Research on the Supergain Properties and Influencing Factors of a Vector Hydrophone Vertical Array in the Deep Sea

Abstract

Increasing array gains is one of the keys to improving underwater targets’ detection capabilities. This paper presents a high-gain approach for a vector hydrophone vertical array (VHVA) that combines white noise gain constraint (WNGC) with vector joint processing to preserve strong robustness and provide noticeable gains. Firstly, this approach treats the VHVA as four independent sub-arrays and achieves sub-array supergains by decorrelating noise using WNGC. The beam outputs of the four sub-arrays are then equated to a single-vector hydrophone, the combination gain of which is obtained by leveraging the strong signal correlation and the weak noise correlation between the sound pressure and the particle velocity. Lastly, the sub-array supergain and combination gain are superposed to provide the spatial gain of the VHVA. It is also summarized that low-frequency signals, coherent noise, accurate elevation-angle estimation, and stable phase differences are required for the VHVA to achieve supergain. The simulation and sea trial confirm that this approach can effectively boost the array gain. The maximum spatial gain in the experiment was increased by 9 dB at a range twice the sea’s depth while operating at a low frequency. This method shows enormous potential for improving the performance of deep-sea target detection.

1. Introduction

Hydrophone arrays are essential underwater acoustic sensors. Compared to a single hydrophone, one benefit of using hydrophone arrays is the considerable array gain. Array gain, which reflects the degree to which an array improves the signal-to-noise ratio (SNR) of a received signal, is a significant parameter for evaluating target-detection capability. Supergain is therefore crucial for efficiently gathering underwater information and enhancing target-detection ranges.

Supergain research initially appeared in the field of antennas and described the use of small-aperture arrays to achieve processing gains greater than those obtained through conventional means. Harrington [1] suggested that array antennas can achieve supergain when the spacing between array elements is narrow enough. Haviland [2] provided an overview of several issues related to antenna supergain. Subsequently, a number of optimizations were progressively performed to address pivotal problems like robustness, compact structure, bandwidth, and so on [3,4,5,6]. As technology advanced, the concept and method of supergain were gradually applied in fields of speech signal processing and acoustic array signal processing [7].

Capon’s renowned minimum variance distortionless response (MVDR) beamforming technique is regarded as the optimal gain method in the case of isotropic noise [8]. However, the performance of MVDR beamforming is severely degraded under non-ideal conditions, such as a small number of snapshots and array manifold mismatch. Diagonal loading [9], statistical class [10,11,12], and white noise gain-constraint methods [13,14] can improve robustness while preserving high processing gain. Nevertheless, it is challenging to determine the value of diagonal loading, and statistical class methods necessitate a priori knowledge of the error’s statistical properties [15]. White noise gain constraints can be addressed numerically using an optimization toolbox, though this increases the computational time.

Superdirectivity techniques have been proposed as a means of achieving supergain. The current superdirectivity approaches mostly rely on phase mode (PM) and eigenbeam decomposition and synthesis (EBDS) models. To obtain the desired directivity, PM technology multiplies the coefficients after orthogonally decomposing the acoustic field by using the symmetry of spherical and circular arrays [16,17]. The robust processing method and the multi-constraint optimization method of the array element domain were gradually applied to the phase modal domain [18,19]. Researchers analyzed the effects of array and acoustic field errors on the method and discussed error-resistant methods [20,21]. The EBDS method was proposed by Ma et al. as a precise, closed-form solution for superdirectivity [22]. Subsequently, Wang et al. published a series of optimization techniques to compromise superdirectivity and robustness [23,24,25]. Overall, superdirectivity methods are mostly applicable to spherical and circular arrays, which limits their application range. Further, superdirectivity and supergain are not exactly equal, since the noise in real marine environments is not completely isotropic or homogeneous.

Vector hydrophones can measure sound pressure and particle velocity information simultaneously. In addition, their intrinsic dipole directivity offers the benefit of enhancing the spatial gain of transducers. Qi et al. employed supergain energy flux beamforming to the pressure−gradient vector hydrophone crossing array with the aim of obtaining a narrower main lobe and lower side lobes [26]. Guo et al. derived an upper limit on the maximum order of modes used for mode synthesis in an isotropic uniform noise field. This method can be generalized to other types of arrays even though it is based on uniform linear and circular arrays [27]. Su et al. [28] computed the combination gain of a vector array in a shallow sea by joint processing of sound pressure and particle velocity. Yu et al. [29] discussed the spatial correlation of isotropic and surface noise vector fields, and they also analyzed the combined SNR output in these noise fields. Liang et al. [30] discussed array gain under various combined forms and validated that combination processing can improve spatial gain by 2–3 dB with experimental data.

In summary, the majority of studies in the literature focused on high-spatial-gain techniques for the scalar array, and little research has been carried out on the supergain of the vector hydrophone vertical array (VHVA). Moreover, the factors that affect VHVA gain have not yet been analyzed and elucidated.

This paper analyzes a supergain approach for a vector hydrophone vertical array. Firstly, the robust white noise gain constraint method was applied to each sub-array of the VHVA. After that, weighting and phase compensation were utilized to construct the equivalent vector element. Lastly, we performed joint processing for sound pressure and particle velocity to improve the VHVA processing gain and determine the parameters that affect spatial gain. Both a simulation and sea trials demonstrated that the method can achieve significant gain improvement for VHVAs in strong noise fields and enhance target-detection performance in deep seas.

The rest of this paper is arranged as follows. Section 2 introduces the basic theory of the high-gain method and combined processing. In Section 3 and Section 4, the simulation and sea trials conducted to validate the reliability and practicability of the method are described. Also, the influencing factors are analyzed. In Section 5, the conclusions are summarized and future research prospects are delineated.

2. Basics Theory

2.1. Received Signal Model of VHVA

The vector hydrophone vertical array is composed of $M$ elements, and the element spacing is $d$. It is assumed that the scalar sensor is non-directional and that $K$ targets are incident into the VHVA at elevation angle ${\phi }_k$ and azimuth angle ${\theta }_k$. The schematic diagram of the signal received by the VHVA is shown in Figure 1.

