LIANG Minshuai, and PENG Linhui

Experimental Study on the Identification of Scholte Waves Based on Acoustic Pressure Field Measurement

LIANG Minshuai, and PENG Linhui*

College of Marine Technology, Ocean University of China, Qingdao 266100, China

Scholte waves at the seafloor interface are generally identified by their velocity features and seismic fields, which are measured using ocean bottom seismometers and geophones. These methods are effective in cases where there is a considerable difference between the velocities of Scholte and acoustic waves in water. However, they are ineffective when the velocities of these two types of waves are close to each other. Thus, in this paper, a method based on acoustic pressure field measurement for identifying Scholte waves is proposed according to their excitation and propagation characteristics. The proposed method can overcome the limitations on the velocities of two types of waves. A tank experiment is designed and conducted according to the proposed method, and an ocean environment is scaled down to the laboratory size. Acoustic measurements are obtained along virtual arrays in the water column using a robotic apparatus. Experiments show that changes in Scholte wave amplitudes, depending on different source depths and propagation distances, are consistent with the theoretical results. This means that Scholte waves generated at the seafloor interface are successfully measured and identified in the acoustic pressure field.

interface waves at seafloor; measurement of Scholte waves; identification of Scholte waves

1 Introduction

In marine settings, the waves trapped near the fluid-solid interface are called Scholte waves (Scholte, 1947). Scholte waves are a type of interface wave that is expected to have a longer traveling path and less transmission loss than waves in water (Nguyen., 2009; Socco, 2010). These characteristics lead that Scholte waves have a great prospect in applications in future ocean explorations and underwater communication. Currently, Scholte waves have been applied in many areas, such as geoacoustic inversion (Ritzwoller and Levshin, 2002; Zywicki and Rix, 2005; Miller, 2012; Godin, 2021) and acoustic source localization (TenCate, 1995). However, some problems in the identification of Scholte waves have been identified, and their resolution requires further study.

Velocity characteristics are mostly applied to identify Scholte waves in existing methods. One method is to use the slow speed of Scholte waves to identify them directly in the time domain. A later arrival signal with high energy can be observed in the time series (Rauch, 1980; TenCate, 1995; Dong, 2020; Cao, 2021). The other method is to identify Scholte waves by their dispersion characteristics, which are extracted using domain transform techniques. It is primarily applied in the inversion of the seabed S-wave structure (McMechan and Yedlin, 1981;Nazarian and Desai, 1993; Park, 1999; Ritzwoller and Levshin, 2002; Zywicki and Rix, 2005; Godin, 2021). When the velocity of Scholte waves differs largely from the acoustic waves in water, these two types of methods can efficiently identify Scholte waves. However, when ve- locities are close to each other, velocity-based methods are ineffective.

The literature we reviewed shows that ocean bottom seismometers (OBSs) (Rauch, 1980; Wang, 2012; Du, 2020) and geophones (TenCate, 1995; Potty and Miller, 2011; Miller, 2012) have been applied to measure the seismic field required for identifying Scholte waves. This kind of equipment mainly detects the seismic wave signal on the seafloor through the coupling between the equipment and the seafloor, which means the equipment’s measurements are susceptible to sediment and seafloor topography (Li and Pan, 2015; Wang, 2019). Compared with seismic measurement equipment, the hy- drophone is immune to technical issues, such as clock drift and seabed coupling (Li and Pan, 2015; Wang, 2019), making it more suitable for measuring underwater acoustic signals.

This paper introduces a method based on the excitation and propagation characteristics of Scholte waves for identifying such waves in an acoustic pressure field measuredhydrophones. Doing so can eliminate the effect of velocity on identification. A scaled-down tank experiment for measurement and identification of Scholte waves is designed and conducted based on the proposed method. First, the principle of this identification method is introduced. The characteristics of Scholte waves that vary with source depth and propagation range are presented in the theoretical analysis. Then, the scaled-down experiment is introduced in detail. Acoustic field properties, measurement analysis, and results from the water tank experiment are presented. Finally, the summary and conclusions are discussed.

2 Theoretical Analysis

The actual deep sea is simplified into a three-layer model consisting of seawater, basalt, and peridotite, as shown in Fig.1. This model is based on Hamilton’s studies (1974) and the results of geological surveys (Heacock, 1977).

Fig.1 Schematic of the marine environment.

