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Ray dynamics in a long-range acoustic propagation experimentFrancisco J. Beron-Vera and Michael G. BrownRosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida 33149

John A. ColosiWoods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543

Steven TomsovicDepartment of Physics, Washington State University, Pullman, Washington 99164

Anatoly L. VirovlyanskyInstitute of Applied Physics, Russian Academy of Science, 6003600 Nizhny Novgorod, Russia

Michael A. WolfsonApplied Physics Laboratory, University of Washington, Seattle, Washington 98105

George M. ZaslavskyCourant Institute of Mathematical Sciences, New York University, New York, New York 10012

~Received 5 September 2002; revised 8 May 2003; accepted 23 June 2003!

A ray-based wave-field description is employed in the interpretation of broadband basin-scaleacoustic propagation measurements obtained during the Acoustic Thermometry of Ocean Climateprogram’s 1994 Acoustic Engineering Test. Acoustic observables of interest are wavefront timespread, probability density function~PDF! of intensity, vertical extension of acoustic energy in thereception finale, and the transition region between temporally resolved and unresolved wavefronts.Ray-based numerical simulation results that include both mesoscale and internal-wave-inducedsound-speed perturbations are shown to be consistent with measurements of all the aforementionedobservables, even though the underlying ray trajectories are predominantly chaotic, that is,exponentially sensitive to initial and environmental conditions. Much of the analysis exploits resultsthat relate to the subject of ray chaos; these results follow from the Hamiltonian structure of the rayequations. Further, it is shown that the collection of the many eigenrays that form one of theresolved arrivals is nonlocal, both spatially and as a function of launch angle, which places severerestrictions on theories that are based on a perturbation expansion about a background ray. ©2003Acoustical Society of America.@DOI: 10.1121/1.1600724#

PACS numbers: 43.20.Dk, 43.30.Cq@RAS#

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I. INTRODUCTION

The chaotic dynamics of ray trajectories in ocean acotics have been explored in a number of recepublications,1–22 including two recent reviews.21,22 In thesepublications a variety of theoretical results are presentedillustrated, mostly using idealized models of the ocean sochannel. In contrast, in the present study a ray-based wfield description is employed, in conjunction with a realisenvironmental model, to interpret a set of underwater acotic measurements. For this purpose we use both ocegraphic and acoustic measurements collected during thevember 1994 Acoustic Engineering Test~the AETexperiment!. In this experiment phase-coded pulse-like snals with 75-Hz center frequency and 37.5-Hz bandwiwere transmitted near the sound-channel axis in the easNorth Pacific Ocean. The resulting acoustic signals werecorded on a moored vertical receiving array at a range3252 km. We present evidence that all of the important ftures of the measured AET wave fields are consistent wiray-based wave-field description in which ray trajectoriespredominantly chaotic.

The AET measurements have previously been analyin Refs. 23–25. In the analysis of Worcesteret al.23 it was

1226 J. Acoust. Soc. Am. 114 (3), September 2003 0001-4966/2003/1

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shown that the early AET arrivals could be temporally rsolved, were stable over the duration of the experiment,could be identified with specific ray paths. These featuresthe AET observations are consistent with those of other lorange data sets.26,27The AET arrivals in the last 1 to 2 s~thearrival finale! could not be temporally resolved or identifiewith specific ray paths, however. Also, significant verticscattering of acoustic energy was observed in the arrivanale. These features of the AET arrival finale are consiswith measurements and simulations~both ray- and parabolic-equation-based! at 250-Hz and 1000-km range27–29 and at133-Hz and 3700-km range.30

The analysis of Colosiet al.24 focused on statisticaproperties of the early resolved AET arrivals. Measuremeof time spread~a measure of scattering-induced pulse broening!, travel time variance~a measure of scattering-inducepulse wander!, and probability density functions~PDFs! ofpeak intensity were presented and compared to theorepredictions based on a path integral formulation as descrin Ref. 31. Travel time variance was well predicted by ttheory, but time spreads were two to three orders of magtude smaller than theoretical predictions, and peak intenPDFs were close to lognormal, in marked contrast to

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predicted exponential PDF. The surprising conclusion of tanalysis was that the measured AET pulse statisticsgested propagation in or near the unsaturated regime~char-acterized physically by the absence of micromultipathsmathematically by use of a perturbation analysis basedthe Rytov approximation!, while the theory predicted propagation in the saturated regime~characterized by the presencof a large number of micromultipaths whose phases aredom!.

In Ref. 25 it was shown that in the finale region of thmeasured AET wave fields, where no timefronts aresolved, the intensity PDF is close to the fully saturatedponential distribution. This result is not unexpected becathe finale region is characterized by strong scattering of brays14 and modes.32

The work reported here seeks to elucidate both the phics underlying the AET measurements, and the causes osuccesses and failures of the analyses reported in Refs.25. We show that a ray-based analysis, in which ideas aciated with ray chaos play an important role, can accountthe stability of the early arrivals, their small time spreads,associated near-lognormal PDF for intensity peaks, thetical scattering of acoustic energy in the reception finale,the near-exponential intensity PDF in the reception finaPart of the success of this interpretation comes about duthe presence of a mixture of stable and unstable ray tratories. Also, an explanation is given for differing intensistatistics in the early and late portions of the measured wfields, and the fairly rapid transition between these regim

The remainder of this paper is organized as follows.the next section we describe the AET environment andmost important features of the measured and simulated wfields. In the simulated wave fields both measured mesosand simulated internal-wave-induced sound-speed pertutions are taken into account. It is shown that even inabsence of internal waves, the late arrivals are not resoand the associated ray paths are chaotic. In Sec. IIIpresent results that relate to the structure of the early porof the measured timefront and its stability. The micromupathing process is discussed, and a quantitative explanafor the remarkably small time spreads of the early ray arals is given. Section IV is concerned with intensity statistiAn explanation is given for the cause of the different wavfield intensity statistics in the early and late portions of tarrival pattern. In the final section we summarize and discour results.

II. MEASURED AND SIMULATED WAVE FIELDS

Figure 1 shows a measured AET wave field in the timdepth plane and ray-based simulations of such a wavewith and without internal waves. Figure 2 is a complemetary display that shows measured and simulated wave fiafter plane-wave beamforming. In this section we descrthe most important qualitative features of measuredsimulated AET wave fields. We begin by reviewing fundmental ray theoretical results so that our ray simulationsbe fully understood. Also, this material provides a foundatfor some of the later discussion. We then briefly describeAET environment and the signal-processing steps that w

J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003

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performed to produce the measured wave field shown in1. As is common practice in ocean acoustics, we treatsoscale ocean structure@whose horizontal length scales aO~100 km! and whose time scales are O~1 month!# determin-

FIG. 1. Measured and simulated AET wave fields in the time–depth plaUpper panel: a typical measurement, shown on a gray-scale plot widynamic range of 30 dB, of wave-field intensity. Middle panel: ray simution with internal waves. Lower panel: ray simulation without internwaves. Wave-field intensity in the simulations is approximately proportioto the local density of dots, each corresponding to a ray.

FIG. 2. Same as Fig. 1 except that plane-wave beamformed wave fieldshown. The beamformed ray simulations assume a dense receivingwhose upper and lower bounds coincide with the bounds of the AET recing array. The reference depth was taken to be the depth of the upperhydrophone,z0520.9 km. The vertical slownessp is defined in the textfollowing Eq. ~7!.

1227Beron-Vera et al.: Ray dynamics

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istically, and internal gravity wave variability@whose hori-zontal length scales are O~1 km! or more and whose timescales are O~10 min! to O~1 day!# using a statistical description. Our treatment of internal-wave-induced sound-spperturbations is briefly described. Finally, we return to dscribing and comparing qualitative features of the measuand simulated wave fields.

A. Theoretical background

Fixed-frequency~cw! acoustic wave fieldsu(z,r ;s) sat-isfy the Helmholtz equation

¹2u1s2c22~z,r !u50, ~1!

where s52p f is the angular frequency of the wave fieand c(z,r ) is the sound speed.~The extension to transienwave fields is straightforward using Fourier synthesis. Itnot necessary to consider this complication to presentmost important results needed below.! We assume propagation in the vertical plane with Cartesian coordinatesz, in-creasing upwards withz50 at the sea surface, and ranger.Ray methods~see, e.g., Ref. 33 for a thorough discussiona broader context! can be introduced when the waveleng2pc/s of all waves of interest is small compared to tlength scales that characterize variations inc. Substitutingthe geometric ansatz

u~z,r ;s!5(j

Aj~z,r !eisTj ~z,r !, ~2!

representing a sum of locally plane waves, into the Heholtz equation gives, after collecting terms in descendpowers ofs, the eikonal equation

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and the transport equation

¹~A2¹T!50. ~4!

