BACKSCATTER FROM ELASTIC OCEAN BOTTOMS:
USING THE SMALL SLOPE MODEL TO ASSESS
ACOUSTICAL VARIABILITY AND UNCERTAINTY
ROBERT F. GRAGG, RAYMOND J. SOUKUP AND ROGER C. GAUSS
Naval Research Laboratory, Washington DC 20375-5350,USA
E-mail: [email protected]
The scattering strength of the ocean bottom as a function of angle and frequency is a
fundamental input for predicting the performance of active sonar systems, particularly in
littoral waters. The small-slope formulation for scattering from the rough water/bottom
interface is by now well established both as a physical theory and as a numerical
algorithm [Gragg et al., JASA 110 (2001)]. In this work, a data-model comparison is
used to address the following questions. How well do the predictions of this elastic
theory agree with data measured at sea? How sensitively does the theoretical prediction
depend on the set of input parameters that characterize (a) the geoacoustics of the bottom
material and (b) the spectrum of the surface roughness? For a given littoral area, how
much needs to be known about these parameters for ASW/MCM purposes? How much
of that could be estimated by remote (e.g., acoustic inversion) methods?
1 Introduction
We first extract the bottom scattering strength (in the 2.0–3.5 kHz band) from LWAD >dH
data sets taken at a pair of nearby littoral sites that have a bare limestone bottom. The
processing techniques complement those of Holland et al. >1H, with special emphasis on
squeezing the widest possible range of grazing angles out of a relatively simple system
(omni source and VLA receiver). We then use small-slope theory to model the scattering,
given a minimal set of input parameters that specify the bottom’s roughness and geoacoustic properties (including shear). Finally, we adjust these inputs, under the control
of a simplex/annealing search algorithm, to maximize the data-model agreement across
frequency and grazing angle (W , ). These geoacoustic inversions yield information on
the importance of each of the parameters in scattering and on their site-to-site variability.
The water/sediment interface is the dominant scattering mechanism often enough
that a solid understanding of it is essential—especially for sand or rock because these
have significant ranges of sub-critical that are important in sonar operation. CST >nH
measurements in the 100–1000 Hz band have supported a variety of W and dependences,
illustrating the need for an improved physical model. The small-slope theory of scattering
from rough interfaces has recently been used to develop such a physics-based formulation
for elastic bottoms >;H. This model typically predicts dependences that are considerably
more complex than the familiar R$tz empirical descriptions (e.g. in Ref. >xH).
Our procedure relies on data with multiple frequencies and a wide range of sub-critical
angles to get enough information to invert for parameters that are rarely measured in-situ;
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N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 187-194.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Figure 1. Test sites described in the text.
e.g. the bottom roughness spectrum. We use LWAD data from the FTE 96-2 experiment
[1, 6, 7]—specifically from Sites Q and C, Fig. 1. MKS units are tacitly used throughout
this article; e.g., “c = 1500” with the “m/s” that specifies the units left implicit.
2
Acquiring and processing the data
Both experiments were essentially monostatic. (The source and receiver were mounted
only 3 m apart on a common cable.) At each site, deployments were made in shallow,
middle and deep configurations (source depths 35 m, 60 or 65 m, and 105 m respectively).
The receiver (a 9-phone VLA cut for 3750 Hz) provided ten beams, five of which are
relevant here. These were steered at 90◦ , 51◦ , 34◦ , 19◦ and 6◦ below horizontal, and
are designated 0 through 4, respectively. At each frequency, sets of 12–15 gated 10-ms
CW waveforms (separated by 3 s) were transmitted. Analysis of longer (50-ms) pulses
from these experiments has already been reported [1, 8]. Our use of these shorter signals
is supported by the weak frequency dependence seen in the scattering data (∼ 1 dB/kHz
across the 240 Hz analysis band).
