UNCERTAINTY IN REVERBERATION MODELLING
AND A RELATED EXPERIMENT
C.H. HARRISON, M. PRIOR AND A. BALDACCI
SACLANT Undersea Research Centre, Viale S. Bartolomeo 400, 19138 La Spezia, Italy.
E-mail: [email protected]
A serious stumbling block in modelling reverberation is the uncertainty in bottom
scattering strength, and in the absence of detailed surveys we are still faced with many
possible physical mechanisms. For instance, bottom penetration at low sound speeds
allows scattering from buried layers and volume scattering which distorts the angle
dependence. Typically in shallow water the propagation fall-off with range is controlled
by ‘mode-stripping’, and the joint scattering and propagation angle dependence has a
profound effect on the reverberation. Some predictions of the reverberation and signalto-noise are made using very simple analytical calculations, and it is shown that there is
a regime where signal-to-reverberation is a constant, independent of range, if we assume
mode-stripping and Lambert’s law! To check these ideas an experiment is proposed to
investigate simultaneously the reverberation intensity and one-way path intensity as
functions of range and vertical angle. By comparison of the two trends vs range it is
possible to estimate the angle dependence of the scattering. The relative importance of
position-dependent scattering strength, bottom shape, scattering law, and propagation
are considered. These are backed up by calculations from a new multistatic sonar
performance model SUPREMO.
1
Introduction
Reverberation is strongly influenced by a number of distinguishable scattering and
propagation mechanisms. Amongst the scattering phenomena are: angle-dependence of
the scattering law, geographic variation of the scattering strength, and local tilt of
bottom facets. Propagation phenomena include range-dependent propagation intensity
and range-dependent arrival angle at the scatterer.
So, can we predict diffuse reverberation when so many things are going on? Some
of these quantities are more susceptible to surveying than others. Some can already be
deduced from existing known data.
From the opposite point of view, given a (non-local) reverberation measurement, is
it possible to tell what mechanism is what? Can we uniquely determine scattering
properties?
On the face of it there is enormous scope for mathematical complication. But in this
paper we try to gain some insight into the relative importance of the mechanisms by first
taking an analytical approach. The importance of a mechanism can be judged by its
impact on the signal-to-reverberation-ratio (SNR) directly. Finally we return to a
numerical model to look at some more detailed effects.
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N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 361-368.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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C.H. HARRISON ET AL.
Anlytical (closed-form) reverberation and implications for SNR
Propagation in shallow water tends to follow a mode-stripping law, culminating in a
single mode at long ranges [1]. The reason for this is that the bottom loss (in dB) almost
always obeys a linear law with angle until the critical angle θc (if it exists) R = αdBθ.
Then the number of reflections is determined by the ray cycle distance rc which for
isovelocity is given in terms of the water depth H by rc = 2H/tanθ. This results in an
exponential decay in range but a gaussian angle distribution exp(−αθ 2r / 2 H ) . By
considering either a sum of eigenrays or a sum of modes [2] the final result for
isovelocity can be shown to be a soluble integral over ray angles
I=
2
rH
θc
∫0
exp( −
αθ 2
r ) dθ =
2H
2π
Hα r 3
erf( αr / 2 H θ c )
(1)
When the square root term in the error function is large (i.e. the gaussian is narrow
compared with the critical angle) the intensity reduces to [1]
I=
2π
Hα r
3
=
20π log(e)
(2)
Hα dB r 3
which is the familiar three-halves law or 15 log(r). Closed-form solutions also exist [2]
with smooth transition to single mode propagation (an exponential decay in range), and
these effects can be seen in some of the later plots here.
For a monostatic sonar the result of two-way propagation with a point target of
target strength ST this becomes
I=
2 π ST
αHr
3
(erf(
α r / 2H θ c )
)
2
(3)
The equivalent reverberation with angle-independent scattering strength SB and
elementary scattering area (determined by the spatial pulse length p=ctp/2 and the spatial
horizontal beam width rΦ ) is
I=
2 π S B Φp
αHr
2
(erf(
α r / 2H θ c )
)
2
(4)
It inevitably falls off more slowly than the target echo because of the widening of the
scattering area with range. Eventually at long range reverberation will win, and at the
transition there is a “reverberation limit”.
Reverberation with Lambert’s Law scatterers behaves differently. Assuming that
the scattering strength is separable in the incoming (θ1) and outgoing (θ2) ray angles
S = µ sin(θ 1 ) sin(θ 2 )
the intensity integrals also separate
(5)
UNCERTAINTY IN REVERBERATION MODELLING
I=
4
r2H 2
θc θ c
∫0 ∫0
sin θ 1 sin θ 2 exp( −
αθ 1 2
2H
r ) exp( −
αθ 2 2
2H
r )dθ 1 dθ 2 µ rΦ p
363
(6)
to give a closed form solution for the reverberation
2
 2 θc