The signal received by the VHVA is written in the form of a matrix:

$$\mathit{X}\left(t\right)=\mathit{A}\left(\theta ,\phi \right)\mathit{S}\left(t\right)+\mathit{N}\left(t\right)$$

(1)

where $\mathit{S}\left(t\right)={\left[s_1\left(t\right),s_2\left(t\right),\cdots s_K\left(t\right)\right]}^T$ and $\mathit{N}\left(t\right)={\left[{\mathit{n}}_1\left(t\right),{\mathit{n}}_2\left(t\right),\cdots {\mathit{n}}_M\left(t\right)\right]}^T$ are the incident signal vector and the noise vector. $s_k\left(t\right)$ is the $k$th transmitted signal. $\mathit{A}\left(\theta ,\phi \right)$ is the array manifold vector, whose expression is

$$\mathit{A}\left(\theta ,\phi \right)=\left[\mathit{a}\left({\theta }_1,{\phi }_1\right)\otimes {{\mathit{u}}_1}^T,\mathit{a}\left({\theta }_2,{\phi }_2\right)\otimes {{\mathit{u}}_2}^T,\cdots \mathit{a}\left({\theta }_K,{\phi }_K\right)\otimes {{\mathit{u}}_K}^T\right]$$

(2)

The symbol $\mathit{u}$ is the unit direction vector of a single-vector sensor, which can be written as

$${\mathit{u}}_k=\left[1,\cos {\theta }_k\sin {\phi }_k,\sin {\theta }_k\sin {\phi }_k,\cos {\phi }_k\right]$$

(3)

The notation $\mathit{a}\left({\theta }_k,{\phi }_k\right)$ is the manifold vector of a scalar sensor array. For the vertical array, the phase-delay differences among elements are solely dependent on the elevation angle and are independent of the azimuth angle. Thus, the manifold vector $\mathit{a}\left({\theta }_k,{\phi }_k\right)$ can be abbreviated as $\mathit{a}\left({\phi }_k\right)$. The time delay among the hydrophones is determined by the element interval, incidence angle, and propagation speed, i.e., $\tau =\left(M-1\right)\cdot d\cos {\phi }_k/c$. For the narrow signal, the time delay can be approximated by a phase shift, and thus, the notation $\mathit{a}\left({\phi }_k\right)$ can be simplified as

$$\mathit{a}\left({\phi }_k\right)={\left[1,e^{-j2\pi d\cos {\phi }_k/\lambda },\cdots ,e^{-j\left(M-1\right)2\pi d\cos {\phi }_k/\lambda }\right]}^T$$

(4)

where $\lambda$ denotes the signal wavelength.

The symbol $\otimes$ represents the Kronecker product. The array manifold of the VHVA can be expressed as the Kronecker product between the array manifold of a scalar array and that of a single-vector sensor.

The time delay for the broadband signal cannot be readily expressed in the time domain since it is difficult to articulate in terms of individual phase shifts. In contrast, its frequency domain representation is more convenient. The frequency domain expression of Equation (1) can be derived by utilizing the Fourier transform.

$$\mathit{X}\left(f\right)=\mathit{A}\left(\theta ,\phi \right)\mathit{S}\left(f\right)+\mathit{N}\left(f\right)$$

(5)

where $f$ represents the frequency.

A VHVA contains M sound pressure channels, M horizontal particle velocity X channels, M horizontal particle velocity Y channels, and M vertical particle velocity channels. Among them, a number of M sound pressure channels can be regarded as a sound pressure sub-array. Similarly, other 3M channels can be viewed as horizontal particle velocity X, horizontal particle velocity Y, and vertical particle velocity sub-arrays. Each of these four separate sub-arrays can adopt beamforming to obtain the spatial angle. Therefore, to achieve elevation angle estimation, the four sub-arrays representing sound pressure and three orthogonal particle velocity components are passed through the beamformers, respectively.

Array gain describes the ratio of the beamforming output SNR to the input SNR of a single hydrophone, of which the formula is as follows:

$$AG=\frac{SN{R}_{out}}{SN{R}_{in}}=\frac{P_s/P_n}{{\sigma }_s^2/{\sigma }_n^2}=\frac{{\mathit{w}}^H{\rho }_s\mathit{w}}{{\mathit{w}}^H{\rho }_n\mathit{w}}$$

(6)

where ${\rho }_s$ and ${\rho }_n$ represent the normalized signal covariance matrix and the noise covariance matrix of each sub-array.

It is demonstrated from Equation (6) that the array gain depends on the signal correlation coefficient matrix, the noise correlation coefficient matrix, and the weighting strategy. Generally speaking, the correlation between the signals of each array element is high after beam compensation, and thus, the array gain is mostly determined by the noise correlation coefficient after weighting compensation. As a result, it is a proven means to boost the array gain by adjusting the weighting methods.

2.2. Supergain Method

2.2.1. Minimum Variance Distortionless Response Method

The MVDR method is a classical high-gain method, the basic idea of which is to ensure that the output power of the target through the beamformer remains constant while the noise output power is minimized. The mathematical expression is as follows:

$$\begin{array}{l}\underset{\mathit{w}}{\min }\hspace{1em}{\mathit{w}}^H{\mathit{R}}_n\mathit{w}\\ s.t.\hspace{1em}{\mathit{w}}^H\mathit{a}\left({\phi }_0\right)=1\end{array}$$

(7)

where ${\mathit{R}}_n$ is the noise covariance matrix of each sub-array and ${\phi }_0$ is the target elevation angle.