For notational simplicity, the subscripts onA and T havebeen dropped in~3! and ~4!. Each term in the sum~2! rep-resents an ‘‘eigenray’’ connecting a source and receiverdiscussed below. More generally, eachAj in ~2! can bethought of as the leading term in an asymptotic expansiopowers ofs21, but the higher-order terms in such an expasion play no role in our analysis.

The solution to the eikonal equation~3! can be reducedto solving a set of ray equations. If a one-way formulatiovalid when r increases monotonically following all rays ointerest~see, e.g., Ref. 22!, is invoked, the ray equations ar

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1228 J. Acoust. Soc. Am., Vol. 114, No. 3, September 2003

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These equations constitute a nonautonomous Hamiltosystem with one degree of freedom;z andp are canonicallyconjugate position and momentum variables,r is the inde-pendent variable,H is the Hamiltonian,L is the Lagrangian,and the travel timeT corresponds to the classical action. Tsystem is nonautonomous becauseH depends explicitly onthe independent variabler. Note also thatp[pz5]T/]z andpr5]T/]r 52H are thez andr components, respectively, othe ray slowness vector. The ray angle relative to the hzontal w satisfiesdz/dr5tanw, or, equivalently,cp5sinw.For a point source, rays can be labeled by their initial slonessp0 . The solution to the ray equations is thenz(r ;p0),p(r ;p0), T(r ;p0). The terms in~2! correspond to eigenrayswhich are solutions to the two-point boundary value problz(r R ;p0)5zR , where the spatial coordinates of the receivare (zR ,r R). The transport equation~4! can be reduced to astatement of constancy of energy flux in ray tubes. Thelution for the jth eigenray can be written

Aj~z,r !5A0uHq21u j21/2e2 im j ~p/2!, ~8!

whereH, the matrix elementq21, and the Maslov indexmare evaluated at the endpoint of the ray, andA0 is a constant.The stability matrix

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In ocean environments with realistic range dependenincluding the AET environment, ray trajectories are predomnantly chaotic. Chaotic rays diverge from neighboring raysan exponential rate, on average, and are characterizedpositive Lyapunov exponent

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trajectories leads to some limitations on predictability. Twill be discussed in more detail below. Additional bacground material relating to this topic can be found in Re21, 22 and references therein.

B. The AET experiment

In the AET experiment a source, suspended at 652depth from the floating instrument platform R/P FLIP in dewater west of San Diego, transmitted sequences of a phcoded signal whose autocorrelation was pulse-like~with apulse width of approximately 27 ms!. The source center frequency was 75 Hz and the bandwidth was approxima37.5 Hz. The resulting acoustic signals were recorded eaHawaii at a range of 3252.38 km on a moored 20-elemreceiving array between the depths of 900 and 1600 m. Acorrecting for the motion of the source and receiving arrincluding removal of associated Doppler shifts, the receivacoustic signals were crosscorrelated with a replica oftransmitted signal~to achieve the desired pulse compressio!and complex demodulated. To improve the signal-to-noratio, 28 consecutive processed receptions, extending ovtotal duration of 12.7 min, were coherently added. The dution over which this coherent processing yields improvsignal to noise is limited by ocean fluctuations, mostly dueinternal waves. The statistics of the acoustic fluctuatiowere shown in Ref. 25 not to be adversely affected by12.7-min coherent integration. Nearly concurrent with tweek-long experiment, temperature profiles in the upperm were measured with XBTs every 30 km along the tramission path. An objective mapping technique was applie23

to these measurements to produce a map of the backgrsound-speed structure~including mesoscale structure! alongthe transmission path. The sampling interval in range ofresulting sound-speed field is 30 km so that no structure whorizontal wavelengths less than 60 km is present infield. More details on the AET experiment, environment, aprocessing of the acoustic signals can be found in Ref. 2

Internal-wave-induced sound-speed perturbations insimulations are assumed to satisfy the relationshipdc5(]c/]z)pz, and the statistics ofz, the internal-wave-induced vertical displacement of a water parcel, are assuto be described by the empirical Garrett–Munk spectr~see, e.g., Ref. 34!. The potential sound-speed gradie(]c/]z)p , the buoyancy frequencyN, and the sound speedcwere estimated directly from hydrographic measurementsing a procedure similar to that described in the AppendixRef. 30; it was found that a good approximation for the AEenvironment is (]c/]z)p /c'(1.25 s2/m)N2. Individual real-izations of the vertical displacementz(r ,z) were computedusing Eq.~19! of Ref. 35 with the variablex used to denotethe range variable, andy5t50. Physically, this correspondto a frozen vertical slice of the internal wave field that icludes the influence of transversely propagating interwave modes. A Fourier method was used to numerically gerate sound-speed perturbation fields within a 32 768-po3276.8-km domain in range—soDx50.1 km and Dkx

52p/3276.8 km. A hard internal wave horizontal wavnumber cutoff,ukxu<kmax52p/1 km, was used in all of thesimulations shown.~Ray calculations were also performe


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energy. The qualitative ray behavior described below diddepend on the manner in which the largekx cutoff wastreated.!

A mode number cutoff ofj max530 was used in all thesimulations shown. Contributions from approximately 230

(215/53215/5330, where the factors of 1/5 are present bcause the assumed wave number cutoff was one-fifth ofassumed Nyquist wave number! internal wave modes wereincluded in our simulated internal-wave-induced sounspeed perturbation fields. A sparse set of measuremensound-speed variance made near the midpoint of the trmission path during the AET experiment24 suggests that theinternal wave energy strength parameterE was close to halfthe nominal Garrett–Munk valueEGM . More extensive andaccurate measurements~see Fig. 3! made in 1998/1999 atwo locations along a path displaced about 800 km tonorth of the AET transmission path favor a value ofE closeto EGM . Ray simulations were performed using bothE50.5EGM andE51.0EGM . All of the simulations shown inthis paper useE50.5EGM . The difference between thessimulations and those performed usingE51.0EGM ~notshown! will be discussed where appropriate.

C. Simulated wave fields

With the foregoing discussion as background, weprepared to discuss our ray simulations in the AET envirment. These are shown in Figs. 1, 2, 4–11, and 14.points plotted in Figs. 1, 2, and 4 were computed by integing the coupled ray equations~5! and~6! from the source tothe range of the receiving array. Several fixed and variastep-size integration algorithms were tested. In the preseof internal waves, convergence using a fixed-step fouorder Runge–Kutta integrator required that the step sizeexceed approximately 1 meter. All ray-tracing calculatiopresented in this paper were performed using doub

FIG. 3. Measured and simulated rms sound-speed fluctuations in theenvironment. The measurements shown were made approximately 100northwest of the source~labeled east! and approximately 1000 km northeasof the receiving array~labeled west!. Simulated rms fluctuations due tinternal waves are shown for bothE50.5EGM and E51.0EGM , as de-scribed in the text.


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FIG. 4. Simulated Lagrangian manifolds in the AEenvironment without~left panel! and with ~right panel!a superimposed internal-wave-induced sound-spperturbation field. The ray density used in the plot othe right and for smallupu in the plot on the left is toosparse to resolve what should, in each plot, be a smounbroken curve that does not intersect itself.

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precision arithmetic~64-bit floating point wordsize!. Thewave-field intensity is approximately proportional to the desity of rays~dots! that are plotted in Figs. 1 and 2. A somwhat more difficult calculation in which the contributionsthe wave field from many rays are coherently added willdiscussed below.

In the presence of internal waves, rays with launangles between approximately65° form the diffuse finaleregion of the arrival pattern arriving after approximate2196 s. The sound speed at the source was approxim1.48 km/s, souw0u<5° corresponds approximately toup0u<0.06 s/km. Steeper rays contribute to the earlier, moresolved arrivals. It is seen in Fig. 1 that our ray simulatin the presence of internal waves slightly underestimatesvertical spread of the finale region, although an ensemaverage over several pulses would be needed to quantifyeffect. Our experience29 suggests that the variability of thspread is of the order of the discrepancy seen in Fig. 1,that some details of the internal wave field may be importin controlling this spread. We note also that a slight undestimate in our ray simulation is expected inasmuch asfractive effects that are not included in this simulation cotribute approximately an additional 200-m vertical spreadenergy at 75 Hz.~This number can be estimated from parbolic equation simulations, examination of lower-ordmodes, or a local Airy function analysis.! With these caveatsthe agreement between simulated and measured vespreads of energy in the finale region is fairly good. A string feature of Figs. 1, 2, and 4 is the contrast betweendiffuse distribution of steep rays~with large p and z excur-sions! in (p,z) seen in Fig. 4 when internal waves apresent and the tightly clustered distributions of the sarays in the time–depth~Fig. 1! and time–angle~Fig. 2! plots.

