1. Introduction
Remote sensing technology is widely applied across various fields, yet single-source remote sensing is no longer sufficient to meet the information acquisition needs in complex environments. Consequently, multi-source information fusion technology gradually became a research hotspot. In polar regions, the transmission of radio and light waves in water is limited, making acoustic waves the primary medium for information transmission. For instance, the propagation distance of light waves in water is limited, typically penetrating only a few meters deep, while radio waves attenuate rapidly in water, rendering them nearly ineffective for transmitting information. Therefore, acoustic waves, with their advantages of long propagation distance and lower attenuation in water, became the main medium for underwater information sensing and transmission in polar regions. Research vessels operating in polar oceans often utilize sonar systems to transmit sound waves for underwater communication and detection. The presence of sea ice cover can scatter and absorb the acoustic waves propagating underneath, thus reducing the efficiency of related work. It is critical to understand the scattering mechanisms of sound waves in the marginal ice zone (MIZ) for ensuring the safety of navigation and other operations in the polar regions. Therefore, continuous attention in Arctic hydroacoustic research is demanded for hydroacoustic studies to elucidate the acoustic scattering characteristics at the ice–water boundary and to unravel the complexities of the scattering processes related to the sea ice [
1].
Hutt [
2] provided a comprehensive review of the underwater acoustic field, emphasizing how the presence of ice cover significantly influences acoustic properties. This foundational understanding led to the development of underwater acoustic techniques for detecting ice cover, which are crucial for improving acoustic communication and navigation beneath sea ice [
3,
4,
5]. McCammom et al. [
6] built upon this by modeling an ice layer as a continuous elastic medium to calculate the reflection coefficient of a layered elastic solid plane wave. Their model, which assumes a fluid half space, offers insights into the relationship between the internal properties of ice and the acoustic reflection at the ice–water interface. Extending this work, Liu et al. [
7] analyzed the scattering coefficient and coherent reflection from a rough, single-layer sea ice cover, highlighting the complexities of acoustic interactions in such environments.
While these studies focus on continuous ice layers, the marginal ice zone (MIZ) presents unique challenges due to the diverse forms of sea ice, such as frazil ice, grease ice, and pancake ice. To address these challenges, several studies from difference aspects are briefly reviewed. Lucieer et al. [
8] introduced novel multiple workflows for processing underwater acoustic bathymetry and backscatter data. Their approach can identify distinct sea ice textures, providing new insights into the acoustic signatures of various ice surfaces. In parallel, Jiang et al. [
9] developed a semi-supervised interactive system for classifying sea ice in dual-polarized RADARSAT-2 imagery. This system leverages machine learning techniques to handle the increasing volume of SAR data, offering a complementary approach to traditional manual interpretation. Empirical observations by Garrison et al. [
10] and Ito et al. [
11] provide further insights into acoustic scattering in icy environments. Garrison et al. measured the backscattering intensity from the bottom of flat, one-year sea ice using a 60 kHz pulse signal, contributing valuable data on acoustic interactions with older, stable ice. Ito et al. focused on detecting backscatter signals from underwater frazil ice and sediment using bottom-mounted acoustic Doppler current profilers with a 300 kHz frequency, shedding light on the acoustic properties in more dynamic ice conditions.
It is a challenge to model the underwater acoustic scattering field beneath partially ice-covered water. Acoustic waves near the ice–water boundary give rises to a multiple-scattered acoustic field due to repeated reflections from the water surface and submerged ice. Accordingly, the problem of sound scattering with at the ice–water interface can be considered as an extension of the target sound scattering problem at the interface, a topic receiving considerable attention. Huang and Gaunaurd [
12,
13] used a classical separation of variables technique to precisely analyze the acoustic scattering by a submerged spherical elastic shell near a free surface, ensuring the boundary conditions on the shell and free surface were satisfied. A virtual source method was employed to examine the acoustic scattering of plane waves around spherical objects at various incident angles within low to medium frequency ranges, where the acoustic scattering fields are solved by expanding the classical modal series of the spherical wave function. The authors further studied the sound scattering by bubble clouds near the sea surface, adapting the classical formulation for evaluating the scattering cross-sections of a single air bubble in a liquid, to a cloud of interacting bubbles near a boundary. The adaption resulted in multiple scattering cross-sections and a series of coupled algebraic equations to determine the coupling coefficients [
14]. Richardson et al. [
15] studied the scattering of high-frequency acoustic signal scattering from seashells to model seafloor backscattering. They conducted experiments on varying sizes of glass beads and shells. Comparison of the measurements with a simple target strength-based model’s predication indicated the potential importance of interface and multiple scattering effects, and the discrete scatterers in backscattering scenarios. Kungl et al. [
16] expanded the theoretical framework by modeling frazil ice in open water as both a sphere and an oblate spheroid. Their work extended the classical Rayleigh analysis to determine the acoustic backscattering cross-section. Their oblate spheroid-based backscattering model is consistent with previous laboratory studies and observations of frazil ice in rivers across a broad frequency range from 125 kHz to 774 kHz.