The above optimization problem can be solved through the Lagrange multiplier method as follows. Firstly, we introduce the transfer operator to construct a cost function. Secondly, we implement the scalar to matrix vector derivation by setting the derivative of the cost function to zero and applying the differential method. Finally, the derivative outcomes are integrated with the constraints to determine the weight vector, expressed as follows:

$${\mathit{w}}_{MVDR}=\frac{{{\mathit{R}}_n}^{-1}\mathit{a}}{{\mathit{a}}^H{{\mathit{R}}_n}^{-1}\mathit{a}}$$

(8)

Obtaining the noise variance matrix directly is often challenging, and thus the sampling covariance matrix $\mathit{R}$ is commonly utilized as a substitute for ${\mathit{R}}_n$. When the noise field satisfies the spatial white noise model, the weights above degenerate into conventional beamforming weights.

The MVDR method has the potential to yield considerable array gain theoretically. However, the performance degrades seriously in the case of limited snapshots and insufficient SNR. Furthermore, signal cancellation may occur when the observation direction of the beam deviates from the actual target direction. The white noise gain constraint (WNGC) method was developed to address the issues of poor robustness.

2.2.2. White Noise Gain Constraint Method

White noise gain may be utilized to evaluate the robustness of a beamformer. Incorporating white noise constraints into error-sensitive beamforming algorithms can enhance their adaptability in intricate settings. This is accomplished by imposing a paradigm constraint on the weighting vectors as outlined in Section 2.2.1.

$$\begin{array}{ll}\underset{\mathit{w}}{\min }& {\mathit{w}}^H\mathit{R}\mathit{w}\\ s.t.& {\mathit{w}}^H\mathit{a}\left({\phi }_0\right)=1\\ & {‖\mathit{w}‖}_2\le \gamma \end{array}$$

(9)

where $\gamma$ is the constraint value and is generally less than $10\log M \mathrm{dB}$. The notation ${‖·‖}_2$ denotes the 2-norm.

The optimization problem can also be solved using the Lagrange multiplier method.

$${\mathit{w}}_{WNGC}=\frac{{\left(\mathit{R}+\lambda \mathit{I}\right)}^{-1}\mathit{a}}{{\mathit{a}}^H{\left(\mathit{R}+\lambda \mathit{I}\right)}^{-1}\mathit{a}}$$

(10)

The above structure shows that the white noise gain constraint is equivalent to the diagonal loading algorithm, but the solution process is more complicated. This convex optimization issue can be transformed into a second-order cone programming (SOCP) problem and addressed using mathematical tools. First, the sampling covariance matrix $\mathit{R}$ is subjected to Cholesky decomposition.

$$\mathit{R}={\mathit{U}}^H\mathit{U}$$

(11)

The first term in Equation (9) can be simplified as ${\mathit{w}}^H\mathit{R}\mathit{w}={\left(\mathit{U}\mathit{w}\right)}^H\left(\mathit{U}\mathit{w}\right)={‖\mathit{U}\mathit{w}‖}^2$ and further written as

$$\begin{array}{l}\underset{\mathit{w}}{\min }\hspace{1em}‖\mathit{U}\mathit{w}‖\\ s.t.\hspace{1em}{\mathit{w}}^H\mathit{a}\left({\phi }_0\right)=1, {‖\mathit{w}‖}^2\le \gamma \end{array}$$

(12)

According to the conditions that need to be satisfied through the SOCP method, the non-negative variable $y_1$ is introduced and the variables $\mathit{y}$ and $\mathit{b}$ meet the following relation:

$$\begin{array}{l}\mathit{y}={\left[y_1\hspace{1em}\mathit{w}\right]}^T\\ \mathit{b}={\left[-1\hspace{1em}{0}_{1\times M}\right]}^T\end{array}$$

(13)

The equation $-y_1={\mathit{b}}^T\mathit{y}$ holds. Substituting Equation (13) into Equation (12) generates the following expression:

$$\begin{array}{l}\underset{\mathit{w}}{\min }\hspace{1em} -y_1\\ s.t.\hspace{1em}{\mathit{w}}^H\mathit{a}\left(\theta \right)=1, ‖\mathit{U}\mathit{w}‖\le y_1, ‖\mathit{w}‖\le \sqrt{\gamma }\end{array}$$

(14)

Three constraints satisfy the zero- and second-order cone forms.

$$\begin{array}{l}1-{\mathit{a}}^H\mathit{w}=1-\left[\begin{array}{cc}0& {\mathit{a}}^H\end{array}\right]\mathit{y}\triangleq {\mathit{c}}_1-{{\mathit{A}}_1}^T\mathit{y}\in \left\{0\right\}\\ \left[\begin{array}{c}{y}_1\\ \mathit{U}\mathit{w}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]-\left[\begin{array}{cc}-1& 0_{1\times M}\\ 0& -\mathit{U}\end{array}\right]\cdot \mathit{y}\triangleq {\mathit{c}}_2-{{\mathit{A}}_2}^T\mathit{y}\in Qcon{e}_1^2\\ \left[\begin{array}{c}\sqrt{\gamma }\\ \mathit{w}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]-\left[\begin{array}{cc}0& 0_{1\times M}\\ 0& 1_{1\times M}\end{array}\right]\cdot \mathit{y}\triangleq {\mathit{c}}_3-{{\mathit{A}}_3}^T\mathit{y}\in Qcon{e}_2^2\end{array}$$

(15)

where $\mathit{c}\triangleq {\left[{{\mathit{c}}_1}^T,{{\mathit{c}}_2}^T,{{\mathit{c}}_3}^T\right]}^T$ and $\mathit{A}\triangleq {\left[{\mathit{A}}_1,{\mathit{A}}_2,{\mathit{A}}_3\right]}^T$. Equation (15) meets the SOCP standard form, and the weighting vector can be solved using the MATLAB toolbox. The estimated weighting $\widehat{\phi }$ is then used to compensate for the phase differences among hydrophones of each sub-array. By coherently superimposing the received signals of each element and incoherently superimposing the noise, this procedure improves the array gain. Following phase compensation, the received signal can be denoted as

$${\mathit{Y}}_i\left(f\right)={\mathit{w}}_{WNGC}{\left(\widehat{\phi }\right)}^H{\mathit{X}}_i\left(f\right), i=p,x,y,z$$

(16)

2.3. Equivalent Vector Element and Joint Processing Method

The VHVA signal after phase compensation comprises four channels, as shown in Section 2.2.2, and is thus regarded as an equivalent vector element. The four channels in Equation (16) satisfy the following relation:

$$\left[\begin{array}{cccc}{\mathit{Y}}_p\left(f\right)& {\mathit{Y}}_x\left(f\right)& {\mathit{Y}}_y\left(f\right)& {\mathit{Y}}_z\left(f\right)\end{array}\right]=\left[\begin{array}{cccc}1& \cos \widehat{\theta }\sin \widehat{\phi }& \sin \widehat{\theta }\sin \widehat{\phi }& \cos \widehat{\phi }\end{array}\right]{\mathit{Y}}_p\left(f\right)$$

(17)

where the notation $\widehat{\theta }$ represents the azimuth angle estimated using the complex acoustic intensifier method [30].