The cause of the complexity seen in Fig. 4 is rchaos;21,22most ray trajectories diverge from initially infinitestimally perturbed rays at an exponential rate. A total48 000 rays, sampled uniformly in launch angle betwe612°, was traced to produce Figs. 1, 2, and 4 in the preseof internal waves. This fan of rays is far too sparse to resowhat should be an unbroken smooth curve—a Lagrangmanifold—which does not intersect itself in phase spa~Fig. 4!. Under chaotic conditions the separation betwe


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neighboring rays grows exponentially, on average, in ranThe complexity of the Lagrangian manifold grows at tsame exponential rate. The Lyapunov exponent is the rerocal of the e-folding distance~see, e.g., Ref. 22!. Finiterange numerical estimates of Lyapunov exponents~hereafterreferred to as stability exponents! are shown as a function olaunch angle in Fig. 5. It is seen that in this environmentflat rays (uw0u&5°) have stability exponents of approxmately 1/~100 km!, while the steeper rays (6°&uw0u&11°)have stability exponents of approximately 1/~300 km!. It fol-lows, for example, that the complexity of the flat ray portioof the Lagrangian manifold grows approximately likexp(r/100 km).

Not surprisingly, stability exponents show some sentivity to parameters that describe the internal wave field. Osimulations show that the stability exponents of rays waxial angles of approximately 5° or less approximatedouble whenj max increases from 30 to 100, while steeprays show very little sensitivity toj max. In contrast, stabilityexponents of rays with axial angles from approximately 8°15° approximately double whenkmax increases from 2p/1000m to 2p/250 m, while the flatter rays show little sensitivity tkmax. With these comments in mind, one should not attatoo much significance to the details of Fig. 5. We note, hoever, that in all of the simulations performed, rays with axangles of approximately 5° or less had higher stability exnents than rays in the 5° to 10° band.

Note that what appears to be temporally resolved ar

FIG. 5. Stability exponents~finite range estimates of Lyapunov exponent!as a function of ray launch angle in the AET environment including aperimposed internal-wave-induced sound-speed perturbation field.


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als in the measured wave field correspond in our simulatito contributions from an exponentially large number of rpaths.~Only a small fraction of the total number is includein our simulations, however, because of the relative spaness of our initial set of rays.! The observation that the travetimes of chaotic ray paths may cluster and be relativstable was first made in Ref. 8.

Simmenet al.14 have previously produced plots similato our Figs. 1 and 4 for ray motion in a deep ocean moconsisting of a background sound-speed structure very slar to ours on which internal-wave-induced sound-speedturbations were superimposed. Although our results are slar in many respects, it is noteworthy that the right panelour Fig. 4 shows more chaotic behavior than is present incorresponding plot in Ref. 14. This difference persists if o3252.38-km range simulations are replaced by simulationthe same range~1000 km! that was used in Ref. 14. Our rayespecially the steeper rays, are more chaotic than thosRef. 14. This difference is likely due to the increased coplexity of our internal wave field over that of Simmeet al.,14 who included contributions from ten internal wavmodes in their simulated internal wave fields.

Figures 6 and 7 show plots of ray depth at the AErange vs launch angle. In Fig. 6 the same rays~excludingthose with negative launch angles! that were used to producFigs. 1, 2, and 4 are plotted. In Fig. 7 two small angubands of the upper panel of Fig. 6 are blown up; thedensity is four times greater than that used in Fig. 6. Fig6 strongly suggests that, even in the absence of intewaves, the near-axial rays are chaotic. Not surprisinglythe presence of internal waves, steeper rays are also prednantly chaotic. Note, however, that Figs. 5, 6, and 7 shthat the near-axial rays are evidently more chaotic thansteeper rays. An explanation for this behavior that relate

FIG. 6. Ray depth vs launch angle in the AET environment without~upperpanel! and with~lower panel! a superimposed internal-wave-induced sounspeed perturbation field. In both panelsDw050.0005°.


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the background sound-speed structures will be givenSec. IV.

In this subsection we have briefly described and copared the gross features of measured and simulatedwave fields. Ray trajectories in our simulated wave fieldsthe presence of internal waves are predominantly chaoticspite of this, Figs. 1 and 2 show that many features ofsimulated wave fields are stable and in good qualitatagreement with the observations. Simulated and measspreads of acoustic energy in time, depth, and angle areerally in good agreement, both in the early and late portioof the arrival pattern. We shall not discuss further the spreof energy in depth and angle, except to note that Figs. 12 show that these spreads can be accounted for usingmethods in the presence of realistic~including internal-wave-induced sound-speed perturbations! ocean structure. In thetwo sections that follow, we shall consider in more detail ttime spreads of the early ray arrivals, and intensity statisin both the early and late portions of the arrival pattern.

III. MICROMULTIPATHS AND TIMEFRONT STABILITY

In this section we consider in some detail eigenratimefront stability, and time spreads. We focus our attenton the early branches of the timefront, where measured tspreads were, surprisingly, only approximately 2 ms, in shcontrast to the theoretical prediction of approximately 1 s24

We focus our attention on the early arrival time spreadscause measured time-spread estimates are available onthese arrivals. In the first subsection we discuss micromupaths, and describe the important qualitative features ofmicromultipathing process. Also, we explain how the micrmultipath travel time spreads of O~10 ms! in our simulationscan be reconciled with O~1 ms! measured wave-field timespreads. In the second subsection a quantitative estima

-FIG. 7. Expanded view of the lower panel of Fig. 6 showing ray depthlaunch angle in two small angular bands in the AET environment includa superimposed internal-wave-induced sound-speed perturbation fielboth panelsDw050.000 125°.


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FIG. 8. Ray depth vs travel time in theAET environment including a superimposed internal-wave-induced soundspeed perturbation field, but only forays with identifiers1137 ~left panel!and 1151 ~right panel!. The pointsplotted are a subset of those plottedthe lower panel of Fig. 1.

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steep ray micromultipath travel time spreads is shown toconsistent with the O~10-ms! spreads seen in our simulations.

A. Micromultipaths and pulse synthesis

It is useful to assign a ray identifier6M to each ray.Here,6 is the sign of the launch angle andM is the numberof ray turning points~wherep changes sign! following a ray.In Fig. 8 rays with two fixed values of ray identifier~1137and1151! are plotted in the time–depth plane. In both casthe points plotted are a subset of those plotted in Fig. 1.seen that the relatively steep1137 rays have a small timspread and form one of the resolved branches of the tifront seen in Fig. 1~similar behavior was noted in Ref. 14!,while the relatively flat1151 rays have a much larger travtime spread and fall within the diffuse finale region of tarrival pattern seen in Fig. 1.

Eigenrays at a fixed depth correspond to the interstions of a horizontal line,z5constant, with the curvez(w0);these intersections are the roots of the equationz2z(w0)50. Although the discrete samples ofz(w0) plotted in Figs.6 and 7 are too sparse to reveal what should be a smcurve, it is evident that the number of eigenrays at almosdepths is very large; because rays are predominantly chathis number grows exponentially in range. In principeigenrays can be found using a combination of interpolaand iteration, starting with a set of discrete samples ofz(w0).In practice, this procedure reliably finds only those eigenrin regions wherez(w0) has relatively little structure. Thesrays have the highest intensity and are the least chaoticmost depths there are many eigenrays with each value oray identifier6M . An incomplete set of eigenrays with identifier 1137, found using the procedure just described,shown in Fig. 9. We shall refer to a set of eigenrays withsame identifier in the presence of an internal-wave-indusound-speed perturbation as a set of micromultipaths.~This


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sed

term is widely used but is rarely unambiguously define!What distinguishes sets of micromultipaths with smallM~e.g., 137!, corresponding to relatively steep rays, from thowith larger M ~e.g., 151!, corresponding to relatively flarays, in the AET environment is that, except near ray turndepths and accidental crossings, the former are temporesolved while the latter are not.@Note that in order for a seof micromultipaths to be temporally resolved it must havetravel time distribution that is separated from neighboridistributions by (D f )21 or more.#

An important~and surprising! feature of sets of micro-multipaths is that they are highly nonlocal in the sense tinterspersed~i.e., having intermediate launch angles! amonga group of micromultipaths are rays withM values that differby several units. We shall refer to this behavior, whichillustrated in Fig. 10, as spectral nonlocality.~Here, the termspectral is used in the sense that an integral representatiothe wave field over ray-like contributions with variablaunch angle is a spectral representation.! In view of theobservation that, in the presence of internal waves inAET environment, ray motion is strongly chaotic and tfunction M (w0) has local oscillations of several units, itremarkable that the early portion of the timefront~see Fig. 1!is not destroyed. As ray launch anglew0 is continuouslyincreased, a continuous curve is traced out in (T,z) at therange of the AET receiving array. Surprisingly, for larglaunch angles the curve that is traced out in (T,z) verynearly coincides with the timefront that would have betraced out in the absence of internal waves—except thathe presence of internal waves the curve includes numecusps at which the direction of motion along the timefroreverses. Spectral nonlocality of micromultipaths is reveain the (T,z) curve by the many direction reversals and tassociated temporary local excursions over several bran

ongt oftheays

FIG. 9. Three segments of the range–depth plane althe AET transmission path showing a subset of a semicromultipaths. The dashed line shows the path ofthree rays with the largest amplitudes; these three rare not resolved on this plot.


en

enlerttebnse-e

halon-tuaye

ofin

t ofx-asilyul-hatn-

theod

aro-

11the-urantul-

e,11he

i-s,av-be-nt

histlyld

antre

uchaln-

s.hslti-

t areured

ill

nseofinhe

a

t o

l: tththed

he

of the timefront, each branch corresponding to a differvalue ofM.

