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1 INTRODUCTION

Modern coastal and port-area monitoring faces

increasing demands due to the growing presence of

small craft, autonomous underwater vehicles, and

other mobile acoustic sources near maritime

infrastructure [7]. Under such conditions, rapid and

continuous monitoring of water bodies becomes an

important component of maritime infrastructure

management, navigation safety, and environmental

acoustic observation.

The main sensor in underwater acoustic monitoring

systems is the scalar hydrophone, which measures

changes in acoustic pressure and provides high

sensitivity and reliability [8, 15]. Recent developments

in thin-cable fiber-optic hydrophone arrays have

demonstrated the ability to maintain sensitivities

above 120 dB over a wide frequency range (20-500 Hz)

and to achieve minimum detectable pressures below

the sea-level ambient noise floor [13].

Underwater monitoring architectures range from

active acoustic sensing configurations to distributed

passive hydrophone arrays. Active acoustic

configurations can simultaneously detect and localize

multiple acoustic targets as they pass through a

monitored region [7]. Passive fiber-optic arrays

deployed in coastal zones can track mobile underwater

and surface acoustic sources, including small vessels

and large ships, over distances of tens of kilometers,

providing multilayer integration with sonar and visual

observation systems [8].

The integration of hydrophones with autonomous

platforms and modern signal-processing algorithms

further enhances acoustic detection and source-

localization capabilities. For example, a Wave Glider-

based system towing a hydrophone array

Energy-Attenuation-Based Passive Localization of

Acoustic Sources in Restricted Aquatic Areas

E. Sabziev

, R. Akhundov

, A. Pashayev

, T. Alizada

, V. Panevski

& D. Dimitrov

Institute of Mathematics, Baku, Azerbaijan

Military Research Institute of the National Defense University, Baku, Azerbaijan

Institute of Metal Science, Equipment and Technologies with Hydro- and Aerodynamics Centre "Acad. A. Balevski"

at the Bulgarian Academy of Sciences, Sofia, Bulgaria

ABSTRACT: The article is devoted to the problem of determining the coordinates of a sound emitter using a

network of scalar hydrophones. The physical operating principle of such a network is that a sound wave carries

energy from the source through the medium, and this energy gradually decreases with distance from the emitter.

The proposed method uses several synchronized scalar hydrophones, each of which measures the total sound

energy at its location. When the measured energy exceeds the ambient-noise threshold, the system interprets this

as the appearance of a new acoustic source in the monitored water area. From the wave energy recorded by

different hydrophones, the relative distances to the source are estimated: the greater the energy attenuation, the

farther the emitter is located. Knowing the exact coordinates of the scalar hydrophones and the calculated

distances, the system determines the position of the sound emitter on the horizontal plane of the water area with

the required accuracy. This approach is based on a predefined model of sound-energy attenuation with distance

and enables reliable localization of the emitter without any active emission from the hydrophone network.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 20

Number 3

September 2026

DOI: 10.12716/1001.20.03.08

604

demonstrated an average signal-to-noise ratio gain of

15.3 dB and enabled the detection of whale

vocalizations against anthropogenic noise [14]. The

application of convolutional neural networks to

spectral-cepstral features obtained from a single

hydrophone makes it possible to estimate range and

localize moving vessels at distances beyond the

capabilities of classical passive acoustic monitoring

methods [6]. Moreover, deep neural networks have

shown high source-localization accuracy in shallow-

water conditions even when information about the

acoustic medium is limited [9].

A promising direction is the fusion of different

sensor types (scalar and vector hydrophones), the use

of distributed optical sensing, and the implementation

of adaptive machine-learning algorithms to improve

system robustness against acoustic interference and

changing oceanographic conditions [15]. Together,

these technologies form a multilayer framework for

underwater acoustic monitoring that is capable of

detecting, localizing, and classifying relevant acoustic

sources in real time and supporting timely operational

decision-making.

2 RESEARCH METHODOLOGY

2.1 Key factors influencing the selected approach

Scalar hydrophones do not generate their own acoustic

pulses. They only record sound vibrations in water and

use them to determine the parameters of noise sources.