The equivalent vector hydrophone has a high SNR due to fully utilizing the array gain of sub-arrays. The sub-array gains calculated from Equation (6) are expressed as $A{G}_{sub}$.

Under the far-field assumption, the sound pressure and particle velocity signals are in phase, whereas the noise phase between them is random. Thus, a joint processing model for sound pressure and particle velocity is constructed by utilizing these characteristics. In general, sound pressure and particle velocity can produce a variety of combinations.

Table 1. Theoretical combination gain.

Number	Form	Combination Gain
1	$p\cdot v_x$	$2\cos \theta$
2	$\left(p+v_x\right)\cdot p$	$\sqrt{0.8}\left(1+\cos \left(\theta -\widehat{\theta }\right)\right)$

lists the more commonly used combinations using the horizontal particle velocity X as an example and the combination gain expression for an isotropic uniform noise field.

In the table, the symbols $\theta$ and $\widehat{\theta }$ represent the searching and estimated azimuth angles. This combination gain, abbreviated as CG, is closely related to the combination form and noise field characteristics. In practice, the signal and noise properties of channels of an equivalent vector hydrophone determine which combination form is best. Combining the sub-array gain and combination gain derives the spatial gain of the VHVA.

$$S{G}_{VHVA}=A{G}_{sub}+CG$$

(18)

The above white noise gain constraint and combination processing is abbreviated as WNGC-CP.

3. Simulation

This section presents a simulation conducted to confirm the WNGC-CP performance. The results are contrasted with the outcomes of CBF and MVDR. The factors affecting the supergain of the VHVA are identified and verified.

3.1. Simulation Settings

The sound speed profile (SSP) was measured in a sea trial in the South China Sea, which is a non-complete deep ocean channel. The sound speed variation and the environmental parameters are shown in Figure 2.

Table 2. Simulation parameters.

Parameter	Value	Parameter	Value
Element number	8	Signal form	Single-frequency signals of 80/100/140 Hz
Element interval	7.5 m	Duration time	30 s
Array aperture	52.5 m	Source range	2.4 km
Array depth	1750–1802.5 m	Source depth	30 m
Sea depth	2000 m	Azimuth angle	120°
Sample rate	8 kHz	SNR	5 dB

lists the critical simulation conditions, in which noise consists of spatial white noise.

The principle of ray theory is simple and the physical meaning is clear, which applies to the distance-independent deep-sea environment in this paper. Thus, we use a bellhop model to simulate the received vector acoustic field and the noise vector field. Specifically, the received signals can be treated as a superposition of multiple eigen-rays, and thus the impulse response of the sound pressure and three particle velocity channels can be derived [31]. Convolution between the channel impulse response and the source signal yields the signal received by the VHVA.

The reference element is the one at the top of the vertical array. For convenience, the sub-arrays for sound pressure, the X and Y axes of horizontal particle velocity, and the vertical particle velocity are designated as P, X, Y, and Z, respectively.

3.2. Simulation Results

The VHVA can be viewed as four independent sub-arrays with identical element numbers. Without loss of generality, each sub-array’s received signal passes through the CBF, MVDR, and WNGC beamformers, respectively. The beam scanning azimuth spectra (BSAS) are displayed in Figure 3a–c, and Figure 3d depicts a sliced view of Figure 3a–c at one moment. It is indicated that the main lobe points to 53° for all methods, with WNGC having the narrowest main-lobe width and lowest side-lobe level. This main-lobe angle is then applied to compensate for the phase differences among hydrophones in sub-arrays and then to calculate the array gain.

Compared to Equation (6), the beam output power spectrum is a method for visualizing the array gain, which provides a more intuitive representation of the level of noise suppression. The improvement in the CBF output power spectrum compared to a single channel is referred to as CBF gain, abbreviated as AG-CBF. Likewise, the enhancement in the output power spectra resulting from MVDR and WNGC is referred as AG-MVDR and AG-WNGC. Figure 3e presents the output power spectra of three methods applied to different sub-arrays. In this paper, the calculation band is defined as ±5 Hz to the left and right of the normalized frequency, within which the output power spectrum is averaged and differenced as the array gain value. Since the noise satisfies the Gaussian white model, both MVDR and WNGC degenerate to CBF with an array gain of 7.4 dB. Therefore, one of the requirements for achieving supergain is coherent noise fields.

Therefore, we redesign the noise field with coherence. Twenty near-sea surface sources with a depth of 10 m are randomly distributed over a range of [10, 40] km and radiate broadband signals. The azimuths of these coherent noise sources are also spatially randomly distributed in the range of [0°–360°]. The SNRs for incoherent and coherent noise are set at −10 dB and −23 dB, respectively. The VHVA receives the signals, coherent noise, and incoherent noise, of which the BSAS of sub-arrays are displayed in Figure 4. We can observe that the elevation angle estimated using the WNGC corrects the errors generated by the MVDR approach and avoids main-lobe energy loss, as depicted in the P sub-array in Figure 4d. Furthermore, the WNGC processing significantly reduces the side-lobe height and maintains a narrow main-lobe width.

