Phase conjugation in the ocean: Experimental demonstrationof an acoustic time-reversal mirrora)
W. A. Kuperman, William S. Hodgkiss, and Hee Chun SongMarine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego,La Jolla, California 92093-0701
T. Akal and C. FerlaSACLANT Undersea Research Centre, La Spezia, Italy
Darrell R. JacksonApplied Physics Laboratory, University of Washington, Seattle, Washington 98105
~Received 4 March 1997; accepted for publication 7 August 1997!
Phase conjugation is a process that has been first donstrated in nonlinear optics1 and more recently in ultrasonilaboratory acoustic experiments.2,3 Aspects of phase conjugation as applied to underwater acoustics also have beenplored recently.4–7 The Fourier conjugate of phase conjugtion is time reversal; implementation of such a process oa finite spatial aperture results in a ‘‘time-reversal mirror.2,3’’In this paper we describe an ocean acoustics experimewhich a time-reversal mirror was demonstrated.
In nonlinear optics, phase conjugation is realized ushigh intensity radiation propagating in a nonlinear mediuEssentially, the incident radiation imparts its own time dpendence on the dielectric properties of the medium. Tincident radiation is then scattered from this time-varyidielectric medium. The resulting scattered field is a timeversed replica of this incident field propagating in the oppsite direction of the incident field. For example, the scattefield that results from an outgoing spherical wave is a sphcal wave converging to the original source point; whenpasses through the origin it has the time reversed signatuthe signal which was transmitted from that point at the orinating time. Clearly, this phenomenon can be thought of aself-adaptive process, i.e., the process constructs a waveof the exact required curvature.~An alternative would be touse a concave spherical mirror with the precise radius
a‘‘Selected research articles’’ are ones chosen occasionally by the EditoChief that are judged~a! to have a subject of wide acoustical interest, a~b! to be written for understanding by broad acoustical readership.
25 J. Acoust. Soc. Am. 103 (1), January 1998 0001-4966/98/10
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curvature of the incident wavefront.! There is an assortmenof nonlinear optical processes which can result in phconjugation.1 In acoustics, however, we need not use tpropagation medium nonlinearities to produce a phase cjugate field.
Because the frequencies of interest in acoustics areders of magnitude lower than in optics, phase conjugatcan be accomplished using signal processing. As in thetical case, phase conjugation takes advantage of reciprowhich is a property of wave propagation in a static mediuand is a consequence of the invariance of the linear losswave equation to time reversal. In the frequency domatime reversal corresponds to conjugation invariance ofHelmholtz equation. The property of reciprocity allows oto retransmit a time reversed version of a multipath disperprobe pulse back to its origin, arriving there time reverswith the multipath structure having been undone.8,9 This pro-cess is equivalent to using the ocean as a matched filter sthe probe pulse arrival has embedded in it the transfer fution of the medium. This process can be extended furthereceiving and retransmitting the probe signal with a sourcreceiver array. Depending on the spatial extent of the arthe above process results in some degree of spatial focuof the signal at the origin of the probe signal.
A time-reversal mirror~TRM! can therefore be realizewith a source–receiver array. The incident signal is receivtime reversed, and transmitted from sources contiguous wthe receiving hydrophones. The time reversal can be accplished in a straightforward way, for example, by using trewind output of an analog tape recorder or by a sim
program that reverses a digitized segment of a receivednal.
An acoustic TRM has already been demonstrated inultrasonic laboratory using an array of source/receiver traducers~SRA!.3 The array length was 10 cm and a sing4-MHz source was placed at a transverse distance of 5together with another receive array. The single source tramitted a probe pulse which was received at the SRA;received pulse was time reversed and retransmitted fromSRA and subsequently received at an array~with the sameorientation as the SRA! near the single source. The resushowed a 15-dB peak at the location of the source relativsidelobes away from the probe source location. Note thatfocal point was at a range one-half the size of the apertu
Phase conjugation~PC! or the implementation of a TRMin the ocean is relevant to recent trends in acoustic sigprocessing which have emphasized utilizing knowledgethe environment, e.g., matched feld processing~MFP!.10
However, MFP requires accurate knowledge of the envirment throughout the propagation path, which of coursedifficult or impossible to obtain. Phase conjugation is anvironmentally self-adaptive process which may therefhave significant applications to localization and communitions in complicated ocean environments. Although the ‘‘fective’’ ocean environment must remain static over the taround time of the PC process, ocean variability on timscales shorter than the turn around time might be compsated for with feedback algorithms. However, an understaing of relevant ocean time scalesvis a visthe stability of thePC process will be required.
In this paper we describe an April 1996 experimentwhich an acoustic TRM was demonstrated in the oceanthis initial experiment, a focal range of about 100 timesSRA aperture was accomplished easily with a 445-Hz prsource, a water depth of the order of 100 m, and a forange of about 6.3 km. Large focal distances are obtainin the ocean because in a waveguide geometry, a SRAimages which increase its effective aperture. Hence, thean advantage to having a waveguide geometry over a ffield environment as was used first in the ultrasonic labotory experiment. Measurements in this first low-frequenocean experiment also suggest a temporal stability of theprocess which is longer than what was expected intuitiveSome quantitative results on this stability are presented.
In the next section we review the relevant theoretiissues including some simulation results leaving the detto an Appendix and appropriate references. Section IIscribes the experiment in which the TRM was demonstraand Sec. III presents the results.
I. BACKGROUND THEORY AND SIMULATION FORTHE TRM EXPERIMENT
The theory of phase conjugationvis a visocean acous-tics already has been presented.4–7 Here we briefly reviewsalient issues using the basic geometry of the TRM expment~shown schematically in Fig. 2!. More detail on theoryis given in Appendix A and some additional details conceing experimental equipment are given in Appendix B.
