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\begin{document}

\title{Tensor-based matched-field processing applied to the SWellEx-96 data}



\author[1]{Fangwei ZHU}%
\author[1]{ZHENG Guangying}%
\author[1]{Xiaowei GUO}%
\author[1]{Fangyong WANG}%
\author[1]{Shuanping Du}%
\author[1]{Linlang BAI}%
\affil[1]{Hangzhou Applied Acoustics Research Institute}%


\vspace{-1em}



  
  \date{May 23, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
This study proposed a matched field source localization method based on
tensor decomposition. By considering the advantages of tensors in
multidimensional data processing, a three-dimensional tensor signal
model of space-time-frequency is constructed, and the signal subspace is
estimated using high-order singular value decomposition (HOSVD). The
source position is estimated by matching the measured data tensor signal
subspace with the replica field tensor signal subspace. The S5 event
data of SWellEx-96 is processed by the proposed tensor-based
matched-field processing (TMFP). The comparison with the results of
conventional matched field processing (MFP) shows that TMFP has a better
suppression effect on ambient noise under low SNR and better source
localization performance.%
\end{abstract}%



\sloppy


Tensor-based matched-field processing applied to the SWellEx-96 data

Fangwei Zhu\textsuperscript{1,2}, Guangying
Zheng\textsuperscript{1,2,3}, Xiaowei Guo\textsuperscript{1,2,3},
Fangyong Wang\textsuperscript{1,2,3}, Shuanping
Du\textsuperscript{1,2,3}, and Linlang Bai\textsuperscript{1,2,3}

\textsuperscript{1} \emph{Science and Technology on Sonar Laboratory,
Hangzhou, 310023, China}

\textsuperscript{2} \emph{Hangzhou Applied Acoustics Research Institute,
Hangzhou, 310023, China}

\textsuperscript{3} \emph{Hanjiang National Laboratory, Wuhan, 430000,
China}

Email: 276454158@qq.com.

This study proposed a matched field source localization method based on
tensor decomposition. By considering the advantages of tensors in
multidimensional data processing, a three-dimensional tensor signal
model of space-time-frequency is constructed, and the signal subspace is
estimated using high-order singular value decomposition (HOSVD). The
source position is estimated by matching the measured data tensor signal
subspace with the replica field tensor signal subspace. The S5 event
data of SWellEx-96 is processed by the proposed tensor-based
matched-field processing (TMFP). The comparison with the results of
conventional matched field processing (MFP) shows that TMFP has a better
suppression effect on ambient noise under low SNR and better source
localization performance.

\emph{Introduction} : Acoustic source localization in an ocean waveguide
is a subject of great interest. Localizing an underwater source target
can serve several applications, from marine biology to anti-submarine
warfare {[}1{]}. Matched field processing (MFP) {[}2-3{]} is the
earliest and the most famous method for localizing the underwater source
target. MFP is based on matching the measured acoustic data with the
dictionary of the replica to estimate the source position. Given that
MFP uses the prior information of the acoustic field, MFP performs
better than the traditional geometry locating methods. MFP is sensitive
to the mismatch of sound speed profile and geoacoustic parameters
{[}4-6{]}.

Recently, tensor signal processing has been a subject of great interest.
Tensor algebra, also known as multilinear algebra, is a natural
extension of classical linear algebra to high dimensions. It is a
mathematical theory that characterizes the linear relationship between
multiple variables and multidimensional data. It has been deeply studied
and widely used in many fields, such as image processing, chemistry, and
artificial intelligence {[}7-10{]}. Tensor is a way of data
representation and provides it with rich theoretical connotation. The
most crucial connotation is to develop effective tensor decomposition
methods under multilinear theory, such as Tucker decomposition and
higher-order singular value decomposition (HOSVD) {[}9{]}. As the name
suggests, this decomposition is the expansion of singular value
decomposition (SVD) under multilinear algebra theory. Compared with SVD,
HOSVD can suppress the noise in multidimensional data samples, reducing
the subspace estimation bias. Given this advantage, the HOSVD of tensor
has been widely used in array signal processing in the last decade
{[}11-13{]}, especially in applying polarization sensor array and
acoustic vector sensor array.

