TinhNguyen Dinh1
-
(Faculty of Radio-Electronic Engineering, Le Quy Don Technical University, Hanoi, Vietnam
tinhnd_k31@lqdtu.edu.vn)
Copyright © The Institute of Electronics and Information Engineers(IEIE)
Keywords
Synthetic aperture sonar (SAS), Multi-receiver SAS, Along-track resolution, Amplitude distribution, SAS image
1. Introduction
Synthetic aperture sonar (SAS) can provide a high along-track (azimuth) resolution
by combining echo signals from consecutive pings when the SAS platform moves along
a straight path to generate a synthetic aperture with a large size [1]. Thanks to this ability, SAS has been used in various applications, such as imaging
the seabed, detection of small objects, and the search for wrecks [2,3]. To improve the mapping rate, multi-receiver SAS has been utilized in many SAS applications
[2-4].
Peak side-lobe ratio (PSLR) and half-power beam width (HPBW) in the beam pattern or
in the point spread function (PSF) are basic parameters for evaluating SAS imaging
quality [5]. To reduce PSLR, some amplitude distributions, such as a Hann window or a tapered
cosine window (Tukey), can be used for SAS arrays [5,6]. However, the effectiveness of these window functions for decreasing both PSLR and
HPBW has not been evaluated in detail. Therefore, such windows may not be optimal
for improving both PSLR and azimuth resolution.
By analyzing the mathematical expression of the beam pattern, and by investigating
the simulation results, this paper proposes a solution to determine the amplitude
distribution for enhancing along-track resolution (i.e., HPBW), reducing PSLR to less
than the required value, and ensuring an integrated side-lobe ratio (ISLR) in imaging
reconstruction. A comparison of the beam patterns generated from the selected windows
can determine the optimal window that reduces both HPBW and PSLR. The improvements
from the proposed solution are demonstrated through simulation of the beam patterns
with the sound velocity profile (SVP) in the seas of Vietnam.
2. Signal Model for Multi-receiver SAS
To study SAS imaging reconstruction algorithms, the received signal model from the
ideal point target is often used [5,7]. Fig. 1 depicts a three-dimensional (3D) geometric model of multi-receiver SAS with N uniformly
spaced receivers at distance d and the ideal point target at P(r,u,h). The axes Ox
and Oz illustrate the range (ground range) dimension and the depth dimension, respectively.
Considering the variation of sound velocity in accordance with depth, the sound trajectory
can be separated into line segments with constant velocity. From the total propagation
time according to meander point A and point B (A and B being the starting point and
the ending point of the propagation trajectory, respectively), the equivalent sound
velocity (ESV) is determined as follows [8]:
where $\tau _{\Sigma }$ is the sum of the propagation time in the short straights
corresponding to fixed velocities from A to B, and AB is the length of the line segment
that joins points A and B.
From (1), ESV $c_{eq\_ T}$ and $c_{eq\_ Ri}$ during emission and reception of sound waves
from the transmitter to the target at P(r,u,h) and from the target to the ith receiver
are respectively calculated according to vertical inclinations $\varphi _{T}$ and
$\varphi _{Ri}$. To focus on determining the amplitude distribution for reduction
of both PSLR and HPBW, the SAS is considered constant motion with velocity v.
At time t = 0, the transmitter is at O(0,0,0). When the transmitter moves to point
T(vt,0,0), the propagation times from the transmitter to the target and from the target
to the receiver are calculated as follows [8]:
where subscript 1 indicates sound propagation from the transmitter to the target,
and subscript 2i is sound propagation from the target to the ith receiver. In (4), $\alpha _{1}$is the angle between the SAS motion direction and the sound propagation
direction from the transmitter to the target, calculated as follows [8]:
In (5),$\Delta _{i}$ is the discriminant of the quadratic equation with variable ${\tau}$$_{\mathrm{2i}}$
generated according to [8] as follows:
The movement of the SAS platform generates the Doppler effect, so the frequencies
of the received signals at P and R$_{\mathrm{i}}$ differ from transmitted frequency
f$_{0}$ [8-10].
Here, $\eta _{1}$and $\eta _{2i}$ are time-stretching factors at point P and the ith
receiver, respectively. In (8), $\alpha _{2i}$is the angle between SAS motion direction and the sound propagation
direction from the target to the ith receiver when receiving the echo signal and is
calculated as follows [8]:
With multi-receiver SAS using a gated continuous wave, the received signal at the
ith receiver after suppressing the scattering from the sea’s surface is determined
as
where $w\left(\tau \right)$ is the windowing function of the transmitted signal, $\tau
_{i\_ mo}$ is the modified signal propagation time expressed as follows [8]:
When multi-receiver SAS uses linear frequency modulation (LFM) pulses, the echo signal
is
where $\gamma $ is the chirp rate in hertz per second.