26 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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A. Experimental geometry
The TRM experiment was performed off the west coof Italy in April 1996 as indicated in Fig. 1. Figure 2 isschematic of the experiment and indicates the types of eronmental measurements that were made. The TRMimplemented by a 77-m source–receiver array~SRA! in 125-m-deep water which was hardwired to the Isola di FormicaGrosseto. The SRA consisted of 20 hydrophones withcontiguously located slotted cylinder sources with a nomiresonance frequency of 445 Hz. The sources were operat a mean nominal 165-dB source level. The received sigwere digitized, time reversed, and after being converted bto analog form, retransmitted. A probe source~PS! of thesame type used in the SRA was deployed from the NAresearch vesselALLIANCE . The probe source together withhydrophone was also used in parts of the experiment atransponder. TheALLIANCE also deployed a vertical 46 element receive array~VRA! spanning 90 m located 6.3 kmfrom the SRA which radio telemetered all individual elemedata back to theALLIANCE . Ideally, the PS should be in thvertical SRA-VRA plane to correspond perfectly to thsimulations in the following section; for practical reasons tPS was placed a few tenths of a km out of this plane. Tdid not introduce a significant error because the bathym
FIG. 1. Location of phase conjugation experiment. A source/receiver a~SRA! was deployed in 125-m-deep water and cabled approximately 1back to a small island, Formica Grande, the northernmost island ofmiche di Grosseto~42° 34.68 N, 10° 52.98 E!. A rf telemetered verticalreceive array~VRA! was deployed in 145-m-deep water approximately 6km west of Formica.
FIG. 2. Experimental setup of the phase conjugation experiment.
26Kuperman et al.: Phase conjugation in the ocean
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was flat in the vicinity of the VRA. Figure 3 is a collection othe sound speed profiles~SSP’s! obtained from theconductivity–temperature–depth probe~CTD! as an indica-tion of the variability over the duration of the experimenThe bottom sound-speed structure as determined from eaexperiments11 is shown in Fig. 4~a!. More details on the in-strumentation and processing are given in Appendix B.
B. Overview of theory
For simplicity in this subsection, we analytically summarize the basics of phase conjugation in a ranindependent waveguide. The simulations and compariswith experimental data presented later on in the paperinclude range-dependent modeling. The source, PS, iscated a horizontal distanceR from the source/receive phasconjugate array, SRA.
1. Harmonic excitation
The acoustic field,Gv(R;zj ,zps), at the j th receiver el-ement of the SRA from the point source PS in Fig. 2determined from the Helmholtz equation12 @assuming a har-monic time dependence of exp (2 ivt)#
¹2Gv~r ;z,zps!1k2~z!Gv~r ;,z,zps!
52d~r2r ps!d~z2zps!, k2~z!5v2
c2~z!, ~1!
wherez is taken positive downward andr5(x,y). Letting rbe the horizontal distance from the probe source, Eq.~1! hasthe far field, azimuthally symmetric normal mode solutifor pressure given by
Gv~r ;z,zps!5i
r~zps!~8pr !1/2 exp~2 ip/4!
3(n
un~zps!un~z!
kn1/2 exp~ iknr !, ~2!
whereun ,kn are the normal mode eigenfunctions and mowave numbers obtained by solving the following eigenvaproblem with well-known boundary conditions:12
FIG. 3. Collection of sound-speed profiles from CTDs taken duringexperiment.
27 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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dz2 1@k2~z!2kn2#un~z!50. ~3!
The mode functions form a complete set~for simplicity weomit discussion of the continuous spectrum though a goapproximation is to use a set of discrete mode functions otained from a waveguide extended in depth and terminaby a pressure release or rigid boundary!
e
FIG. 4. Single frequency simulation of phase conjugation for the geomeof Fig. 2 for a probe source located at a depth of 40 m and a range ofkm. ~a! Sound-speed profile. The density,r and attenuation,a ~in dB/wavelength! of the bottom two layers are also given.~b! Simulation for a20-element SRA. Note the sharp focus in depth.~c! Simulation for only thebottom 10 elements of the SRA.
27Kuperman et al.: Phase conjugation in the ocean
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un~z!un~zs!
r~zs!5d~z2zs!, ~4!
and satisfy the orthonormality condition
E0
` um~z!un~z!
r~z!dz5dnm , ~5!
wherednm is the Kronecker delta symbol.The received field at the source/receiver array~SRA! at
rangeR from PS with source/receive elements at depthszj ,is Gv(R;zj ,zps). The phase conjugation process consistsexciting the SRA sources by the complex conjugate ofreceived field,Gv* (R;zj ). The resulting acoustic field transmitted from theJ sources satisfies the wave equation,
¹2Ppc~r ,z!1k2~z!Ppc~r ,z!5(j 51
J
d~z2zj !Gv* ~R;zj ,zps!,
~6!
where the ranger is with respect to the SRA. Using Greenfunction theory, the solution of Eq.~6! is the volume integralof the product of the Green’s function as specified by Eq.~1!and the source term of Eq.~6!. For a vertical line of discretesources, the integral reduces to a sum over the sourcetions,
Ppc~r ,z;v!5(j 51
J
Gv~r ;z,zj !Gv* ~R;zj ,zps!, ~7!
whereR is the horizontal distance of the SRA from PS andris the horizontal distance from the SRA to a field point.
Note that the magnitude squared of the right-hand s~rhs! of Eq. ~7! is the ambiguity function of the Bartletmatched-field processor10 ~with an appropriate normalizatiofactor! where the data are given byGv(R;zj ,zps) and thereplica field byGv(r ;z,zj ). In effect, the process of phasconjugation is an implementation of matched-field proceing where the ocean itself is used to construct the repfield. Or, alternatively, matched-field processing simulathe experimental implementation of phase conjugationwhich a source/receive array is used. To demonstratePpc(r ,z) focuses at the position of the probe source, (R,zps),we simply substitute Eq.~2! into Eq.~7! which specifies thatwe sum over all modes and array sources
Ppc~r ,z;v!'(m
(n
(j
um~z!um~zj !un~zj !un~zps!
r~zj !r~zps!AkmknrR
3exp i ~kmr 2knR!. ~8!
For an array which substantially spans the water columnadequately samples most of the modes, we may approximthe sum of sources as an integral and invoke orthonormaas specified by Eq.~5!. Then the sum overj selects outmodesm5n and Eq.~8! becomes
Ppc~r ,z;v!'(m
um~z!um~zps!
r~zps!kmArRexp ikm~r 2R!. ~9!