Assume that all layers in the marine environment are isotropic media. Subscripts= 1, 2, 3 are used to represent the acoustic parameters in seawater, basalt, and peridotite, respectively;ρrepresents density;1is seawater sound velocity;c2andc3are compressional wave velocities (P wave); andc2andc3are shear wave velocities (S wave). Both1and2are interface depths in a two-layered seabed.

A time harmonic plane wave is considered, and time dependence is assumed as e?i. Subsequently, all potentials can be represented as follows:

where2πis the angular frequency,=/is the acoustic wavenumber, andZ() andZ() are the depth- dependent functions for compressional and shear waves, respectively. Substituting Eq. (2) into Eq. (1), the general solutions for potentials can be obtained by the free surface condition and radiation condition at infinity.

In the equations above,k2?2?2ξ2, and1=/1are the wavenumbers in water;k=/c(= 2, 3) are the com- pressional wavenumbers, andk=/c(= 2, 3) are shear wavenumbers in the elastic seabed. In addition,,1,2,1,2,, andare undetermined coefficients for potentials.

Here, it is assumed thatis displacement andis stress tensor. Thus,uis the horizontal displacement,uis the vertical displacement,Tis the normal stress, andTis the shear stress.

The displacements are determined based on the relation between the velocity potentials and displacements as follows:

According to the stress-strain constitutive relation, the following equations can be obtained:

The boundary condition at the interface between water and basalt (=1) is satisfied by the continuity of the normal displacement, normal stress, and zero tangential stress.

The boundary condition at the interface between basalt and peridotite (=2) is satisfied by the continuity of displacements and stresses:

Substituting the potential equation (Eq. (3)) into boundary conditions Eqs. (6) – (7), a system of linear equations can thus be obtained:

Elements in the first matrix represent the known coefficients in the equations of the boundary conditions. Elements in the first matrix are a function of, such that Eq. (8) can be simplified as follows:

whererepresents a zero vector, andrepresents the undetermined coefficients for potentials and cannot be zero. Given that Eq. (9) is homogeneous, non-zero solutions for coefficients in vectorexist only when the determinant of matrix()(det(()) = 0) is zero. Therefore, the values ofthat make det(()) = 0 are the eigenvalues for this question. Once an eigenvalue has been found, the coefficients incan be calculated by solving the linear equations. Many methods can be used to determine thevalues, such as the Newton method (Hall, 1983) and bisection (Jensen, 2011), among others. To accurately determine the wavenumber, the KRAKENC program (Porter, 1992) is used. Substituting the wavenumberinto Eq. (8), coefficients in matrixcan be calculated. Then, the normal stresses,T, can be computed by the relations between the stress and potentials. Acoustic pressure in water is simply the negative ofT, according to the definition. Eventually, the pressure model function in water can be obtained using the theory mentioned above.

The parameters of the ocean environment (Jensen, 2011) are presented in Table 1. As shown in Fig.2(a), the dispersion curves of phase velocities for this environment can be computed using the KRAKENC program. The dash- ed lines in the figure present these normal modes that are not shown, and the dotted lines are the shear wave velocities of peridotitec3and water acoustic velocity1. The curves in Fig.2(a) illustrate that the Scholte wave and normal modes in water exist in this model. There is no cutoff frequency for the Scholte wave, which is also called the zeroth mode; however, all normal modes have cutoff frequencies.

Table 1 Parameters of the marine environment

Fig.2 (a) Dispersion curves of the Scholte wave and normal modes in the marine environment; (b) pressure mode functions for a 10 Hz source.

The phase velocities of the Scholte wave and normal modes at 10 Hz are listed in Table 2. The data indicate that the velocities of the Scholte wave and normal modes are extremely close, which means that the Scholte wave in the field cannot be identified using velocity features. In this paper, the proposed method can identify the Scholte wave in conditions without the effect of velocities, and the identification principle is introduced here.

Table 2 Phase velocities of the Scholte wave and normal modes at 10 Hz

The pressure field for a single point source can be represented as a sum of the normal modes. As such, pressure can be written as:

whereΨ() is a mode function,Φ() is a mode coefficient, and the subscriptis an order of modes. Here, the variable0() is used to represent the amplitude of the Scholte wave in the acoustic field. We assume that there is no continuous spectrum so that the modes form a complete set. Applying the operator to this sum, the coefficientsΦ() can be calculated, where() is the density of media.

Mode analyses based on the elastic normal mode are applied to the acoustic field. The pressure mode functionsΨ() at a source frequency of= 10 Hz are illustrated in Fig.2(b), and the normal stressTis illustrated in basalt and Peridotite. The mode distribution of the Scholte wave indicates that the Scholte wave energy exists in the water pressure field. Combining with the solution of mode coefficientsΦ() in normal-mode theory, this indicates that the excitation intensity of Scholte waves0() may be affected by source depths and propagation distances.