Figure 5 displays the beam output power spectra of four sub-arrays. Under this circumstance,

Table 3. Array gains of four sub-arrays of VHVA processed using three methods at 80 Hz and 100 Hz.

Gain (dB)	CBF	MVDR	WNGC	CBF	MVDR	WNGC
Frequency (Hz)	80			100
P sub-array	10.77	11.44	11.43	7.20	6.82	6.79
X sub-array	14.66	16.20	16.21	7.40	8.68	8.79
Y sub-array	9.09	9.58	9.59	8.30	8.28	8.29
Z sub-array	10.15	10.51	10.47	5.23	3.76	3.85

lists the processing gains at 80 Hz and 100 Hz because of the poor 140 Hz SNR. The AG−WNGC shows a 0.5–1.6 dB increase over the AG−CBF at a low frequency of 80 Hz. As the frequency rises, the supergain methods lose their gain advantage, except for the X sub-array. The Z sub-array even experiences a slight decrease in AG−WNGC. This is due to the high noise correlation among the hydrophones of the Z sub-array after phase compensation. As is known to us, array gain can be defined as signal gain minus noise gain. In this case, the signal gain is close to the ideal value, while the noise gain is positive, which somewhat weakens the Z sub-array gain. The simulation results show that the WNGC method has little effect on vertical particle velocity but can enhance the array gain for sound pressure and horizontal particle velocity. Thus, we only take into account the combination of $p.v_x$ and $p.v_y$ in the subsequent joint processing.

As shown in Figure 6 and

Table 4. Combination gains with the form of $p.v_i$ at 80 Hz and 100 Hz.

Gain (dB)		CG−CBF	CG−MVDR	CG−WNGC	CG−CBF	CG−MVDR	CG−WNGC
Frequency (Hz)		80			100
Combination	P. Vx	5.18	6.47	6.64	4.64	6.03	6.07
	P. Vy	5.27	5.97	5.98	4.80	5.44	5.58

, the four phase-compensated channels are approximated as an equivalent vector hydrophone, and the combination gain is computed using the combined form of $p.v_i,i=x,y$. It is worth noting that the combination gain is the difference between the combined power spectrum and the P channel power spectrum of the equivalent vector element. The combination gains of the three methods are represented by the notations CG−CBF, CG−MVDR, and CG−WNGC. It is evident that at 80 Hz, the CG−WNGC can reach a maximum value of 6.6 dB, while the CG−CBF is about 5.2 dB. As the frequency increases to 100 Hz, the CG−WNGC remains higher than the CG−CBF, indicating that the combined processing and WNGC processing can lead to a significant and stable combined gain. It is also observed that the combination gain of $p.v_x$ is slightly higher than that of $p.v_y$ due to the higher SNR. Hence, the form of $p.v_x$ is utilized to calculate the spatial gain of the VHVA.

Figure 7 shows a comparison of the maximum array gains of VHVA after WNGC and combination processing at various frequencies and the CBF gains for the sound pressure array. The two are referred to as WNGC−CP−VHVA and CBF−PHVA, respectively. CBF−PHVA shows a 10.77 dB gain at 80 Hz, whereas WNGC−CP−VHVA can reach a gain of 21.3 dB. The results reveal that the conventional method can achieve spatial gains exceeding $10\log M$ dB in low-frequency coherent noise environments, whereas WNGC−CP processing could increase the gain by 10 dB or more. As the frequency rises, the value of WNGC−CP−VHVA decreases but still offers a gain advantage over CBF−PHVA. Hence, the low-frequency signal is the second requirement for obtaining supergain.

As the receiving distance increases, the SNR gradually decreases, and the performance of the WNGC−CP method also gradually deteriorates. The WNGC−CP method struggles to accurately estimate the elevation angle beyond 6 km with the aforementioned SNR condition. These errors prevent full compensation for the phase difference among hydrophones, leading to a loss of array gain. Consequently, the accuracy of elevation angle estimation is crucial for achieving optimal gain.

The stability of the phase difference between the sound pressure and particle velocity is essential for realizing high gain. Previous research suggests that a VHVA can be equivalently regarded as a single-vector sensor, with combination gain tied to the phase characteristics of sound pressure and particle velocity. Generally speaking, results from multiple processes are averaged to increase accuracy. Hence, the stability of the phase difference between sound pressure and particle velocity significantly impacts the ultimate combination gain.

To simplify the arithmetic, signals from one single-vector hydrophone’s axes are analyzed directly. The simulation parameters are kept unchanged except that the incoherent SNR and coherent SNR are adjusted to 0 dB and −5 dB. Constant phase errors and time-varying random errors are added to each channel to assess their impact on combination gain. Figure 8a depicts the phase difference fluctuation between sound pressure and particle velocity over time for a reception distance of 2.4 km and a signal frequency of 80 Hz. Cases 1–3 represent error-free, constant phase error and varying phase-over-time scenarios. Figure 8b displays the combination gain corresponding to the three cases. It is evident that a constant phase difference only alters the phase relationship between channels and does not affect array combination gain. Time-varying random errors deteriorate the combination gain, resembling “noise” in terms of phase features. Note that the $p.v_x$ combination gain is greater than the $p.v_y$ combination gain at this point, which is caused by randomly distributed vector noise sources. Consequently, the sound pressure and particle channels must maintain a steady phase difference, which is crucial for achieving optimal gain.

4. Sea Trial

4.1. Experiment Settings

The sea trial was conducted in the South China Sea in 2022. A vector hydrophone vertical array, composed of eight elements with an interval of 7.5 m, was deployed at a depth of 1750 m–1802.5 m. The sound speed profile was measured using the conductivity temperature depth (CTD) apparatus, as shown in Figure 2. After traveling a predetermined distance, the motor was turned off and the sound source was suspended to a depth of thirty meters underwater. The global positioning system (GPS) was installed on both the VHVA and the ship to record the real-time position. Figure 9 presents the schematic diagram of the experiment. From the element depth recorded with the TD sensors and the azimuth angle of each vector hydrophone, it can be determined that the array was vertical during the experiment. Specific parameters for the sea trial are listed in

Table 5. Experimental parameters.