The individual terms change sign rapidly with mode numbHowever, for the field at PS,r 5R, the closure relation ofEq. ~4! can be applied approximately~we assume that the
28 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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r.
kn’s are nearly constant over the interval of the contributimodes! with the result thatPpc(r ,z)'d(z2zps). Figure 4 isa simulation of the phase conjugation process using Eq.~7!for a probe source at 40-m depth and at a range of 6.3from a 20 element SRA as specified in Fig. 2 verifying tabove discussion. Range-dependent bathymetry was usethe input to an adiabatic mode model13 for the specificsound-speed profile taken from the ensemble of profilesFig. 3 and a bottom sound-speed structure shown in Fig.~a!which includes a low speed layer as has been ascertaexperimentally.11 Notice that the focusing in the vertical iindicative of the closure property of the modes. As a maof fact, for an SRA with substantially fewer elements, we sthat the focusing still is relatively good. For example, F4~c! also shows a result for the bottom 10 elements ofSRA which are below the thermocline.
2. Pulse excitation
In this experiment a 50-ms pure-tone pulse with cenfrequency 445 Hz was used for the probe transmission.can Fourier synthesize the above results to examine pconjugation for pulse excitation. Here, in the context of thexperiment, we remind the reader that phase conjugatiothe frequency domain is equivalent to time reversal intime domain. Thej th element of the SRA receives the folowing time-domain signal, given by Fourier synthesis of tsolution of Eq.~1!:
P~R,zj ;t !5E Gv~R;zj ,zps!S~v!e2 ivt dv, ~10!
where S(v) is the Fourier transform of the probe sourpulse. This expression incorporates all waveguide effectscluding time elongation due to multipath propagation. Fconvenience, take the time origin such thatP(R,zj ;t)50outside the time interval~0,t!. Then the time reversed signathat will be used to excite thej th transmitting element of theSRA is P(R,zj ;T2t) such thatT.2t. This condition isimposed by causality; the signal has to be completelyceived before it can be time reversed. Then
P~R,zj ;T2t !5E Gv~R;zj ,zps!S~v!e2 iv~T2t ! dv
5E @Gv* ~R;zj ,zps!eivTS* ~v!#e2 ivt dv,
~11!
where the sign of the integration variable,v, has been re-versed and the conjugate symmetry of the frequency-domGreen’s function and probe pulse has been used. The qtity in brackets in Eq.~11! is the Fourier transform of thesignal received by thej th SRA receiver element after timreversal and time delay. Hence there is an equivalencetime reversal and phase conjugation in their respective tand frequency domains.
Noting that the bracketed quantity in Eq.~11! is thefrequency-domain representation of the signal retransmiby the j th element of the SRA, Fourier synthesis can be uto obtain the time-domain representation of the field pduced by the TRM. Using Eq.~7!,
28Kuperman et al.: Phase conjugation in the ocean
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Ppc~r ,z;t !5(j 51
J E Gv~r ,z,zj !Gv* ~R,zj ;zps!eivT
3S* ~v!e2 ivt dv. ~12!
This expression can be used to show that the TRM produfocusing in time as well as in space. Focusing in time occbecause a form of matched filtering occurs. To understthis, examine the TRM field at the focus point@that is, taker 5R, z5zps in Eq. ~12!#. Neglecting density gradients, recprocity allows the interchange Gv(R,zps ,zj )5Gv(R,zj ,zps). Then the time-domain equivalent of E~12! is
Ppc~r ,z;t !51
~2p!2 E (j 51
J F E Gt81t9~R,zj ,zps!
3Gt8~R,zj ,zps!dt8GS~ t92t1T!dt9, ~13!
where the time-domain representations of the Green’s fution and probe pulse are used. Note that the Green’s funcis correlated with itself. This operation is matched filterinwith the filter matched to the impulse response for propation from the probe source to thej th SRA element. Thisoperation gives focusing in the time domain, that is, itduces the time elongation due to multipath propagation.8 Thesum over array elements is a form of spatial matched filing, analogous to that employed in the Bartlett matched-fiprocessor.10 In addition, this sum further improves temporfocusing as the temporal sidelobes of the matched filterseach channel tend to average to zero which also is analoto broadband matched-field processing results.14 Finally,note that the integral overt9 in Eq. ~13! is a convolution ofeach matched-filtered channel impulse response withtime-reversed and delayed probe pulse. As a consequethis pulse isnot matched filtered, for example, a linear Fup-sweep will appear as a down-sweep at the focus andnot be compressed.
Figure 5~a! shows a simulation for a 50-ms rectangupulse with center frequency 445 Hz for the same geomused in Fig. 4~a! as received at the SRA and Fig. 5~b! showsthe pulse as transmitted to a plane at a range of 6.3 kmrange of PS. Four sources were excluded from the simulabecause these phones were not used in the experiment.the temporal focusing; that is, the 50-ms pulse disperseabout 75 ms at the SRA but the time reversed pulse receat the VRA is compressed~focused! to 50 ms as opposed texhibiting even further time dispersion. On the other haFig. 5~c! shows a pulse 500 m outbound of PS~i.e., the VRAis at the same location but PS is 500 m closer to the SR!.The pulse is not spatially focused and it is temporally mdiffuse than the result for the focal spot.
3. Properties of the focal region
A detailed discussion of the spatial and temporal factaffecting the focus is given in Appendix A. The primaresult is that the TRM focus is robust, provided the SRadequately samples the field in the water column. First,focus tends to depend primarily on the properties of
29 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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ocean near the focus and tends to be independent of~thepossibly range-dependent! properties of the medium betweethe SRA and the focus. Temporal changes in the mediumto, for example, surface waves and internal waves degrthe focus, but this degradation will be tolerable if the avera~or coherent! Green’s function is not severely reduced bthese time variations. Generally, the shape of the focuapproximated by the field that a point source placed at
FIG. 5. Simulation of a 445-Hz, 50-ms transmitted pulse for the geometrFig. 2 for a probe source located at a depth of 40 m.~a! Pulse received onthe SRA at range of 6.3 km from PS. There is a temporal dispersion of a75 ms and significant energy throughout the water column.~b! The focus ofthe time reversed pulse at the VRA. There is pulse compression back toriginal transmitted 50-ms duration as well as spatial focusing in depth~c!Vertical and temporal distibution for a pulse 500 m outbound of PS~theVRA is at the same location but PS is 500 m closer to the SRA!.