The finite element method was applied in the current study to simulate the ocean acoustic field in the constructed model to verify the above analysis. The environmental para- meters in the simulation are in line with those in Table 1. The Scholte wave amplitude in the pressure field is extracted using the mode decomposition method. The source frequency is= 10 Hz, and a vertical receiving array is set to take samples at an interval of 10 m at a depth from 0 to 3500 m. Fig.3(a) shows the Scholte wave amplitudes (with different source depths) at a 3000-m horizontal distance away from the source. Furthermore, Scholte wave amplitudes can vary with range during propagation. Furthermore, when the source depth is SD = 2950 m, the Scholte amplitudes vary from the propagation distance, as shown in Fig. 3(b).

Fig.3 (a) Scholte wave amplitude versus source depth; (b) Scholte wave amplitude versus propagation distance.

The results in Fig.3 indicate that the excitation amplitude of Scholte waves is affected by source depths, and the received amplitude is affected by the excitation amplitude and propagation distances. The Scholte wave can be identified in the water pressure field using these two characteristics. Thus, a water tank experiment is designed according to this principle.

3 Tank Experiment

3.1 Acoustic Field Analysis for the Laboratory Environment

The ocean model is scaled down to laboratory size according to the similarity principle. Basalt and peridotite in the seabed are modeled by brass and iron slabs because they have close velocities. Table 3 lists a scale model that is presented at a scale of 1:5000, where the water depth and source frequency have been appropriately modified at 0.6 m and 50 kHz, respectively. Table 3 lists the media parameters and sizes.

Table 3 Media parameters and sizes in the tank experiment

Fig.4 shows the pressure mode functionsΨ() of the Scholte wave and normal modes for a 50 kHz source in the water tank. As can be seen, the mode functions in the two environments are proven to be consistent by comparing the results in Figs.2(b) and 4. Moreover, it reveals that the parameters for the scale model experiment are correct.

Fig.4 Pressure mode functions for a 50 kHz source.

3.2 Experiment Settings

The scale model experiment is performed in a water tank (:3 m ×:2 m ×:1 m). Reflection waves from the tank’s walls are absorbed by absorbing wedges. The tank is filled with water to a height of 0.6 m, where the sound speed in water is measured to be 1485 m s?1. A spherical transducer, the source level of which is measured at approximately 140 dB (re 1 μPa m V?1) in the frequency range used in the experiment, is used as the source. An RHCA-7 hydrophone is used as the receiver. The sensitivities of hydrophones are about ?210 dB (re 1 V μPa?1) in the frequency range from 20 – 100 kHz. The vertical array, including 119 receiver elements, is obtained by the synthetic aperture method. The source and receiver hydrophones are positioned in water using a robotic apparatus (with an accuracy of 0.01 mm), which allows for accurate positioning. The hydrophone measures depth from 0.5 – 59.5 cm, with a depth interval of 0.5 cm. Fig.4 shows the receiver positions in the form of red circles, and the configuration of the experiment equipment is shown in Fig.5.

Fig.5 Diagram of the experimental system setup.

3.3 Experimental Data Analysis

The source radiates ten cycles of sine waves at a frequency of= 50 kHz in 1-s time intervals. Fig.6(a) shows the output signal of a power amplifier that is used as a reference signal for a measurement with a vertical array of 119 hydrophones. Meanwhile, Fig.6(b) illustrates the received waveform of a hydrophone, representing the tem- poral correlation between the received sound pressure by the hydrophone and the reference signal. By the temporal correlation, the virtual receiving array plays the same role as the real receiving arrays of the same length. Therefore, the received waveforms by the virtual array are used to obtain the complex sound pressure required for identifying Scholte waves. To eliminate the effect of reverberation in the water tank, received waveforms are truncated by assuming that the pulse duration of the received signal from the source is almost the same for each hydrophone. The truncated waveform is shown in Fig.6(c).

Through the fast Fourier transform of truncated waveforms, we obtain the complex sound pressure for the hy- drophone at different positions. The sound pressure at 50 kHz is extracted to compose the matrix(), including 119 elements in the depth direction. Finally, the amplitudes of Scholte waves in the acoustic pressure field can be obtained using the mode decomposition method based on the orthogonality.

