MODELING PROPAGATION AND REVERBERATION
SENSITIVITY TO OCEANOGRAPHIC AND
SEABED VARIABILITY
KEVIN D. LEPAGE
SACLANT Undersea Research Centre, Viale San Bartolomeo 400, 19138 La Spezia, Italy
E-mail: [email protected]
The propagation of bottom and oceanographic variability through to the variability of
acoustic transmissions and reverberation is evaluated with a simple adiabatic model
interacting with Gaussian distributed uncertainty in a narrow band. Results show that
there is significant sensitivity of time series and reverberation uncertainty to different
types of environmental uncertainty. For propagation over uncertain bottoms, we show
that it is that later part of the time series, corresponding to the highest angle energy
reflecting most often off the surface and bottom, which is most sensitive to bottom
uncertainty. This implies that the higher reverberation from the highest grazing angles
is also the most uncertain. Conversely, it is the lowest angle arrivals which are most
sensitive to uncertainty in the sound speed profile. These controlling principles are
intuitive and are predicted in closed form with the theory.
1 Introduction
The effects of oceanographic and seafloor variability on acoustic propagation and reverberation in shallow water waveguides are of interest in the context of sonar performance
uncertainty. In shallow water the downward refracting nature of the waveguide causes
significant bottom interaction, so to the extent that the general background properties of
the bottom sediments are unknown, significant variability in the forward propagation of
energy to and from scatterers is to be expected. The same is of course also expected
from oceanographic variability. In order to propagate uncertainty in the background properties of the ocean and the bottom through to uncertainty in acoustic propagation and
reverberation an acoustic modeling approach is required which treats the propagation and
reverberation stochastically. One common approach is to use one of the available “high
fidelity” acoustic models to compute realizations of propagation or reverberation over a
sample of oceanographic or bottom variability. This Monte-Carlo approach has the advantage that as long as the acoustic models are accurate and the underlying samples from
which the oceanographic or bottom ensemble is drawn is known, then all the statistics of
the desired property may be estimated. However, more rapid insight into the controlling
parameters of oceanographic and bottom variability can be gained from a simpler, lower
fidelity approach which is derived on more restrictive assumptions regarding both the
distributions from which the bottom variability is drawn and on the simplicity of the
acoustic propagation.
Here the lower fidelity approach is taken to understanding the effects of bottom vari353
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 353.360.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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K.D. LEPAGE
ability on acoustic propagation and reverberation. We begin with the simplest useful
parameterization of waveguide variability as a Gaussian distributed process which is adequately described by second order statistics and a correlation length scale. We then
evaluate the effects of this variability on adiabatic propagation and reverberation in a
narrow band. Using this approach it is possible to derive closed form expressions for
the expected value of received intensity [1,2]. The results show that significant understanding of the sensitivity of temporal propagation and reverberation to different types of
environmental variability can be gained using the closed form expressions.
2
Examples of variability in propagation
We evaluate the variability of propagation in the 140 m deep shallow water waveguide
illustrated in Fig. 1. A downward refracting sound speed profile overlies a slow sediment
layer 5 m thick with a background sound speed of 1482 m/s, a density of 1 g/cm3 and bulk
attenuation of 0.06 dB/λ. The sediment lies over a 1562 m/s basement with a density of
1.8 g/cm3 and an attenuation of 0.1 dB/λ. Time series excited by a 20 m source operating
at 500 Hz with 25 Hz of bandwidth are predicted over the full water column at a range of
15 km for ideal propagation and for propagation through oceanographic and over bottom
variability.
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Figure 1. Shallow water environment used in the study.
The first case we study is propagation through internal waves. These are characterized
vertically by the EOF mode shapes illustrated in the left panel of Fig. 2 and horizontally
by the correlation length scales shown in the right hand panel. The most energetic
EOF shapes have the longest correlation length scales and the fewest number of vertical
oscillations, while the weakest have the shortest correlation length scales and the most
vertical oscillations. Horizontal correlation length scales range from over 3 km for the
first EOF to less than 100 for the ninth. The standard deviation of the sound speed defect
associated with each EOF (not shown) ranges from 2.5 m/s for the first EOF to 0.13 m/s
for the ninth.
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MODELING PROPAGATION AND REVERBERATION UNCERTAINTY
Correlation length scale of shallow water EOFs
EOF mode shapes for SOFAR (1 GM)
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Figure 2. Mode shapes of the first 6 EOFs for the shallow water sound speed profile (left) at 1
GM. The right plot shows the corresponding correlation length scales.