The nonlocality of a set of micromultipaths in thrange–depth plane, which we shall refer to as spatial nocality, is seen in Fig. 9. Note that there are significant diffences in the upper and lower turning depths of the plomicromultipaths, and that these rays are spread in rangesignificant fraction of a ray cycle distance. In simulatiousingE51.0EGM , not shown, the spatial nonlocality, espcially in range, of a set of micromultipaths is much strongthan that shown in Fig. 9.

The result of performing a ray-based synthesis of wappears in the measurements to be an isolated arrivashown in Fig. 11. This involves finding a complete setmicromultipaths, including their travel times and Maslov idices, and coherently summing their contributions. Unfornately, owing to the predominantly chaotic motion of rtrajectories in the AET environment in the presence of int

FIG. 10. Ray identifier vs launch angle for a subset of the rays thatplotted in Fig. 6.

FIG. 11. Upper panel: normalized ray intensity vs travel time for a subsethe eigenrays~micromultipaths! with identifier 1137 that connect the AETsource and a receiver at depth 1005 m at the AET range. Middle panecorresponding Maslov indices vs travel time. Lower panel: envelope ofwaveform synthesized by coherently adding the contributions from allmicromultipaths shown~solid curve!, and the normalized envelope of thAET source waveform~dashed line!. The solid circles in the upper anmiddle panels identify the three rays~two of the three solid circles fall ontop of one another! whose intensity is greatest; open circles identify tremaining rays.


t

o--d

y a

r

tis

f

-

r-

nal waves, it is extremely difficult to find a complete setmicromultipaths. Indeed, the constituent ray arrivals usedthe Fig. 11 synthesis do not constitute a complete semicromultipaths. This is less important than might be epected because standard eigenray finding techniques elocate the strongest micromultipaths; the missing micromtipaths in the Fig. 11 synthesis are highly chaotic rays thave very small amplitudes. An interesting result of this sythesis is that the micromultipath-induced time spread insynthesized pulse is only about 1 ms, which is in very goagreement with the AET measurements.24,25 This approxi-mately 1-ms spread is the difference in the width~approxi-mately 25 ms! of the transmitted pulse, two cycles of75-Hz carrier, and the width of the coherent sum of micmultipath contributions~see Fig. 11!. Note that the travel-time spread among the micromultipaths found—aboutms—is much greater than the resulting spread of the synsized pulse. The principal cause of this difference in osimulations was that the travel-time spreads of the dominmicromultipaths were much smaller than the total micromtipath travel-time spreads.

The qualitative features of micromultipath travel timamplitude, and Maslov index distributions shown in Fig.are typical of distributions found at other locations along tearly portion of the timefront in our simulations usingE50.5EGM . Micromultipath travel-time spreads were typcally 10 to 15 ms, with only a few of these micromultipathwhose travel-time spread was typically less than 2 ms, hing dominant amplitudes. One surprising aspect of thishavior is the very small travel-time spread of the dominamicromultipaths. We speculate that the explanation for tbehavior is that the dominant micromultipaths are frequenorganized locally by a simple caustic structure. This wouaccount, in addition, for the observation that the dominmicromultipaths have Maslov indices that differ by no mothan one unit.

Simulations ~not shown! using E51.0EGM result insynthesized steep ray pulses that are spread in time mmore than is shown in Fig. 11. With the stronger internwave field, the total micromultipath travel-time spread icreases by approximately a factor of 2—to about 25 mMore importantly, however, the dominant micromultipathave travel-time spreads comparable to the total micromupath travel-time spread, leading to synthesized pulses thaspread by 5 to 10 ms, rather than 1 to 2 ms. Thus, measearly arrival time spreads are in better agreement withE50.5EGM simulations than withE51.0EGM simulations.The question of which value of the internal wave strengthEin our simulations gives the best fit to the observations wbe revisited in the next section.

An interesting and unexpected feature of our simulatiois that the Maslov indexm was consistently lower than thnumber of turnsM made by the same ray. The distributionM2m typically had a peak at 3 to 4 units and a width, withwhich all but a few outliers were contained, of 3 units in tE50.5EGM simulations ~see Fig. 11!; these numbers in-creased slightly in theE51.0EGM simulations. The Maslovindex was computed by counting zeros ofq21 ~9! followingray paths. The quantityq21 was computed by solving the

re

f

heee


o

wiulotrlstongdeie

.

arrgthv

mnmth

mthac

ontin

waf

iktioinsoe-,

-nicofilelus-herical-un-ray.

es-iderf

ver,fromwillcat-d

gen-

thefol-entisthewn

dro-

the, asherns,

xial

e of
coupled ray-variational equations~5!, ~6!, ~7!, ~11!, and~12!.The discrepancy betweenM and m in our simulations ismildly surprising because it is known36 that in range-independent environments these numbers differ by no mthan 1 unit.
B. Micromultipath travel time spreads

In the preceding subsection we saw that associatedthe chaotic motion of ray trajectories is extensive micromtipathing, and that the micromultipathing process is bspectrally and spatially nonlocal. Surprisingly, on the eabranches of the timefront the micromultipathing procecauses only very small time spreads and does not leadmixing of ray identifiers. The foregoing discussion centerion Fig. 11 described how micromultipath travel-time spreaof 10 to 15 ms can be reconciled with measured wave-fitime spreads of 1 to 2 ms. To complete this picture itnecessary to explain why the micromultipath travel-timspreads of the early~steep! ray arrivals are only 10 to 15 msWe shall now provide such an explanation.

Computing time spreads is conceptually straightforwusing ray methods. In a known environment one finds a laensemble of rays with the same identifier that solvessame two-point boundary value problem; the spread in tratimes over many ensembles of such rays, each ensecomputed using in a different realization of the environmeis the desired quantity. The constraint that all of the coputed rays have the same fixed endpoints complicatescalculation. In the following we exploit an approximate forof the eigenray constraint that simplifies the calculation;approximate form of the eigenray constraint is first-ordercurate in a sense to be described below.

Before describing the manner in which the eigenray cstraint is imposed, it is useful to describe the physical setin which the constraint will be applied. Figure 12 showsuHuvs r following a moderately steep ray~axial angle approxi-mately 11°! in a canonical environment37 on which the pre-viously described@based on the measured AETN(z) profile#internal-wave-induced sound-speed perturbation fieldsuperimposed. It is seen thatH(r ) consists of a sequence oessentially constantH segments, separated by near-step-ljumps which are caused by the sound-speed perturbaThe near-step-like jumps occur at the ray’s upper turnpoints. A simple model that captures this behavior—thecalled apex approximation31—assumes that the transition rgions can be approximated as step functions. This is

FIG. 12. uHu vs range following a ray emitted on the sound channel awith a launch angle of 11° in a canonical environment with an internwave-induced sound-speed perturbation field superimposed.


re

th-hys

a

slds

deeelblet,-is

e-

-g

s

en.g-

of

course, an idealization asH(r ) must be a smooth functionfor any smoothc(z,r ). ~The validity of the apex approximation is linked to the anisotropy and inhomogeneity of oceainternal waves; details of the background sound-speed prare not important. For this reason we have chosen to iltrate this effect using a background canonical profile ratthan the AET environment.! Figure 13 shows a schematdiagram of two rays. One ray has undergone two internwave-induced apex scattering events; the other ray is anscattered ray with the same launch angle as the scattered

Our strategy to build the eigenray constraint into antimate of travel-time spreads can now be stated. We consa scattered and an unperturbed~that is, in the absence ointernal-wave-induced scattering events! ray with the sameray identifier; the rays shown schematically in Fig. 13 haidentifier 13. In addition to having the same ray identifiethe scattered and unperturbed rays are assumed to startthe same point and end at the same depth, but theygenerally end at different ranges. We assume that the stered ray has travel timeTr and is one of many scattereeigenrays with the same ray identifier at ranger. We estimatethe travel timeT(r ) of an unperturbed eigenray at rangerwhose ray identifier is the same as that of the scattered eiray. It will be shown thatdT(r )5Tr2T(r ) vanishes to first-order if the apex approximation is exploited. Becausesame result applies to all of the scattered eigenrays, itlows that the time spread vanishes to first-order, independof r, if the apex approximation is exploited. Note that thresult is not expected to apply, even approximately, tolate near-axial rays where the apex approximation is knoto fail.