An acoustic wave is energy propagating from a source

through the surrounding medium. Several critically

important factors influence the propagation of acoustic

waves, each of which changes the speed of sound in its

own way.

2.2 Parameters of the water environment

Physicochemical properties of water (salinity,

temperature, density). As the concentration of

dissolved salts increases, as the liquid is heated, and as

its density grows, the propagation speed of acoustic

vibrations increases.

Internal currents in the water column. Currents

disturb the rectilinear propagation of sound, causing

refraction and distortions.

Sea depth and seabed characteristics (silt, sand,

clay, gravel, etc.). As depth increases, hydrostatic

pressure grows, which accelerates wave transmission

toward the seabed. If bottom sediments have low

reflectivity, a significant part of the acoustic energy is

absorbed.

2.3 Parameters determined by the acoustic emitter itself

Frequency range and amplitude level of acoustic

waves. As wave frequency increases, absorption in the

medium grows, which leads to a decrease in

transmitted energy.

Distance to the scalar hydrophone. As the source

moves farther away, sound disperses in the

surrounding medium, as a result of which its energy at

the receiver (the hydrophone) decreases.

Scalar hydrophones based on piezoelectric

elements are used to record acoustic vibrations in

water. To ensure the reception of signals from all

directions, a scalar hydrophone consists of an array of

piezoelectric elements placed in a single housing.

Hereinafter in this article, the term scalar hydrophone

refers to devices designed to receive acoustic waves

arriving from all directions [15].

3 PROBLEM STATEMENT

For monitoring the water areas of seaports and coastal

zones in order to detect newly appearing acoustic

sources, it is proposed to deploy a network of scalar

hydrophones at optimal points on the seabed that do

not create significant obstacles to navigation. It is

assumed that the scalar hydrophones in this network

have synchronized internal clocks and that the

information transmitted to the server includes the

precise time of signal registration.

There are several approaches to determining the

position of an object in a water area on the basis of the

acoustic signal it emits. The location of an object is

determined from the difference in arrival times of

acoustic pulses at scalar hydrophones distributed at

different points of the seabed [11]. However, this

method is applicable only in the case of irregular

acoustic emissions; otherwise, the signal registration

time becomes ambiguous.

This paper considers the problem of source

localization in the case of repeated acoustic impulses.

It is assumed that the internal clocks of the scalar

hydrophones are precisely synchronized, that the

receivers transmit information to the server with

minimal delay, and that the coordinates of the

receiving nodes on the seabed are known in advance.

Anthropogenic acoustic sources, such as underwater

vehicles, surface vessels, and port-related equipment,

generate characteristic noise patterns at each point of

the water area during motion and operation. Natural

sounds produced by aquatic fauna (fish, marine

mammals, etc.) are also present, but their level is

usually substantially lower than the level of

anthropogenic noise.

The principal characteristic of an acoustic wave is

its energy, which decreases during propagation both

with distance and depending on the properties of the

medium. It is assumed that sound is absorbed by

seawater in such a way that the sound intensity

decreases exponentially with distance [5]. Therefore, if

the acoustic-wave energy at the source is equal to E,

then at a distance R its value is estimated by the

formula [2]:

E E R



−−

=  

(1)

where



is the (average) sound attenuation coefficient

in water, calculated on the basis of the sound frequency

f, the (average) temperature of the medium T, and the

propagation depth z:

27 17

4.9 10 e





−+



−



=  

(2)

605

According to Huygens' principle, waves propagate

along straight lines in a homogeneous medium [3].

Let us assume that a closed marine area is being

monitored (for example, port waters), and that its

water may be considered homogeneous. Let each

scalar hydrophone record the energy of acoustic waves

passing through its observation point. Consequently, if

the acoustic-noise sources in the area operate at a

certain stable power, each hydrophone will record a

stable background-noise level. In this paper, it is

assumed that, at the time instants tj=j∆t, ∆t>0,

j=0, 1, 2, … the deviation of the background-noise

energy recorded by the scalar hydrophones serves as

the basis for calculating the coordinates of the newly

appeared acoustic source.

