,thresholdingalsolimitsthe
data availableat long range, which eventuallylimits an
inversion'seffectiveness
at long range.
at least$(N/2) due to the Cartesiangrid'sgenerallack of
symmetryabout an arbitrary array axis. If all observations
s had array axes symmetric to the grid, i.e.,
q•s=(s-1)•r/4, the number of nonidentical equations
would be limited to S(N/2), due to right-left ambiguity.]
When properly constrained,these equationscan be
solved
for scattering
coefficient
bkby matrixinversion.
12
Instead,we choosean iterativeoptimizationprocedure,becausesuch proceduresare extremelyefficientin both formulationand solutionof problemsinvolvinga largenumber of intricatelyrelatedparameters.They are ideal for our
situationsincethe imagesthat we needto invert typically
contain
between
N= 104to 106pixelsandaretheresultof
a highly convolutedprocess.
Becausethe problem is linear, we are guaranteedto
obtain the correct solution via an iterative random-search
We initialize our scatteringcoefficientestimate•[m,n]
at some value within an allowable range Crmi
n
<b[m,n] <Crma
x for all grid points m--1,2,3,...,Nx and
for local minima, if the problemis properly constrained.
This can also be seen in Eq. (9). The function we are
minimizing is parabolic in the parametersto be varied.
This insuresthat the minimum found in the parameter
searchspaceis a globalminimum.Besidesbeingextremely
n-1,2,3,...,Ny. For eachobservation
s, we construct
rep-
efficient, this method is also convenient for another reason.
lica dataRs[too,no]
with our estimatedscatteringcoefficient
as in Eq. (4), replacingQs[mo,no]with Rs[too,no]and
or[re,n]with b[m,n]. We then computea least-squares
cost
function,which sumsthe differencesquaredbetweenreplica data and true data for eachobservationand grid point,
Sinceit is essentiallythe sameasusingsimulatedannealing
at zero temperature,it is easilytransformedto a simulated
annealingalgorithm.If there is uncertaintyin the observation geometry,i.e., the ship'slocationand array heading
are not preciselyknown, theseparameterscan be included
in the search.The problem then becomesnonlinearand a
simulatedannealingapproachis necessaryto escapelocal
minima in the parametersearchspace.
II. INVERSION
E(a) = •] •]
s=l
m=l
(gs[m,n]
--Rs[m,n]
)2.
(8)
n:l
Minimization of this cost function with respectto the
him,n] leadsto a seriesof S linearequationsfor eachgrid
point. This can be seenby expressingthe replica data in
Eq. (8) in terms of him,n]. For notationaleconomywe
convert[m,n] dependence
to an indexk= 1,2,3,...,N,where
N=NxNy. We alsointroducea siftingfunctionBs,,,,k
which defines the beamwidth in discrete space
•c= 1,2,3,...,N for observations. It is unity insidethe scattering areas about fixed grid point k and its ambiguous
imagepoint k', and zero outsidetheseregions:
o__O
-aaj
s=l
k=l
x
x
x
\
\
1, (9)
Os,
k- a,,f
s,,,Bs,,,,•:
.
I
tc=l
I
v
(x-x c)
wherej = 1,2,3,...,N. This leadsto the linear equations
l
l
l
l
N
l
l
Qs,
j = • t•,rf
s,,rBs,,r,j
(10)
l
l
l
l
l
l
for each observations=1,2,3 ....,S at each grid point
l
l
l
j = 1,2,3,...,N.