The travel timeT(r ) of the eigenray in the unperturbeocean can, of course, be computed numerically, but this pvides little help in deriving an analytical estimate ofdT.Instead, consider a ray in the unperturbed ocean withsame launch angle as one of the scattered eigenraysshown in Fig. 13. In general, the range of this ray at treceiver depth, after making the appropriate number of tuwill be r 0Þr . But, T(r ) can be estimated fromT(r 0). Thisfollows from the observation that ray travel time~assuming apoint source initial condition, for instance! is a continuouslydifferentiable function ofzandr, with ¹T5p, wherep is theray slowness vector. Because of this propertyT(z,r ) can be

s-

FIG. 13. Schematic diagram showing a twice apex-scattered ray~heavyline! and the path that the same ray would have followed in the absencapex scattering events~light line!.


thde

c-

omn-senc

.

e

ndthbease,

p

gve

th

-

ec

nt

om-n of

ed, toxanme

tioned

pen-e ofturerayn-

,

-aint.el-on-

s

-ays.ays.rger, to-

tly

e

ken-

nli-ray

expanded in a Taylor series. If we consider rays withsame ray identifier and fix the final ray depth to coinciwith the receiver depth, then

T~r !'T~r 0!1pr~r 0!~r 2r 0!, ~14!

wherepr52H is ther component of the ray slowness vetor. @More generally,T(z,r ) consists of multiple smoothsheets that are joined at cusped ridges; in spite of this cplication, the property of continuous differentiability is maitained, even in a highly structured ocean. For our purpowe need only consider one sheet of this multivalued fution. Note also that our application of~14! in the backgroundocean makes this expression particularly simple to use# Itfollows from ~14! that

dT~r !5Tr2T~r !'Tr2T~r 0!2pr~r 0!~r 2r 0! ~15!

is, correct to first-order, the travel-time difference at rangrbetween eigenrays with and without internal waves.

To evaluatedT(r ), we shall assume that the backgrouenvironment is range independent. Also, consistent withuse of the apex approximation, we may treat the perturenvironment as piecewise range independent. Within erange-independent segment it is advantageous to make uthe action-angle variable formalism~see, e.g., Refs. 11, 22or 38!. In terms of the action-angle variables (I ,u), H5H(I ) and the ray equations are

du

dr5

]H

]I5v~ I !, ~16!

dI

dr52

]H

]u50, ~17!

and

dT

dr5I

du

dr2H~ I !5Iv~ I !2H~ I !, ~18!

where the physical interpretation ofH as2pr is maintained.The angle variable can be defined to be zero at the upturning depth of a ray, and increases by 2p over a ray cycle~double loop!. It follows that the frequency v(I )52p/R(I ), whereR(I ) is a ray cycle distance. Integratinthe ray equations over a complete ray cycle then giR(I )52p/v(I ) andT(I )52p(I 2H(I )/v(I )), with I con-stant following each ray. In the apex approximationI jumpsdiscontinuously at each ray’s upper turning depth; forperturbed ray R(I 1DI )'2p/v(I )22pDIv8(I )/(v(I ))2

and T(I 1DI )' 2p(I 2H(I )/v(I ))12pDIH (I )v8(I )/(v(I ))2, wherev8(I )5dv(I )/dI. Note that, like these expressions, Eqs.~14! and ~15! are first-order accurate inDI .

Consider again Eq.~15! and Fig. 13 withI equal toI 0 ,I 1 , andI 2 in the left, center, and right ray segments, resptively. The left segment gives no contribution todT as theray has not yet been perturbed. In the center ray segme

Tr2T~r 0!'2p~ I 12I 0!H~ I 0!v8~ I 0!

~v~ I 0!!2, ~19!

and


e

-

s,-

ed

chof

er

s

e

-

2pr~r 0!~r 2r 0!'H~ I 0!F22p~ I 12I 0!v8~ I 0!

~v~ I 0!!2G .

~20!

These terms are seen to cancel. Note that if additional cplete cycle ray segments are added to the center sectiothe ray, Eqs.~19! and ~20! are unchanged except thatI 1

2I 05DI 1 is replaced by( i 51n DI i5I n2I 0 ; again, the two

terms cancel. In the final~incomplete cycle! ray segment thedifference between the termsTr2T(r 0) and pr(r 0)(r 2r 0)can be shown to beO((DI )2); the terms do not exactlycancel because theu values of the perturbed and unperturbrays are generally not identical at the receiver depth. Thusfirst-order inDI , dT50, independent of range, if the apeapproximation is valid. This simple calculation providesexplanation of why, in spite of extensive ray chaos, the tispreads of the early AET ray arrivals are quite small.

Several comments concerning the preceding calculaare noteworthy. First, we note that although it was assumthat the background sound-speed structure is range indedent, the preceding argument also holds in the presencslow background range dependence, i.e., with strucwhose horizontal scales are large relative to a typicaldouble-loop length. Adiabatic invariance in such enviroments guarantees that whileH is not constant following raysbetween apex scattering events,I is nearly constant. Secondafter n random kicks,I 12I 0 in ~19! and ~20! is replaced byI n2I 0'An(DI )rms, and the magnitude of~19! under AETconditions isO(1 s). This is an example of a travel timespread estimate that fails to enforce the eigenray constrThis calculation shows that the difference between travtime estimates that do and do not enforce the eigenray cstraint can be quite significant.~The constrained estimate irefined below.! Third, it should be emphasized that Eqs.~19!and ~20! apply ~approximately! only to the steep rays because the apex approximation applies only to the steep rWe have not addressed the time spreads of near-axial rNote, however, that Fig. 8 shows that time spreads are lafor flatter rays. Fourth, the above calculation shows thatlowest order inDI , there is no internal-wave-scatteringinduced travel-time bias if the apex approximation is stricapplied. And fifth, the arguments leading to Eqs.~19! and~20! apply whether the scattered rays are chaotic or not.

A relaxed form of the apex approximation in which thaction jump transition region has widthDu gives a nonzerotravel-time-spread estimate. In the transition region, tafor convenience to be 0<u<Du, one may choose a Hamiltonian of the formH5H(I 2su), wheres5DI /Du, andH5H(I ) elsewhere. It follows that in the transition regioI (u)5I 01su and I 5constant elsewhere. A simple generazation of the above calculation then gives for a completecycle

Tr2T~r 0!'~2p2Du!~ I 12I 0!H~ I 0!v8~ I 0!

~v~ I 0!!2

1Du

2~ I 12I 0!, ~21!


deet

ur

12.

ns

awe

ar

edter.adeahaiverbettivpowe

suomd

oae-

en

lonear-astial

ic-imeisun-icro-alum-o-ex-alal

rgere-eri-in-

ics

deli-

esls.us-

-tur-andin

inrim-

llynly

ap-thed be

u-totely.t ther-aleentsid-centonrontse

ET

and

2pr~r 0!~r 2r 0!'H~ I 0!F2~2p2Du!~ I 12I 0!v8~ I 0!

~v~ I 0!!2G ,

~22!

so the sum@recall Eq.~15!# is

dT'Du

2~ I 12I 0!. ~23!

It should be emphasized that this expression is first-oraccurate inDI 5I 12I 0 , but that no assumption about thsmallness ofDu has been made. As noted above, incomplcycle end segment pieces giveO((DI )2) corrections todT.If Du has approximately the same value at each upper tthen one has, aftern upper turns, correct toO(DI )

dT'Du

2 (i 51

n

~ I i2I i 21!5Du

2~ I n2I 0!

'Du

2An~DI !rms. ~24!

Consistent with the numerical simulations shown in Fig.Du'0.8 radians andDI'4 ms. ~We, and independently FHenyey @personal communication, 2002#, have confirmedthat these estimates also apply under AET-like condito!With these numbers andAn58, appropriate for AET,~24!gives a time-spread estimate of approximately 13 ms, inproximate agreement with the numerical simulations shoin Fig. 11. Note that~24! does not preclude a travel-timbias. A cautionary remark concerning the use of~24! is thatVirovlyansky38,39 has pointed out that, owing to seculgrowth, theO((DI )2) contribution todT may dominate theO(DI ) contribution at long range.