αθ 2
I = 
r ) dθ  µ rΦp
θ exp( −
0
2H
 rH

2
4µ
= 2 3 Φp 1 − exp( −α rθ c 2 / 2 H )
α r
∫
(
)
(7)
In Eq. (3) we saw that the mode-stripping three-halves range law resulted in r −3 in
two-way propagation when the gaussian was much narrower than the critical angle.
Interestingly the Lambert’s law reverberation in Eq. (7) shows exactly the same range
dependence in this case; the usual proportionality of scattering area to range that alters
the range-dependence in Eq. (4) is exactly counteracted by mode-stripping in this new
integral. At shorter ranges where the critical angle truncates the gaussian the error
function of Eqs. (3) and (4) is replaced by a (1−exponential) term.
(a)
(b)
Figure 1. Target (two-way propagation) and reverberation: flat bottom, 100m depth, (a) 1000 Hz,
(b) 100 Hz. Both frames show general formula and HF approximation for comparison. The source
term and the µΦp terms have been set to unity in this example.
Another interesting point is that the dependence of reverberation on bottom loss α
is stronger than that of the target. In fact low bottom loss benefits the reverberation more
than the target so the SNR is improved by high bottom loss!
Figure 1 shows the behaviour of these formulae for target and reverberation at
1 kHz and 100 Hz. Note that µΦp has been set to unity since we are only interested in
shapes of the curves. The regime of constant SNR is seen at long ranges where target
and reverberation are parallel. Single mode effects are seen in the right hand frame.
Although these formulae assume isovelocity it is easy to see that refraction simply
introduces the possibility of ducts at low angles; angles steep enough to hit the
boundaries and therefore reverberate are still attenuated in more or less the same way
[2].
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C.H. HARRISON ET AL.
It is shown in [2] that these formulae can be extended to slowly varying rangedependent environments by using ray invariant H sin(θ) = constant [3] to cope with the
angle changes. The results are written in terms of water depth at the source Hs and
receiver Hr, the shallowest depth Hc, and an intermediate ‘effective’ depth Heff [2,3]
defined in terms of the depth profile H(r).
r
dr '
0
H 3 (r ' )
H eff = ( H r 2 H s 2 / r ) ∫
(8)
(a)
(b)
Figure 2 Signal-to-reverberation ratios: flat bottom, 100 m depth, (a) 1000 Hz, (b) 100 Hz. Blue
line is large critical angle approximation; green line is arbitrary critical angle; red line is no
approximation.
The target echo becomes
I=
2
2π
α r 3 H eff


