XXII Session of the Russian Acoustical Society
Session of the Scientific Council of Russian Academy of Science on Acoustics
Moscow, June 15-17, 2010
A.L. Virovlyansky, L.Ya. Lyubavin, A.Yu. Kazarova
PENETRATION OF SOUND SIGNALS INTO THE SHADOW ZONE AT MEGAMETER RANGES
IN THE OCEAN DUE TO SCATTERING AT RANDOM INHOMOGENEIES
Institute of Applied Physics RAS
46 Ulyanov St., Nizhny Novgorod, 603950 Russia
Tel.: (7-831) 416 4784; Fax: (7-831) 436 5745
E-mail: [email protected]
The distribution of the sound field intensity within a time-depth plane at an observation range of 1500 km is considered.
It is assumed that the sound field is excited by a point source of a pulse signal. In the absence of sound speed fluctuations the distribution has a regular shape and it is localized at not very large depths. For each travel time there is a
limiting depth below which the sound does not penetrate. This means that in the time-depth plane there is a shadow
zone. In a series of field experiments American acousticians discovered that due to scattering at sound speed fluctuations, sound pulses at megameter ranges penetrate into the shadow zone (that is at large depths). Moreover, signals
recorded in the shadow zone form stable insonified areas abutted on the lower cusps of the timefront (the distribution of
ray arrivals in the time-depth plane) in an unperturbed waveguide. In our work this effect is explained from the viewpoint of a stochastic ray theory derived for statistical description of the chaotic ray dynamics in a randomly inhomogeneous underwater sound channel (USC). It is shown that a simple statistical description of the penetration of rays into a
shadow zone can be derived by analyzing the diffusion of the action variable of the ray path due to scattering at random
inhomogeneities. The formation of stable peaks in the shadow zone is closely related to a well known phenomenon of
clusterization of travel times of chaotic rays.
z, km
In the present work we study the influence of refractive index fluctuations on the distribution of
sound energy in the time-depth plane at long range propagation of sound pulses in a deep ocean. In the geometrical optics approximation the temporal structure of a pulse signal in the USC is usually characterized by
the so-called timefront representing ray arrivals in the time-depth plane. As an example we present a timefront computed at a range of 1500 km in a range-independent waveguide with a sound speed profile shown
in Fig. 1. This timefront is presented in the upper panel of Fig. 2.
The timefront represents two broken curves shifted relative each other
0
along the time-axis and consisting of segments. Each segment is formed by
contributions from rays with the same identifier ± M , where ± is the sign of
1
the ray launch angle and M is the number of ray turning points [1]. A smooth
solid curve in the upper panel of Fig. 2 indicates the lower border of the un2
perturbed timefront. The portion of t-z plane below this border does not contain ray arrivals and in what follows will be called the shadow zone.
3
The sound scattering at inhomogeneities of USC causes the penetration of sound signals into the shadow zone. This phenomenon was discovered
4
by American acousticians in a series of field experiments in the 1990-s. Later
1.5
1.51
1.52
1.53
1.54
1.55
c, km/s
on a special experiment for a detailed study of this effect was conducted using
Fig. 1. Unperturbed sound
vertical receiving arrays to record pulse signals [2]. The analysis of data obspeed profile.
tained in this experiment showed that a physical mechanism of sound penetration into the shadow zone is based on the sound scattering at internal-wave-induced inhomogeneites. Earlier,
it was thought that this mechanism cannot explain the effect quantitatively and different mechanisms were
considered.
The lower panel in Fig. 2 depicts a timefront computed in the presence of refractive index fluctuations caused by random internal waves with statistics determined by the empirical Garrett-Munk spectrum.
Although ray paths in this environmental model exhibit a chaotic behavior, an early portion of the timefront
formed by steep rays looks approximately the same as in the unperturbed waveguide. Segments present in
this part of timefront become fuzzy, but they as earlier are formed by rays with equal identifiers. There are
two smooth solid lines in the lower panel. One of them (it is the same line as in the upper panel) indicates the
upper border of the shadow zone in the unperturbed waveguide, while the other one indicate the border of the
shadow zone in the presence of perturbation.
