STATISTICS OF THE WAVEGUIDE INVARIANT
DISTRIBUTION IN A RANDOM OCEAN
DANIEL ROUSEFF
Applied Physics Laboratory, College of Ocean and Fishery Sciences,
University of Washington, 1013 NE 40th St., Seattle, WA 98105, USA
E-mail: [email protected]
Brekhovskikh and Lysanov popularized the concept of an ocean “waveguide invariant”
in the second edition of their book. The showed how acoustic intensity, mapped in
range and frequency, would exhibit streaks of high correlation, and they further
described the slope of these striations in terms of an invariant parameter. In shallow
water, the description works best at moderate frequencies and when the sound speed
profile changes only gradually in range and depth. In more complicated scenarios, the
concept of a scalar invariant can be generalized to allow a distribution of values.
Previously, the effects of shallow water internal waves on the distribution were studied
by numerical simulation. In the present work, equations are derived relating the
statistics of the waveguide distribution to the statistics of the random ocean. The
formulation makes use of a stochastic coupled normal mode model to describe the
acoustic propagation.
1
Introduction
Assume an acoustic source is transmitting in shallow water with the resulting field
measured on a horizontal array. At sufficient range, the ocean acts as an acoustic
waveguide supporting the propagating acoustic modes. Assume the array is oriented at
end-fire from the source and that the measured field is processed incoherently. If the
measured intensity is plotted versus frequency and distance along the array, the resulting
image will exhibit striations, nearly parallel contours of relatively high intensity. The
observed striations are a consequence of interference between the propagating acoustic
modes. Chuprov [1] related the slope of the striations, dω dr , to the range r and the
frequency ω via the parameter β:
β=
r dω
.
ω dr
(1)
Chuprov showed how, in some sense, beta is an invariant quantity. Brekhovskikh and
Lysanov popularized the concept of an ocean “waveguide invariant” in the second edition
of their book [2].
Equation (1) is useful for studying contour plots of measured data. For analytical
studies, it is preferable to express beta directly in terms of either the intensity I or
369
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 369-376.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
370
D. ROUSEFF
quantities related to the acoustic modes. One can also write the waveguide invariant beta
as
β =−
r ∂I ∂r
d (1/ v )
=−
,
d (1/ u )
ω ∂I ∂ω
(2)
where u is the group velocity and v the phase velocity of the pertinent acoustic modes.
The ocean environment will never be perfectly independent of range. Range
dependence in the water column might be caused by ocean internal waves. Roughness at
the water-sediment interface or in the sea surface would introduce range-dependence, as
would variability in the sediment. Random variability in the environment will introduce
random variability in the acoustic field with consequent effect on the propagating
acoustic modes. From Eq. (2), the waveguide invariant beta will also be a random
quantity. Often there is a statistical model for the randomness in the ocean. One would
like to relate the statistical models for the environment to a statistical description for
observable acoustic quantities. In the present case, the goal is to estimate relevant
statistics for the waveguide invariant.
Unfortunately, the two forms for the waveguide invariant given in Eq. (2) are not
amenable to a statistical analysis. The simplest quantity of interest would be the mean of
beta, <β>. It follows from Eq. (2) that <β> is given by the average of a ratio. The
average of a ratio is not something that is commonly calculated; only in special cases, for
example, can the average of a ratio be replaced by the ratio of averaged quantities [3].
Recently, the waveguide invariant was reformulated as a distribution. Rather than
assign a single scalar, a distribution of values is produced; effectively, one calculates the
“beta content” of a set of measurements. Previously, the effects of shallow water internal
waves on the distribution were studied by numerical simulation [4]. In the present work,
we show how the formulation is also useful in theoretical studies. We outline the
derivation of a formula for the first moment of the waveguide distribution. The
derivation makes use of a stochastic coupled normal mode model to describe the acoustic
propagation.