29Kuperman et al.: Phase conjugation in the ocean
FIG. 6. Experimental results for probe source PS and VRA at same range.~a! The pulse data received on the SRA for PS at depth of 40 m.~b! The datareceived on the VRA from the time reversed transmission of pulses shown in~a!. The VRA is 40 m inbound from the focus as determined by DGPS.~c! Thepulse data received on the SRA for PS at depth of 75 m.~d! The data received on the VRA from the time-reversed transmission of pulse shown in~c!. TheVRA is 40 m outbound from the focus as determined by DGPS.
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focus generates after nonpropagating modes are subtraThus if absorption or scattering tends to eliminate high-ormodes, the focus will be comprised of the remaining loworder modes and will be relatively broader. Very roughly, tvertical width of the focus will be equal to the water dep~or depth of the duct! divided by the number of contributingmodes if the sound speed~in the duct! is not strongly depen-dent on depth.
The TRM focus is also robust with respect to arrshape4 provided the shape does not change betweenprobe reception and time reversed transmission. This perty makes it unnecessary to know the exact shape ofTRM array and offers a considerable advantage over contional beamforming.
II. EXPERIMENTAL DEMONSTRATION OF A TRM INTHE OCEAN
An assortment of runs was made to examine the stture of the focal point region and the temporal stability of tprocess. Here we will be reporting on three types of expments~note that range refers to the distance from the SR!:
~1! Demonstration of the time-reversal mirror~TRM! inthe ocean. The probe source~PS! is moved from shorterrange to a longer range past the VRA. At each PS rangsends out a 445-Hz, 50-ms pulse on the even minute.pulse is received at the SRA, time reversed and retransm
30 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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five times~once every 10 s! starting at the odd minute. Thisignal is received at the VRA and data from all channelsrecorded. Note that when PS is at the same range ofVRA, the data recorded at the VRA are a vertical slice of tfocal range as indicated in the simulation for a harmosource in Fig. 4. Figure 5~b! is a simulation of the expecteresults at that range. When PS is closer than the VRA,VRA data correspond to a measurement beyond the forange and vice versa when PS is beyond the VRA.
~2! Stability of TRM. PS is at the VRA range whichmeans that we are measuring the vertical profile of the foregion. A 50-ms, 445-Hz pulse is sent out once and the Sretransmits the same time reversed signal every 10 s foextended period. Here the goal is to determine how lonsingle probe signal remains a valid phase conjugate probethe specific ocean environment and source location. Thresults are constrained by the limitations of the actual expment.
~3! Acoustic ping pong. The probe source with colcated receiver now acts as a tranponder. The SRA transa 50-ms water column filling signal to the transponder whis at a depth of 75 m. The transponder retransmits theceived signal~no time reversal! to the SRA which then transmits the time-reversed signals from the full array. This comences an acoustic ping pong iteration between SRA andwith PS acting as a transponder~SRT!.
30Kuperman et al.: Phase conjugation in the ocean
FIG. 7. Out of focus data received on the VRA from the time-reversed transmission of pulses with PS at a depth of 40 m.~a! PS is outbound 600 m.~b! PSis outbound 200 m.~c! PS is inbound 200 m.~d! PS is inbound 500 m.
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A. Demonstration of TRM in the ocean
The vertical receive array VRA was deployed at a randetermined by DGPS, of 6.24 km from the SRA and tprobe source PS was deployed at two different depths, 4and 75 m. Figure 6 shows the pulse as received on the Sand VRA for both source depths. The data at the SRA acombination of signal and noise. A 233-ms window wdigitized and time reversed for transmission to the VRWhen the VRA and PS have the same range~experimentallywithin 40 m by a DGPS measurement! to the SRA, we seethe focusing as predicted in Sec. I for a probe source at 4depth and similar results for a probe source at 75-m deClearly, we have implemented a time-reversal mirror focing at the range and depth of the probe source.
Figure 7 shows the result as we sweep through the fopoint. Note that because of the way the experiment had toperformed, we are actually keeping the VRA fixed achanging the range of PS An alternative way to presentfocusing effect which displays the sidelobes off the mpeak is shown in Fig. 8. The solid line with circles is thnearest to the focal region. Here we see the sidelobes invertical becoming large as we move away from the foregion.
B. Stability of TRM
The variability of the sound-speed structure in the wacolumn is indicated in Fig. 3 which contains a collection
31 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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sound speeds derived from CTD’s at different positions atimes throughout the experiment. A thermistor chain placat the position indicated in Fig. 2 reveals the varying teperature structure as shown in Fig. 9. In addition, thereinformation concerning wave heights from the waveridshown in Fig. 2. The time series of the rms waveheight
FIG. 8. The energy over a 0.3-s window as a function of depth for variranges from the focal region. The depth of the probe source was 40 m1means VRA is outbound from the focus~PS!.
31Kuperman et al.: Phase conjugation in the ocean
4.
FIG. 9. Thermistor chain data. The contours from the top down are 15, 1and 14.2 °C.
32 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
6, FIG. 10. Surface waveheight measurements from the waverider.
FIG. 11. Results on stability of the focal region.~a! Pulse arrival structure at VRA for probe source at 40 m depth averaged over 1 h.~b! Pulse arrival structureat VRA for probe source at 75 m depth averaged over 2 h.~c! Mean and standard deviation of energy in a 0.3-s window for 40-m probe source.~d! Mean andstandard deviation of energy in a 0.3-s window for 75-m probe source.
32Kuperman et al.: Phase conjugation in the ocean
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shown in Fig. 10. Although the time series of the enviromental data does not have the temporal and spatial resolufor an exhaustive comparison of theory and data, thestability data collection periods show qualitative agreemwith a first order analysis of the nature of the fluctuations
Basically, as shown in Appendix A, theory predicts ththe mean field dominates the focal region with fluctuatiobeing a diffuse phenomenon, becoming more apparent afrom the focus. That is, if one considers the total field tocomposed of a mean field and a fluctuating field, it ismean field which has the coherence properties which pduce the focusing whereas the fluctuating field is a formsignal-generated noise.
Two stability data collection periods for the probsource depths of 40 m and 75 m were made for 1 h and 2 h,respectively~the lengths of the runs were dictated by expemental circumstance!. The Julian day and times of the stbility runs for SD575 m and SD540 m were J114 15:11–
FIG. 12. Simulation of vertical profile of the mean field at the focal ranfor different values of surface roughness.~a! Probe source at 40 m.~b! Probesource at 75 m.