Figure 3. Variability of acoustic propagation in the presence of internal waves. Top: Unperturbed
waveforms at 500 Hz. Middle: Expected value averaged over all internal wave realizations. Bottom:
Standard deviation of intensity over 50 realizations normalized by the average intensity from the
middle plot.
The expected value of the intensity received at a range of 15 km as a function of
depth and time is illustrated in Fig. 3. The top panel shows the expected value in the
absence of variability, i.e. the intensity in an ideal waveguide. The middle panel shows
the expected value of the intensity averaged over all internal wave realizations. The
bottom panel shown the standard deviation of the intensity estimated over an ensemble
of 50 Monte-Carlo realizations of an internal wave field generated using the EOFs, dB
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K.D. LEPAGE
3
EOF mode shapes for bottom
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EOF 1
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Figure 4. Mode shapes of the first 20 EOFs for sediment sound speed defects conforming to a
power law distribution. The right plot shows the corresponding correlation length scales.
re the expected value of the intensity from panel 2. The first panel showns that the
time series in a shallow water waveguide is a decresendo with the lowest order modes
arriving first followed by the surface-bottom multiples. With a 1562 m/s basement sound
speed, the length of the coda is restricted to 300 ms at 15 km range. The middle and
bottom panel show that the first arrivals are the least predictable, while the later arrivals
are highly predictable. This is understood from the fact that the earliest arrivals travel
along the direction of the largest correlation length scale of the internal wave induced
variability, while the steeper surface-bottom multiples travel increasingly along the shorter
vertical correlation length scale of the variability. This is the predominant characteristic
of shallow water propagation through internal wave fields.
We now investigate how bottom variability affects the propagation. In Fig. 4 the
EOFs and horizontal length scales of a 1 m/s rms bottom sound speed defect realization
are shown. The realization was generated from an anisotropic two dimensional power
spectrum conforming to a power law asymptotic to k −3 with horizontal and vertical
correlation length scales of "x = 500 m and "z = 0.3 m. As with the internal waves,
the most energetic EOFs have the fewest vertical oscillations and the longest correlation
length scale. However, the roll-off rate of the correlation length scale from a nominal
value of 500 m is much slower, as the underlying process has only the one horizontal
correlation length scale. The power in the various EOFs (not shown) also falls off more
slowly, with a standard deviation of 0.35 m/s for the first EOF and 0.09 for the twentieth.
In Fig. 5 the variability in the bottom is propagated through to the acoustic uncertainty. As before, the top panel shows the unperturbed time series. The middle panel
shows the expected value of the intensity averaged over the ensemble of bottom sound
speed perturbations conforming to the EOF decomposition shown in Fig. 4 with a 20 m/s
sediment sound speed standard deviation. Here it seen that it is the later arrivals, which
interact more strongly with the bottom, which are uncertain. The bottom panel, which
shows the ratio of the standard deviation of the intensity to its expected value, also shows
that the later arrivals can have an uncertainty which equals their average intensity.
Finally, we evaluate the effect of fluctuations of attenuation in the sediment with a
standard deviation of 0.018 dB/λ. The result is shown in Fig. 6, where it is seen that
MODELING PROPAGATION AND REVERBERATION UNCERTAINTY
357
Figure 5. Variability associated with 20 m/s standard deviation in the sediment sound speed.
Figure 6. Variability associated with 0.018 dB/λ standard deviation in the background attenuation
of the sediment.
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K.D. LEPAGE
Figure 7. Reverberation variability caused by water column sound speed perturbations associated
with unity Garrett-Munk internal wave spectrum.
while the expected value of the intensity is the same as the unperturbed value, the standard
deviation can again approach the expected value itself, especially at late times, while at
early time the standard deviation can be as much as 10 dB lower. The two results for
the sensitivity of received intensity to bottom perturbations clearly show that it is the
later arrivals, corresponding to the bottom interacting modes or the steepest ray paths
with the most boundary interactions, which are the most sensitive to bottom uncertainty.
This intuitively obvious result can be quantified using the closed form expressions as a
function of the acoustic normal mode properties of the waveguide and the EOF modes
shapes, correlation length scales and power.
3
Examples of variability in reverberation
We here show some preliminary results for variability in coherent reverberation for the
same waveguide and source characteristics as were used to study variability in propagation. Reverberation in the case of uncertainty in water column or sediment properties
has two sources of variability about the mean intensity: the first is standard deviation
of reverberation due to the fact that the intensity is a pseudo-random process typically
distributed Rayleigh in amplitude for Gaussian distributed scatterer amplitudes [3,4], and
the second is associated with the uncertainty in the underlying watercolumn or sediment
properties themselves for fixed realizations of scatterer distributions. Here we evaluate
the latter alone, i.e. the standard deviation of the reverberation intensity for fixed scatterer
realizations due to watercolumn variability alone.