2
Waveguide invariant distribution
The experimental observable is the intensity measured at depth z over a horizontal array
of length L = rmax − rmin oriented at end-fire from the source. Consider the measurements
in the frequency band ω min < ω < ω max . Let I(r, ω) be the intensity as measured over
the finite-length receiving aperture. Define I! (κ ,τ ) as the two-dimensional Fourier
transform of the intensity:
I! (κ ,τ ) =
ω max rmax
I (r , ω ) exp[i(κ r + ωτ )]dr dω .
∫
∫
ω
min
rmin
Define the waveguide invariant distribution as
(3)
STATISTICS OF THE WAVEGUIDE INVARIANT DISTRIBUTION
B
Eφ = (2π ) −2
∫
2
I! ( K cos φ , K sin φ ) K dK .
371
(4)
−B
Note that ˜I (K cos φ , K sin φ ) is the transform evaluated along a line passing through the
origin in Fourier space and oriented at angle φ from the κ axis. The integral has been
truncated at some maximum spatial frequency of interest B. The ramp filter K
emphasizes the higher spatial frequencies. Integrating Eφ over all angles would yield, in
the sense of Parseval’s Theorem, the total energy in the original image [4]. The remaining
task is to relate the angle φ to β. Assume that the range to the midpoint of the array rmid
is large compared to the length of the array L, and that the center frequency ω mid is large
compared to the bandwidth. Neglecting the effects of the finite data window, we obtain
the mapping [4]
β = − rmid (ω mid tan φ ) .
(5)
In this approach, beta is treated not as a single number but rather as a distribution. The
output of the processing is the distribution plotted versus β. This distribution might be
sharply peaked around a single value in which case the traditional notion of β as a scalar
would be reasonable.
Figure 1 shows a sample calculation of the waveguide invariant distribution. The
scenario involves a Perkeris waveguide of depth 70 m. The sound speed in the water
column and in the sediment is 1480 and 1580 m/s, respectively. The bottom loss is
0.1 dB/λ. Both the source and receiver are at depth 50 m. The frequency band extends
from 400 to 420 Hz and the range is 10 km. The plot is normalized so that the average
value of the distribution over the interval shown is one. The distribution is sharply peaked
around the canonical shallow water value β = 1; see the analysis in Brekhovskikh and
Lysanov [2]. If there is a significant sound speed profile and the source and receiver are
appropriately positioned, the location of the peak can shift or be lost altogether; see [4]
for examples of internal wave effects.
distribution
8
6
4
2
0
0
1
2
beta
3
4
Figure 1. Sample calculation of the waveguide invariant distribution.
372
3
D. ROUSEFF
First moment of the waveguide invariant distribution
In this section, we first express the waveguide invariant distribution in terms of the
acoustic normal modes. The resulting expression is a double summation over mode pairs
where the weights for the terms depend on the mode amplitudes. Next, the master
equations for the stochastic mode amplitudes are outlined. Finally, we take a formal
average to get the mean waveguide invariant distribution.
3.1
Modal Formulation for the Waveguide Invariant Distribution
The starting point is a modal representation for the pressure field observed at depth z and
at range r from the source
p (r , ω ) =
∑A
m (r )Ψ m ( z )
(ξ m r )1/ 2 ,
(6)
m
where Ψm and ξm are the eigenfunction and wavenumber, respectively, of mode m for
the unperturbed background sound speed profile c 0 (z) . For the special case of a rangeindependent medium, the modal amplitude Am (r ) would be proportional to the
eigenfunction evaluated at the source depth zs :
Am ( r ) = Ψ m ( zs ) exp[i (ξ m + iα m ) r ] . (Range-independent environment.) (7)
For this special case, there is no coupling of energy between the acoustic modes between
the source and the receiving array. The modal attenuation is α m accounts for energy loss.
In the more general case considered here, we allow stochastic mode coupling and defer
the specific master equations for Am (r ) to the next section. To simplify the analysis, we
assume the modes propagate without coupling or loss over the extent of the aperture.
This is reasonable when the extent of the aperture L is small compared to the distance to
the source. With the array extending for rmin < r < rmax , then over the array
Am ( r ) = Am (rmin ) exp[iξ m ( r − rmin )] .