4 MATHEMATICAL MODEL

Consider a conditionally bounded water area in which

N scalar hydrophones are installed, indexed as

i=1, 2, 3, …, N. It is assumed that the depth of this sea

region is negligibly small compared with its horizontal

dimensions. In this case, a two-dimensional model is

sufficient for solving the problem.

We introduce a Cartesian coordinate system and

denote the position of the i-th hydrophone by the point

(xi, yi). At the initial time t0 each hydrophone recorded

the background-noise energy E0i. At a later time tj, j>0

the recorded energy became equal to Eji  E0i, and the

difference

 Eji-E0i is associated with the

appearance of a new acoustic source. Let the unknown

source coordinates be (x, y), and let the initial acoustic-

wave energy be E. Then, based on Eq. (1), for each

hydrophone we obtain the approximate relation

( ) ( )

1, 2, 3, , .

x x y y



− − + −



− + −

=

(3)

Here



is the attenuation coefficient, and the symbol

"≅" means that the equality holds with a certain

measurement error. This approximation is sufficiently

accurate provided that the source is detected by at least

four hydrophones.

Thus, the problem reduces to finding (x, y) such that

the set in Eq. (3) is satisfied simultaneously, where

, E and



are known positive quantities.

Equations (3) are transcendental equations. If we

denote





then after substitution

( ) ( )

, 1, 2, 3, , ,

i i i

w x x y y i N



= − + − = 

(4)

they can be written as

exp ,

i i i





and its positive solution can be expressed "explicitly"

in terms of the Lambert W function [4]:

( )

W 1, 2, 3, , .

w i N



 = 

(5)

However, in practice it is preferable to use iterative

numerical methods, for example Newton's method [1],

since the function W itself is also evaluated iteratively.

Thus, we shall assume that for each i=1, 2, 3, …, N

the values



i have been computed by one method or

another, and the problem reduces to finding x, y such

that the set in Eq. (4) is satisfied simultaneously.

Finding an approximate solution (x, y) that best

satisfies this system can be formulated as minimizing

the following functional by the least-squares method

[10]:

( )

( ) ( )

x y x x y y





=  − − + −







(6)

which is a positive nonlinear function of the

unknowns. Finding its minimum reduces to finding

the common zeros of the functions



and



typically using iterative numerical algorithms:

( ) ( )

w x x

x x y y



−



− − + − =





− + −



( ) ( )

w y y

x x y y



−



− − + − =





− + −



5 EXAMPLE OF CALCULATING THE

COORDINATES OF AN ACOUSTIC SOURCE

Let four scalar hydrophones be installed in the

monitored shallow-water area, and let their

coordinates be given in nautical miles:

( ) ( )

1.00, 0.20 , 0.90, 0.10

0.20,1.00 , 0.70,0.85

− − −

−

Assume that the acoustic source is located at the

point

( )

0.15, 0.25 .A

For the numerical example, let the initial acoustic-

wave energy and the attenuation coefficient be equal,

respectively, to E=1.00,



=0.12 nautical miles

-1

According to the adopted energy-attenuation model,

the additional energy recorded by the i-th hydrophone

is computed by the formula

( )

exp ,



=−

where the distance from the source to the i-th

hydrophone is determined by the expression

( ) ( )

i i i

r x x y y= − + −

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For the true source position A0(0.15, 0.25), the

distances to the hydrophones and the corresponding

ideal values of the recorded additional energy are

given in Table 1.

Table 1. True distances and ideal energy values

Hydr.

Coordinates

True distance

Ideal energy

A₁

(−1.00, −0.20)

1.23491

0.565422

A₂

(0.90, −0.10)

0.82765

1.321832

A₃

(0.20, 1.00)

0.75166

1.617254

A₄

(−0.70, 0.85)

1.04043

0.815360

Taking into account random deviations not

exceeding 1%, we simulate the following scalar-

hydrophone data:

0.569945 1.313901

1.625340 0.808022

These values correspond to the following relative

errors

0.8%, 0.6%, 0.5%, 0.9%.+ − + −

To reconstruct the distances from the source to the

hydrophones, we use the transformation through the

Lambert W function. First, denote





and then compute

( )

W ,

i i i





Table 2. True and reconstructed distances

Hydr.