 erf{ α rH eff θ c H c } S
T

2
Hr Hs 


(9)
Reverberation with angle-independent scatterers becomes
I=
2
2π
α r 2 H eff


 erf{ α rH eff θ c H c } S Φp
B

2
Hr Hs 


(10)
and reverberation with Lambert’s law becomes
I=
4H s 2
α 2 r 3 H eff
2
(1 − exp{−θ
2
c
)
H c 2α rH eff /(2 H r 2 H s 2 )}
2
µ Φp
(11)
Surprisingly, propagation is only weakly affected because, for instance, the
increased bottom loss up-slope tends to be counteracted by the concentration of flux into
a shallower water depth. The main effect on Lambert’s law reverberation is the change
of scattering angle rather than incident intensity, which manifests itself in Eq. (11) as the
(Hs/Heff)2 term. Up-slope reverberation is therefore generally stronger than down-slope,
UNCERTAINTY IN REVERBERATION MODELLING
365
as one might guess. However the difference is small; for instance on a uniform slope
halving the depth results in only a 2.5 dB increase, and doubling it results in a 3.5 dB
decrease (Heff is just the average of depths at source and receiver for a uniform slope).
3
A proposed experiment
The fact that we expect different range dependences according to the scattering law
angle dependence suggests a possible experimental way of determining the scattering
law. Two related methods are proposed, and it is hoped that they will be tried this year.
There is one experimental arrangement but two sets of measurements.
3.1
Compare Reverberation and Propagation Range-Dependence
In principle, referring to Fig. 1 a single shot and a single monostatic hydrophone are
sufficient to obtain the entire reverberation curve. To obtain the ‘target’ curve we need
to move the source and receiver apart, measuring transmission loss as we go. Doubling
the dBs gives a ‘point target’ echo level. The parallel region on the right of the figure
shows that we expect the same range dependence for ‘target’ and ‘reverberation’ if there
is Lambert’s law. Otherwise we need to try different analytical forms for the law until
the trend is explained. Uncertainty in bottom reflection loss is unimportant for this ratio.
Given ideal experimental conditions, no numerical modelling is required. However one
can think of many practical problems such as finding an environment that is rangeindependent for, say, 50 km radius (in all directions) from the source. In addition there
is the problem of ducted propagation. The above analysis is strictly for isovelocity,
although as noted, there is not much difference for reverberation with or without
refraction, but we need to be careful that, if, for instance, there is downward refraction
the source and receiver are either both in the duct or both out. Also we need to
remember that we are interested in the bottom scattering law rather than the surface one.
These latter problems are controlled and possibly solved by using a vertical array that
can separate ducted and completely reflected arrivals.
3.2
Reverberation and Propagation Angle-Dependence
If we use a vertical array as receiver in the same experiment we have access to vertical
arrival angle for the reverberation as well as for the propagation. This means we have
access to the integrands of Eq. (1) and (partially) Eq. (6). In more general terms we can
write the angle- and range-dependent propagation and reverberation responses as
dI P (θ , r ) = P(θ , r ) dθ
(12)
dI R (θ 1 , r ) = s (θ 1 ) P(θ 1 , r ) dθ 1 ∫ s(θ 2 ) P (θ 2 , r ) dθ 2 = s (θ 1 ) P(θ 1 , r ) dθ 1 F (r ) (13)
where s(θ) is the unknown scattering strength and P(θ,r) and F(r) are initially unknown
functions (F(r) is defined by the right hand side of Eq. (13)). The array measurements
with moving source (dIP) determine P completely. The single shot monostatic
reverberation measurement (dIR) determines s×P×F, but we now know P, and s depends
only on θ while F depends only on r. This means that s and F are separable without a
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C.H. HARRISON ET AL.
model! Taking each range in turn, the ratio of the responses always has the angledependence of the scattering strength s but with a range-dependent multiplier F.
dI R (θ , r )
= s (θ ) F (r )
dI P (θ , r )
(14)
If we take logs of both sides to obtain, say A(θ , r ) = B(θ ) + C ( r ) , then by taking the
average over all ranges we get an estimate of the function B(θ) , i.e. s*(θ). Similarly we
obtain an estimate of F*(r) by taking the average over all angles, but there is still an
unknown numerical factor which could be attributed to either term. Introducing a factor
γ such that s=s*γ and F=F*/γ we can use the definition of F implicit in Eq. (13) to solve
for γ .
γ2 =
F * (r )
∫
s * (θ ) P(θ , r ) dθ
(15)
Thus we obtain absolute, separated values for s(θ) and F(r). Although γ appears to be a