Figure 8 shows that in the AET environment simulatnear-axial ray time spreads are greater than simulated sray time spreads. We do not fully understand this behaviopossible explanation for this behavior is that time spreincrease as rays become increasingly flat owing to the brdown of the apex approximation. But, one would expect tthis trend should be offset, in part or whole, by the relatsmallness of internal-wave-induced sound-speed pertutions near the sound-channel axis. In addition, we have ssome evidence that an additional factor may be importanthe AET environment. Namely, we have observed a posicorrelation between travel-time spreads and stability exnents; stability exponents in the AET environment are shoin Fig. 5. We have chosen not to dwell on flat ray timspreads in this paper because there are no AET meaments of these spreads to which simulations can be cpared. It is clear, however, that the issues just raised neebe better understood.

IV. WAVE-FIELD INTENSITY STATISTICS

In this section we consider the statistical distributionthe intensities of both the early and late AET arrivals. Recthat experimentally the early arrival intensities have beshown24,25 to approximately fit a lognormal probability density function~PDF!, and the late arrival intensities have be


er

e

n,

,

.

p-n

epAsk-t

a-enine-

n

re--

to

flln

shown25 to fit an exponential distribution. The late arrivaexponential distribution is not surprising as this distributiis characteristic of saturated statistics. The early arrival nlognormal distribution is surprising, however, inasmuchtheory31,24 predicts saturated statistics, i.e., an exponenintensity PDF. It has been argued40 that the theory can bemodified in such a way as to move the early arrival predtion from saturated to unsaturated statistics. The latter regis characterized by a lognormal intensity PDF. This fixconceptually problematic inasmuch as, in this theory, thesaturated regime is characterized by the absence of mmultipaths, which seriously conflicts with the numericsimulations presented in the previous section where the nber of micromultipaths is very large. In this section we prvide self-consistent explanations for both the late arrivalponential distribution and the early arrival lognormdistribution. The challenge is to reconcile the early arrivnear-lognormal intensity PDF with the presence of a lanumber of micromultipaths. Some of the arguments psented are heuristic, and some build on the results of numcal simulations. A complete theoretical understanding oftensity statistics has proven difficult.

Our approach to describing wave-field intensity statistbuilds on the semiclassical construction described by Eq.~2!.At a fixed location it is seen that the wave-field amplitudistribution is determined by the distribution of ray amptudes and their relative phases. Note that both travel timand Maslov indices influence the phases of ray arrivaComplexities associated with transient wave fields and catic corrections will be discussed below.

An important observation is that in the AET environment, including internal-wave-induced sound-speed perbations, simulated geometric amplitudes of both steepflat rays approximately fit lognormal PDFs. This is shownFig. 14. Previously, it has been shown20 that ray intensities ina very different chaotic system also fit a lognormal PDF;that system single-scale isotropic fluctuations are supeposed on a homogeneous background.~Note that all powersof a lognormally distributed variable are also lognormadistributed, so ray amplitudes have this property if and oif ray intensities have this property.! The apparent generalityof the near-lognormal ray intensity PDF suggests that itplies generally to ray systems that are far from integrable;arguments presented in Ref. 20 suggest that this shoulthe case.

In the two subsections that follow, the intensity distribtions of the early and late AET arrivals, correspondingsteep and flat rays, respectively, will be discussed separaThe phase assumptions made in these subsections limianticipated validity of the results presented to ‘‘interior’’ potions of the early resolved timefront branches and the finregion, respectively. Thus, accidental crossings of differresolved branches of the timefront are excluded from coneration, as are regions near turning points where adjatimefront branches merge. Similarly, the narrow transitiregion separating the early resolved branches of the timeffrom the finale region is excluded from consideration. Therestrictions are consistent with the manner in which the Aintensity statistics were computed.24,25 Also, it should be


ic

n

thrah

ioieoitm

usousial-

ak

bynidsee

sig-mi-

Fatedig.of

ngethe

oseityo dor-

is

e

ln-to

l

gw

ual

ge

he

fedi-

tri-an

i-p

ngtions

noted that our implicit assumption that intensity statistshould be independent of position (T,z) in each of theseinterior regions is consistent with both our simulations athe observations.24,25

A. Early arrivals

We saw in the previous section that the micromultipathat make up one of the early arrivals have the sameidentifier and have a very small spread in travel time. Tdominant micromultipaths were seen~see Fig. 11! to have atime spread of approximately 1 ms, which is a small fractof the approximately 13-ms period of the 75-Hz carrwave. Also, our simulations show that the Maslov indicesthe dominant micromultipaths differ by no more than 1 unThese conditions dictate that interference among the donant micromultipaths is predominantly constructive. Becatravel-time differences are so small, the pulse shape shhave negligible influence on the distribution of peak intenties. In a model of the early AET arrivals consisting ofsuperposition of interfering micromultipaths, the micromutipath properties that play a critical role in controlling pewave-field intensities are thus:~1! their amplitudes have anear-lognormal distribution;~2! the dominant micromulti-paths have negligible travel-time differences; and~3! thedominant micromultipaths have Maslov indices that differno more than 1 unit. It should be noted that very differebehavior would have been observed if: the source bandwwere significantly more narrow, as this would have caumicromultipaths with different ray identifiers to interfer

FIG. 14. Cumulative probability density of ray intensity in the AET envronment including a superimposed internal-wave-induced sound-speedturbation field~solid lines!. Upper panel: flat rays~uwu<5°! sampled uni-formly in launch angle. Middle panel: steep rays~6°<uwu<11°! sampleduniformly in launch angle. Lower panel: eigenrays with1137 identifierconnecting the AET source and a receiver at depth 1005 m at the AET raIn all three panels the dashed lines correspond to lognormal distribuwhose mean and variance approximately match those of the simulatio


s

d

sy

e

nrf.i-eld-

tthd

with one another; or the source center frequency werenificantly higher as phase differences between interferingcromultipaths would then have been significant.

An additional subtlety must be introduced now: the PDof the intensities of the constituent micromultipaths thmake up a single arrival is not identical to the PDF describin Ref. 20 and shown in the upper and middle panels of F14. The latter PDF describes the distribution of intensitiesrandomly ~with uniform probability! selected rays leavingthe source within some small angular band and whose rais fixed. This PDF is biased in the sense that it overcountsmicromultipaths with large intensities and undercounts thwith small intensities. Unbiased micromultipath intensPDFs can be constructed from the biased PDFs shown. Tso, consider a manifold~a smooth curve in phase space coresponding to a fan of initial rays! which begins near (z0 ,p0)and arrives in the neighborhood of (zr ,pr) at ranger. Sam-pling in fixed steps of the differentialdp0 ~as was done toproduce the upper and middle panels of Fig. 14; thisequivalent to uniform random sampling! leads to a highlynonuniform density of points on the final manifold, sinc(z0 ,p01dp0) propagates to (zr1q21dp0 ,pr1q11dp0). Thegreaterq21, the lower the density of points locally at finarange. A uniform sampling in final position is achieved istead by considering the initial conditions that would lead(zr1dzr ,pr1q11/q21dzr). Its density of points on the initiamanifold can be deduced from its initial condition, (z0 ,p0

1dzr /q21). It is necessary to sampleq21 times more denselyon the initial manifold in order to achieve uniform samplinin dzr at ranger. To account for this effect, we need to knothe PDF for q21 with uniform initial sampling. Roughlyspeaking, the PDF of the absolute values of the individmatrix elements ofQ have the same form as foruTr(Q)u,apart from a shift of the centroid that is lower-order in ranthan the leading term. From the results of Ref. 20, the~bi-ased in the sense described above! probability thatq21 fallsin the interval betweenx andx1dx is

r uq21u~x!5A 1

2pr ~ n2nL!

1

xexpF2~ ln~x!2nLr !2

2r ~ n2nL! G ,x>0. ~25!

Here,nL is the true Lyapunov exponent, andn is equal tonL

plus an additional contribution due to the fluctuations in tvalues ofq21 over an ensemble of rays. The new~unbiased inthe sense described above! PDF,r8, for uniform sampling indzr is related to the previous one by

r uq21u8 ~x!5

x

^x&r uq21u~x!

5A 1

2pr ~ n2nL!