Furthermore, advanced numerical methods were developed to resolve scattering problems with complicated target surfaces. Simon et al. [
17] adopted the finite element method to model acoustic wave propagation scattered by the ice–water interface, where the ice cover is modeled as both an elastic medium and a pressure release surface. Chen et al. [
18] proposed a boundary condition method for calculating the acoustic reflection coefficient of the layered ice–snow model based on the conditions between two adjacent layers. The plate element method is used to model complex interface shape and calculate the scattering field of the target with an incident plane wave [
19,
20]. Abewi [
21] modeled a target as an assembly of triangular facets to solve the scattering problem from a target in a waveguide. The Kirchhoff approximation was used to compute scattering from each facet analytically for each set of incident and scattered plane waves, and solutions were then combined coherently.
This study focuses on the impact of polar sea ice surfaces on underwater acoustic wave propagation, aiming to enhance the accuracy of underwater information sensing and transmission and to provide theoretical support for the application of multi-source information fusion technology in polar environments. Specifically, this paper examines the acoustic scattering phenomena due to partial sea ice coverage in the MIZ, particularly brash ice, with notable three-dimensional shapes. For analytical tractability, following Kungl et al. [
16], we simplify the complex ice–water interface by treating individual ice floes as spherical scatterers with acoustic resistance, hereafter called ice balls. This approach enables a detailed analysis of the principal scattering waves and provides insights into the acoustic field under rough interface. This simplification inherently limits the model’s ability to capture the irregular geometries, and variable surface roughness of various ice types will, consequently, not be fully represented in the real MIZ. Nonetheless, this approach provides a necessary foundation for extending the analysis to more complex and realistic ice–water interfaces. We further extend this model to account for multiple, randomly distributed ice balls and explore both uniform and non-uniform distributions. This comprehensive analysis is crucial for understanding the acoustic behavior in real MIZ conditions, where ice floes are rarely uniform. The theoretical models are then validated through controlled water tank experiments, with results showing good agreement between simulated and experimental data, reinforcing the reliability of the proposed models. These findings have direct implications for acoustic detection and communication in polar regions, laying the groundwork for future research and operational strategies in these challenging environments.
The structure of this paper is addressed below. In
Section 2, we investigate the sound scattering of a spherical ball floating in the water. By accounting for principal scattering paths, the acoustic scattering field of an ice–water interface is simulated using the plate element method. Moreover, we conduct acoustic scattering experiments with real ice balls in a water tank for validation. In
Section 3, the model is further extended for general conditions, including multiple ice balls with different sizes, and to report the relevant validations by laboratory experiments. The discussion and conclusion are given in
Section 4.
2. Single Ice Ball Model
For an inelastic target, the acoustic scattering field is primarily governed by the geometric shape of the target surface, while the surface impedance characteristics can be treated as a geometric scattering field. In the case of high-frequency waves (i.e.,
, where
indicates the wavenumber, and a indicates the radius of the ice ball) [
22], the scattered waves are inclined to follow the patterns of geometric scattering. In fact, geometrically scattered waves often form the essence of an acoustic scattering field. Given that the surface of an ice ball is non-rigid, the resulting acoustic scattering field contains both internal elastic echoes and geometric reflections. Our focus primarily lies on geometric scattering, and an additional reflection coefficient attributable to the resistive surface of an ice ball. This reflection coefficient is influenced by the angle and frequency of the incident wave, and the properties of the ice surface.
The plate element method is utilized to calculate the surface element integral of complex target surfaces. By applying Green’s theorem, the area integral of acoustic scattering by a target surface is transformed into a line integral, subsequently simplified as a simple algebraic summation. For each small surface element, its acoustic scattering field is solvable through classical acoustic methods. The aggregate acoustic scattering field is represented by the vector summation of the acoustic fields emanating from all elements. Hence, the plate element method is appropriate for simulating the underwater acoustic scattering field of intricate targets. The precision of calculation improves with finer element subdivision yet incurs higher computational costs.
2.1. Acoustic Scattering under Ice–Water Interface with a Single Ice Ball
The acoustic scattering field for an ice ball floating on water is more complex than that in the free field. This complexity comes from multiple scatterings between the target and the water surface, resulting in a complicated echo structure of acoustic scattering around the target, which in turns influences the target strength. The Kirchhoff approximation formula for the target scattering problem near the interface [
23,
24] is discussed further below,
where
is the scattering potential function at the receiving point
in the semi-infinite space with vector radius
,
is the outer normal direction,
is the potential function at the scattering point
on the surface, and
is an approximation of the Green’s function in the semi-infinite space:
where
indicates imaginary unit,
is reflection coefficient of the interface,
is the included angle between the incident wave direction and the interface normal direction,
indicates distance from the sound source
to
, and
indicates distance from the virtual source
of
to
.