Parameter	Value	Parameter	Value
Element number	8	Signal form	Four single-frequency signals of 70/100/130/160 Hz
Element interval	7.5 m	Duration time	30 s
Array aperture	52.5 m	Horizontal range	1.4 km/9.2 km
Array depth	1750–1802.5 m	Source depth	30 m
Sea depth	2000 m	Sample rate	8 kHz

.

4.2. Experiment Results

As mentioned earlier, the signal frequency, coherent noise, accurate elevation angle, and noise characteristics are factors that affect supergain, which is demonstrated experimentally in this subsection.

4.2.1. Horizontal Range of 1 km

The VHVA was used to receive the acoustic signal. Figure 10 presents the spectrogram of the signal received by the first vector hydrophone, from which we can find the line-spectrum components of 130 Hz and 160 Hz. Due to the high energy of low-frequency noise, the line spectra at 70 Hz and 100 Hz are submerged in the noise. There exists a strong interference of 90 Hz in the Z channel, which is caused by cable disturbance and does not affect the signal components during processing.

The BSAS of the three algorithms are shown in Figure 11, from which it can be found that all maximum power points to 38° for the three methods, and the WNGC has the narrowest beamwidth of the three with a relatively low side-lobe height. The high-power grating lobe at the angle of 137° is due to the processing frequency of 130 Hz being greater than the center frequency corresponding to the element spacing. Thus, the main-lobe angle at 38° is used to compensate for the phase differences among hydrophones.

As mentioned earlier, the white noise gain can characterize the beamformer’s robustness. Figure 12 displays the spatial distribution of white noise gain for the CBF and MVDR techniques. The location of the angular grooves reflects the estimated elevation angle, and the depth of the grooves indicates the robustness of the method. The estimated elevation angles for each sub-array are approximately 40°, with symmetrical angles along the array normal direction reflecting the grating lobe position. In addition, the grooves of the X and Z sub-arrays are shallower compared to those of the P and Y sub-arrays, indicating marginally greater robustness and lower processing gain. The green solid line in Figure 12 demonstrates that the CBF method can maintain superior robustness in the spatial-angle domain. The white noise gain constraint refers to the situation where the white noise gains associated with the angles in full space exceed a particular value. This loss of white noise gains improves the robustness of the angle estimation.

Figure 13 displays the output power spectrum normalized at 70 Hz following phase difference compensation. One can observe that the CBF significantly reduces the noise and obtains an appreciable processing gain. MVDR and WNGC further suppress the noise of the frequency of interest, leading to a higher processing gain than CBF. This high gain can also be attributed to the elevation angle energy distribution. As depicted in Figure 11d, the beam energy processed by WNGC is more concentrated in the main lobe and leaks to the side lobes less, thereby resulting in a higher spatial gain.

Table 6. Array gains of four sub-arrays of VHVA processed with three methods at 70 Hz, 100 Hz, 130 Hz, and 160 Hz.

Frequency (Hz)	Sub-Array	CBF (dB)	MVDR (dB)	WNGC (dB)	Frequency (Hz)	Sub-Array	CBF (dB)	MVDR (dB)	WNGC (dB)
70	P	11.69	14.65	14.79	130	P	13.25	13.11	12.38
	X	9.38	11.48	11.58		X	10.69	11.11	10.84
	Y	11.37	13.55	13.62		Y	14.04	15.28	14.89
	Z	7.24	6.29	6.36		Z	6.87	5.03	4.84
100	P	9.34	10.70	10.64	160	P	9.37	9.61	9.55
	X	7.40	7.92	7.67		X	9.30	9.82	9.92
	Y	6.96	9.32	9.14		Y	10.28	10.79	10.65
	Z	4.60	0.31	0.37		Z	4.98	3.92	3.91

intuitively lists the array gains of four sub-arrays processed using three methods. Due to the small elevation angle and low frequency at this horizontal range, the conventional processing method can also achieve a processing gain higher than $10\log M$ dB. One can observe that the P and Y sub-arrays have high AG−CBFs of 11.5 dB, followed by the X and Z sub-arrays. MVDR processing enhances the array gain by decorrelating the noise among hydrophones, and thus, the AG−MVDR of the P, X, and Y sub-arrays is further increased by 2–3 dB. The WNGC method improves robustness while maintaining the same array gain as AG−MVDR. The insensitivity of the array gain of the Z sub-array to high-gain methods is mainly due to the noise characteristics in the vertical direction.

We roughly adopt the noise of 69 Hz for analysis since it is challenging to extract the noise information from the 70 Hz received signal. Specifically, on the one hand, the received noise of the Z sub-array shows a weak and positive correlation and thus weakens the array gain, as presented in Figure 14. On the other hand, the high-gain method is ineffective against the weakly correlated noise. Consequently, the AG−CBF and AG−MVDR of the Z sub-array are lower than those of the other sub-arrays. Moreover, the WNGC method does not enhance the processing gain of the Z sub-array, which is consistent with the simulation results. This approximate substitution is based on the condition that the noise is flat near the frequency band of the signal of interest and can only qualitatively illustrate the effect of the vertical correlation of the noise on the array gain.

To confirm the effect of frequency on the array gain, Table 6 lists the processing gain of each method at the frequencies of 100 Hz, 130 Hz, and 160 Hz. The advantage of the WNGC in terms of high gain gradually fails as the frequency increases. The WNGC method at 100 Hz improves by 1.3 dB compared to the CBF. At 130 Hz and 160 Hz, the AG−WNGC is the same as AG−CBF. The high gain appearing at 130 Hz in Table 3 is due to the high SNR, which can also be observed in the spectrogram of Figure 10. There is a significant drop in the performance of the Z sub-array because of the weak and positive correlation of the noise.

Therefore, in low-frequency situations, WNGC processing outperforms the conventional method by achieving more robust angle estimation and higher array gain. One of the requirements for the VHVA to obtain supergain is a low frequency.