1

xexpF2~ ln~x!2 nr !2

2r ~ n2nL! G ,x>0, ~26!

where the factorx accounts for the extra counting weight oq21, and ^x& just preserves normalization and is calculatusing Eq.~25!. This calculation shows that the unbiased mcromultipath ray intensity PDF also has a lognormal disbution; the only change relative to the biased PDF is

er-

e.ns.


ng

4th

dindledAeale

teandilya

icha

rlynaslytheronlee

ite,

ios

le

eedoo

oniopl

e

-te

re-

y.eldthat

on-es.

-dn

ourof

nal

is-etrice

l

if-by

ve-

ateby

iorics

sureng-

u-

increase in the mean fromnLr to nr . Because lognormalityis maintained, this correction represents only a trivial chato the problem.

Approximate lognormality of the constrained~eigenray!PDF of ray intensity is shown in the lower panel of Fig. 1This PDF was constructed using the same eigenrayswere used to produce Figs. 9 and 11. The corresponunconstrained ray intensity PDF is shown in the midpanel of Fig. 14.~Two constraints—fixed receiver depth anfixed ray identifier—are built into the lower panel PDF.very similar constrained PDF results if only the receivdepth constraint is applied, provided ray launch angleslimited to the ‘‘steep’’ ray band used to construct the middpanel.! It should be noted that the constrained~eigenray!PDF shown in the lower panel of Fig. 14 was construcfrom numerically found eigenrays; because weak eigenrare difficult to find numerically they are undercounted athe PDF is biased. Because this bias is confined to the tathe distribution corresponding to the very weak eigenrawhose contribution to the wave field is expected to be smit is not expected to alter the argument that follows—whassumes only that micromultipath amplitudes are nelognormally distributed.

We return now to the problem of simulating the eaAET arrivals. It is tempting to think that because the costituent micromultipath amplitudes have a near-lognormdistribution, the sum of the micromultipath contributionshould also be near-lognormally distributed. Unfortunatethis is generally not the case. Consider, for example,special case in which phase, including Maslov index, diffences are negligible. Then, all micromultipaths interfere cstructively and peak wave-field amplitudes can be modeas the sum of many lognormally distributed variables. Bcause all moments of the lognormal distribution are finthe central limit theorem applies. Under these conditionssufficiently many contributions are summed, the distributof the sums—the wave-field amplitudes—would be a Gauian.

To simulate the statistics of the early AET arrivals~re-call Fig. 14 and the accompanying discussion!, we have usedseveral variations of a simple model. An arrival was modeas a sum ofnm interfering micromultipaths whose:~1! am-plitudes are lognormally distributed; and~2! phases,sTi

2m ip/2 mod 2p, have a PDF with a clearly identifiablpeak. Micromultipath contributions were coherently addThe peak intensity of the sum—whose travel time is nknown a priori—was then recorded. Using an ensemble104 peak intensities, a peak-intensity PDF was then cstructed. Peak-intensity PDFs constructed in this fashwere found to be very close to lognormal; a typical examis shown in Fig. 15. In this examplef 575 Hz, theTi ’s wereidentical, m iP$ j , j 11% with equal probability ~note thatchoice of the integerj is unimportant!, and nm55. Othercombinations of distributions forTi ~either a Gaussian or thlimiting case of a delta distribution!, m i ~taken either from$ j , j 11% or $ j 21,j , j 11% with equal probability!, and theparameternm ~between 2 and 100! were tested. These simulations showed that, provided the phase constraint no


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above was satisfied, a near-lognormal peak-intensity PDFsulted.

Two points regarding this simple model are noteworthFirst, this model does not constitute a theory of wave-fipeak-intensity statistics, but it does serve to demonstrateour ray-based simulations of the early AET arrivals are csistent with the measured distribution of peak intensitiSecond, simulations~not shown!, performed with E51.0EGM yield sets of dominant micromultipaths that violate assumption~2!; phases are uniformly distributed ansumming micromultipath contributions yields a distributioof peak intensities that is not close to lognormal. Thus,simulations suggest that a near-lognormal distributionearly arrival peak intensity requires a relatively weak interwave field.

A complication not accounted for in the preceding dcussion is the presence of caustics. At caustics geomamplitudes~8! diverge and diffractive corrections must bapplied. At short range~on the order of the first focadistance—a few tens of km in deep ocean environments! weexpect that intensity fluctuations will be dominated by dfractive effects. The entire wave field should be organizedcertain high-order caustics which leads to a PDF of wafield intensity with long tails.41–43 In spite of the importanceof diffractive effects at short range—and probably alsovery long range—we believe that, in the transitional regimdescribed above, intensity fluctuations are not dominateddiffractive effects. This somewhat counterintuitive behavcan be understood by noting that in the vicinity of caustthe importance of diffractive corrections to~8! decreases asthe curvature of the caustic increases. A quantitative meaof caustic curvature is the second derivative of the Lagraian manifold ~recall Fig. 4! z9(p) at a local extremum.

FIG. 15. Cumulative probability density of peak wave-field intensity simlated using a simple model as described in the text~solid lines!. The samesimulated cumulative density is compared to exponential~upper paneldashed line! and lognormal~lower panel dashed line! cumulative densityfunctions. This figure should be compared to Fig. 14 in Ref. 24.


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~Higher-order caustics can be treated using an extensiothe same argument.! Under chaotic conditions the curvatuof caustics increases, on average, with increasing rangethe Lagrangian manifold develops structure on smallersmaller scales. Thus, the fraction of the total number of mtipaths that require caustic corrections decreases, on avewith increasing range. This is true even as the numbecaustics grows exponentially, on average, in range. Thisgument leads to the somewhat paradoxical conclusion tprior to saturation at least, we expect that the importancecaustic corrections decreases with increasing range.

B. Late arrivals

Recall that in the finale region of the AET wave fieldsets of micromultipaths with different ray identifiers, all coresponding to near-axial rays, are not temporally resolvedeach (T,z) in the finale region the wave field can be modelas a superposition of micromultipath contributions with radom phases. The quadrature components of the wave fieeach (T,z) have the form of sums of terms of the form

xi5ai cos~f i !, ~27!

and

yi5ai sin~f i !, ~28!

where f i is a random variable uniformly distributed o@0,2p!. The distribution ofai is close to lognormal, but acorrection must be applied to account for pulse shapemany of the interfering micromultipaths partially overlaptime. This correction is unimportant inasmuch as the cenlimit theorem guarantees that, provided the distributions oxi

and yi have finite moments, the distributions of sums ofxi

and yi converge to zero mean Gaussians. Thus, wave-fiintensity is expected to have an exponential distribution, csistent with the observations. The comments made eaabout caustics apply here as well.

C. Intensity statistics discussion

The question of what causes the transition fromstructured early portion of the AET arrival pattern to tunstructured finale region deserves further discussion. Rethat the early resolved arrivals have small time spreadspeak intensities that are near-lognormally distributed, whthe finale region is characterized by unresolved arrivalsnear-exponentially distributed intensities. In both regionstime spreads and intensity statistics are consistent with eother inasmuch as in our simulations a near-lognormal insity distribution is obtained only when there is a preferrphase, while the exponential distribution is generated wphases are random, i.e., when phases are uniformly disuted on@0,2p!.

With these comments in mind, it is evident that the mimportant factor in causing the transition to the finale regis the increase in internal-wave-scattering-induced tispreads as rays become less steep; as time spreads incneighboring timefront branches blend together andphases of interfering micromultipaths get randomized. Ttrend toward increasing time spreads as rays become


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steep is evident in Fig. 8. The surprising result is thatscattering-induced time spreads of the steep arrivals arsmall; we have seen that this can be explained by makingof the apex approximation. As noted at the end of the preous section, we do not fully understand the cause of the trtoward larger time spreads as rays flatten. In the finale reginternal-wave-scattering-induced time spreads exceedtime difference between neighboring timefront branches twould have been observed in the absence of internal waFigure 1 shows that these time gaps decrease as rays beincreasingly flat. Indeed, this figure shows that even inabsence of internal waves there would not have beenresolvable timefront branches in the last half-second or sthe arrival pattern. Some loss of temporal resolution inmeasurements is of course due to the finite source bawidth, but without internal waves~or some other type ofocean fluctuations! wave-field phases would not be randominstead, a stable interference pattern would be observed

Finally, we note that an interesting feature of our rsimulations is that the near-axial rays have much higherbility exponents @typically about ~100 km!21# than thesteeper rays@typically about~300 km!21#; see Fig. 5. Also,note that Figs. 6 and 7 strongly suggest that the near-arays in the AET environment are much more chaotic thansteeper rays. The likely cause of the relative lack of stabiof the near-axial rays in the AET environment is the bacground sound-speed structure. This topic is discussed intail in Ref. 44. There it is shown that ray stability is largecontrolled by a property of the background~which is as-sumed to be range independent! sound-speed profile,a5(I /v)dv/dI; ray instability increases on average with icreasinguau. Here,I is the ray action variable and 2p/v(I ) isthe ray cycle~double loop! distance; recall Eqs.~16!–~18!.The dependence ofa on I can be replaced by dependenceaxial ray anglewax because, in an environment with a singsound-speed minimum,I is a monotonically increasing function of wax . Plots ofa vs wax in five 650-km block range-averaged sections of the AET environment are shown in F16. ~The choice of averaging over 650-km blocks in rangeof course, arbitrary. Range averaging should be done lochowever, because the local sound-speed structure mavery different than that which results after averaging overentire propagation path.! The relatively large near-axial ravalues of uau seen in this figure are consistent with thstrongly chaotic nature of these rays seen in Figs. 5, 6, anUnfortunately, the measured wave-field intensity statisticsthe AET finale region are not very sensitive to the near-axray intensity PDF; as noted above, the argument leadinthe expectation that wave-field intensity in the finale regshould have an exponential distribution holds for a very laclass of ray intensity distributions. Thus, we are not awareany way that the AET measurements can be used to tesfinding that the near-axial rays have larger stability expnents, on average, than the steeper rays.

