USING A FACETED ROUGH SURFACE ENVIRONMENTAL
MODEL TO SIMULATE SHALLOW-WATER SAS IMAGERY
A.J. HUNTER, M.P. HAYES AND P.T. GOUGH
Acoustics Research Group, Dept. Electrical and Computer Engineering,
University of Canterbury, Private Bag 4800, Christchurch, New Zealand
E-mail: \a.hunter,m.hayes,p.goughi@elec.canterbury.ac.nz
Synthetic Aperture Sonar (SAS) is an extension of the conventional side-looking sonar
technique for higher resolution underwater imaging. In SAS, as with conventional sonar,
it is often difficult to obtain ground-truth data to compare with the reconstructed imagery. Thus, realistic simulation is invaluable for development of SAS algorithms. SAS
simulation models are complicated by the required coherent processing and the larger
beamwidths compared with conventional sonar of the same resolution. In particular,
the large beamwidths and high-resolution imagery require the acoustic response of large
insonified areas to be simulated down to a very small scale. In this paper, we present
a simulation model capable of producing realistic SAS imagery of three-dimensional
shallow-water environments. The simulation is conducted in the temporal frequency
domain and is based on a faceted representation of the sea-floor and targets, with the
sea-floor facets having a roughness component generated using a fractional Brownian motion processes. The acoustic response from the facets is determined using the
Kirchhoff method, which is extended for fast simulation using facets with small-scale
roughness. Occlusions and multiple scattering are resolved using the ray-tracing technique of geometric optics. Results so far indicate we have constructed an algorithm
capable of modelling 3-D targets on a rough sea-floor that appear similar to actual data
recorded here and elsewhere.
1 Introduction
Synthetic Aperture Sonar (SAS) differs from conventional side-looking sonar by using
both the magnitude and phase of the acoustic response as well as the forward motion
of the sonar platform in order to artificially synthesise a larger aperture. This coherent
processing allows apertures many times the size of the physical aperture to be generated. Consequently, SAS can achieve improved range-independent azimuth resolution
over conventional sonar >dH.
Successful reconstruction of SAS imagery relies on the assumption that the sonar
platform moves at a known speed along a linear path. Obviously in the case of a freetowed platform, this is not always the case. High-precision navigation hardware and
autofocus techniques are used to determine and correct any deviation from the ideal path.
It is useful in developing these techniques to have ground-truth data to compare with
the reconstructed imagery. Similarly the recent trend in bathymetric SAS research also
benefits from ground-truth data. However, in underwater acoustics this data is often
difficult to obtain. Thus, a realistic simulation model is required to test and calibrate
481
N.G. Pace and F.B. Jensen (eds.), Impact of Littoral Environmental Variability on Acoustic Predictions and
Sonar Performance, 481-488.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
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Along track
direction
Towfish velocity
v
y
z
x
Range
(across-track) direction
Figure 1. Typical shallow-water SAS imaging geometry.
reconstruction and autofocus algorithms.
SAS simulation models are complicated by the required coherent processing and the
larger beamwidths compared with conventional sonar of the same resolution. In particular
the large beamwidths and high resolution imagery require the acoustic response of large
insonified areas to be simulated down to a very small scale. Rather than to be comprehensive, the goal of the simulation model presented here is to generate reasonable echo
data for a broadband widebeam sonar that incorporates aspect dependent scattering, shadowing, sea-floor reverberation, and towfish motion. Aspects such as multiple scattering
have been neglected in favour of computational speed.
The approach we use is to model the sea-floor and targets as a collection of facets,
small enough so that we can use the Fraunhofer approximation of Fourier optics [2]. However, unlike other SAS simulations [3], we also model coherent sea-floor reverberation by
assigning a correlated rough surface to each facet.
2
Imaging geometry
The typical shallow-water SAS imaging geometry is illustrated in Fig. 1. This is a sidelooking geometry whereby the free-towed platform travels along an ideally linear path
parallel with the strip-map imaged region. When imaging in shallow water environments,
the platform sits roughly mid-water to give the best sea-floor coverage. In order to
determine the orientation of the transducer beam patterns with respect to the imaged
targets, the orientation and location of each transducer must be determined as the sonar
moves along-track. Due to the free-towed platform, this is complicated by the additional
SAS SIMULATION MODEL
483
motions of pitch, roll, yaw, heave, surge, and sway.
Assuming a single projector and a single hydrophone transducer located at t xp =
t
( xp , t yp , t zp ) and t xh = (t xh , t yh , t zh ), respectively, with respect to the towfish coordinate frame, the transducer positions in the world coordinate frame are given by
xp (t) = xt (t) + w Rt (t) t xp ,
xh (t) = xt (t) + w Rt (t) t xh ,
(1)
(2)
where w Rt (t) is the rotation matrix describing the orientation of the towfish with respect
to the world coordinate frame and xt (t) is the displacement of the towfish in world
coordinates.