In a broad sense, matched field processing (MFP) can also be considered
a beamformer. Thus, this study uses the beamformer based on tensor
decomposition for reference, and it uses the HOSVD of tensor for matched
field processing to improve the source localization performance.

\emph{Construction of the tensor signal model:} A vertical line array
with elements is used to receive the broadband sound field; the element
dimension, frequency dimension, and snapshot dimension can be expanded
into a third-order tensor (a snapshot is a sampling of all the elements
in the time domain). The third-order tensor received data can be
expressed as the sum of the three-mode product of the array Green's
function tensor , the multi-snapshot source model , and the ambient
noise tensor, as shown in Fig.1.

\textbf{Fig. 1} \emph{The diagram of the tensor signal model.}

, (1)

where denotes the number of array elements, denotes the number of
frequency points, denotes the number of sources, and denotes the number
of snapshots. The elements in the array Green's function tensor denote
the sound field Green's function from the n-th element to the k-th
source at the m-th frequency point.

\emph{Derivation for tensor-based matched field processing (TMFP)} : The
multidimensional space-time-frequency signals received by the array are
reconstructed into tensor signals. HOSVD decomposes the tensor signals
to solve the tensor signal subspace, and the position of the sound
source is estimated by utilizing the matching estimator. Since the
matrix's singular value decomposition (SVD) expanded in each tensor
dimension can further suppress the noise to obtain a more accurate
signal subspace, the source localization accuracy is improved.

Applying HOSVD {[}9{]} to the third-order tensor, as follows:

, (2)

Where is the kernel tensor of tensor, and are the tensor products of
three dimensions, respectively.

Then, SVD is applied to the 3-mode expansion matrix of the tensor:

, (3)

, (4)

, (5)

where is the left singular matrix of the 1-mode expansion of the tensor,
is the singular value matrix of the 1-mode expansion of the tensor, is
the right singular matrix of the 1-mode expansion of the tensor, is the
left singular matrix of the 2-mode expansion of the tensor, is the
singular value matrix of the 2-mode expansion of the tensor, is the
right singular matrix of the 2-mode expansion of the tensor, is the left
singular matrix of the 3-mode expansion of the tensor, is the singular
value matrix of the 3-mode expansion of the tensor, and is the right
singular matrix of the 3-mode expansion of the tensor.

By truncating the left singular matrix of the n-mode expansion of the
tensor, the signal correlation matrix is composed of the column vectors
of the left singular matrices corresponding to the top M largest
singular values, that is, , , and . The noise correlation matrix is
composed of the column vectors of the left singular matrices
corresponding to the remaining smaller singular values, that is, , , and
.

Considering that the kernel tensor can be denoted as follows:

. (6)

The truncated kernel tensor can be expressed as follows:

, (7)

where is the kernel tensor of the truncated signal.

The third-order tensor output data can be approximately expressed as
follows:

. (8)

Thus, the tensor signal subspace of the third-order tensor output data
can be obtained, as follows:

. (9)

The ambiguity surface for source localization can be defined based on
higher-order singular value decomposition and inner tensor product. For
the hypothesized source range and depth , the ambiguity surface for
source localization is defined as follows:

, (10)

where represents the inner product of the tensor in the third dimension,
represents the normalized Green's function matrix, and represents the
inner product of tensor and Green's function matrix in the first and
second dimensions.

To compare the performance of TMFP and MFP, the normalized broadband MFP
I {[}15{]} (based on summation) output of the Bartlett processor and
broadband MFP II {[}16{]} resulting from maximum likehood estimation,
given by,

, (11)

, (12)

where is the Green's function vector of the receiving array
corresponding to the -th frequency point under the hypothesized source
range and depth ; is the number of frequency points; is the trace
operation; is the cross-spectral density matrices corresponding to the
-th frequency point, which is expressed as follows:

, (13)

where is the -th snapshot data corresponding to the -th frequency point.