387
XXII Session of the Russian Acoustical Society
Session of the Scientific Council of Russian Academy of Science on Acoustics
Moscow, June 15-17, 2010
Figure 3 presents results of
numerical simulation of
pulse signal propagation in
the parabolic equation approximation. The computa2
tion has been performed
for the same realization of
perturbation as in Fig. 2. In
4
Fig. 3 we depicted the
same borders of the sha0
dow zone as in Fig. 2. The
comparison of results presented in Figs. 2 and 3 confirms that the timefront
2
properly (at least, qualitatively) describes the distribution of sound energy not
4
only
in
a
rangeindependent
waveguide,
-7
-6
-5
-4
-3
-2
-1
0
but in a range-dependent
t, s
waveguide, as well. Both,
Fig. 2. Distribution of ray arrivals in the time-depth plane at the range of observation
ray-based and parabolic
1500 km in the unperturbed (upper panel) and perturbed (lower panel) waveguides.
equation based, calculations show that in the presence of random internal waves the lower border of the insonified zone is shifted to
the bottom by 300-500 m. The main part of the sound energy penetrated in the shadow zone is located between the two smooth solid lines shown in the lower panels of Figs. 1 and 3. Our objective in this work is to
investigate the distribution of field intensity in this area. A magnified view of a part of this area is shown in
Fig. 4.
Numerical simulation (as
well as field experiments)
0
demonstrates that the
signals penetrating into
the shadow zone due to
2
scattering at random inhomogeneities form a
rather regular intensity
distribution. In particular,
4
they form compact inso0
nified areas. In Fig. 4
these areas are numbered
from 1 to 14. Each of
these areas represents an
2
“extension” of a cusp
(caustics) in the lower
part of the unperturbed
timefront where two
4
timefront segments have
-7
-6
-5
-4
-3
-2
-1
0
a common endpoint. Pot, s
sitions of these insonified
Fig. 3. Distribution of the sound field amplitudes in the time-depth plane at the range of areas practically do not
observation 1500 km in the unperturbed (upper panel) and perturbed (lower panel) wadepend on a particular
veguides.
realization a random perturbation. This phenomenon is a manifestation of a fundamental relationship between ray travel times and
identifier of ray paths. The point is that if a perturbed (chaotic) and an unperturbed rays have equal identifier
z, km
z, km
z, km
z, km
0
388
Moscow, June 15-17, 2010
XXII Session of the Russian Acoustical Society
Session of the Scientific Council of Russian Academy of Science on Acoustics
z, km
and their vertical coordinates at the observation range do not differ significantly, then these rays have close
travel times. A quantitative formulation of this statement is discussed in detail Refs. [1,3]. In accord with this
general rule, each of 14 insonified areas in Fig. 3 is formed by rays with identifiers (they differ by unity) corresponding to two segments whose “extension” represents the area.
The simplest quantitative description of the
effect under consideration (sound energy pe2
netration into the shadow zone) can be obtained in the scope of the ray-based description. Take a point source radiating a sound
2.5
pulse s (t ) . The intensity of the sound field at
an observation point calculated by an incohe3
rent ray summation is
13 14
11
9
3.5
7
5
12
u (t , z ) = ∑A2j s (t − t j ) ,
2
2
10
8
where
6
4
4
A=
3
1
(1)
j
2
C
∂z / ∂p0
is a ray amplitude in the small-angle approximation, C is a constant depending on a source
Fig. 4. A magnified view of a portion of lower panel from Fig.
power. Coordinate z and travel time t at the
3.
observation range are functions of ray starting
momentum p0 = tan χ , where χ is a launch angle. Equation (1) can be rewritten as
-6
-5
-4
-3
t, s
u (t , z ) = ∫ dp0 δ ( z − z ( p0 )) s (t − t ( p0 )) .
2
2
Consider the field intensity averaged over a rectangular area in the t-z plane with sizes ∆ z and ∆ t ,
along the z- and t-axis, respectively. It is assumed that ∆ t >> T0 , where T0 is an effective pulse length. Under assumption that s (t ) is normalized in such a way that ∫dt s ( t ) = 1 , we can formally replace
2
2
s(t − t ( p0 )) by T0 δ (t − t ( p0 )) . As a result we get an expression for the smoothed field intensity presented
in the form which is convenient for the use in numerical simulations. Namely, if a fan of N>>1 rays with
starting momenta p0 uniformly filling some interval with a step size δ p0 is traced numerically, then the
smoothed intensity can be presented as
J (t , z ) ≡
1
∆t ∆ z
∫
t +∆t / 2
z +∆ z / 2
t −∆t / 2
z −∆ z / 2
dt ′∫
2
dz ′ u (t ′, z ′) =
C
δ N δ p0 ,
2π∆ t ∆ z
where δ N is the number of rays arriving at the area over which the intensity is averaged.