Now consider a scatterer centred at a position xs in world coordinates (typically this
would be on the sea-floor at (xs , ys , −D) where D is the water depth). The ranges from
the target to the projector and hydrophone are given by
rps (t) = |rps (t)| = |xp (t) − xs | ,
rhs (t) = |rhs (t)| = |xh (t) − xs | ,
respectively. From these the direction cosines are found using
"T
!
cos βps (t) = w Rt (t)t Rp · r̂ps (t),
!
"T
cos βhs (t) = w Rt (t)t Rh · r̂hs (t),
(3)
(4)
(5)
(6)
where the matrices t Rp and t Rh describe the respective rotations of the projector and
hydrophone axes in the coordinate frame of the towfish. The direction cosines describe the
orientation of the beam patterns with respect to the imaged targets and are used together
with the range expressions to evaluate target responses in the acoustic model.
3
Acoustic scattering model
The acoustic model employs the Kirchhoff scattering method and resolves occlusion by
ray-tracing [4]. The Kirchhoff method is selected due to its computational simplicity.
However, it is still too demanding for SAS simulations of complicated environments.
Thus, the method is extended with an efficient approximation for scattering from rough
facets.
3.1 The Kirchhoff Method
The Kirchhoff method (also known as the tangent-plane method or the method of physical
optics) gives a good first-order approximation to the field scattered by a rough interface,
provided the surface is comparatively smooth compared to the wavelength [5]. At high
frequencies this method is easily extended to include the effects of shadowing and multiple
scattering using the ray-tracing technique of geometric optics.
The penultimate step in the Kirchhoff method is the assumption that the rough surface
can be modelled locally by a planar interface that is independent of the rest of the surface.
Thus, over a local region the total field Ψ is given by
Ψ = Ψi + Ψs = (1 + R) Ψi ,
(7)
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A.J. HUNTER ET AL.
where Ψi and Ψs are the incident and scattered fields, and R is the reflection coefficient
that is a function of the incident field angle (we ignore mode conversion and subsea
penetration of sound). The acoustic response of a planar surface element Σ can then be
determined by substitution of Eq. (7) into the Helmholtz-Kirchhoff equation [2]
#
(1 − R) G∇Ψi · n̂ − (1 + R) Ψi ∇G · n̂ dS
(8)
Ψs =
Σ
where G is the Green’s function, Σ is the domain of the planar surface element, and
n̂ is the surface normal on Σ. If we assume that the medium above the sea-floor is
homogeneous we can use the free space Green’s function for G. We usually operate in
shallow water harbour conditions where the assumption of a homogeneous medium is
sufficient in winter when the temperature gradient is small. Otherwise it is necessary to
use high frequency approximations for G given an estimate of the sound speed profile [4].
3.2 Acoustic Response of a Faceted Target
The objects and surfaces in a typical underwater environment (sea-floor and targets) can
be conveniently represented as a collection of facets. The acoustic response of these
facets can then be determined using the Kirchhoff method. Provided the facets are small
enough, the reflection coefficient R can be considered a constant for each facet. Then,
assuming a homogeneous medium, the field at a point several wavelengths away from
the facet can be approximated by
$
%
Ψs (xh |xp , f ) ≈ j2π(f/c) (1 − R) cos βps · n̂ − (1 + R) cos β hs · n̂
#
× G(x − xh , f )G(xp − x, f ) dS
(9)
Σ
where xp and xh are the positions of the respective projector and hydrophone transducers.
Now provided each facet is in the far-field of the sonar transducers, we can employ the
Fraunhofer approximation to simplify the integral of Eq. (9) to a 2-D Fourier transform.
The resultant echo signal from each visible facet thus has the form
Es (xh |xp , f ) = j2π(f/c)S (f) G(xs − xh , f )G(xp − xs , f)
!
"
×Bp (f /c) cos β ps Bh ((f /c) cos β hs )
!
!
""
×Bs (f /c) cos β ps + cos β hs Ks (βps , βhs ),
(10)
where S(f ) is the temporal spectrum of the transmitted signal, xs is the centre of the facet,
Bp , Bh , and Bs are the beampatterns of the projector, hydrophone and facet respectively,
and Ks (βps , βhs ) describes the scattering amplitude from the facet given by the factor
in square brackets in Eq. (9).
The beampatterns are obtained by taking the Fourier transform of the aperture functions. In the cases of typical rectangular transducers the beampattern is the familiar sinc
function. However, it is not possible to tesselate an arbitrary object into rectangular facets.
Thus, triangular facets are employed instead.
The total scattered field is determined by summing the responses from each of the visible facets at each receiver position, where the visibility of each facet is determined using
ray-tracing. This models the effects of occlusion but is only valid for high frequencies.