\emph{SWellEx-96 data results:} The SWellEx96 experiment was conducted
near San Diego, CA in the spring of 1996. A vertical line array (VLA)
was used to record the acoustic field. The VLA contained 21 hydrophones
spanning a depth from 94 m to 212 m of the water column, spaced at 5.6
m.

In this section, we analyze the first 55 min of data recorded by the
vertical line array after the start of event S5, in which the two towed
sources travelled along an isobath of a mildly sloping environment
toward the VLA at a distance from 8.6 km to 1.0 km. The depth
corresponding to the shallow source was 9 m, and the depth corresponding
to the deep source was 54 m. The data corresponding to the shallow
source analyzed here involved three tonal signals at 109, 127, and 145
Hz. The data corresponding to the deep source analyzed here applied
three tonal signals at 112, 130, and 148 Hz.

In event S5, the data sampling frequency is 1500Hz, and the data
analysis starts from the first minute and is processed every 3 minutes.
The duration of data intercepted in each processing is 54.6 s, and the
intercepted data is divided into 9 snapshots. The adjacent snapshots
overlapped by 50\%, and the duration of each snapshots was 10.9 s.
According to the tensor construction method in Section 2.1, the data
tensors for shallow and deep sources are constructed with dimensions of
21\selectlanguage{ngerman}×3×9.

For the replica field generation, the input water depth is approximately
216 m, the sound speed profile is a typical downward refracting sound
speed profile, and the marine environment parameters are available
online in Ref. 14. The normal mode program KRAKEN {[}17{]} was used to
generate the acoustic field (or Green functions) for the 21-element VLA
with frequencies of 109, 112, 127, 130, 145, and 148Hz. Then, the
tensor-based MFP and conventional MFP I and MFP II are used to locate
the shallow source and deep source.

The comparisons of localization results for the SwellEx-96 towed sources
with MFP I, MFP II and TMFP are shown in Fig. 2.

Fig. 2a gives the variation in the range estimation errors of the
shallow source versus time. The range estimation errors are small for
TMFP, MFP I and MFP II. Figure 2b shows the variation in the depth
estimation errors of the shallow source versus time. The depth
estimation results by TMFP are similar to that by MFP I and MFP II, and
the depth estimation errors are less than 10m.

Fig. 2c gives the variation in the range estimation errors of the deep
source versus time. TMFP can accurately estimate the source range
throughout the tow period studied, where as MFP I and MFP II has large
errors in range estimation at 1 and 13mins. Fig. 2d shows the variation
in the depth estimation errors of the deep source versus time. As the
time changes, the range from source to VLA decreases, the depth
estimation results of TMFP, MFP I and MFP II show a gradual deepening
trend, and the maximum depth estimation error is up to 15m.

The reason is that the generation of replica field does not incorporate
the bathymetry of real marine environment, which leads to the depth
estimate offset.

Comparing the processing results of Figs.2b and 2d, the processing
results of TMFP, MFP I and MFP II can accurately distinguish whether the
source is a surface source or a submerged source.

By comparing the results of Fig. 2(a--d), the processing performance of
TMFP is slightly better than that of MFP I and MFP II for the VLA data
in event S5.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image73/image73}

\end{center}
\end{figure}

\textbf{Fig. 2} \emph{Comparison of localization results for the
SwellEx96 towed sources with MFP I, MFP II and TMFP. (a) Range
estimation error and (b) depth estimation error for the shallow source;
(c) Range estimation error and (d) depth estimation error for the deep
source.}

To illustrate the advantages of TMFP compared with MFP I and MFP II, we
give the variation in range and depth estimation ambiguity surface
versus processing time of TMFP, MFP I and MFP II, respectively. The data
processing procedure is as follows:

(1) For segment data , the ambiguity surfaces , and can be obtained by
applying TMFP, MFP I and MFP II.