i
l
l
When the numberof equationscoupledto a particular
set of grid pointsis greaterthan or equalto the numberof
grid pointscoupled,the equationsfor thesegrid pointsare
properlyconstrained.Satisfactionof this criterionfor a set
of grid pointsdependsupon the observationgeometry,local resolution,and integration threshold.Satisfactionof
this criterionfor all grid pointsis more stringentthan having the overallnumberof nonidenticalequations,which is
at least $(N/2),
exceed the overall number of unknowns,
N. [We note that the numberof nonidenticalequationsis
986
d. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
l
FIG. 3. Syntheticobservationgeometrywith towed-arraycourseindicated by the dotted lines and arrows. The coordinateaxis shownis at the
center of the region displayed in Fig. 1, i.e., at m½=x/Ar=81,
n½=y/Ar=81, where tick spacingis Ar. Monostaticobservationlocations are givenby asterisks.Array axisorientationsare within 4- 5* of the
tracksshown.Specificarray positionsand orientations(x/Ar, y/Ar, cps)
are (mc+4,nc+l,5*),
(mc+2,nc+3,55*),
(mc--l,nc+4,114*),
(mc-- 3,nc+2,133'), respectively,for the s- 1,2,3,4observations.
NicholasC. Makris:Imagingreverberationvia inversion
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,
-12
-5O
FIG. 4. Syntheticreverberationmaps 101og(Qs)for the s= 1,2,3,4observations.
Outsidethe interior white lines,the resolutionis poorerthan one grid
increment.All data to be usedfor scattercoefficient
estimationarewithin the exteriorwhite line,i.e., poorresolutiondatafrom endfirebeamsandbeyond
a thresholdrange are excluded.
To implement the inversionnumerically,we perturb
theestimated
scattering
coefficient
&j at a singlegridpoint
j by a randommultiplicative
factore suchthat •.
- e&j,andOrmi
n < •;- < O'ma
x. (Specifically,
wechoose
e
= 10(2x-1)3
where
0<X<Iisa random
variable.
Thisis
analogousto the methodusedin Ref. 13 for additiveper-
original value. We continue this processuntil each grid
point hasbeenperturbed,i.e., for j = 1,2,3...,N. That comprisesthe first global iteration. We then iterate globally
until the costfunction reachesa minimum valuegmin with
respect
to the &j. As discussed
in thelastparagraph,
the
resulting
&j will bethesolution
for a properly
constrained
turbations.
Whene&j< O'mi
nor O'ma
x• E•'jweset•) equal problem. If the problem is not properly constrainedfor
toO2min/E•'j
or02maxlEa'j,
respectively.)
Thescatter
coeffi- certain grid points,resolutionwill be poorer than the grid
cient Cr
k remainsthe samefor all other grid points,i.e., for
k- 1,2,3...,N but k-7kj. We then computea new costfunction E', which is identicalto E in all terms exceptthose
size and specklingwill occur in that portion of the image.
Also, ambiguity will be reduced, but may not be elimi-
containing
thegridpointj. Whenk= j, &j is replaced
by
theperturbed
value&). If theperturbed
costfunction
E' is
lessthan the originalcostfunctionE we acceptthe pertur-
In comparingE' to E it is inefficientand prohibitively
time consumingto computeeachcostfunctionas shownin
Eq. (8). Since only a single test grid point has been
bation, i.e., if AE=E'--E<O.
changed
thereis no difference
between
many.correspond-
987
Otherwise we retain the
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
nated.
NicholasC. Makris:Imagingreverberationvia inversion
987
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-35.0
-66.5
-98.0
FIG.7. Scattering
strength
estimate
• fromadBaverage
ofreverberation
FIG. 5. The numberof reverberationmapsavailableper grid point,
ing terms in each summation.Therefore,insteadof sum-
maps.
•s[ rn-- mo,n-- no]=6 [ rn-- too,n--no]
mingoverall Nx andNygridpointsforeachperturbation,
only a smallerinfluenceregion•s[rn--too,n--no] aboutthe
test grid point [m0,no]needbe considered.
In general,the
influenceregionis approximatelyequalto the integration
regionof Eq. (7) for a givenobservation
s. For observation
pointsrelatedby the array's fight-left ambiguity,the influenceregionmustbe largeenoughto accountfor the lack
of perfectreciprocityabout the array's axis.As notedearlier, this occursfor array orientationswhich are not symmetric with respectto the Cartesiangrid.
The runningtime of the algorithmfor eachglobaliteration is then proportionalto
•
s=l
•
rno=l no=l
•
rn=l
•s[m--mo,n-no].