Measured energy

True distance ri

Estimated distance

A₁

0.569945

1.23491

1.23034

A₂

1.313901

0.82765

0.83002

A₃

1.625340

0.75166

0.74987

A₄

0.808022

1.04043

1.04487

After the distances have been reconstructed, the

source coordinates are found by minimizing the

residual functional

( )

( ) ( )

i i i

x y r x x y y



= − − + −







Solving this least-squares problem gives the

following estimate of the acoustic-source coordinates:

( )

0.14863, 0.24875 .A

Let us compare the obtained point with the true

source position:

( )

0.15, 0.25 .A

The distance between the true and calculated

positions is equal to

( )

( ) ( )

, 0.14863 0.15 ² 0.24875 0.25 ² 0.00186d A A = − + − =

Since 0.00186 nautical miles ≈ 3.44 m, the

obtained error can be considered acceptable for the

demonstration numerical example. Therefore, even in

the presence of small random errors in energy

measurements, the proposed method makes it possible

to estimate the coordinates of an acoustic source in a

restricted water area with sufficient accuracy.

Table 3. Summary of the comparison between the true and

calculated source positions

Indicator

Value

Comment

True point

A₀(0.15, 0.25)

Specified in the

computational experiment

Calculated

point

A(0.14863, 0.24875)

Obtained by the least-squares

method

Error

0.00186 nmi≈ 3.44 m

Approx.

It should be noted that in a full-scale

implementation, the values E and



should be refined

based on calibration measurements, and the obtained

coordinates should be checked for robustness to noise,

multipath propagation, and spatial heterogeneity of

the water medium.

6 RESULTS

In real monitoring of a shallow-water marine area,

mobile or newly appearing acoustic sources can often

be identified by the spectrum of the acoustic waves

they emit. After the source coordinates are determined

using the hydrophone network, the signal frequency is

calculated, and then, using Eq. (4) (for example, at i=N),

the attenuation coefficient \alpha is refined:

( ) ( )

x x y y





− + −

For a shallow-water region, it is convenient to take

the water level as z=0. Given the known average water

temperature T and using Eq. (2), one can determine the

oscillation frequency f of the sound wave emitted by

the source.

7 DISCUSSION

The numerical example obtained above shows that the

proposed energy-based localization scheme is

computationally feasible and can produce a small error

under moderate energy-measurement inaccuracies. At

the same time, this result should be regarded not as a

field validation of a ready-to-use system, but as a

demonstration of the internal consistency of the model

in a controlled computational scenario. The example

uses a known hydrophone geometry, a specified

attenuation coefficient, a single source, and a relatively

small measurement error. Therefore, the main

conclusion is that the selected mathematical apparatus

is suitable for further experimental verification, while

its practical accuracy must be evaluated under real

hydroacoustic conditions.

A significant advantage of the considered approach

is its passive nature. Scalar hydrophones do not

generate their own acoustic pulses and do not increase

the acoustic load on the water environment. This

makes the method potentially convenient for long-

607

term monitoring of restricted water areas, port zones,

and sites with increased requirements for

environmental and operational safety. In addition, the

method uses the spatial distribution of measured

energy and therefore can be applied as an additional

localization channel in cases where methods based

solely on the difference in signal arrival times become

ambiguous due to repeated, continuous, or poorly

separable acoustic impulses.

At the same time, the proposed method should not

be regarded as a universal replacement for TDOA,

vector hydrophones, or active hydroacoustic tools [12,

15]. Its applicability is determined by the type of

source, the signal-to-noise ratio, the stability of spectral

features, the density of sensor placement, and the

degree of homogeneity of the water medium. The most

natural role of energy-based localization is to

complement other methods: it may serve as a

preliminary estimation mechanism, a confirmation

channel, or part of a hybrid system combining energy

features, time delays, spectral classification, and, when

appropriate sensors are available, the direction of

arrival of the signal.