Fig. 1. A 3D model for multi-receiver SAS.
3. Determining an Amplitude Distribution Improving Along-track Resolution and Reducing
PSLR for Multi-receiver SAS
With the combination of received signals from successive pings, the beam pattern of
the synthetic array when steering to point P$_{0}$(r$_{0}$,u$_{0}$,h$_{0}$) is defined
as
where $s_{i}\left(t\right)$ is the signal at the ith receiver at time t (y = vt),
and $\psi _{i}\left(t,r_{0},u_{0},h_{0}\right)$ is the phase distribution provided
for echo signals at the ith receiver and the mth ping so that these signals are in
phase when steering at point P$_{0}$(r$_{0}$,u$_{0}$,h$_{0}$). The phase distribution
is determined according to the average ESV as follows [8]:
where $\eta _{10\_ pro}$, $\tau _{10\_ pro}$, $\eta _{2i0\_ pro}$, and $\tau _{2i0\_
pro}$are calculated in ways similar to expressions (1) to (4) with the ESV chosen based on the average value. Subscripts 10\_pro and 2i0\_pro illustrate
the parameters for processing images during emission and reception of signals, respectively.
Expression (12) represents the beam pattern of the synthetic array with uniform distribution,
which provides a high PSLR when reconstructing the SAS image in the azimuth direction.
To decrease the PSLR, we can use windows for the synthetic array. Based on separation
of the synthetic array into a physical array in a pulse repetition interval and a
virtual array in the pings, the beam pattern of the synthetic array is represented
as
where a$_{i}$ and a$_{m}$ are the amplitude distributions according to the physical
array and the virtual array, respectively.
From (14), we can choose amplitude distributions a$_{m}$ and a$_{i}$, and analyze
the beam pattern to determine the amplitude distributions for reducing HPBW and satisfying
PSLR with a value less than the required value. With a general formula to change the
PSLR, it is necessary to investigate both distributions and select two suitable amplitude
distributions. This requires a large number of calculations and a large amount of
analysis.
Since the size of the synthetic array is mainly determined by the number of pulse
repetition intervals, an amplitude distribution based on a virtual array plays a major
role in generation of the PSLR and HPBW. Therefore, this study focuses on investigating
the amplitude distribution in pings from a$_{m}$ and analyzing the synthetic beam
pattern. Based on the analysis and comparison of beam patterns from simulation results
in MATLAB, this study can determine the optimal amplitude distribution that decreases
HPBW and provides a PSLR value that is less than the required value.
4. Simulation Results
4.1 Simulation Results of the Beam Pattern in the Azimuth Plane
To highlight the effectiveness of separation into two windows in determination of
the amplitude distribution for reducing HPBW and PSLR to less than the required value,
this study considers multi-receiver SAS configured as seen in Table 1. The parameters in Table 1 were chosen to avoid grating lobes of the synthetic beam pattern.
Assume it is necessary to observe the two targets at positions (20 m, 8 m, 44.6 m)
and (20 m, 8 m, 42 m) in the Oxyz coordinate system in Fig. 1. This study uses two SVPs obtained from geographic coordinates (17$^{\circ}$03'07''N,
107$^{\circ}$27'14''E) and (17$^{\circ}$03'09''N, 107$^{\circ}$27'17''E) in the seas
of Vietnam. Fig. 2 depicts the SVPs derived via SWIFT SVP on 12 April 2021 and 15 April 2021, respectively.
The positions of the observation points are chosen from the beam width of each element
in the physical array.
To deal with the variation of sound velocity according to depth with various real
SVPS, we can use the average ESV to reconstruct SAS images [8,11]. The average ESV (sound velocity in processing the images) is derived from the minimum
ESV and the maximum ESV [8], which correspond to the initial inclinations at the zero value and the maximum value,
respectively, when reconstructing SAS images [11].
With the minimum inclination angle of 0$^{\circ}$ and the maximum inclination angle
of 80$^{\circ}$ relative to the vertical, the average equivalent sound velocities
for beamforming corresponding to the two SVPs are 1527.01 m/s and 1529.17 m/s.