( 11)
n=l
Thisreduces
to a minimum2N•.N•Swhentheresolution
is
equal to the grid size for all observations
and grid points,
i.e., when
+6[m--m•)(s),n--n•(s) ]
for all s, n0, m0, where [m•(s),n•(s)] is the ambiguous
point for [n0,m0]and observations. This minimum run
time per globaliterationis equalto the overallrun time of
an averageof reverberationmaps.As we will showin the
next section,however,averaginggenerallygivesan unsatisfactory estimate.
III. SIMULATION
RESULTS
A. Observation geometry and reverberation maps
Simulationsare preformedfor the typical monostatic
observationgeometrysketchedin Fig. 3. The ship'scourse
is indicatedby dashedlinesand arrows.The straightline
segmentsform a star pattern, consistingof four towedarray tracks.Asterisksindicatethe locationof the observations, where the array axis is roughly parallel to the
track. To demonstratethe generalityof the method, the
tracks are chosento be asymmetricwith respectto the
-15.0
"Cost_Function"
-28.5
-30
-60
-42.0
0
500
1000
1500
Global
2000
Iteration,
2500
3000
3500
4000
i
FIG.6. Scattering
strength
estimate
•; froma linear
average
ofreverber-
FIG. 8. The costfunctionEi versusglobaliteration i for the scattering
ation maps.
coefficient inversion.
988
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
NicholasC. Makris:Imagingreverberation
via inversion
988
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TABLEI. Statistical
comparison
between
estimated
scatter
coefficients
b andoriginal
coefficients
•r.Here,theestimate
results
froma linearaverage
of
reverberation
maps.
Statistics
arecomputed
separately
foreachof fourdecades
in •r andforthesame
respective
gridpoints
in b. Statistics
areonly
computed
forthe21055gridpoints
where1< fl, seeFig.9. Thepercentage
of thisnumber
foreachdecade
isindicated.
Linear
average
10-5<a
< 10-4
10-4<a
< 10-3
32.30
60.81
%
p•,•
0.041
(b)/(a)
10-3<a
< 10-2
10-2<a< 10-]
6.78
0.11
0.33
0.88
0.28
5.01
2.58
14.19
2.11
differentfrom the actualones.Artificial
Cartesiangrid.Also,to allowfor deviations
typicalin real are significantly
experiments,
the arrayaxeshavebeenrandomlyperturbed featuresof highscatteringstrengthalsoappeardueto imperfectremovalof right-left ambiguity.
from the track by rotationsof within 4-5ø'
To quantifythesedifferences,
andsotestthe accuracy
A syntheticreverberation
map 10log(Qs[m,n]),where
m=l,2,3,...,Nx, n=l,2,3,...,Ny, for each observation, of the estimate,we computethe ratio of the meansand
s= 1,2,3,...,S,is shownin Fig. 4. Note the decreasein resolutionfor beamsapproachingendfireand for distantscat-
teringareas,as well asthe right-leftambiguityaboutthe
array'saxis.The interiorwhitecontourencompasses
data
with spatialresolution
equalto thegridincrement,
i.e.,the
scatteringareascontaina singlegrid point so that •/= 1.
Outsidethis contour,resolutionis poorer,i.e., the scattering areascontainmultiplegrid pointsand 1< •/. Due to
discretization,thesecontoursvary with observationlocation and ambiguous
lobesare not symmetricwith respect
to the array axes.The exteriorwhite contoursencompass
datato be usedin subsequent
scattercoefficient
estimation.
These data are within the selectedTImax
= 10 integration
number threshold and are also limited by a maximum
rangefrom each observation
location(Xs,Ys)suchthat
Rmax=83Ar=124.5km, which spansabout two deep
oceanconvergence
zones.Data outsidethesebordersare
consideredto have resolutiontoo poor to be useful.
Due to thresholding,
the mappingQs[rno,no]
may not
normalizedcovariancefor the originaland estimatedscat-
tering coefficients.