The accuracy of the method depends substantially

on the physical assumptions adopted in the sound-

propagation model. This work uses a two-dimensional

approximation for a shallow-water area and an

effective energy-attenuation model that includes the

dependence on distance. In a real environment, the

recorded energy is influenced not only by geometrical

spreading and absorption, but also by reflections from

the surface and the bottom, scattering by

inhomogeneities, bottom losses, refraction, water

motion, and spatial variations in temperature and

salinity. Therefore, in practical implementation, the

attenuation parameter should preferably be treated as

an effective or locally calibrated parameter rather than

as a constant quantity that is identical for the entire

water area.

The acoustic background is of particular

importance. In restricted water areas it is rarely

stationary: it is affected by navigation, port equipment,

wind, rain, biological sources, reflections from

structures, and short-term anthropogenic noise.

Consequently, the detection of a new acoustic

component should be based not on a fixed threshold,

but on an adaptive assessment of the background level

in selected frequency bands. A practically useful

approach is preliminary spectral or spectral-cepstral

extraction of ranges characteristic of the type of source

under consideration, followed by the calculation of

energy features precisely in these ranges. This

approach can reduce the number of false alarms and

improve the robustness of coordinate estimation.

The stability of the solution is also affected by the

geometry of hydrophone placement. Although the

model requires detection of the source by several

spatially distributed receivers, the mere presence of

four hydrophones does not guarantee good

conditioning of the problem. If the sensors are

arranged almost along one line, if the source lies

outside the covered zone, or if the measured energies

differ only slightly, the residual functional may have a

broad minimum region. Therefore, when designing the

network, it is necessary to analyze the sensitivity of the

solution to measurement errors, control the residual

discrepancy, estimate the confidence region of the

coordinates, and choose the hydrophone placement so

as to minimize geometric uncertainty in the most

important zones of the water area.

A separate limitation is associated with the possible

presence of multiple sources. The energy recorded by

a scalar hydrophone is a total quantity; therefore, when

several objects operate simultaneously, a simple single-

source model may yield biased or physically incorrect

estimates. In such cases, additional signal-separation

procedures are required: clustering by spectral

features, temporal tracking of sources, extraction of

dominant frequency components, use of several time

windows, or integration with vector-hydrophone data

[12, 15]. Without such separation, energy-based

localization should be applied with caution and

accompanied by an uncertainty indicator.

From a practical point of view, the proposed

approach should be developed in stages. At the first

stage, calibration experiments should be carried out

with sources of known power and known coordinates

in order to estimate the effective attenuation

parameters and the influence of sensor geometry. At

the second stage, series of computational and field tests

are required at different noise levels, depths,

temperature conditions, and seabed types. At the third

stage, energy-based localization can be included in a

multilayer monitoring system in which the result is

presented not only as a single calculated point, but also

as a probable-position region, a confidence level, and a

recommendation on the need for additional

verification. Such a cautious approach makes the

proposed method more realistic and increases its value

for future systems of passive monitoring of restricted

water areas.

8 CONCLUSION

The article presents a method for detecting and

localizing acoustic sources in restricted water areas,

including port water areas, using a network of scalar

hydrophones. The proposed solution provides

continuous monitoring of the acoustic background of

the water area: when a new source appears, a sharp

jump is observed in the acoustic energy measured by

the scalar hydrophones. Since the amplitude of the

received signal at each hydrophone depends on its

distance from the source, the system accounts for these

differences during data processing. The method is

based on fundamental physical laws, which makes it

possible not only to detect a new acoustic source but

also to determine its coordinates. The deployment of

the proposed scalar-hydrophone network is

particularly relevant for improving continuous

acoustic monitoring and operational safety in port

zones: it can effectively detect newly appearing

acoustic sources within water-area boundaries and

thereby support timely technical assessment and

operational response.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the financial support

provided by the NATO Science for Peace and Security

608

Programme under Project G7475. This work has benefited

from the expertise and constructive feedback of our research

partners and colleagues, whose contributions were

important for its successful completion. We also thank the

administrative and technical staff, whose dedicated work

ensured the uninterrupted implementation of the project.

The views expressed herein are those of the authors and do

not necessarily reflect the official position of NATO.

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