33 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
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17:07 and J114 18:47–19:47, respectively. Figure 11 shthe results of these runs. These plots indicate that the fowas considerably more stable for the deep probe sourcesus the shallower probe source and that the focus is brofor the shallower probe source.
Simulations using representative rms wave heights frFig. 10 and the environment of the experiment with a normmode rough surface mean field scattering theory15 are shownin Fig. 12. The results indicate that surface scattering dnot have a significant impact on the focal region for thparticular environment. On the other hand, examinationthe environmental data indicates that the probe source ashallower depth was at the bottom of the thermocline whthe water column variability was the greatest. As derivedAppendix A, we expect the focusing phenomenon to be msensitive to the environment at the endpoints of the expmental geometry. The tentative conclusion is that the fltuations in this case were caused by sound-speed fluctuain the water column, but more analysis and finer sampvolume data are required.
FIG. 13. Acoustic ping pong between a transponder at 75 m depth arange of 6.24 km from the SRA. The waterfall plot shows the energy i0.3-s window at the VRA~which is at the same range as the transponder! asa function of depth for each of the 15 round trips. There were two minubetween each round trip.
FIG. 14. Sound-speed profiles. The solid line was the optimum profile frthe inversion process. The dashed line is thermistor chain derived sospeed closest in time to the data shown in Fig. 6~a!,~b!.
33Kuperman et al.: Phase conjugation in the ocean
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the
eRA:
C. Acoustic ping pong: iterative focusing
The purpose of the acoustic ping pong experiment isdemonstrate that focusing can be iteratively improved. Thas already been demonstrated and explained in earlierfield multiscatterer, ultrasonic experiments.16–18 Basically,since a TRM returns signals to their origin in proportiontheir original relative strengths, repeating the process aond time will reduce the level of the focused field for thweaker signals versus the stronger signals, and so on.theoretical explanation is in terms of eigenvalues and eigvectors of the time-reversal operator. Eventually, onlystrongest signal~or that part of the field corresponding to thlargest eigenvalue! is focused.
In this experiment, ping pong was initiated and kept ging for 15 round trips. Figure 13 is a waterfall plot of thenergy in a 0.3-s window of the pulses received on the Vwhich was at the same range as the transponder. Theremin between each round trip.
These results show the increased focusing brought aby the iteration process. However, this single source resunot completely analogous to the free-space multiscatteresults in Refs. 16–18. Rather, it depends on the particTRM array-data eigenvector structure in the specific waguide environment. A paper with a detailed explanationthis process is in preparation.
FIG. 15. Backpropagation using data from the SRA for the probe sourca depth of 40 m.~a! From the inversion process.~b! From the profile mea-sured at the time of the experiment.
34 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
oise-
c-
hen-e
-
Ae 2
utise-ar-f
III. EXTRAPOLATING THE EXPERIMENTAL RESULTS
We have demonstrated that a time reversal mir~TRM! can be implemented in the ocean and that its permance is consistent with theory. In this section we uscombination of data and theory to gain some additionalsight into the potential usefulness of this process. In partilar, we examine:
~1! its potential as a tool for inversion;~2! whether a smaller aperture or few source/receiver e
ments would still be effective for producing a TRM.
Further, we use item 1 to help estimate the TRM perfmance of a smaller SRA.
A. TRM applied to sound speed inversion
Empirical orthogonal functions19 ~EOFs! about the meanof the profiles shown in Fig. 3 were constructed. It was thfound through trial and error that the mean profile was sficient to provide the optimum focusing using simulated bapropagation from the SRA data. That is, the coefficient offirst term of an EOF expansion was negligible with respecthe expected accuracy of the sound-speed profiles. A plothis result compared to ‘‘CTD 11’’ which was used in thsimulations in Sec. I is shown in Fig. 14. This procedureakin to matched field tomography10,20,21 except that moreinformation is available because of the vertical array at
atFIG. 16. Backpropagation using data from every other element of the S10-element TRM.~a! Probe source at a depth of 40 m.~b! Probe source at adepth of 75 m.
34Kuperman et al.: Phase conjugation in the ocean
FIG. 17. Backpropagation produced TRM from a five-element SRA for the probe source at 40 m.~a! Elements 1, 5, 9, 13, 17 as numbered from the top.~b!First quarter of SRA.~c! Second quarter of SRA.~d! Third quarter of SRA.
FIG. 18. Backpropagation produced TRM from a five-element SRA for the probe source at 75 m.~a! Elements 1, 5, 9, 13, 17 as numbered from the top.~b!Elements 12, 14, 16, 18, 20.~c! Third quarter of SRA.~d! Lowest quarter of SRA.
35 35J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998 Kuperman et al.: Phase conjugation in the ocean
ede
omnt
ntnte
ximaelhliz
litla
singreli-
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Theayshet re-on-ra-notwas
nd-usedthe
ationrraye ofpacethe
g istiveve
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RA
focal distance. Also shown in Fig. 14 is the sound-speprofile taken closest in time to the experimental runs undiscussion.
Figure 15 shows backpropagation results initiated frSRA data using~a! the profile obtained from the inversioand~b! the profile taken at the time of the TRM experimenClearly, the single experimental profile does not represerange-independent profile descriptive of the experimeacoustic results, whereas the profile derived from the invsion represents an adequate range-independent approtion to the structure of the water column. These resultsalso meaningful in the context of mismatch in matched-fiprocessing. The experimental results indicate that a matcfield processor using the measured profile would not locathe source.
B. Reduced and sparse aperture TRM
A reduced aperture SRA would enhance the practicaof an ocean TRM. We have already shown through simu
FIG. 19. Energy strength of five-element SRA backpropagation to the V~a! Probe source at 40 m.~b! Probe source at 75 m.
36 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
dr
.a
alr-
a-redede
y-
tion of a harmonic source in Fig. 4~c! that we can expect thephase conjugation process to remain effective as a focuprocedure as aperture is reduced. We should be able toably estimate the focal properties of a TRM using data fra subset of source/receiver elements and simulations ofbackpropagation using the effective soundspeed proshown in Fig. 14 found from the inversion. Of course, diremeasurement for the sparse arrays would best study thispect of the TRM, but such data were not taken in this expment.
