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Abstract

The ocean ambient noise is one of interference fields of underwater acoustic channel. The design and use of any sonar system are bound to be affected by ocean ambient noise, so to research the spatial correlation characteristics of noise field is of positive significance to improving the performance of sonar system. Only wind-generated noise is considered in most existing ambient noise models. In this case, the noise field is isotropic in horizontal direction. However, due to those influencing factors, like rainfall, ships and windstorm, etc. for a real ocean environment, noise field becomes anisotropic horizontally and the spatial structure of ambient field also changes correspondingly. This paper presents a spatial correlation of the acoustic vector field of anisotropic field by introducing Von Mises probability distribution to describe horizontal directivity. Closed-form expressions are derived which relate the cross-correlation among the sound pressure and three orthogonal components of vibration velocity, besides, the influence of the non-uniformity of noise field on the correlation characteristics of noise vector field was analysed. The model presented in this paper can provide theoretical guidance for the design and application of vector sensors array. Furthermore, the achievement could be applied to front extraction, Green’s function extraction, inversion for ocean bottom parameters, and so on.

Keywords

Horizontally stratified medium noise vector field spatial correlation Von Mises distribution

Introduction

Ambient sound in the ocean is a combination of natural and man-made sounds. Various geophysical processes including wind, rain and bubbles,^1,2 as well as geological processes including earthquakes,³ as well as man-made noise generated radiating from offshore,⁴ are the primary sound sources in the frequency range from a few hundred Hertz to 50 kHz. Such noise interacts with sea surface, sea floor and water column in propagation process and forms a complex interference background field. It disturbs sonar equipment’s detection and serves as a major factor that restricts the passive sonar performance. Making full use of the time–frequency–space statistical properties of underwater noise can effectively improve the performance of sonar equipment. In addition, since underwater noise field is a random process in which lots of noise sources interact with ocean environment, it is inevitable that the spatial structure of ambient noise field includes some ocean environment parameter information, so the ocean ambient noise can be inverted to return information such as bottom sound speed,^5–7 water attenuation,⁸ and so on. Besides, a more recent application is noise interference, whereby the Green’s function for acoustic propagation between two points in the ocean is recovered from the cross-correlation function of the noise fluctuations observed at two points,^9,10 which can be applied to inversion and passive fathometer.^11,12 Over all, it is of important practical significance to research the spatial characteristics of ambient noise field.

If we only consider ideal wind-generated noise in horizontal stratified waveguide, noise field is isotropic in horizontal direction, but actual ocean ambient noise consists of not only surface wind-generated noise, but also biological noise, sailing noise, and so on. The uneven spatial distribution of noise sources makes noise field appear non-isotropic in horizontal plane.¹³ Only wind-generated noise source was considered in most of the existing models,^14,15 with the non-uniformity of noise field in horizontal direction ignored, which might result in errors between actual result and theoretic result. Walker and Buckingham¹⁶ introduced Von Mises circular distribution from directional statistics to represent the noise directionality in the horizontal plane, whilst the vertical component is consistent with a surface distribution of vertical dipoles. An analysis of the coherence and cross-correlation of the noise at two horizontally aligned sensors in deep ocean is developed. Barclay and Buckingham¹⁷ improved the traditional spatial correlation of deep water by using Von Mises circular distribution function to incorporate non-uniform noise components caused by trench shading and ship; simulations by advanced model are in agreement with the experimental data. Walker¹³ made a further improvement and complement to Walker and Buckingham¹⁶ and Barclay and Buckingham,¹⁷ proposing a spatial correlation of arbitrarily directive noise in the depth-stratified shallow ocean.

The study mentioned above concentrates on sound pressure field characteristic of ambient noise, while little attention is given to vibration velocity field. Compared to traditional hydrophone arrays, the acoustic vector sensors array has lots of advantages, for example, better direction of arrival (DOA) estimation performance for a given aperture, full azimuth/elevation estimation with a linear array and they can be used in a sparse configuration with uniform geometry. Much work is currently being done on the development of velocity sensors.¹⁸ Therefore, the research of spatial correlation characteristics of noise vector field has great significance. In this regard, Abdi and Guo¹⁹ developed a statistical framework for mathematical characterization of different types of correlations in acoustic vector sensor arrays. Closed-form expressions for correlations between the pressure and velocity channels of the sensors are derived, but his research focused on vector correlation of signal field but not noise field, besides, only influence of part channel parameters such as mean angle of arrivals and angle spreads was incorporated in his model, while other ocean physical parameters such as speed, density and attenuation of water and bottom were ignored. Huang et al.,²⁰ Huang and Guo²¹ and Ting²² have done some research and exploration on correlation of acoustic vector field for isotropic noise field where only wind-generated noise was considered. Both of their models are some different from actual ocean environment. Therefore, this paper will present a more realistic spatial correlation of acoustic vector model of arbitrarily directive noise field based on the work of Walker¹³ and Huang et al.,²⁰ Huang and Guo²¹ and Ting.²² Closed-form expressions are derived which relate the cross-correlation between sound pressure and vibration velocity after a series of deductions. Compared to the model in Walker,¹³ the model presented in this paper not only can forecast the correlation of pressure field, but also can predict cross-correlation characteristics of sound pressure and vibration velocity. This result can provide theoretical references for the design of vector hydrophone array and make it possible to detect targets in a low SNR environment. Then, the spatial characteristics of horizontally isotropic and non-isotropic noise vector fields are contrastively studied. Finally, influences of the non-uniformity of noise field in horizontal direction on the spatial correlation characteristics of noise vector field are summarized.