We computethesestatisticsindependentlyfor setsof grid pointscorresponding
to 10-dBscattering strengthbins in the original image, and only
examinegrid points where the number of mappings
ft[rn,n]> 1. The resultsare in TableI, wherewe use
((a-
(a)) (a- (a)))
P&r-•/((•__
(•))2)
((O.__
(O.))
2)
(12)
to definethe normalizedcovarianceso that 0•p&,• 1. We
seethat the correlationis only high for a smallportionof
the regionimaged,where high amplitudescattercoefficients are found. The mean value is estimated best for these
highamplitudecoefficients,
but is stillonlywithina factor
of 3 of the true coefficients.
C. Estimate via dB average
B. Estimate via linear average
We next computethe scatteringstrengthestimateresultingfrom a dB averageof reverberation
maps,i.e., we
add all mappings10log(Qs[rn,n])and again divide by
ft[rn,n].The resultingimage,Fig. 7, hasbetterright-left
ambiguityreductionthan the linearestimate,but still resembles
the originalimageonlypoorly.The comparison
is
againmadequantitative
by computingcrossstatistics,
as
We firstshowthe scatteringstrengthestimateresulting
from a linear averageof reverberation
maps.We add all
increased,
theyare still low, exceptfor a smallfractionof
the regioncorresponding
to the highamplitudescatterers.
mappings
Qs[mo,no]
of gridpoint[rn0,n0]
anddivideby the
numberof mappings,O<ft[rno,no]
<S, for that grid point.
However, the mean estimateis ordersof magnitudeworse
than in the linear average.
existat grid point [rn0,n0]for observation
s. To quantify
this for the given observation
geometry,the numberof
mappings
per grid pointft[rn,n]is shownin Fig. 5.
indicated in Table II. While correlationshave generally
The resulting
scattering
coefficient
estimates
b[rn,•n]
are
converted to scattering strengths via 5•[rn,n]
D. Estimate via global inversion
= 10log(b[rn,n]),for rn= 1,2,3,...,Nx,
andn= 1,2,3,...,Ny.
The globalinversion
methodprovides
a solutionwith
The resultingimage,in Fig. 6, bearsa poorresemblance
to
costfunctionEmin•E4ooo--Eo(2.77XlO-5),
the actualscatteringstrengthimagein Fig. 1. In general, minimum
globaliterationnumberi appears
the magnitudeand shapeof estimatedscatteringfeatures wherethe corresponding
TABLEII. Statistical
comparison
between
dBaverage
estimate
andoriginal
scatter
coefficients.
SeeTableI caption
forfurtherdetails.
dB
Average
%
10-5<rr
< 10-4
32.30
10-4<rr
< 10-3
60.81
Pt,•
0.059
989
d. Acoust.
Soc.Am.,Vol.94, No.2, Pt. 1, August1993
(b)/(a)
9.25X
10-4
0.53
8.34X
10-4
10-3•
<rr< 10-2
10-2•<rr
< 10-•
6.78
0.11
0.81
0.26
5.27X
10-3
1.68X
10-2
Nicholas
C. Makris:
Imaging
reverberation
viainversion 989
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as a subscript.Here, the costfunctionis givenin termsof
the initial "energy"E0, whichis obtainedby settingthe
replicadata Rs[m,n] to zero in Eq. (8). The costfunction
is plottedasa functionof globaliterationin Fig. 8.
The scatteringstrengthestimatefor the globalinversionis shownin Fig. 9, wherethe white contourcontains
grid pointswhere 1< ft. As expectedfrom the dramatic
decrease
in the costfunction,the globalinversionimageis
nearlyidenticalto the true imagein Fig. 1. The right-left
ambiguity of the data has been removed and the ocean
basinis resolvedto the grid incrementin areaswhere reverberationmappingsof suchhigh resolutiondo not exist.
This is confirmedquantitativelyin Table III, where near
perfect correlation is found betweenthe inversionestimate
and the actual scatter coefficients. The mean values are
identical.