Three weighting vectors are utilized to compensate for the phase difference among the hydrophones of each sub-array, and the VHVA is further equivalent to a single vector element. Considering the poor performance of the Z channel, the combination of the pressure and horizontal particle velocity channels with the form of $p.v_i$ is used to calculate the combination gain. From Figure 15 and

Table 7. Combined gains with the form of $p.v_i$ at 70 Hz, 100 Hz, 130 Hz, and 160 Hz.

Frequency (Hz)	Combination	CG−CBF (dB)	CG−MVDR (dB)	CG−WNGC (dB)
70	P. Vx	2.78	5.51	5.74
	P. Vy	2.95	4.70	4.87
100	P. Vx	3.85	4.35	4.60
	P. Vy	0.92	3.56	3.18
130	P. Vx	3.27	3.72	3.16
	P. Vy	1.22	2.06	1.19
160	P. Vx	3.63	4.03	4.16
	P. Vy	1.26	0.84	0.65

we can see that the CG−CBFs for the $p.v_x$ combination are about 3 dB, while they are 1 dB for the $p.v_y$ combination. The CG−WNGC is slightly higher than CG−MVDR. Moreover, CG−WNGC is slightly higher under low-frequency conditions than high-frequency, and the maximum value can reach 5.74 dB. The reason for the difference in the CG−WNGC at various frequencies is the extent of phase characteristics among the channels of the equivalent vector element.

Figure 16 shows the phase difference between the sound pressure and the horizontal particle velocity after CBF weighted compensation of the received signal and the received noise at each frequency. Figure 16a visualizes the fluctuation in the signal phase difference around the mean value. Due to the high noise in the low-frequency target line spectrum, there is phase difference fluctuation under the cooperation of the signal and noise. Therefore, the range of phase difference fluctuation in the 70 Hz and 100 Hz frequencies is up to 50°. As the frequency increases, the signal in the received data becomes more “pure”, and the phase difference between sound pressure and particle velocity is controlled within 20°, resulting in an increase in conventional combination gain compared to low frequencies. The energy proportion of the low-frequency line spectrum signals is lower, and the gain enhancement effect obtained through WNGC processing is better. Therefore, the WNGC combination gain is higher under low-frequency conditions.

Analogously, the noise near the desired frequency is used to approximately analyze the combination gain. Normally, the noise phase varies randomly within $\left[-\pi ,\pi \right]$. At higher frequencies, the random undulation of the noise phase difference is not obvious, and there is a certain correlation between the sound pressure and particle velocity Y component of the noise, as shown in Figure 16b. The noise suppression ability of the combined processing in the form of sound energy flow is reduced. Therefore, the combination gain brought by the joint processing of sound pressure and the horizontal particle velocity X component is greater than that brought by the sound pressure and the horizontal particle velocity Y component.

The above analysis proves that another factor affecting the supergain of the vector array is the phase relationship between the sound pressure and the particle velocity.

The spatial processing gain of the VHVA can be viewed as the sum of the sub-array and combination gains. Table 6 and Table 7 demonstrate that the X sub-array has the highest AG−WNGC and the combination gain of the P and X channels is larger. Therefore, their sum denotes the maximum spatial gain of VHVA.

As depicted in Figure 17, the maximum gain of WNGC−CP−VHVA can reach 20.53 dB at 70 Hz, which can be improved by 8.8 dB compared with CBF−PHVA. As the frequency increases, the vector array’s spatial gain is still significantly higher than that of the sound pressure array conventional processing, but the improvement value gradually degrades. At the frequency of 160 Hz, which is greater than the half-wavelength frequency, the spatial gain of VHVA can only be improved by 4.3 dB. When the SNR of the received signal is strong and the energy of each line spectrum is similar, the vector array gain corresponding to low-frequency targets with frequencies less than half the wavelength of the array is higher. As the frequency increases, the vector array gain gradually decreases, and the advantage of high gain also gradually weakens.

Consequently, the white noise gain constraint proves to be the superior choice for applications requiring high precision and robustness in elevation angle estimation. Using the signal correlation and noise uncorrelation properties, the joint processing of sound pressure and particle velocity improves the combination gain. Thus, the combination of the two significantly effectively enhances the spatial gain of the vector hydrophone vertical array.

4.2.2. Horizontal Range of 9.2 km

The feasibility of the white noise gain constraint and the of sound pressure−particle velocity joint processing method under weak SNR conditions is discussed below. Figure 18 depicts the spectrogram at a receiving range of 9.2 km. Compared to the case of 1.4 km, the SNRs of four line spectra have all decreased. Particularly, the 70 Hz and 100 Hz line spectra have been completely drowned in the noise. Therefore, we performed elevation angle estimation based on the frequency of 160 Hz. The estimation results of the CBF, MVDR, and WNGC methods are shown in Figure 19.

From the BSAS we can find that the main-lobe energy is concentrated at 104° and the performance of the supergain method decreases. The former occurs because 9.2 km is beyond the detection range of the direct-wave zone, and the arrival sound lines consist mainly of upgoing waves reflected by the seafloor. The latter is because SNR is reduced, so the MVDR is no longer robust and the improvement in WNGC is limited. As presented in Figure 19d, although the main lobe of MVDR and WNGC is narrower than that of CBF, the side lobe is significantly upgraded. Moreover, the projection of the signal on the Z axis is small because of the distance. The Z-axis noise is mainly distributed on the sea surface above the array, and the projection component of the noise is large according to the cosine relationship. Consequently, the SNR of the vertical particle velocity is quite poor, and the main lobe of the beamforming can no longer reflect the target direction correctly. Actually, the elevation angle estimation performance of each sub-array is superior under close-range conditions. As the range increases, the performance of each sub-array decreases, especially for the Z sub-array. Hence, only the sound pressure and horizontal particle velocity channels are analyzed below.