In summary, we believe that the observed nelognormal PDF of wave-field intensity for the early resolvAET arrivals is transitional between fluctuations dominatby caustics at short range and saturation at long range, wphase differences will be larger. Surprisingly, phase diff


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FIG. 16. Stability parametera vs axialray angle in five 650 km block rangeaverages of the AET environment.
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ences among the micromultipaths that form the earlysolved AET arrivals are very small so that saturation hasbeen reached. The late-arriving AET energy, on the othand, is characterized by interfering micromultipaths wrandom phases, leading to an exponential PDF of wave-fiintensity. Here, the underlying near-lognormal PDF of rintensity is obscured. The arguments that have beensented to explain the intensity fluctuations of both the eaand late AET arrivals assume that diffractive effects doplay a dominant role. Although our theoretical understandof many of the issues raised in this section is clearly incoplete, it is worth emphasizing that our ray-based numersimulations of intensity statistics, in which ray trajectoriare predominantly chaotic, are in good qualitative agreemwith the AET measurements. We do not believe that tagreement is accidental.

V. DISCUSSION AND SUMMARY

In this paper we have seen that in the AET environmeincluding internal-wave-induced sound-speed perturbatioray trajectories are predominantly chaotic. In spite of extsive ray chaos, many features of ray-based wave-field silations were shown to be both stable and in good agreemwith the AET measurements. Simulated and measuspreads of acoustic energy in time, depth, and angle wshown to be in good agreement. It was shown that associwith the chaotic motion of ray trajectories is extensive mcromultipathing, and that the micromultipathing processhighly nonlocal; the many micromultipaths that add at treceiver to produce what appears to be a single arrival msample the ocean very differently. It was shown, surprisingthat on the early arrival branches the nonlocal micromupathing process causes only very small time spreadsdoes not lead to a mixing of ray identifiers. A quantitatiexplanation for the cause of the very small time spreadsthe early ray arrivals was presented. Partially heuristic exnations for the near-lognormal and exponential distributioof wave-field intensities for the early and late arrivals,spectively, were provided.


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The fact that one is able to accurately simulate mawave-field features using ray methods under conditionswhich ray trajectories are predominantly chaotic may sprise some readers. Chaotic motion is, after all, intimatlinked with unpredictability. This apparent paradox is recociled by noting that while individual chaotic ray trajectorieare unpredictable beyond some short predictability horizdistributions of chaotic ray trajectories may be quite staand have robust properties.45 Consider, for example, the motion of a ray whose initial depthz0 and anglew0 are knownin the AET environment, including a known internal-wavinduced sound-speed perturbation. At the range of the Areceiving array the depthz, anglew, travel timeT, and eventhe number of turnsM made by this ray are likely to be, foall practical purposes, unpredictable. In contrast, at the srange the distribution in (z,w,T,M ) of a large ensemble orays with the same initial depth and whose launch angleswithin a 1° band centered onw0 is very stable in the sensthat essentially the same distribution is seen for any laensemble of randomly chosen~with uniform probability!rays inside this initial angular band. It is the latter propethat allows us to make meaningful predictions using ansemble of chaotic rays in a particular realization of the intnal wave field. In addition, ray distributions with similar sttistical properties are observed using different realizationsthe internal wave field, suggesting that these statistical prerties of rays are robust.

Our exploitation of results that relate to ray dynamicincluding ray chaos, leads to a blurring of the distinction this traditionally made between deterministic and stochawave propagation problems. The ray dynamics approachphasizes the distinction between integrable and nonigrable ray systems, corresponding to range-independentrange-dependent environments, respectively. In genrange-dependent environments at least some ray trajectwill exhibit chaotic motion. The oceanographic origin—mesoscale variability, internal waves, or something else—the range-dependent structure is not critical. An importconceptual insight that can come only from exploitation


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results relating to ray chaos is that generically phase spapartitioned into chaotic and nonchaotic regions. This mixtcontributes to a combination of limited determinism and costrained stochasticity.

For those who wish to exploit elements of determinisin the propagation physics for the purpose of performdeterministic tomographic inverses, the results that we hpresented have important implications. We have seen,example, that, in spite of extensive ray chaos, many rbased wave-field descriptors are stable and predictable,should be invertible. A less encouraging but important obsvation is that although the travel times of the steep eaarrivals are stable and can be inverted, the spatial nonlocof the micromultipaths limits one’s ability to invert forange-dependent ocean structure, as was first pointed oRef. 8.

For those who wish to understand and predict wave-fistatistics, our results also have important implications. Thcomments are based on our analysis of the AET measments, but are expected to apply to a large class of lorange propagation problems. First, we have seen that sand unstable ray trajectories coexist and that this influenwave-field intensity statistics. Second, we have seen thatcromultipathing is extensive and highly nonlocal. Tstrongly nonlocal nature of the micromultipathing processsignificant because:~a! this process cannot be modeled usia perturbation analysis which assumes the existence oisolated background ray path; and~b! the effects of this pro-cess will not, in general, be eliminated as a result of losmoothing by finite frequency effects. Third, we have sethat the appearance of stochastic effects is not entirely duinternal waves. We have seen, for example, that, even inabsence of internal waves, near-axial rays in the AET enronment are chaotic. Recall that we noted in the Introductthat part of the motivation for the present study was toderstand the reasons underlying the finding of Colosiet al.24

that the theory described in Ref. 31 fails to correctly predthe AET wave-field statistics. This failure is not surprisingview of the observation that this theory is based on assutions that either explicitly violate, or lead to the violation oall three of the aforementioned properties of the scattewave field. Note also in this regard that the combinationextensive micromultipathing, small time spreads, and lognmally distributed intensities that characterizes the early Aarrivals is not consistent with any of the three propagatregimes identified in this theory.

Although we have argued that a ray-based wave-fidescription which relies heavily on results relating to rchaos can account for all of the important features ofAET measurements, it should be emphasized that our ansis has some shortcomings. Recall that our simulations uE50.5EGM resulted in steep arrival time spreads and peintensity statistics in good agreement with measuremebut that simulations usingE51.0EGM did not, althoughthere are indications from direct measurements thatE51.0EGM is a more appropriate choice. We have not pvided a rigorous argument to establish the connectiontween a near-lognormal ray intensity PDF and a nelognormal wave-field intensity PDF. Indeed, a comple


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theory of wave-field statistics which accounts for completies associated with ray chaos is lacking. This is closlinked to the subject of wave chaos,22 which is currently notwell understood. Also, we have only briefly discussedspreads of energy in depth and angle, and more work neto be done on quantifying time spreads.

Finally, we wish to remark that the importance of ramethods is not diminished by recent advances, both theoical and computational, in the development of full wamodels, such as those based on parabolic equations. Thter are indispensable computational tools in many appltions. In contrast, the principal virtue of ray methods is ththey provide insight into the underlying wave propagatiphysics that is difficult, if not impossible, to obtain by another means. The results presented in this paper illustratestatement.

ACKNOWLEDGMENTS

We thank Fred Tappert, Frank Henyey, and Dave Palmfor the benefit of discussions on many of the topics includin this paper, and the ATOC group~A. B. Baggeroer, T. G.Birdsall, C. Clark, B. D. Cornuelle, D. Costa, B. D. DushaM. A. Dzieciuch, A. M. G. Forbes, B. M. Howe, D. Menemenlis, J. A. Mercer, K. Metzger, W. Munk, R. C. SpindeP. F. Worcester, and C. Wunsch! for giving us access to theAET acoustic and environmental measurements. This wwas supported by Code 321 OA of the U.S. Office of NaResearch.

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