We note that both high and low valuesare found with
equal precisionin the inversion.However, low values are
more statisticallysensitiveto the successive
approximationssincethey are givenlessweightin the costfunction,
when consideredindividually.As a result,more iterations
may be necessaryto reducethe variance of the estimatefor
lower values.This is shownin Fig. 10 and Table IV, where
the scattercoefficient
estimateandcrossstatistics
are provided for intermediate iterations. For inversions with real
data, this meansthat if the costfunctionreachesan early
minimum,i.e., it is reducedby lessthan an orderof magnitude,a statisticalanalysisis necessary
to interpretthe
estimate.(One possibleway of reducingthe number of
iterations necessaryto reach a minimum is to make the
perturbationdecreasein magnitudeas a functionof theit-
eration.This has not beendonefor the presentanalysis,
but we have performedsomesimulationsalong this line
which showthat the idea is promising.)
Evenif thereare no reverberation
mappings
for a particular grid point, due to thresholdingor nearnessto a
border,the scatteringcoefficient
canstill be determinedby
globalinversionif the scattering
areasof adjacentmapped
pointsleak onto the unmappedpoint enoughtimes.Comparingthe numberof mappingsper gridpoint,Fig. 5, with
the inversionestimate,in Fig. 9, we note that this is the
casefor a varietyof grid pointsat the imageperiphery.It
-33.3
't
ß
.
'-.
: /'"'x.:,-.\..
..•....:
, ., .
•.,.•,•.:.,....,.
,.:
,
..
., -.',..'":. ::.'"
-49.0
FIG. 9 Scattering
strengthestimate• from a globalinversion
of reverberation maps, with correspondingglobal minimum
Emin•E4ooo--Eo(2.77
X 10-5).Thewhitelineencompasses
gridpoints
where1<
990
J. Acoust.Soc.Am.,Vol.94, No.2, Pt. 1, August1993
Nicholas
C. Makris:Imaging
reverberation
viainversion 990
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 18.38.0.166 On: Mon, 12 Jan 2015 18:32:33
TABLE III
Statistical comparisonbetween global inversion estimate and original scatter coefficientsEstimate has correspondingcost function
approximately
at globalminimum,
Emin=Enooo=Eo(2.77X
10-5).SeeTableI caption
for furtherdetails.(Forgridpointswhere3 < f•,pC,
o= 1.0when
10-5<c< 10-n.)
Global
inversion
10-5<o'< 10-4
10-4<o' < 10-3
10-3<c < 10-2
10-2<0' < 10-1
32.30
60.81
6.78
0.11
0.93
1.00
1.00
1.00
1.00
1.00
1.00
1.00
is also clear that the leakageis insufficientto properly estimate scatteringstrength for some other peripheral grid
points.
IV. DISCUSSION
We have developeda method for imaging the ocean
basin via a simultaneousinversionof multiple reverbera-
tion measurementsmade at differing spatial locations.By
optimal useof the data, this methodremovesthe right-left
ambiguity associatedwith line-array data and producesas
fine a resolutionas possible.The method has been tested
with a typical monostatic observationgeometry where it
has proven to be far superiorto data averagingfor simulations involving reverberation from scattering sites
-17.6
E29=EoxlO
-2
-33.3
-49.0
E202=Eoxl
0-3
E640=Eoxl
0-4
FIG. 10.Intermediate
scattering
strength
estimates
•; froma global
inversion
of reverberation
mapsforindicated
iterations.
991
d. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August 1993
NicholasC. Makris:Imagingreverberationvia inversion
991
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. TABLE IV. Statisticalcomparisonbetweenfour intermediateglobalinversionestimatesand originalscattercoefficients.
SeeTable I captionfor further
details.