Figure 16 shows the results of an adiabatic mode mobackpropagation of time-reversed pulse data from evother element of the SRA. We see that for both PS depthsfocal region remains prominent for the ten-element SRA. Walso present some results for an assortment of five elemarrays in Figs. 17 and 18. Figure 19 shows a prediction ofvertical profile of the energy strength of these results whuse five element subsets of SRA elements. The resultsextended in depth to show the fields near the boundaries.key thing to notice is that there are some very small arrwhich still produce significant concentration of sound in tdesired focal region. This probe source depth-dependensult has practical ramifications for active sonar system ccepts in which one desires to minimize boundary reverbetion at the range of the target. These results areconclusive for the 40-m probe source depth because thatthe depth of more or less maximum variation of the souspeed profile. Hence, the sound-speed inversion resultin the backpropagation calculation might be the cause ofpoorer focusing of the shallow source.
IV. CONCLUSIONS
We have constructed a time-reversal mirror~TRM! inthe ocean and hence demonstrated that phase conjug~PC! is realizable in the ocean using a source–receive aand rather simple signal processing. The waveguide naturthe ocean enhances the focusing properties over a free-senvironment because the boundaries in effect enlargeTRM aperture through its images. The degree of focusinin excellent agreement with theory. Furthermore, an effecTRM need not be a full water column array. We also hainvestigated the stability of the PC processvis a visoceanfluctuations and measurements suggest a relatively longbility of the PC process. Future studies will be aimed atdetailed relationship between ocean variability and theprocess and an investigation into the possibility of usingfor inverting for the ocean environment. In addition,should be straightforward to experimentally confirm predtions of the focal size versus SRA aperture.
ACKNOWLEDGMENT
This research was supported by the Office of Naval Rsearch Code 321US, Contract No. N00014-96-D-0065.
APPENDIX A: FACTORS AFFECTING THE FOCUS
In interpreting the results of the 1996 phase conjugatexperiment, a primary issue is degradation of phaconjugate focusing. Such degrading influences can be
.
36Kuperman et al.: Phase conjugation in the ocean
udttanas
dbe
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ofb
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vided into static and dynamic categories, the former incling propagation and array structure effects and the laincluding effects due to the time-varying ocean surfacevolume. The object of study is the field produced by a phconjugate source–receiver array~SRA!, which can be writ-ten in the general form
Ppc~r ,z;v!5(j 51
J
G2~r ,r j !G1* ~r j ,r s!. ~A1!
In Eq. ~A1!, Ppc(r ,z;v) is the field produced at the fielpoint, r5(r ,z), by the phase-conjugate array with prosource placed atr ps5(R,zps). The sum is over theJ ele-ments of the SRA whose position vectors are denotedr j
5(0,zj ). Following the convention used in the main texhorizontal ranges are measured from the SRA. Propagafrom the probe source to the array is described byGreen’s functionG1(rn ,r ps), while propagation from the array to the field point is described byG2(r ,rn). The sub-scripts 1 and 2 allow for the possibility that time variationthe ocean might cause changes in the Green’s functiontween the probe and phase-conjugate transmission cyDuring either propagation cycle, the ocean is assumed to‘‘frozen’’ in the sense that it behaves as a time-invarialinear system. In this view, the Green’s function is tfrequency-dependent system transfer function for acoupropagation between any two points in the ocean. Thequency argument of the Green’s function used in the mtext is suppressed here for convenience, but it becomesportant in treating pulsed transmissions.
1. Phase conjugation in static environments
The factors that control phase-conjugate focusingstatic environments will be examined by considering a geral nonuniform, nonadiabatic waveguide. The conditions‘‘ideal’’ phase-conjugate focusing in such a waveguide wbe derived and this will implicitly identify the factors thadegrade focusing. To simplify the discussion, only vertiphase-conjugate arrays will be considered. The main obtive is to generalize Eq.~9! of the main text to the rangedependent case, using the approach given by Sideriuset al.22
in connection with the ‘‘guide source’’ concept. In this aproach, small regions near the probe source and SRAassumed to be range independent, but the larger regiontween is allowed to have arbitrary range dependencebathymetry and sound speed. Losses are neglected andbe discussed later in qualitative terms.
The Green’s function for the probe field near the prosource is approximated using range-independent normodes.
Gv~r ,z;R,zps!5(n
an~zps!un~R,z!
Akn~R!ur 2Rueikn~R!ur 2Ru. ~A2!
Similarly, the Green’s function for the probe field at the SRis written in the form
Gv~0,zj ;R,zps!5(n
bn~zps!un~0,zj !
Akn~0!Reikn~0!R. ~A3!
37 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
-erde
one
e-es.bet
tice-inm-
n-r
l
lc-
ree-
inill
eal
The modal eigenfunctions in the vicinity of the probe sourand SRA are denotedun(R,z) andun(0,z), respectively. Thecorresponding eigenvalues arekn(R) and kn(0). TheseGreen’s functions do not bear the subscripts 1 and 2 induced earlier because a time-invariant environment is unconsideration. The subscriptv is used here in the same senas in the main text. The mode amplitudes for the near-souGreen’s function are
an~zps!5ie2 ip/4
A8pr~zps!un~R,zps!, ~A4!
and the mode amplitudes for the Green’s function nearSRA are given by the linear transformation
bm~z!5(n
Umnan~z!. ~A5!
For convenience, it is assumed that there are the same nber of modes near the source and near the array, so thatUmn
is a square matrix. Cases for which these numbers are simbut not equal can be treated by discarding high-order moThe matrixUmn includes any mode coupling that is duethe range dependence of the ocean and is defined in suway as to be independent of source depth. Furthermorethe extent that absorption loss in the water column and sfloor can be neglected,Umn is unitary.
The field produced by the SRA is
Ppc~r ,z;v!5(j 51
J
Gv~r ,z;0,zj !Gv* ~0,zj ;R,zps!. ~A6!
The Green’s function for propagation from thej th arrayelement to the field point (r ,z) can be expressed in terms othe Green’s function for propagation in the opposite diretion by using reciprocity:
Gv~r ,z;0,zj !5r~z!
r~zj !Gv~0,zj ;r ,z!. ~A7!