To illustrate the effect of the amplitude distribution according to the physical array
on the synthetic beam pattern, this section selects amplitude distributions for a$_{i}$
by using a uniform window, a Chebyshev window with side-lobe attenuation (SLA) of
-25 dB, a Hann window, and a Gaussian window with standard deviation ${\Sigma}$ =
2.5. The amplitude distribution for virtual array a$_{m}$ is chosen as the uniform
window. With phase distribution (13) and the selected amplitude distributions, when
steering the main beam to positions (20 m, 8 m, 44.6 m) and (20 m, 8 m, 42~m), the
synthetic beam patterns in the azimuth dimension using MATLAB are illustrated in Fig. 3.
Fig. 3 shows that with the different amplitude distributions of a$_{i}$, PSLR and HPBW in
the synthetic beam patterns are almost unchanged. In other words, the amplitude distributions
of the physical array do not significantly change the synthetic beam patterns of multi-receiver
SAS. Therefore, this study chooses the uniform distribution for a$_{i}$ to simplify
the calculations.
From the choice of a$_{i}$ as the uniform distribution, the amplitude distributions
based on the virtual array were selected and investigated to determine the optimal
distribution for reducing HPBW and PSLR to less than the required value. To decrease
PSLR to less than -20 dB, selected amplitude distributions a$_{m}$ for investigation
included the Chebyshev window with SLA = -21.2 dB, the Taylor window with nearly-constant-level
side-lobes adjacent to main-lobe $\overline{n}$= 5, and a maximum side-lobe level
ASLL = -20.8 dB [12], the Gaussian window with ${\Sigma}$~=~1.41, and the Kaiser window where ${\beta}$
= 2.45. With the distributions, the synthetic beam patterns when steering the main
beam to positions (20 m, 8 m, 44.6 m) and (20 m, 8~m, 42 m) in MATLAB are shown in
Fig. 4. From this figure, Table 2 shows the determined parameters of the synthetic beam patterns, including HPBW (along-track
resolution), PSLR, and ISLR, along with the amplitude distributions.
Table 2 shows that with the same PSLR values, the Chebyshev window provides the narrowest
HPBW compared with the other windows. These values of HPBW achieved from the Chebyshev
window are 1.72 cm and 1.65 cm when steering to (20 m, 8 m, 44.6 m) and (20 m, 8~m,
42 m), respectively. The selected windows also generate ISLR values less than -17
dB, which satisfy the requirement for image processing [13]. Thanks to separation of the amplitude distributions of the synthetic array into
distributions according to the physical array and virtual array, it is possible to
determine the amplitude distribution, which reduces the PSLR to less than the required
value, and generates the narrowest HPBW. Determination of the optimal amplitude distribution
is only based on investigation of distribution a$_{m}$ and comparison of simulation
results generated from the distributions of virtual arrays.
To demonstrate the effectiveness of the proposed solution more clearly, two synthetic
beam patterns in the azimuth plane derived by the amplitude window in [5] and by the optimal amplitude window from the proposed solution were compared. The
amplitude distribution from [5] is a Hann window, whereas to obtain the PSLR as the Hann window, the amplitude distribution
from the proposed solution is a Chebyshev window with SLA~=~\hbox{-}33~dB. The two
beam patterns generated from these distributions are depicted in Fig. 5.
Fig. 5 shows that with the same PSLR of approximately - 30 dB (the distribution from the
proposed solution produced PSLR = -30.03 dB, whereas the distribution from [5] produced PSLR = -30.02 dB), the distribution from the proposed solution generates
an HPBW of 2.07 cm, while the distribution from [5] generates an HPBW of 2.61 cm. When considering ISLR in the beam pattern, the distributions
from the proposed solution and the Hann distribution provided ISLR values of -41.72
dB and -46.47 dB, respectively. These values are much smaller than the required value
of -17 dB, so the difference does not affect SAS image quality. Therefore, despite
virtually the same PSLR, the amplitude distribution from the proposed solution improves
along-track resolution compared with the conventional solution [5].
Fig. 2. Sound velocity profiles at (a) (17°03'07''N, 107°27'14''E); (b) (17°03'09''N, 107°27'17''E).
Fig. 3. Synthetic beam patterns with amplitude distributions ai when steering the main beam to (a) (20 m, 8 m, 44.6 m); (b) (20 m, 8 m, 42 m).
Fig. 4. Synthetic beam patterns with amplitude distributions am when steering the main beam to positions: (a) (20 m, 8 m, 44.6 m); (b) (20 m, 8 m, 42 m).
Fig. 5. Synthetic beam patterns from the amplitude distributions.