Figure 20 depicts the spatial gains of the three sub-arrays compensated by different weights when the normalized frequency is 100 Hz. The processing gain at 100 Hz, 130 Hz and 160 Hz is intuitively listed in

Table 8. Array gains of four sub-arrays of VHVA processed using the three methods at 100 Hz, 130 Hz, and 160 Hz.

Frequency (Hz)	Sub-Array	CBF (dB)	MVDR (dB)	WNGC (dB)
100	P	2.85	4.00	3.99
	X	3.92	4.44	4.37
	Y	2.22	2.56	2.32
130	P	3.74	5.11	4.69
	X	4.76	5.05	4.63
	Y	4.11	3.38	2.07
160	P	5.77	4.56	5.20
	X	6.94	7.23	7.26
	Y	6.65	3.12	2.30

. In this instance, there is no positive processing gain at 70 Hz. Overall, the processing gain is significantly decreased compared to 1.4 km. The maximum array gain of CBF is only 6.9 dB for 160 Hz, and the gain is reduced to about 3 dB at 100 Hz with lower SNR. The MVDR and WNGC methods can improve the array gain by about 1 dB at the low frequency of 100 Hz. However, as the frequency increases, the supergain methods are ineffective. A performance deterioration is displayed in the Y sub-array, as listed in Table 8. This phenomenon can also be seen from the BSAS of the Y sub-array in Figure 19d, in which a high energy output exists on the side lobe of 88° and near the main lobe of 104°. This high energy leads the signal energy to leak to the side lobe, thus weakening the main-lobe energy.

Further, implementing combination processing into three beam outputs of the sub-array yields the combination gain. Table 8 indicates that the X sub-array has a greater processing gain than the Y sub-array. Additionally, the phase difference between the P and X channels of the equivalent vector hydrophone is smaller. Hence, the combined form of $p.v_x$ is utilized.

Table 9. Combined gains with the form of $p.v_x$ at 100 Hz, 130 Hz, and 160 Hz.

Frequency (Hz)	Combination	CG−CBF (dB)	CG−MVDR (dB)	CG−WNGC (dB)
100	P. Vx	3.21	3.66	3.67
130	P. Vx	6.59	6.93	7.27
160	P. Vx	5.75	5.62	5.87

presents the combination gain under the form of $p.v_x$, from which we can deduce that the maximum combination gain at each frequency is generated from the WNGC method. In this case, the CG−WNGC is about 3.7–7.3 dB. The slight increase in gain compared to 1.4 km is because the signal phase difference between the sound pressure and the particle velocity of the equivalent vector element is more stable at this range.

The spatial processing gain of the VHVA is depicted in Figure 21b. For 100 Hz, the maximum gain of WNGC−CP−VHVA is 8.04 dB, which is 5.2 dB higher than that of CBF−PHVA. As the frequency gradually increases to 130 Hz and 160 Hz, both the WNGC−CP−VHVA and CBF−PHVA are improved due to the gradual enhancement in the SNR. The difference between the two is about 8 dB. In this case, the larger spatial gain occurs at high frequency for two main reasons. Firstly, the high-frequency signal has a higher SNR with less noise and interference. Secondly, the phase difference between the pressure and particle velocity of the high-frequency line spectra in the equivalent vector element remains more stable.

On the whole, the white noise gain constraint−combined processing method can effectively improve the spatial gain. Even in strong noise environments, this method can obtain much higher spatial gain than the conventional method for the scalar array.

4.3. Results Discussion

The WNGC−CP method presented in this work has been verified for its applicability and validity through simulations and sea trials. Moreover, the contributing factors to the high gain of the VHVA are further clarified and validated.

Firstly, the signal frequency is generally lower than the center frequency corresponding to the half wavelength, which allows for greater sub-array gain. As shown in Table 6, for the P sub-array, the sub-array gain at 70 Hz is enhanced by about 2–5 dB in comparison to higher frequencies. With increasing frequency, the effectiveness of the supergain technology decreases.

The second factor is the accuracy of the elevation angle estimation. Precise elevation angles can completely compensate for the phase difference among the array elements so that the signals are superimposed in phase and the sub-array gain is increased. In Figure 19d, the Z sub-array can no longer determine the elevation angle accurately. The array gain will be negative if the angles associated with the maximum beam energy (i.e., 90°) are utilized to compensate for the phase difference.

A coherent noise environment is also one of the indispensable conditions for optimal gain. Figure 3e and Figure 5 demonstrate that the WNGC beamforming gain deteriorates to conventional array gain when the background noise satisfies the Gaussian white model. Moreover, as the proportion of coherent noise components increases, the potential for enhancement through supergain methods similarly escalates. Real marine environmental noise is usually dominated by ship- and wind-generated noise, both of which are coherent noise.

Finally, the signals from the sound pressure and particle velocity channels must have a stable phase difference in order to achieve high combined gains. As illustrated in Figure 8, the time-varying random phase difference alters the phase relationship between sound pressure and particle velocity signal, significantly reducing the combined gain.

In the noisy environment of this trial, azimuth has a relatively small influence on the vector array gain. In future research, we will explore the influence of target azimuth angle on combination gain under given marine ambient noise and discuss the combination gain under different combined forms. In addition, the following research will focus on the supergain features of vector arrays under long-range conditions.

5. Conclusions

This paper discusses a supergain approach for the vector hydrophone vertical array and analyzes the implementation conditions for supergain. This method employs the white noise gain constraint to decorrelate the noise and offers significant and robust sub-array gains. Afterwards, the VHVA is equivalent to a single vector element by weighted compensation, to which the joint processing is utilized to realize the combination gain. The superimposition of sub-array supergain and combination gain constitute the spatial gain of the VHVA. This paper further elucidates that the low-frequency signal, coherent noise field, accurate elevation angle compensation, and phase stability of sound pressure and particle velocity are the requirements for attaining the supergain of a VHVA. The trial data show that this approach can improve processing gain by 9 dB at low frequency and close range. The simulation and sea trials both validate that this approach offers higher spatial gain than conventional scalar array methods in noisy environments. This method presents a novel idea for weak target detection in deep-sea scenarios.