For the condition of monostatic transducer, the potential function of incident acoustic wave
consists of the direct wave and the interface reflected wave, which shares the same format as the Green’s function in Equation (2):
For targets with a non-rigid surface, such as the ice discussed in this paper, in the case of high frequencies where the local plane wave approximation is applicable (
). We denote the surface reflection coefficient as
V(
β) and the acoustic impedance as
. The following equations describe the behavior at the target surface:
We have an approximation under far-field condition:
where
and
. Substituting Equation (4) into Equation (5), we can obtain
Substituting Equations (5) and (6) into Equation (1), and considering the relationship:
, where
is water density and
is sound speed in water, it yields the physical acoustic formula of the scattering field beneath the resistive surface target at the water surface as
The surfaces of the target and water can be discretized into a number of small elements
M using the plate element method. This approach enables us to effectively approximate the integral of the acoustic scattering problem. As the number of plate elements
M approaches infinity, the scattering potential function can be written as
For the single ice ball model with a monostatic transducer, we select three principal scattering paths for sound waves emanating from source
S. The choice of these paths is detailed below and is depicted in
Figure 1 to help visualize each path.
Path 1: A sound wave from S is reflected off the ice ball’s spherical surface and returns to S. This trajectory represents the target echo in the free field, unaffected by the water surface, and is commonly considered as the primary contributor to the overall acoustic scattering field [
22,
25].
Path 2: A sound wave from S initially reaches the water surface, then contacts the target surface following reflection off the water, and then reverberates back to S after the spherical reflection; alternatively, the sound wave from S first reaches the spherical surface of the target, and subsequently the water surface, before finally reaching S. Both trajectories exemplify double-station scattering from S to S’.
Path 3: A sound wave from S is reflected by the water surface to reach the spherical surface of the ice ball. Subsequently, it is reflected from the spherical surface to the water surface and then back to S. This path represents a single-station echo of S’ relative to the ice ball.
Potential functions corresponding to the selected paths are:
Path 1:
where
.
Path 2:
where
.
Path 3:
where
.
In a scenario employed with a bistatic configuration, we present four principal scattering paths, such as the derivation of the scattering potential function under the monostatic transducer. These paths are described below and illustrated in
Figure 2.
Path 1: considering the target echo in the free field without interface effect, that is, the sound wave from S reaches R after being reflected by the ice ball.
Path 2: Starting from S, the wave reaches the ice ball surface, then the water surface, and finally R. This is equivalent to the path arriving at the mirror image point of the receiving position R’ through the water surface. The reflection coefficient is added due to the reflection at the water surface.
Path 3: Similar to the path 2, is considered. The path can be viewed as the starting point at the S’ and reaching R after being reflected by the ice ball surface.
Path 4: Both the incident and reflected waves from the ice ball are reflected by the water surface, which can be viewed as waves emitted from S’ to R’ after being reflected by the ice ball surface. Due to the twice reflections at the water surface, the interface reflection coefficient becomes a quadratic term .
The relevant potential functions of scattering field for the biostatic transducer case:
Path 1:
where
.
Path 2:
where
.
Path 3:
where
.
Path 4:
where
.
The target strength of an object is defined as
where
indicates the incident sound intensity at the object, and
indicates the scattered sound intensity at
from the center of the object. We can rewrite Equation (16) in terms of the potential function as:
Substituting the principal potential functions mentioned above into Equation (17), we can obtain the target strength in the acoustic scattering field of a single ice ball model.
2.2. Simulation of the Signal Ice Ball Model
We conduct numerical experiments to simulate the acoustic scattering field of an ice ball floating on water. The experimental layout is depicted in
Figure 3.
Figure 3a depicts a schematic vertical cross-section of an ice ball floating in still water, where
denotes the ball radius, and
denotes the vertical distance from center point
to the water surface. Applying the Archimedes principle of a floating ice ball on water,
, for a given ball radius
can be solved from the following derivation,
where
indicates the ice volume,
denotes the partial volume of the ice ball that is above water, and
indicates the ice density.
The portion of the ball exposed above the water surface is excluded in the analysis since it does not affect the underwater acoustic scattering field. The underwater portion of the ice ball surface is divided by the element segmentation through the Delaunay triangulation method, as illustrated in
Figure 3b.
Figure 3c shows a schematic of the single ice ball model employing a bistatic configuration with a transmitting transducer and a receiving transducer. In this layout, the horizontal direction y is parallel to the water surface. The source
and receiver
are located on an arc with a radius
, and centered at
. The grazing angle
is defined as the angle between the original sound wave direction and the horizontal direction. While the receiving angle
γ is the angle between
and the horizontal direction, varying between 0° and 180°.