SAS SIMULATION MODEL
485
3.3 Scattering from Rough Facets
High resolution SAS imagery is obtained using large beamwidths and to model seafloor
reverberation we must consider large insonified areas down to a very small scale. Therefore, representing surface roughness for a typical shallow-water environment using the
simple Kirchhoff method becomes computationally impractical due to the extremely large
number of facets required. Instead we use larger facets and assume that the roughness on
each facet is small so that no points on the facet are occluded by each other and that multiple scattering is negligible. We can then determine a frequency dependent beampattern
for each facet. This can be rapidly computed for each facet using FFTs and interpolated
for each field point.
4
Sea-floor model
A realistic model of the sea-floor and its roughness is important for modelling the seafloor reverberation. We model the sea-floor as a two-dimensional fractional Brownian
motion process since this has been shown to be a good model of natural surfaces over
a wide range of scales [6]. The correlated surface roughness was generated using the
midpoint displacement algorithm [7], adapted for an arbitrary structure function (incremental variance function). This can generate fractional Brownian motion processes more
accurately than frequency domain methods often employed [8]. It also is better suited for
generating very large rough surfaces.
We assume an isotropic sea-floor roughness although it is straightforward to extend the
midpoint displacement algorithm to model anisotropic roughness. It is also straightforward
to adapt the algorithm to use the von Kármán correlation function proposed by Goff and
Jordan [9] to model large scale sea-floor roughness.
5
Implementation
The described simulation model has been implemented in the Python programming language with C implementations for the computationally intensive routines (Fourier transforms, interpolations, etc.). Python is a freely available modern scripting language suited
for interactive development of algorithms [10].
An advantage of the facet model is that it is ideally suited for parallel computing—the
response from each facet can be computed independently and then linearly combined. We
run our simulations on a cluster of PCs running the Mosix kernel, a variation of the Linux
kernel that can dynamically migrate processes across the cluster to distribute the load.
6
Results
To validate the model, a number of seafloor scenes were modelled with a variety of targets.
For example, Fig. 2 shows a rendered image of a rough seafloor with a superimposed
cylindrical target and four cubical targets. The cylinder and cubes were modelled using a
collection of triangular facets (either smooth or rough). The sea-floor also was modelled
using triangular facets but with rough surfaces generated from a fractional Brownian
motion process with a Hurst parameter of 0.35 using the midpoint displacement algorithm.
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A.J. HUNTER ET AL.
−8
−8.5
(m)
−9
−9.5
−10
−10.5
−11
3
2
1
0
26
25
−1
24
23
−2
(m)
22
−3
21
20
(m)
Figure 2. Example of a typical sea-floor with fractional Brownian motion statistics with a Hurst
parameter of 0.35 with a smooth cylindrical target and four smooth cubical targets.
The sonar we simulated has the parameters of the KiwiSAS-III sonar [11]. This
is a free-towed, short range, synthetic aperture sonar that operates in two frequency
bands 20–40 kHz and 90–110 kHz giving a range resolution in each band of nominally
35 mm. The sonar has a single projector with three vertically displaced hydrophones (for
interferometric bathymetry [12]), each of the order of 300 m in length giving a theoretical
along-track resolution of 150 mm. Fig. 3 shows the pulse compressed echo data simulated
for this sonar at 100 kHz using the sea-floor/target model shown in Fig. 2. The rough seafloor model can be seen to produce speckle with a size commensurate with the resolution
of the sonar. The scattering from the targets has generated a few highlights and some
shadowing of the sea-floor. After synthetic aperture reconstruction, the shadows become
more apparent as shown in Fig. 4.
7
Conclusion
A simulation model has been proposed based on the Kirchhoff approximation. The simple
Kirchhoff approximation has been modified such that surfaces with roughness may be
implemented rapidly. The model provides aspect dependent scattering, occlusions, and
coherent speckle from rough surfaces. However, there are a number of limitations with
the model. The small-slope approximation is violated when the surface roughness has
correlation lengths shorter than the wavelength [5, 6]. Multiple scattering, low frequency
diffraction effects around targets, and sea-surface scattering are neglected. Future enhancements are to correct some of these problems; for example, we envisage modelling
multipath from strong scatterers using recursive ray-tracing and to incorporate scattering
from the sea-surface [13].
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3
2
along−track (m)
1
0
−1
−2
−3
21
22
23
24
range (m)
25
26
27
Figure 3. Simulated pulse compressed echo data for scene shown in Fig. 2.
3
2
along−track (m)
1
0
−1
−2
−3
21
22
23
24
range (m)
25
26
27
Figure 4. Synthetic aperture reconstructed image for scene shown in Fig. 2.
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Acknowledgements
Alan Hunter thanks the University of Canterbury for his Doctoral Scholarship.
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