To study the effect of source depth, ten positions are set evenly for the source in the depth range from 0.1 – 0.55 m; the hydrophone array is at a horizontal distance of 0.6 m away from the source. The result in Fig.7(a) shows the nor- malized amplitudes of the Scholte wave and normal modes for a source depth of SD = 55 cm. It is demonstrated that the Scholte wave has the largest amplitude. Furthermore, the amplitudes of each mode in the experiment are in excellent agreement with the theoretical calculations. The Scholte wave amplitudes in the experiment field that vary with source depth are presented in Fig.7(b), and the theory is a simulation using the finite element method. The results in Fig.7 reveal that the excitation amplitudes of Scholte waves in the experiment are consistent with the theoretical calculations.

To study the effect of propagation distance, the source is maintained at a depth of SD = 0.55 m. Seven positions are set evenly for the receiving array within the horizontal range of 0.15 – 1.05 m away from the source. Scholte wave amplitudes in the experiment field that vary with the prop- agation distance are presented in Fig.8. As can be seen, Scholte wave amplitudes decrease as the propagation distance increases. The changes in Scholte wave amplitudes in the experiment are consistent with theoretical calculations. Furthermore, this study shows that the Scholte wave can be measured and identified by the method proposed in this paper.

Fig.6 (a), Reference signal for a vertical array measurement; (b), the received waveform of a hydrophone with a source depth of SD = 55 cm (temporal correlation between the received sound pressure and the reference signal); (c), the truncated waveform adopted for identifying Scholte waves.

Fig.7 (a), Normalized amplitudes of the Scholte wave (0th order) and normal modes for a source depth, SD = 55 cm; (b), Scholte wave amplitude versus source depth.

Fig.8 Scholte wave amplitude versus propagation distance.

4 Discussion

In existing methods, Scholte waves are often identified by their velocity features and the seismic field; however, the method will not be effective when the velocities of Scholte and acoustic waves in the water are close to each other. This paper proposed an identification method for Scholte waves based on their characteristics of excitation and propagation, unlike existing methods. For this purpose, the sound pressure in water is applied to extract the composition of Scholte waves. The experimental results in Figs.7 and 8 illustrate that changes in Scholte wave amplitudes depending on different source depths and propagation distances are consistent with the theoretical calculations. Therefore, the proposed method can effectively identify the Scholte wave without the limitation on velocities.

There are, however, certain discrepancies in the details between the experiment and the calculation in this study. Therefore, the error analysis is performed. As the mode decomposition method is adopted to obtain the amplitudes of Scholte waves in this paper, the orthogonality between Scholte and normal modes waves is essential to ensure the accuracy of the results. Theoretically, the upper limit of the integral in Eq. (11) is infinity, but only the range of water depth can be measured in practice; thus, modes cannot be strictly orthogonal. The orthogonal coefficients of Scholte mode (0th mode) and each mode in the water column are shown in Fig.9. As can be seen, the Scholte mode is not strictly orthogonal to higher order normal modes. Therefore, this could be the cause of errors in the amplitudes of Scholte waves.

The very-low-frequency sound studied in this paper can penetrate the unconsolidated sediment and generate the Scholte wave on an elastic seabed; hence, the effect of sed- iment is not considered in the paper. Studies in the future will adopt a more complex acoustic ocean model.

Fig.9 Orthogonal coefficients of the Scholte mode (0th mode) and each mode in water.

5 Conclusions

In this paper, a method based on the excitation and prop- agation features of Scholte waves is proposed to identify Scholte waves using their acoustic pressure field. A series of laboratory experiments with the scaled model of the elastic ocean bottom are designed according to the actual ocean environment and then performed to measure and identify Scholte waves at the seafloor. The study results show that changes in Scholte wave amplitudes depending on different source depths and propagation distances are consistent with the theoretical results. Thus, the Scholte wave at the seabed interface is measured and identified by means of the acoustic pressure field in water.

Compared with existing methods, the proposed method is not limited by the velocities of Scholte waves and acoustic waves in water. The pressure field in water is measured by a hydrophone, and its measurement is not affected by complex seabed conditions. Thus, it has an obvious advantage over OBSs and geophones. Finally, this paper proposes a new approach for detecting and identifying Scholte waves in the actual ocean environment in the future.

Acknowledgements

This study was funded by the National Natural Science Foundation of China (No. 11474258), and the State Key Laboratory of Acoustics (No. SKLA202206).

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? Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023

. E-mail: penglh@ouc.edu.cn

(Edited by Xie Jun)