To compare with laboratory experimental data addressed in
Section 2.3 later, we choose the model parameters unless otherwise specified, including
cm,
,
. We note that salinity levels in both ice and water are critical in affecting the underwater sound speed and refraction, and the participation of the submerged ice surface in the underwater acoustic scattering. In our model, the salinity effects are implicitly introduced by the choice of model parameters, such as the densities of ice and water, sound speed, and absorption coefficient. Model parameters are selected to satisfy specific salinity levels in the calculation of target or scattering intensities within our model.
Applying the incident wave on the ball surface and under far-field conditions, we can calculate the target strength against receiving angle
by Equation (17). For
45°,
Figure 4 shows that the target strength of the ice ball fluctuates greatly when the receiving angle is less than 60° or greater than 120°. The fluctuations can be attributed to the occurrence of the additional wave paths caused by multiple scattering between the ice ball and the water surface. The latter results in lager differences in the wave phase of the echo signal for each path. Because of the independences of the path-related bright regions and the signals received at each angle, the target strength curve does not show symmetry. In the bright spot model, the presence of the water surface increases the inherent bright spot from the free-field case, thus resulting in a complex echo structure of the floating ice ball.
2.3. Water Tank Experiment
To validate the theoretical simulation mentioned earlier, we conducted a laboratory experiment in a water tank to measure the target strength in a single ice ball model. The laboratory experiment layout is depicted in
Figure 5. Here, an ice ball is centrally positioned on the water surface in a water tank. The dimensions of the water tank were 1.4 m in length, 1.2 m in width, and 1.2 m in height, and the water depth was 1.1 m. Sound-absorbing wedges were attached to both sides of the tank walls to further reduce the wall scattering noises. The transmission signal is a CW pulse signal at a frequency of 90 kHz, with a single continuous wave duration of 0.2 ms and a signal interval of 1 s. This configuration of long pulse interval and short duration aids in the distinction between the object backscattering signal and the wall scattering noise. Transmitting the transducer sound source level and the receiving transducer sensitivity were tested in prior, and the results are detailed in the
Appendix A.
The transducers were mounted in a pre-made curved steel frame mounted at the tank bottom. Some of these transducers are observed as black boxes in
Figure 5. Given the potential of deviations from the intended design angles during the fabrication of the steel frame and the installation of the transducers onto the frame, we measured the angle of each transducer relative to a reference plane after they were placed in the slots. A high-precision digital inclinometer was used for this purpose, aligned with the vertical axis, and multiple measurements were taken to determine the final installation angles of the transducers. We use a transmitting transducer with grazing angle
θ = 30° and an array of five receiving transducers with receiving angles
γ = 30°, 45°, 60°, 73°, and 90°, respectively. The angles of the receiving transducer and the distances between the receiving transducers and the ice ball center are given in
Table 1. The experimental duration was short enough to ignore the ice melting in the cool water. Therefore, we did not measure the ice ball’s mass or volume post-experiment to estimate the melting impact. To minimize experimental error, we replicated each experiment four times by measuring the scattering acoustic signals. Steady-state values from the measured signal waveforms are selected to obtain the averages.
The target strength can be written as
where
is the distance between the receiving transducer and the ice ball surface, and
is the distance from the sound source point to the ice ball surface,
is gravity acceleration,
indicates the valid sound pressure value at the receiver, and
is the reference source pressure in water, then the acoustic pressure level is calculated through
. Additionally,
is the acoustic pressure sensitivity of the receiver where
indicates the open-circuit voltage output by the transducer.
By substituting the measured data into Equation (19), we calculate the target strengths at the five selected receiving angles. The measured target strengths are depicted as blue dots in
Figure 6 along with the theoretical simulation (red curve), as a function of the receiving angle. To quantify the agreement between the experimental and simulation results, the average of absolute errors across the receiving angles is calculated as 2.91 dB. The bias is considered acceptable within the scope of our study for validating the single ice ball model discussed in
Section 2.
Note that errors in our study can arise from both theoretical modeling and experimental procedures. Theoretically, we model the ice ball as an elastic sphere with resistance, simulating wave scattering within a semi-infinite domain. This simplification may introduce discrepancies between the modeling and the measurements, which contributes to the errors. Experimentally, errors can arise from several sources. For instant, the positioning of the ice ball and the transducers in the laboratory setup may deviate from ideal conditions. Additionally, the ice ball may not be perfectly spherical with the homogenous reflection feature due to mold constraints and laboratory environmental conditions with even slight melting potentially affecting the acoustic wave scattering. Experimental errors could also come from factors including the precise positioning the transducer, the distance between the transducer and the target, variations in the transducer’s sound source level, and the instruments’ sensitivity.