(8)
The experimental observable is the intensity I (r , ω ) measured along the end-fire
array in the frequency band ω min < ω < ω max . It follows that
I (r , ω ) ≡ pp* =
∑B
lm
exp(i ∆lm r ) ,
(9)
lm
where ∆ lm = ξl − ξm is the difference in wavenumber between a pair of modes. The
weightings
Blm = Al ( rmin ) Am* ( rmin )Ψ l ( z )Ψ m ( z ) exp[ −i (ξl − ξ m ) rmin )] (ξlξ m r 2 )1 2
(10)
STATISTICS OF THE WAVEGUIDE INVARIANT DISTRIBUTION
373
are derived from Eqs. (6) and (8).
To show how the output of the processing algorithm in Sect. 2 depends on the modal
composition of the intensity, substitute Eq. (9) into Eq. (3) and combine with Eq. (4).
Rearranging terms yields
ωmax
B
Eφ = (2π ) −2
∑∑ ∫
K dK
lm l ′m ′ − B
∫
ω
min
ω max
dω
∫
ω
min
rmax
dω ′
rmax
∫ ∫ dr ′B
*
lm Bl ′m′
dr
rmin
rmin
(11)
× exp[i ( ∆lm r − ∆l ′m′ r ′)]exp(iK sin φω d + iK cos φ rd )
where the difference coordinates ω d = ω − ω ′ and rd = r − r ′ .
With some manipulation, Eq. (11) can be reduced to a relatively simple formula for
the distribution. The derivation closely parallels that for the range-independent case
given by Rouseff and Spindel [5]. In the present work, we merely outline the derivation
and present the final result; the interested reader is referred to [5] for the details.
The derivation starts by considering the range integrals in Eq. (11). These are
rewritten in terms of the difference and average coordinates, rd and ra = (r + r ′) / 2 ,
respectively. As before, we assume the length of the array L is small compared to the
distance to the source which allows us to simplify the limits of integration. Neglecting the
weak range-dependence in Eq. (10), the range integrals can then be evaluated. The
integration over rd yields a delta function that can then be used to evaluate the
integration over K.
The derivation proceeds with the frequency integrals. It becomes useful to expand
the wavenumbers about the center frequency ω mid = (ω max + ω min ) 2 . The phase contains
terms like
ξ m (ω ) ≈ ξ m (ω mid ) +
dξ m
ω
(ω − ω mid )
.
(ω − ω mid ) = mid +
dω
vm
um
(12)
where the phase velocity v m and the group velocity um are evaluated at the center
frequency. Similar expansions can be developed for the other terms in the phase. It is
also useful to define the local invariant β lm
β lm = − (1/ vl − 1/ vm ) (1/ ul − 1/ um ) ,
(13)
which makes explicit the dependence of the invariant on the specific pair of mode indices.
Equation (13) can be viewed as the finite difference approximation to Eq. (2). With these
expansions, one can determine the stationary phase points of the remaining integrals.
One finds that for most combinations of mode indices, the stationary phase points are
outside the regions of integration. The dominant contribution occurs when l = l′ and
m = m′ . Retaining only these terms, the frequency integrals can be evaluated in terms of
the Fresnel integrals S and C [6]. The final result is
374
D. ROUSEFF
Eβ ≡ Eφ
dβ
dφ
−1
= 12 Lω mid
∑β
2
lm
β Blm [ FC2 + FS2 ] ,
(14)
lm
where the terms involving Fresnel integrals
FC = C (γ + ) + C (γ − ), FS = S (γ + ) + S (γ − ),
(15)
have arguments
12
γ ± = ω mid rmid glm ( βπ ) 
 1 Q −1 ± ( β − β lm )  .
2

(16)
In Eq. (16), Q = ω mid (ω max − ω min ) is the usual “quality factor” that arises when
analyzing resonating systems, and glm = 1 ul − 1 um is the difference in group slowness.