In terms of mode amplitudes,
Gv~r ,z;0,zj !5r~z!
r~zj !(
n
cn~z!un~0,zj !
Akn~0!reikn~0!R, ~A8!
where the mode amplitudes,cn(z), are
cm~z!5(n
Umnan~z!eikn~R!~r 2R!. ~A9!
Note that the mode amplitudes,cn(z), are essentially thesame as thebn(z), but with the source range coordinashifted byr 2R.
Equations~A3! and ~A8! can be inserted in Eq.~A6! toobtain an expression for the phase-conjugate field in a randependent waveguide:
Ppc~r ,z;v!5r~z!
ARr(m,n
cm~z!Dmnbn* ~zps!
Akm~0!kn* ~0!ei @km~0!2kn* ~0!#R,
~A10!
where
37Kuperman et al.: Phase conjugation in the ocean
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Dmn5(j 51
Jum~0,zj !un~0,zj !
r~zj !. ~A11!
In the ideal case, the array spans the entire water coluwith elements having uniform spacing,da , and the modaleigenfunctions have negligible amplitude in the bottom.this case, the sum over array elements in Eq.~A11! approxi-mates the orthogonality integral for modal eigenfunctio@Eq. ~5!#, andDmn da can be taken equal todmn . This idealcan be approached quite closely in the environment of1996 experiment. Using the environmental parametersfined in Fig. 4~a!, and considering only the first 12 modes,array with 36 elements with spacingda53.33 m and with theshallowest element 4.44 m below the surface gives diagoelements inDmnda that are within 3% of unity and off-diagonal elements that are of order 0.03 or less. Themode is an exception; it has a small diagonal element astrapped in the first sediment layer and not adequasampled by the array. This is of no consequence, asmode is very lossy and does not contribute to propagatThe element placement of the actual array gives smadiagonal elements of about 0.5 with a few off-diagonal ements as large as 0.3.
Returning to the derivation of the conditions for idephase-conjugate focusing, takeDmn5dmn /da in Eq. ~A10! toobtain
Ppc~r ,z;v!5r~z!
daARr(m,n
Qmnam~z!an* ~zps!eikm~R!~r 2R!,
~A12!
where
Qmn5(l
UlmUln*
kl~0!e22J@km~0!#R. ~A13!
Losses due to absorption and scattering are detrimentaphase-conjugate focusing, as they cause attenuationhigher-order modes, yielding a blurrier focus than wouldpossible with lower loss. Furthermore, this blurring will increase as the range between the source and the arracreases owing to the strong range and mode number dedence of attenuation. Thus in defining the ideal case, loare set to zero and the mode coupling matrix,Umn , is takento be unitary. If the mode dependence ofkl(0) in Eq. ~A13!is neglected,
Qmn5dmn
km~0!, ~A14!
and the phase-conjugate field for an ideal array in a lossenvironment can be approximated as
Ppc~r ,z;v!5(n
un~R,z!un~R,zps!eikn~R!~r 2R!
8pr~zps!kn~0!daARr.
~A15!
Apart from inessential factors, this expression is the samEq. ~9! of the main text which was derived for the rangindependent case. Even though Eq.~A15! represents theideal case, it illustrates properties that actual phase-conjuarrays may possess, provided they are not too far from id
38 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
n
s
ee-
al
stislyisn.st-
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as
teal.
One such property is independence of the focus patternthe distance between the probe source and the array~whenabsorption can be neglected and apart from the cylindrspreading factor 1/ARr!. Even more strikingly, the focusfield is independent of the~possibly range-dependent! envi-ronment between the focus and the array~see examples presented by Sideriuset al.22!. That is, the focus depends onon the local properties of the water column and sea flooris not affected by bathymetry or range-dependent waterumn properties in the region between the array and the foprovided the latter do not change apprcciably during the tpropagation cycles. This means that, in the ideal case, pconjugation is not affected by time-invariant forward scatting due to bathymetry, fronts, etc. It also implies that,simulations of phase-conjugate focusing, it is importantaccurately model the ocean in the vicinity of the focus, bless accuracy is required for the more distant parts ofpropagation path. One important reservation must be adat this point. The derivation above is essentially two dimesional in that cross-range spatial variation of the oceanneglected. Static out-of-plane scatteringwill degrade phaseconjugate focusing if one-dimensional vertical arraysused. Planar or volumetric arrays of sufficient aperture,the other hand, will not suffer due to static out-of-plane sctering.
The invariance seen in the ideal case is similar to tpredicted for an ideal, closed phase-conjugate surface ar4
which produces a strictly invariant focal field that resembthe original field of the probe source, except that the phaconjugate field is a standing wave. In the present case,probe source field~including only propagating modes! isgiven by Eq.~A2! which can be put in the form
G~r ,z;R,zps!5ie2 ip/4
r~zps!A8pur 2Ru
3(n
un~R,z!un~R,zps!eikn~R!ur 2Ru
Akn~R!.
~A16!
Apart from a difference in spreading loss and an ovephase difference, Eqs.~A15! and ~A16! are quite similar.There is a slight term-by-term difference owing to differinfactors involving modal eigenvalues, but the primary diffeence is in the propagation phase factor. The source fipropagatesaway from the source location while the phasconjugate field propagatespast the source location in thedirection away from the array.
2. Phase conjugation in time varying environments
Time-dependent forward scattering due to surfaceinternal waves causes change in the propagation charactics of the medium in the time interval between the probe aphase-conjugate transmission cycles with attendant degrtion of phase-conjugate focusing.5 In discussing scatteringfrom a general point of view, it is convenient to decompothe Green’s function into coherent and incoherent parts:
Ga~r ,r 8!5G~r ,r 8!1dGa~r ,r 8!. ~A17!
38Kuperman et al.: Phase conjugation in the ocean
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The subscripta takes on the values 1 and 2 for the probe aconjugate transmission cycles, respectively. The coherenmean, Green’s function,G(r ,r 8) is not assigned a subscripbecause the random time variations are assumed to betionary in the statistical sense. It will be assumed that sucient time has elapsed between the probe and conjutransmission cycles that variations in the two Green’s futions are uncorrelated.