After beamforming, power spectra were obtained using Fourier transforms of length
equal to the ping duration, the successive transforms proceeding to the end of the time
series with 90% overlap. A frequency band representing the total energy about the zeroDoppler peak was selected and a time series was formed for each ping using only the
energy in that band. The pings’ direct arrivals were then temporally aligned and then
averaged (before conversion to dB), producing a single reverberation curve for each beam
and frequency bin. Integration over the zero-Doppler spectral peak yielded the total
returned power over time and beam. Transmission loss terms to and from the scattering
patch were obtained from the geometric spreading loss along each ray path. The computed
beam pattern and ray trace were used to calculate the scattering patch area. With these
inputs, bottom scattering strength was calculated from the sonar equation as a function
of beam, f and θ.
BACKSCATTER FROM ELASTIC OCEAN BOTTOMS
189
Figure 2. Scattering strength vs grazing angle at 3 kHz for a deep deployment at Site Q. The
two types of corrections mentioned in the text are illustrated. Multipath designations B, SB, BSB
etc. correspond to Fig. 3.
Figure 3. Sketch of multipaths that produce ambiguous returns. (See Fig. 2.)
As a final step, the data were processed to correct for (a) multipaths (as described in
detail in Ref. [9]) and (b) reverberation decay over the pulse length. Figure 2 illustrates
the situation, with multipaths labeled according to their boundary interactions (bottom and
surface are abbreviated B and S). These are most prominent in the “Phone Data” curve,
which comes from a single hydrophone. Two examples of uncorrected data are included
for illustration. There is a sizable disparity between the uncorrected levels for beam 2
on either side of the BS/SB arrival. However, after correction for multipath effects, these
levels agree and are consistent with beam 3. For beam 0, reverberation slope corrections
become significant at high θ.
3
Geoacoustic inversion
The geoacoustic parameters of the problem are the densities and sound speeds of the
two media. For the water, these are ρw and cw (both real); for the bottom, they are
the density ρb and the complex-valued compressional and shear speeds, cp and cs . We
assume the water/bottom interface to have an isotropic, power-law, roughness spectrum
of the form S(k) = w2 /(h0 k)γ2 with 2 < γ2 < 4 (which corresponds to a fractal
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dimension between 1 and 2). We take the essentially arbitrary [4] reference length h0 to
be 1. This leaves ρw , cw , w2 , γ2 , ρb , cp , and cs as model inputs. Before inverting for
their values, we impose two conditions. We fix the parameters whose values are in no
real doubt, ρw = 1000 (nominal sea water) and cw = 1487 (from in-situ measurement),
and vary the others within appropriate bounds. We also examine the data to identify a
grazing angle θ̂ that appears to mark the onset of sub-bottom scattering, and then limit our
data/model comparisons to θ < θ̂. Inversion is then a matter of quantifying the data-model
deviation across all the experimental frequencies and allowed grazing angles by devising
a cost function Φ, and then searching the appropriate region of the parameter space for
the minimum of Φ(w2 , γ2 , ρb , cp , cs ). All that remains is to specify the cost function,
the search algorithm, and the search region. As with most inversion techniques, some
uniqueness problems are to be expected. One should anticipate fairly precise evaluations
for the parameters that strongly affect scattering, but only rough estimates for the rest.
We examined several cost functions based on the calculated theory-minus-data deviations over the experimental θ and f ranges. The simplest choice, an RMS average
deviation (in decibels), performed relatively poorly. We concluded that this was due to (i)
the high concentration of data points at low angles (the median data angle is only about
20◦ ), and (ii) an unexplained ripple that persists in some of the data at higher angles.
To counter these effects, we inserted a bin-averaging step in which the data points are
assigned to bins of width ∆θ ≈ 4◦ , bin values are computed by averaging (of the signed
deviations, not their absolute values), and finally the bin values are RMS averaged to
form the cost. This proved an effective remedy, and was adopted as our final working
definition for Φ.