function of r it ought to be flat. Knowing F(r) we can also extract bottom reflection
parameters such as α and hence geoacoustic parameters by well-known methods.
4
Reverberation examples using the model SUPREMO
Returning to the original question of which phenomena need to be modelled we
illustrate some of them here using a new multistatic sonar performance model
SUPREMO [4]. The message is that there are many mechanisms that we can model, but
given experimental measurements it may still not be possible to identify them.
Figure 3. Monostatic and bistatic reverberation from a square seamount modelled with
SUPREMO. Ambiguous beam returns can be seen on the right of each frame.
For example, a black and white photograph of craters on the moon could be interpreted
as geographic changes in scattering strength rather than sloping facets!
UNCERTAINTY IN REVERBERATION MODELLING
4.1
367
Scattering: Bottom Topography Constructed from Multiple Sloping Facets
Figure 3 shows the reverberation response of a monostatic and a bistatic towed array
sonar to a square seamount. One can easily see a bright front side, dark back side, and
additional ambiguous image. SUPREMO has a “plug-in” propagation section, and this
demonstration used only a range-independent model since the path on the way out is
indeed range-independent.
Figure 4. Lambert’s law (lower line – red), Lommel-Seeliger (middle line – green), and angleindependent (upper line – blue) reverberation with SUPREMO.
Figure 5. Monostatic and bistatic reverberation from the Ragusa Ridge (depth contours overlaid).
Reverberation is underestimated because SUPREMO included a tilted-facet bottom but used a
“plug-in” range-independent propagation model.
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4.2
C.H. HARRISON ET AL.
Scattering Law Angle Dependence
Figure 4 shows the effect of three different scattering angle laws (Lambert, LommelSeeliger, and angle-independent) on an otherwise horizontal seabed.
4.3
Realistic Bathymetry
Figure 5 shows the effect of tilted-facet scatterers on reverberation from the Ragusa
Ridge south of Sicily. Although echoes follow the contours their intensities are
underestimated because SUPREMO has been run with a range-independent propagation
model. The analytical calculations above suggest that we can probably ignore the
additional effects of the increased bottom loss up-slope since there is a counteracting
concentration of flux in the shallower water. However, they also suggest that the raw
multiplier (Hs/Hr)2 would be an overestimate. This could be confirmed with a rangedependent propagation module at a future date.
5
Conclusions
There are a number of scattering and propagation effects that are mathematically
distinguishable, for instance, the scattering law angle dependence, local tilt of bottom
facets, scattering strength variation, ray arrival angle (as modified by water depth), and
propagation intensity. We have used an analytical approach to find first order
dependences, then a numerical approach to look at detailed mechanisms.
The implications for coping with environmental uncertainty when predicting
reverberation are that scattering is more important than propagation effects. Subjectively
ranking the effects we have scattering law, local bottom tilt, and absolute scattering
strength. Range-dependent propagation intensity effects are probably less important
although the accompanying (predictable) arrival angle change is important. An
experiment is proposed that could help to solve the scattering law problem by deducing
the angle-dependence and absolute value directly from long range VLA measurements.
The same ranking of effects also has implications for the design of models such as
SUPREMO. For instance, one could spend a lot of computational effort on rangedependent propagation modelling when a range-independent code with a simple
correction term for depth profile could conceivably do just as well. The pay-off would
be that trials planning and operational research studies using such models could then
deal with moving platforms and targets in range-dependent environments with
acceptable computation times.
References
1. Weston, D.E., Intensity-range relations in oceanographic acoustics, J. Sound and Vib. 18,
271–287 (1971).
2. Harrison, C.H., Reverberation and signal-excess with mode-stripping and Lambert’s law.
Rep. SR-356, SACLANT Undersea Research Centre (2002).
3. Weston, D.E., Propagation in water with uniform sound velocity but variable-depth lossy
bottom, J. Sound and Vib. 47, 473–483 (1976).
4. Harrison, C.H., Prior, M. and Baldacci, A., Multistatic reverberation and system modelling
using SUPREMO. In Proc. 6th European Conference on Underwater Acoustics, Gdansk,
Poland (2002).