(2) The peak position of the ambiguity surfaces is found. The depth
dimension ambiguity function and range dimension ambiguity function
corresponding to the peak position are extracted. and are for TMFP, and
are for MFP I, and are for MFP II.

(3) Based on the results of 19 times of processing, the ambiguity
surfaces in depth and range for MFP I, MFP II and TMFP are shown in
Figs. 3 and 4, Fig. 3 shows the ambiguity surface for shallow source
with a depth of 9 m, Fig. 4 shows the ambiguity surface for deep source
with a depth of 54 m.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image84/image84}

\end{center}
\end{figure}

\textbf{Fig. 3} \emph{Variation in the ambiguity surface versus time for
shallow source. The depth and range ambiguity surfaces are shown for
TMFP in (a) and (d) respectively, for MFP I in (b) and (e), and for MFP
II in (c) and (f), where the solid black line denotes the real source
range and depth.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image85/image85}

\end{center}
\end{figure}

\textbf{Fig. 4} \emph{Variation in the ambiguity surface versus time for
deep source. The depth and range ambiguity surfaces are shown for TMFP
in (a) and (d) respectively, for MFP I in (b) and (e), and for MFP II in
(c) and (f), where the solid black line denotes the real source range
and depth.}

Observing Figs. 3 and 4, comparing the depth estimation ambiguity
surface of TMFP, MFP I and MFP II, the main lobe width of depth
estimation ambiguity surface for MFP II is the narrowest, but the
background interference of the ambiguity surface for TMFP is lower than
for MFP I and MFP II, especially under the condition of low
signal-to-noise ratio (the range from source to VLA in the first 30min
is large). The suppression performance of TMFP on ambient noise is
better than that of MFP I and MFP II.

\emph{Summary and conclusions:} Comparing the range estimation ambiguity
surface obtained by TMFP with the range estimation ambiguity surface
obtained by MFP I and MFP II, the main lobe width of range estimation
ambiguity surface for MFP II is the narrowest, but the background
interference of the ambiguity surface obtained by TMFP is lower than
that obtained by MFP I and MFP II, and the suppression performance of
the ambient noise under a low signal-to-noise ratio is better. The
reason is that the singular value decomposition of the matrix expanded
in each dimension of the tensor can obtain a more accurate tensor signal
subspace and then realize the suppression of the ambient noise.

This study draws on the advantages of tensors in multidimensional data
processing and applies tensor decomposition to broadband matched field
sound source localization processing for the first time. A
space-time-frequency three-dimensional tensor signal model is
constructed, and then a matched field sound source localization method
based on tensor decomposition is proposed. The performance of TMFP with
MFP I and MFP II is compared by processing the VLA data recorded in
event S5 of SWellEx-96. The results show that TMFP has a better
suppression effect on ambient noise than MFP I and MFP II. Especially
under a low signal-to-noise ratio, given the advantage of tensor
decomposition in signal subspace estimation, the advantage of TMFP is
more evident than that of MFP I and MFP II. Therefore, the TMFP
processors could be used in real applications because of better
performance. Finally, it needs to be mentioned that one can develop an
adaptive TMFP with higher resolution (similar to MVDR beamformer)
{[}18{]}.

\emph{Acknowledgments:} This research was funded by Science and
Technology on Sonar Laboratory foundation, Grant No. 2022-JCJQ-LB-031-02
and Youth Elite Scientists Sponsorship Program by CAST, Grant No.
YESS20200330.

? 2021 The Authors. \emph{Electronics Letters} published by John Wiley
\& Sons Ltd on behalf of The Institution of Engineering and Technology

This is an open access article under the terms of the Creative Commons
Attribution License, which permits use, distribution and reproduction in
any medium, provided the original work is properly cited.

\emph{Received:} xx January 2021 \emph{Accepted:} xx March 2021

doi: 10.1049/ell2.10001

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