Derivation of the spatial properties of noise vector field

Consider the horizontally homogenous, depth stratified ocean medium where the sound speed $c = c (z)$ , and density $σ = σ (z)$ , can vary with depth. Then, the velocity potential function of noise field can be expressed as¹⁵

ψ (ρ, z) = \int d^{2} ρ' S (ρ') G (ρ', ρ, z, z_{s})

(1)

where z_s is the depth of the surface noise sheet and

ρ

and z denote the horizontal and depth coordinates, respectively.

G (ρ, ρ', z, z_{s})

is the depth-dependent Green’s function,

S (ρ')

is the source distribution function, using

g (k, z, z_{s})

to denote the Fourier component of the depth-dependent Green’s function at horizontal wavenumber k,

g (k, z, z_{s})

and

G (ρ, ρ', z, z_{s})

satisfy the relation as follows¹⁵

G (ρ', ρ, z, z_{s}) = (\frac{1}{2 π})^{2} \int d^{2} k g (k, z, z_{s}) \exp [i k (ρ' - ρ)]

(2)

It is important to note that k is a quantity relevant to azimuth angle, that is the intensity coming from different directions is different. If the noise field is isotropic in horizontal direction, $g (k, z, z_{s})$ is only determined by absolute value of k and does not depend on direction of wavenumber. In this situation, the K/I ambient noise model¹⁵ is recovered.

Euler equation shows the relationship between particle vibration velocity and velocity potential function by $u = - \nabla ψ$ . For ease of derivation subsequently, sound pressure and three orthogonal components of velocity potential function are expressed in terms of vectors

U = [p u_{x} u_{y} u_{z}]

(3)

where p denotes the sound pressure, and

u_{x}, u_{y}, u_{z}

denote the three orthogonal components of velocity field, respectively. If consideration is given only to small-amplitude single frequency noise, constant factor

ω σ_{1}

can be ignored and velocity potential function can be expressed in terms of sound pressure, where ω denotes sound wave frequency and

σ_{1}

denotes seawater density. So the cross spectral density (CSD) function of combination-type three-dimensional vector hydrophone can be expressed as

W (ρ_{1}, z_{1}, ρ_{2}, z_{2}) = á U^{T} (ρ_{1}, z_{1}) U (ρ_{2}, z_{2}) ñ = [W_{11} W_{12} W_{13} W_{14} W_{21} W_{22} W_{23} W_{24} W_{31} W_{32} W_{33} W_{34} W_{41} W_{42} W_{43} W_{44}] = [W_{pp} W_{{pv}_{x}} W_{{pv}_{y}} W_{{pv}_{z}} W_{v_{p}} W_{v_{v_{x}}} W_{v_{v_{y}}} W_{_{v_{v_{z}}}} W_{_{v_{p}}} W_{v_{v_{x}}} W_{v_{v_{y}}} W_{v_{v_{z}}} W_{v_{p}} W_{v_{v_{x}}} W_{v_{v_{y}}} W_{v_{v_{z}}}]

(4)

Suppose noise sources are uncorrelated, the correlation function of noise source intensity at any two points $ρ$ and $ρ'$ satisfies the relationship: $á S^{*} (ρ') S (ρ) ñ = Q (ρ) δ^{(2)} (ρ' - ρ)$ , where $á ñ$ denotes expectation and * indicates complex conjugation. When far-field approximation is employed, the spatial correlation function of source intensity can be expressed as $Q (k' - k) \to (2 π)^{2} Ω (\overset{\land}{k}) δ^{(2)} (k' - k)$ , where $\overset{\land}{k}$ denotes the direction of k, and $Ω \overset{\land}{k}$ is an arbitrary scalar quantity representing the different power from direction k. Equations (1) to (3) are substituted into equation (4), noting that the Fourier transform of Dirac function satisfies $(1 / 2 π)^{2} \int d^{2} ρ \exp [i (k' - k) \cdot ρ] = δ^{(2)} (k' - k)$ , so equation (4) can be simplified as