Table 1. The parameters of the multi-receiver SAS.
|
Parameter
|
Value
|
Unit
|
|
Carrier frequency (f0)
|
100
|
kHz
|
|
Platform velocity (v)
|
1.5
|
m/s
|
|
Distance between transmitter and first receiver (d1)
|
0.03
|
m
|
|
Distance between two adjacent receivers (d)
|
0.02
|
m
|
|
Number of receivers (N)
|
64
|
element
|
|
Pulse repetition interval (TR)
|
0.2
|
s
|
|
Number of pings (M)
|
68
|
|
Table 2. The Parameters of Synthetic Beam Patterns with the Amplitude Distributions.
|
|
Steering to (20 m, 8 m, 44.6 m)
|
Steering to (20 m, 8 m, 42 m)
|
|
Amplitude distribution
|
HPBW (cm)
|
PSLR (dB)
|
ISLR (dB)
|
HPBW (cm)
|
PSLR (dB)
|
ISLR (dB)
|
|
Chebyshev (SLA = -21.2 dB)
|
1.72
|
-20.18
|
-23.93
|
1,65
|
-20.11
|
-23.68
|
|
Taylor ($\bar{n}$= 5, ASLL = -20.8 dB)
|
1.81
|
-20.02
|
-29.56
|
1,73
|
-20.07
|
-29.63
|
|
Gaussian (α = 1.41)
|
1.90
|
-20.01
|
-34.76
|
1,81
|
-20.02
|
-34.94
|
|
Kaiser (β = 2.45)
|
1.93
|
-20.18
|
-35.75
|
1,85
|
-20.06
|
-36.04
|
4.2 Analysis of PSF
To visually observe the SAS image generated from the proposed solution, the following
section considers the image of the point target at (20 m, 8 m, 44.6 m) corresponding
to the SVP obtained at geographic coordinates (17$^{\circ}$03'07''N, 107$^{\circ}$27'14''E).
Assume that the SAS has the configuration in Table 1 and uses an LFM pulse with a width of 10 ms and a bandwidth of 20 kHz. A Hann window
is also applied for matched filtering to suppress side-lobes in the range direction,
as detailed in [5]. In the azimuth direction, the Chebyshev window (SLA~=~\hbox{-}33 dB) is chosen for
the virtual array to reduce PSLR as described in Section 4.1. With the two amplitude
windows in the range and azimuth directions, the PSF (or image of the point target)
derived by the algorithm in the frequency domain [8] is illustrated in Fig. 6.
Fig. 6 shows there is good convergence in the SAS image at the position of the point target,
and there are no side-lobes larger than -30 dB in both the range and azimuth directions
(the values of PSLR in both directions are less than -30 dB). In the ground-range
dimension, the image trail is wider than in the azimuth dimension because the ground-range
is smaller than the depth, and the change in ground-range has a small impact on the
slant-range. When the ground-range is greater than the depth, the resolution in the
range dimension is approximately equal to that in the slant-range dimension (depending
on the bandwidth of the transmitted signal).
Ambient noise generated by tides, waves, marine life, surface disturbance, seismic
disturbance, ocean turbulence, thermal noise, and shipping and wind noise can reduce
the contrast in SAS images [6,17]. At 100 kHz, ambient noise effecting SAS is reduced considerably [6,17,18]. Therefore, with the SAS configuration in this study, ambient noise is ignored to
determine the amplitude distribution for improving along-track resolution of multi-receiver
SAS.
When the SAS platform has translational motion (heave, surge, and sway) and angular
motion (yaw, pitch, and roll), it is necessary to compensate for the motion in order
to obtain high-resolution SAS images [14]. The feasible solution is to utilize the sonar itself as a navigational sensor (micro-navigation)
combined with the inertial navigation system [15,16]. With the configuration of the multi-receiver SAS in this paper, the solution using
INS has not yet been considered in order to concentrate on determining an amplitude
distribution enhancing the along-track resolution. This issue will be considered in
more detail in future work.
Fig. 6. Image of point target at (20 m, 8 m, 44.6 m).
5. Conclusion
This paper proposed a solution to determine an amplitude distribution improving the
along-track resolution and reducing PSLR to a value smaller than the required value
for multi-receiver SAS. The proposed solution also decreases the number of investigations
by only considering the amplitude distribution according to pings in a virtual array.
The simulation results demonstrated the effectiveness of the proposed solution with
real data on sound velocities.
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Author
Nguyen Dinh Tinh received his B.E. in Electronics-Telecommunications and an M.E.
in Radar Navigation Engineering from Le Quy Don Technical University, Vietnam, in
2008 and 2012, respectively. He has been a lecturer at Le Quy Don Technical University
since 2009. His research interests include antennas, signal processing, synthetic
aperture sonar, and sonar engineering.