MODELING PROPAGATION AND REVERBERATION UNCERTAINTY
359
Figure 8. Reverberation variability caused by water column sound speed perturbations associated
with twice the intensity of a Garrett-Munk internal wave spectrum.
In Fig. 7 the deviations of the reverberation intensity from the ideal case are shown for
water column sound speed perturbations caused by internal waves whos EOF properties
are shown in Fig. 2. The top panel shows the reverberation intensity as a function of
time and depth as received on a monostatic vertical line array. As with the propagation
studies, the source depth is 20 m and the source pulse has a center frequency of 500
Hz, but in this case the bandwidth is 50 Hz, giving a “resolution cell” on the seafloor of
15 m. The reverberation is caused by interaction with scatterers on the sediment-water
interface with an rms amplitude of 1 and a correlation length scale of 0.25 m.
The middle panel shows the expected value of the reverberation intensity averaged
over the ensemble of oceanographic variability caused by the internal waves. One sees
that most of the details of the depth-time arrival structure are preserved, with the exception
of some higher frequency modulations at mid-depth between 6 and 8 s, and less deep
nulls at late time. The bottom panel shows the standard deviation of the intensity dB
re the expected value of the intensity from the second panel, estimated by a MonteCarlo average over 10 oceanographic realizations. With such a small sample size the
true spatial structure of the standard deviation is not resolved: however it is a sufficient
sample size to see that the reverberation time series at early time is highly stable or robust
to oceanographic variability, while at late time is is less so, with the standard deviation
often exceeding the mean value itself, especially in regions where the average intensity
is lower.
In Fig. 8 the effects on the reverberation are evaluated for a doubling of the intensity
of the internal wave spectrum. The middle panel shows the expected value of the now
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K.D. LEPAGE
significantly modified coherent structure of the reverberation. The results show that the
coherent structure is all but totally smoothed out at late time, and that even at early times
from 2 s onward the average intensity is significantly reduced from the unperturbed results.
The normalized standard deviation of the reverberation, shown in the bottom panel, also
shows that the uncertainty of the reverberation intensity is significantly enhanced over the
1 Garrett-Munk case in Fig. 7, a result consistent with the reduced coherent component
shown in the middle panel.
4
Conclusions
Two theories for the uncertainty of time domain propagation and reverberation in the
presence of waveguide variability have been derived based on the method of normal
modes, the narrow band approximation, perturbation expansions of the deviations of the
wavenumbers and modal slownesses to the environmental perturbations, and an EOF
decomposition of the oceanographic or bottom variability. The results for time series
show that the character of the temporal development of uncertainty is highly dependent
on the spatial distribution of the environmental uncertainty. For oceanographic variability
associated with internal waves, the results show that it is the earliest, lowest angle arrivals
which are the most uncertain. This is consistent with the understanding that the correlation
length scales of the environmental variability are several orders of magnitude greater
in the horizontal direction then they are in the vertical direction, causing uncertainty
to accumulate more rapidly for low angle propagation. Conversely, results for bottom
variability show that it is the latest arrivals, corresponding to the highest order surfacebottom multiples, which are most sensitive to the bottom variability, consistent with
intuition.
Results obtained with the new theory of reverberation variability show promise for
evaluating the relative sensitivity of reverberation to oceanographic and bottom variability. Preliminary results, restricted to evaluating the sensitivity of reverberation to
oceanographic variability, show that the coherent, predictable structure of the reverberation time series is reduced as the oceanographic variability is increased, and that the
standard deviation of the reverberation is correspondingly increased. Future work will be
directed to further exploiting the model to evaluate the relative sensitivity of reverberation
to oceanographic and bottom uncertainty.
References
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Acoust. Soc. Am. 92, 1408–1419 (1992).
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cumulative effects of water column variability, J. Comp. Acoust. 9, 1455–1474 (2001).
3. LePage, K.D., Bottom reverberation in shallow water: Coherent properties as a function of
bandwidth, waveguide characteristics, and scatterer distributions, J. Acoust. Soc. Am.
106, 3240–3254 (1999).
4. Abraham, D.A., Modeling non-Rayleigh reverberation. Rep. SR-266, SACLANT Undersea
Research Centre, La Spezia, Italy, 1997.