This condition was very likely satisfied in the 1996 expement with respect to scattering by surface waves, which hcorrelation timescales on the order of seconds, while the tbetween transmission cycles was measured in minuteshours. Internal waves have relatively long correlation timscales, but the longer transmission intervals~several minutesto a few hours! of the experiment were most likely sufficiento produce decorrelation of fluctuations in volume scatteri
Combining Eqs. ~A1!, ~A17! and ~A18!, the meanphase-conjugate field is
The covariance,K j j 8(r ), is proportional to the correlationbetween the incoherent field at elementsj and j 8 of the arraywith a unit point source situated atr . In deriving Eq.~A20!,free use was made of reciprocity~which allows interchangeof the two arguments of the Green’s function! and stationar-ity ~which means thatdG1 anddG2 have identical statistics!.
Equations ~A19! and ~A20! are general and includthree-dimensional scattering~i.e., in-plane and out-of-planescattering!. They lead to two general conclusions regardifocusing in the 1996 experiment for those cases in whsufficient time elapsed between the two transmission cycFirst, the mean focus field, that is, the focus field averaover many independent probe-conjugate-transmission cyis obtained by using the coherent Green’s function in plof the actual~random! Green’s function. Second, and moimportant, the field near the focus does not fluctuate apciably, that is, it is well approximated by the mean focfield. This conclusion is supported by careful inspectionEq. ~A20!, which shows that the variance of the phasconjugate field is not localized near the focus, but is sprdiffusely in range and depth. Thus near the focus, the m
39 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
dor
ta--te-
veende
.
hs.ds,e
e-
f-dn
field dominates, unless scattering is strong enough to dimish the mean Green’s function to such a degree that focuis essentially destroyed.
To see that the field variance is unfocused, it is necsary to discuss each term in Eq.~A20! The first term can beviewed as being proportional to the intensity of a phaseversed retransmission of the incoherent field producedscattering of the probe transmission. This retransmissionbe directed back toward the the scatterers responsible foincoherent component of the probe field, and thesespread over the entire volume and surface of the ocean. Slarly, the second term is proportional to the intensity prduced at the source location by a coherent retransmissiothe phase-reversed incoherent field produced from a fictitisource placed at the field point~reciprocity is being used inthis interpretation!. Again, this retransmission will be diffusand will not peak as the field point approaches the soulocation. The last term in Eq.~A20! is more difficult to as-sess. It is a double sum over all array elements of the prodof covariances due to sources placed at both the field pand probe source location. If scattering and propagationvery complicated in a spatial sense, these covariancesnot be strongly dependent upon the source locations. Thathe incoherent field produced by these sources does nottain information on the source location. If this is the case,covariances will be largely independent ofr andr ps , and thelast term of Eq.~A20! will not peak asr approachesr ps .
APPENDIX B: HARDWARE DESCRIPTION
The phase conjugation~time reversal mirror! experimentwas carried out in April 1996 off the northwest coast of ItaAs shown in Fig. 2, a source–receiver array~SRA! was de-ployed in 125-m-deep water and cabled approximately 1back to a small island, Formica di Grosseto~42° 34.68 N, 10°52.98 E!. A rf telemetered vertical receive array~VRA! wasdeployed in 145-m-deep water approximately 6.3 km wesFormica and used to measure the structure of the acoufield across the water column. The R/ALLIANCE received this rf telemetered data stream and adeployed a source–receive transponder~SRT! ~echo re-peater! which also was used as a probe source~PS!.
The vertical source array portion of the SRA consistof 24 slotted cylinder sources spaced 3.33 m apart~totalaperture 76.6 m!. The sources have a resonance at appromately 445 Hz and a 3 dBbandwidth of approximately 35Hz as shown in Fig. B1. Thus the SRA sources were serated by approximately one wavelength at their centerquency. Each source was hardwired individually back totransmit control system on Formiche di Grosseto via a mtiple twisted pair umbilical cable. The transmit control sytem synthesized the low-level analog signals for each souand these then were amplified prior to coupling ontoumbilical cable. Based on a nominal driving level of 10VRMS, the nominal source level of the transducers was 1dB re: mPa.
In addition to the vertical source array, the SRA icluded a colocated~i.e., physically strapped together! verti-cal receive array consisting of 48 hydrophones spacedthe separation of the source array transducers. The time
39Kuperman et al.: Phase conjugation in the ocean
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.
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ries from each array element was sampled atf s51.5 kHzusing 24 bit A/D converters, multiplexed onto a single digidata stream, and cabled back to Formiche di Grosseto vseparate coaxial umbilical cable.23 The shore-based digitadata acquisition system archived the data stream and enacapturing short segments of the array time series~from the24 hydrophones colocated with the source array transduc!for time reversal and retransmission by the transmit consystem. Due to high-level contaminants observed in fouthe time series, these channels were set to zero duringretransmission process.
The rf telemetered vertical receive array~VRA! con-sisted of 64 hydrophones in a nested configuration ove90-m aperture.24 A 46 element subset of these hydrophonwith 2-m spacing was used to generate the results discuin the main text. The time series from each array elemwas sampled atf s51.2 kHz, multiplexed onto a single digital data stream, and sent via rf telemetry to the RALLIANCE for both quick-look analysis and archival puposes.
Last, the source–receive transponder~SRT! ~echo re-peater! and probe source~PS! consisted of a slotted cylindetransducer identical to those used in the source array, awas operated at the same nominal source level of 165 dBre:1 mPa. When used as an echo repeater, the SRT includseparate receiving hydrophone to sample the acoustic fiethe depth of the source. In this case, a short segment oreceived time series containing a SRA transmission wastured, amplified, and retransmitted~without time reversal!.When used simply as a source, the SRT transmitted a 50445-Hz pulse which probed the multipath structure ofchannel. In this case, the SRA received the temporallyspatially spread transmission, time reversed and amplithe 24 time series, and retransmitted them from the sou
FIG. B1. Transmitting voltage response~TVR! versus frequency for one othe slotted cylinder source array transducers.
40 J. Acoust. Soc. Am., Vol. 103, No. 1, January 1998
la
led
rslfhe
ased
nt
it
aathep-
s,edd
ce
array transducers. By allowing the R/V ALLIANCE to tow thePS slowly through the range between the SRA and VRA,focal region of the phase conjugation process could be sied and these results are discussed in the main text.
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