Besides an estimate based on the direct numerical ray tracing in a particular realization of the sound
speed field, an analytical estimate of δN can be obtained using a stochastic ray theory developed in Refs.
[1,3]. This approach is based on the fact that the ray paths in the environmental model under consideration
exhibit chaotic dynamics. Ray paths with close starting parameters diverge exponentially with range and rapidly become practically independent. At megameter ranges the ray chaos is well developed and an averaging over ray starting parameters (in our case over starting momentum p0) can be interpreted as a statistical
averaging. Then the probability that a ray arrives at a given (large enough) area of the phase space weakly
depends on a particular realization of random perturbation. In our problem this means that if the selected
smoothing scales ∆ z and ∆ t are large enough then the quantity δN and, hence, the value of the smoothed
intensity weakly depends on a particular realization of random internal waves generating the fluctuations of
refractive index. Results of numerical simulations are consistent with this expectation.
The evaluation of statistical characteristics significantly simplifies if the ray trajectories are described using the Hamiltonian formalism expressed in terms of canonical variables action-angle. The point is
that the range dependence of a fluctuating constituent of the action variable may be approximated by a ran389
Moscow, June 15-17, 2010
XXII Session of the Russian Acoustical Society
Session of the Scientific Council of Russian Academy of Science on Acoustics
dom Wiener process representing the simplest mathematical model of diffusion. In the scope of this approximation the statistics of chaotic rays may be described analytically.
This statistical approach was used for obtaining average intensities for 14 areas of the t-z plane
shown in Fig. 4. In this evaluation we exploited the fact that each of these areas is formed by contributions
from rays whose identifiers correspond to the two segments associated with the area. Therefore, the quantity
δN was estimated as Nα, where α is a probability that an arbitrary ray at the range of observation has one of
the two identifiers and its vertical coordinate is located between the two smooth curves shown in Figs. 2, 3,
and 4. All possible values of the starting momentum are considered as equiprobable.
0.2
1
0.18
2
11
7
8
0.16
9
5
12
J
3
4
0.14
10
6
13
14
0.12
0.1
-6
-5
-4
-3
-2
t, s
Fig. 5. Mean intensity of the wave field within 14 areas of the t-z plane
shown in Fig. 4. Predictions of the stochastic ray theory (squares) are
compared to results of parabolic equation based simulation (circles) and
to results of direct numerical ray tracing (triangles).
In Fig. 5 the ray-based estimates are compared with results obtained by numerical solving of the parabolic equation at a set of carrier frequencies (the pulse signal was synthesized out of these solutions). It is
seen that the ray approach allows one to predict properly the order of magnitude of the smoothed intensity.
Since the incoherent summation does not account the interference of rays, we hardly could expect a more
accurate prediction. It should be emphasized that the ray approach also explains the effect of formation compact insonified areas within the shadow zone. Some aspects of this effect can be explained even quantitatively. The ray approach also properly predicts the lower border of the insonified portion of the t-z plane.
The work was supported by the Program "Fundamentals of acoustic diagnostics of artificial and natural media" of Physical Sciences Division of Russian Academy of Sciences, the Grants No. 10-02-00228 and
08-05-00596 from the Russian Foundation for Basic Research, the Federal Program "Scientific and Scientific-educational brainpower of innovative Russia" (Contract N 02.740.11.0565) and Leading Scientific
Schools grant N 3700.2010.2.
1.
2.
3.
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L. J. Van Uffelen et al. The vertical structure of shadow-zone arrivals at long range in the ocean// J. Acoust. Soc.
Am. 2009. V. 125. P. 3569-3588.
A. L. Virovlyansky, A. Yu. Kazarova, and L. Ya. Lyubavin. Statistical description of chaotic rays in a deep water
acoustic waveguide// J. Acoust. Soc. Am. 2007. V. 121. No. 5. P. 2542-2552.
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