Since the parameter space is seven-dimensional, some algorithm more efficient than
an exhaustive search was called for. We chose the ”amebsa” algorithm—essentially a fast
downhill simplex method that efficiently negotiates narrow valleys in parameter space
and is augmented with simulated annealing to prevent trapping by local minima [10, 11].
Given a suitable empirically chosen cooling rate, the technique usually “freezes” into a
final state of near-minimum Φ well within 50 temperature steps.
We chose the parameter search region based, as far as possible, on archival records
for the geoacoustics of limestone and on roughness data from rocky sea floors. For convenience, we first changed from using the shear speed Re(cs ) itself as a search parameter,
and used the compressional/shear speed ratio ξ = Re(cp )/ Re(cs ) instead. This does not
materially affect the operation of the search algorithm and is more convenient in two respects. Hamilton has concluded from his analysis of a large collection of data sets dealing
with saturated marine limestone that, although Re(cp ) and Re(cs ) vary considerably, their
ratio is to be found in the interval |ξ −1.90| < 0.06, “within 95% confidence limits” [12].
One can relax this empirical statistical rule somewhat, allowing a larger variance about
the mean value ($ξ% = 1.90) by taking |ξ − 1.90| < n × 0.06 with n > 1. We use
n = 5, and are thus dealing with the range 1.6 < ξ < 2.2 (Table 1).√(Since n < 12,
this also automatically respects the physical requirement [13] ξ > 2/ 3.) The limits
on Re(cp ) come from the observation that the compressional critical angle in our data
appears to lie within the range 70◦ < θp < 73◦ . The low and high values for ρb embody
the range reported in Hamilton’s references [14, 15]. The bounds on γ2 and w2 reflect
our experience with seafloor spectra. Reference [16] reports γ2 ≈ 2.64 for a scarp of the
Mid-Atlantic Ridge (MAR). The range in Table 1 includes this value. The range of w2
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Table 1. Inversion parameters and their high and low values (MKS units).
low
high
w2
0.0003
0.0009
γ2
2.4
3.0
ξ
1.6
2.2
Re(cp )
4348
5086
ρb
2400
2800
Im(cp )
−300
−5
Im(cs )
−900
−5
is chosen to undershoot the MAR value [16], w2 ≈ 0.0021, because the present bottom
is expected to have lower relief. The high values of Im(cp ) and Im(cs ) correspond to
attenuations reported for pure, homogeneous, water-saturated limestone samples (Table I
of Ref. [14] ). The low values (corresponding to compressional and shear attenuations of
approximately 0.5 and 7 dB/m/kHz, respectively) are essentially guesswork inspired by
the saprolitic (weathered-in-place) nature of the bottom [17].
4
Results
Figure 4 plots the data at all four experimental frequencies for site C along with a simulation curve produced using the optimal environmental parameters from the inversion.
The curve is black where the data-model fit is calculated (min(θ) < θ < θ̂ = 70◦ )
and gray elsewhere. The critical angles θp , θs are calculated from the optimal values of
Re(cp ), Re(cs ). As the frequency increases, the scattering strength, σ, in the figure rises
by ∆σ ≈ 3dB at small θ, consistent with the dependence σ ∝ f 4−γ2 to be expected
from small-slope computations [4]. The fit is quite good: Φ ≈ 0.5 dB and the maximum
absolute data/model deviation is only ∆ ≈ 4 dB. There are large slope corrections near
θp because the first derivative of the reverberation time series changes abruptly there. (In
measuring the slope at such a discontinuity, we obtain not a consistent value among different frequencies and sites but rather an artifact that indicates that a change in a scattering
mechanism has affected the derivative of the time series.) The marked data-model divergence as θ exceeds θp suggests the onset of significant sub-bottom volume scattering at that
point. The optimal values of the parameters w2 , γ2 , ρb , Re(cp ), Im(cp ), Re(cs ), Im(cs )
are listed in the figure along with the associated values of Φ and ∆. These numbers
are simply the raw output of the algorithm presented without any regard for significant
figures. “γ2 = 2.8258,” for example, does not mean that we have determined the spectral
exponent to four decimal places. We have obtained very similar theoretical curves even
using subsets of the allowed angles, e.g. with θ̂ = 50◦ . However, attempts to invert using
still smaller subsets, e.g. with θ̂ < 45◦ or using only angles in the range 45◦ < θ < 70◦ ,
have had little success. Evidently, inversion in this environment requires data over a
fairly broad range of angles and at least a modest bandwidth. Otherwise, the system
point seems too easily lured into physically dubious regions of the parameter space.