Intermediate
iterations
%
10-5<rr< 10-4
32.30
10-4<rr< 10-3
10-3•<rr< 10-2
10-2•<rr< 10-1
60.81
6.78
0.11
0.44
0.80
0.17
0.49
0.80
0.98
0.39
0.86
1.00
1.00
0.97
0.99
1.00
1.00
1.00
1.00
(El2--E0X 10-1)
p•,,,
(b)/(a)
0.030
2.09
Pt,,,
(b)/(o')
0.12
1.26
Po,,
(b)?(rr)
0.47
1.02
pt,,,
(&)/(a)
0.78
1.00
0.33
1.15
(E29--EoX10-2)
0.65
1.01
(E202--EoX
10-3)
0.95
1.00
(E64o=EoX
10-4)
0.99
1.00
broadly distributedin spaceand scatteringstrengthmagnitude. In situationswhereonly isolatedscatterersof high
amplitudeare present,and thesedo not interferein ambiguousbeams,data averagingmay suffice.However, this situation is the exceptionfor most ocean environments.A
more robust method, such as the inversion presented,is
generallynecessary.
While the method is currently applicableto many
long- and short-rangemonostaticand bistaticexperiment
geometries,it cannotbe appliedarbitrarily. For example,a
seriesof separateinversionsis necessaryto determinethe
angular dependenceof bottom reverberation.In each inversion,the separationbetweenobservations
mustbe small
enoughto maintaina similarorientationto the regionbeing imaged. This insuresthat the differencesbetweenobservationangleare small with respectto variationsin the
angular dependencebeing measured.
Alternatively, it is possibleto reformulatethe inversionto searchdirectlyfor regionalangulardependence
via
a singleglobalinversion.However,this approachrunsinto
difficultiesfor oceanenvironmentsdominatedby negative
excessdepth and ruggedbathymetrydue to extremevariationsin transmissionlossthat can make spatiallydiver-
that can be exploitedto help resolvefight-left ambiguity
for a singleobservation,asis demonstratedin an upcoming
article.
4 Thepresent
inverse
method
canbereadilymodifiedto exploitthesesituations,
8 andprovidea means
of
directly estimating the regional angular dependenceof
scatteringvia a singleglobal inversionof spatially divergent observations.
No attempt has been made to accountfor statistical
variationsin reverberationthat may arise at differingobservationlocations.While suchstochasticphenomenon
are
possiblein real data, they are not relevant to this initial
formulation.Our goal is principallyto showthat thereare
ways to make optimal use of towed-arraydata that are
independentof the often stochasticnature of the reverberation. And, even if high variance exists, an ensembleof
observationswith redundantgeometrymay be usedin the
presentformulationto make the problemstatisticallyoverdetermined and so reduce the variance by the number of
redundant observations.Alternatively, stochasticscan be
incorporatedby searchingfor probabilitydistributionsto
parametrizeseafloorscatteringrather than meanscattering
coefficientvalues. If the reverberationis inherently detergentobservations
independent
ofeachother.
4Forexample, ministic,i.e., the scatteringsurfacesare much largerthan a
regions of extremely high transmissionloss, or shadow wavelengthand approximatelyplanar, or the varianceis
zones,may exist where no useful scatteringinformation sufficientlylow, only observationssufficientto removeambiguity and obtain the desired resolution would be recan be extracted.Under these circumstancesscalingby
quired, as in the simulationspresented.
transmissionlossis not appropriateand redundantinforFinally, we contendthat it is possibleto determinethe
mationvital to ambiguityresolutionmay be lost.A simple
optimal
setof observationsneededto imagethe oceanbotway to circumventthis problemis to invert observations
tom
with
methodsrelated to thosepresented.That is, assufficientlyclose that their respectivetransmissionloss
suming
some
or no knowledgeof regionalbottom scatterfunctionsare approximatelythe sameand to excludedata
ing,
and
given
constraints upon our resolution, ship
whentransmission
lossexceeds
a thresholdmaximum.Opmaneuverability,time, site locations,etc., we could invert
timally resolved linear reverberation would then be the
for the optimalcourseof the researchvessel.This hasbeen
output of the inversionrather than scatteringcoefficient.
On the other hand, variations in transmission loss
referred
toasthe"traveling
acoustician
problem."
14Qual-
alongambiguousradialscan be usedmore advantageously itatively, the problemis related to the much simplerprobthan has been suggested.