Equation (14) expresses the waveguide invariant distribution as a double summation
of terms involving the modal constituents. The behavior of the term in the square
brackets is instructive. When the local invariant β lm equals β, this term is maximized and
potentially makes a strong contribution to the distribution. As β lm diverges from β , the
contribution is reduced. Note that each term in the double summation is weighted by its
2
corresponding Blm . These weightings depend on the mode amplitudes and also contain
the depth-dependence in the result; see Eq. (10). For the range-independent problem
when Eq. (7) applies, Rouseff and Spindel [5] used Eq. (14) to study the effect of source
depth on the distribution. The more general case of a range-dependent environment is
considered in this report.
3.2
Stochastic Coupled Mode Equations
It remains to specify the mode amplitudes Am (r ) in Eq. (6). We assume the range
dependence is primarily due to random variability in the medium and use the formalism
developed by Dozier and Tappert [7]. Dozier later extended the technique to include
bottom loss, an important consideration for propagation in shallow water [8]. Creamer
used and extended these results to study scintillation in shallow water waveguides [9].
Here, we summarize some of their results pertinent for calculating statistics of the
waveguide invariant distribution.
Using the quasi-static and narrow-angle approximations, the mode amplitudes can be
shown to satisfy [6-8]
∂Am
− i(ξ m + iα m ) Am = −i
∂r
∑ρ
mn ( r ) An
,
(17)
n
where ρmn contains the effects of mode coupling. Note that if there is no coupling, the
right hand side of Eq. (17) is zero and the solution to the differential equation reduces to
STATISTICS OF THE WAVEGUIDE INVARIANT DISTRIBUTION
375
Eq. (7) as it must. The mode coupling terms are related to the perturbations in the index
of refraction δ c by
ρ mn =
k2
ξ mξ n
∫ dz δ c(r, z) c ( z)Ψ
0
m ( z )Ψ n ( z )
.
(18)
The perturbations δ c would typically be modeled as a Gaussian random process. This
lets us relate the autocorrelation of the mode coupling terms
Rmn ( r − r ′) = ρmn ( r ) ρ mn (r ′)
(19)
to a particular model for the fluctuations in the medium. These fluctuations might be due
to internal waves or other types of environmental variability.
In this short
communication, we do not consider specific statistical models for the environment.
3.3
Fourth Moment Equations
The average waveguide invariant distribution follows immediately from Eq. (14). Since
all the randomness is contained in the weightings Blm ,
Eβ = 12 Lω mid
∑β
lm
β
Blm
2
[ FC2 + FS2 ] .
(20)
lm
From Eq. (10),
Blm
2
= Al (rmin )
2
Am ( rmin )
2
Ψ l ( z )Ψ m ( z )
2
2
(ξl ξ m rmid
).
(21)
Note that the average waveguide invariant distribution depends on the fourth moment of
the pressure or, more precisely, the second moment of intensity. Making the Markov
approximation, Creamer [8] developed the state equations for the second moment of
intensity. The state matrix contains projections of the autocorrelation Eq. (19) as well
terms related to the modal attenuation α m . Following Dozier, Creamer diagonalized the
state matrix, but this is not necessary for our purposes. One can write the solution to the
state equation in terms of the matrix exponential [10]. This lets us calculate the moments
in Eq. (21) and hence the average waveguide distribution given by Eq. (20).
4
Discussion
The waveguide invariant distribution is a generalization of the usual scalar invariant. In
our previous work, we did numerical simulations to study how the distribution was
affected by ocean internal waves [4]. In the present work, we derived an expression for
the average distribution when random variability in the medium introduces acoustic mode
coupling. The result is quite general and should apply at moderate frequencies whenever
376
D. ROUSEFF
the variability in the medium can be modeled as a Gaussian random process. In our
future work, we will consider specific models for the ocean variability and do detailed
numerical calculations of the mean waveguide invariant distribution.
Acknowledgements
A portion of this research was conducted while the author was a Senior Visiting Fellow in
the Department of Applied Mathematics and Theoretical Physics at the University of
Cambridge, Cambridge, England. The author thanks Dr. Barry Uscinski and the staff at
DAMTP for their hospitality. The author also thanks Dr. Lewis Dozier for providing a
copy of reference [8]. This work was supported by the United States Office of Naval
Research.
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