As noted above, inversion results are rarely, if ever, 100% repeatable. To investigate
this tendency, we first ran a series of fifty independent inversions for Site C. The outcome,
in Fig. 5, provides a look at how successfully our inversion process determines that site’s
parameters. The results are displayed as • symbols in separate vertical panels (one for each
of the seven parameters), with the fifty repetitions arranged along the horizontal. In each
panel, the solid horizontal line marks the parameter’s mean value, and the dashed lines are
one standard deviation away. (The third panel also has dotted lines to mark Hamilton’s
nominal interval for limestone.) It is clear from this that the spectral parameters w2 and
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R.F. GRAGG ET AL.
Figure 4. Site C data (symbols) and associated inversion result (curves).
Figure 5. Results of a series of 50 independent inversions for Site C.
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BACKSCATTER FROM ELASTIC OCEAN BOTTOMS
γ2 are both well determined (though in retrospect we might have extended the search
range for w2 a bit higher). So is Im(cs ), which means that the shear attenuation is
definitely substantial (presumably due to the saprolitic nature of the rock). The ξ ratio is
also well determined, though definitely somewhat above Hamilton’s range. The remaining
three—ρb and the real and imaginary parts of cp —are not well determined. They are not
among the principal parameters that determine the scattering strength of this bottom (for
2.0 kHz≤ f ≤3.5 kHz and θ < θ̂, at least).
Similar computations were also done for Site Q, with a similar outcome: w2 , γ2 , ξ,
and Im(cs ) were well determined, but ρb , Re(cp ), and Im(cp ) were not. However, the
results (summarized in Table 2) do exhibit some site-to-site variability. In particular, the
spectral strength w2 more than doubles over the 10 miles between the two sites.
Table 2. Summary of inversion results for the two sites.
Site Q
Site C
5
mean
w2 × 104
4.11
γ2
2.59
ξ
2.04
Re(cp )
4691
ρb
2630
Im(cp )
-172
Im(cs )
-721
std. dev
mean
0.34
8.86
0.05
2.80
0.11
2.14
271
4733
139
2723
92
-141
90
-825
std. dev.
0.24
0.04
0.06
252
113
116
63
Discussion
We have been able to invert acoustic backscatter measurements for the essential parameters
in interface scattering from sea floors. One key to the success of this effort has been
the use of data covering a wide range of grazing angles and a moderate frequency band.
Of equal importance has been the availability of a physical theory of scattering (namely,
small-slope) that allows both large roughness on the interface and elasticity in the bottom.
The physical basis of such scattering models supports extrapolation in angle and frequency
and also provides a foundation for relating geophysical variability to acoustic variability.
Our results suggest that the geoacoustic parameters that are important for scattering
in typical ASW scenarios could well be evaluated in-situ by inverse methods. Site-to-site
variability is an important consideration here; however, preliminary indications from our
geographically very sparse data sets indicate that the variability problem may not be too
severe. (The spectral strength w2 changes by only about 3 dB over the 10 miles between
Sites Q and C, for example.) Extending the technique to the MCM realm, where bottom
penetration is more important, would probably require the inclusion of a volume scattering
module in the scattering algorithm (a step which we have under active consideration).
Acknowledgements
This work was supported by the Office of Naval Research.
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