Thesevariationsoften provide lem of refiningthe array's navigationby includingit as a
additionalenvironmentalsymmetrybreakinginformation free parameterin the reverberationinversion.Both• prob992
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
NicholasC. Makris:Imagingreverberationvia inversion 992
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 18.38.0.166 On: Mon, 12 Jan 2015 18:32:33
lems are nonlinear, and are most efficientlysolvedwith a
techniquelike simulatedannealing.
6R.A. Wagstaft,
M. R. Bradley,
andM. A. Herbert,"TheReverberation
ACKNOWLEDGMENTS
7N. C. Makris,J. M. Berkson,
W. A. Kuperman,
andJ. S. Perkins,
Array Heading Surface,"in OceanReverberation,editedby D. Ellis, J.
Preston, and H. Urban (Kluwer, Dordrecht, 1993).
"Ocean-Basin Scale Inversion of Reverberation Data," in Ocean Rever-
The author would like to thank W. A. Kuperman for
motivatingthis work by suggestingan inversionof ONR
SpecialResearchProgram reverberationdata as well as J.
M. Berkson,M. C. Collins, and J. S. Perkins for many
useful comments
and discussions. The author would
also
like to thank B. E. Tucholke for making SRP bathymetry
available.
beration, edited by D. Ellis, J. Preston, and H. Urban (Kluwer,
Dordrecht, 1993).
8A paperonmapping,
inverting,
andprojecting
high-resolution
directpath reverberationonto bathymetry for Mid-Atlantic Ridge data is in
preparationfor submission
to J. Acoust. Soc.Am.
9Thebathymetry
shown
is subsampled
fromtheSRPGeophysical
Survey of 1992 which had Principal InvestigatorBrian E. Tucholke and
Co-Principal InvestigatorsMartin C. Kleinrock and Jian Lin, all from
the WoodsHole OceanographicInstitution.
•J. M. Berkson,
N. C. Makris,R. Menis,T. L. Krout,andG. L. Gibian,
lop.M. OgdenandF. T. Erskine,"Low-frequency
Surface
andBottom
"Long-range measurementsof sea-floor reverberation in the MidAtlantic Ridge area," in Ocean Reverberation,edited by D. Ellis, J.
Preston, and H. Urban (Kluwer, Dordrecht, 1993).
ScatteringStrengthsMeasuredUsing SUS Charges,"in OceanReverberation, edited by D. Ellis, J. Preston, and H. Urban (Kluwer,
Dordrecht, 1993).
2F. T. Erskine,G. M. Bernstein,
S. M. Brylow,W. T. Newbold,R. C.
• B.D. Steinberg,
Principles
ofAperture
andArraySystem
Design
(Wiley,
Gauss,E. R. Franchi, and B. B. Adams, "Bathymetric Hazard Survey
Test (BHST Report No. 3), ScientificResultsand FY 1982-1984 Processing,"NRL Report 9048 (1987).
•2G. Strang,
Linear,4lgebra
andIts,4pplications
(Academic,
NewYork,
j. R. Preston,
T. Akal,andJ. M. Berkson,
"Analysis
ofbackscattering
data in the Tyrrhenian Sea," J. Acoust. Soc.Am. 87, 119-134 (1990).
N. C. MakrisandJ. M. Berkson,
"Long-Range
Backscatter
fromthe
Mid-Atlantic Ridge," submittedto J. Acoust. Soc.Am.
5R. A. Wagstaft,"Iterativetechnique
for ambient-noise
horizontaldirectionalityestimation from towed line-array data," J. Acoust. Soc.
Am. 63, 863-869 (1978).
993
J. Acoust.Soc. Am., Vol. 94, No. 2, Pt. 1, August1993
New York, 1976).
1980).
13W. A. Kuperman,
M.D. Collins,
J. S.Perkins,
andN. R. Davis,"Optimal Time Domain Beamformingwith SimulatedAnnealingincluding
applicationof a priori information,"J. Acoust.Soc.Am. 88, 1802-1810
(1990).
•4Privatecommunication
withW. A. Kuperman.
A paperonthe"traveling acousticianproblem"is in preparation.
NicholasC. Makris:Imagingreverberationvia inversion
993
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