W (ρ_{1}, z_{1}, ρ_{2}, z_{2}, z_{s}) = (\frac{1}{2 π})^{2} \int d^{2} k Ω (\overset{\land}{k}) g^{T} (k, z_{1}, z_{s}) g (k, z_{2}, z_{s}) \exp [ik (ρ_{2} - ρ_{1})]

(5)

where

g (k, z_{1}, z_{s}) = [g_{p} g_{x} g_{y} g_{z}] g_{p} = - ig (k, z, z_{s}) g_{x} = - {ik}_{x} g (k, z, z_{s}) g_{y} = - {ik}_{y} g (k, z, z_{s}) g_{z} = - \partial g (k, z, z_{s}) / \partial z

(6)

In equation (6), g is the normal-mode expression for the Green’s function in horizontally stratified media, and $g_{p}, g_{x}, g_{y}, g_{z}$ denote Fourier components of the depth-dependent Green’s function of the sound pressure and three orthogonal components of particle vibration velocity. φ and $θ_{0}$ denote azimuth angle of horizontal wave number vector and the azimuth angle of array, respectively (azimuth angle of array is defined as the angle between the projection of array onto the horizontal plane and axis). To integrate equation (5), we need to separate φ and k as shown below

E (φ) = [1 \cos φ \sin φ 1 \cos φ \cos^{2} φ \cos φ \sin φ \cos φ \sin φ \cos φ \sin φ \sin^{2} φ \sin φ 1 \cos φ \sin φ 1] K (k) = [1 k k - i k^{*} | k |^{2} | k |^{2} - {ik}^{*} k^{*} | k |^{2} | k |^{2} - {ik}^{*} i ik ik 1]

(7)

Equation (5) can be written as

W = \frac{1}{2 π} \int_{0}^{\infty} I (d, k) \cdot [g^{*} (k, z_{1}, z_{s}) g (k, z_{2}, z_{s}) K (k) k] d k

(8)

$I (d, k)$ is shown as follows

I (d, k) = (1 / 2 π) \int_{0}^{2 π} Ω (φ) \exp [idk \cos (φ - θ_{0})] E (φ) d φ

(9)

where d is the distance between sensors.

Ω (φ) = \exp [u \cos (φ - γ)] / J_{0} (iu)

is Von Mises probability distribution density function. Von Mises distribution has been used successfully in many fields.^22–25 Von Mises distribution is also called circular normal distribution, which is a function used to describe directional statistics. u is the angular width parameter and used to measure the angular width of the Von Mises peak, γ is the bearing of the peak parameter decided by the orientation of noise source, J₀ is the first-order Bessel function. When

u > 0

, Von Mises distribution density function is unimodal and symmetrical about the bearing of the peak. The larger u is, the sharper the figure is, and the closer it is to γ (see Figure 3). Walker¹³ and Ting²² introduced Von Mises distribution into ocean ambient noise field to describe the spectral intensity of ambient noise at different azimuth angles. When

u = 0

, Von Mises distribution shows a uniform distribution in azimuth angles. In other words, the noise field is isotropic in horizontal direction. The bigger the value of angular width parameter u is, the more obvious the difference of wave intensity from different directions is. The directivity of realistic ambient noise field in horizontal plane can always be approximated by the combination of a finite number of Von Mises distributions. Once the horizontal directivity is expressed in Von Mises distribution function, equation (8) can be expressed analytically, which may be easily and efficiently evaluated numerically, besides, this simplification makes the influence of analysis of the various physical quantities on the correlation properties much simpler and more intuitive.

It can be seen from equations (7) and (9) that, when $u = 0$ , owing to the orthogonality of trigonometric function, $I_{12}, I_{13}, I_{24}, I_{34}$ are equal to zero, that is sound pressure and vertical vibration velocity are not correlated to the two components of horizontal vibration velocity. But when $u > 0$ , every element in $I (d, k)$ is a non-zero value. That is to say, for the horizontally non-isotropic noise field, there is certain correlation between any two components of the sound pressure and three orthogonal components of the particle velocity. This is a significant difference between horizontally isotropic noise vector field and non-isotropic noise vector field.

By using the integral property of special function (the derivation process is listed in Appendix 1), the various elements of $I (d, k)$ can be derived as shown below

I_{11} = I_{44} = I_{14} = I_{41} = J_{0} (P_{d} (k)) / J_{0} (iu) I_{22} = \frac{J_{0} (P_{d} (k)) - \cos 2 χ J_{2} (P_{d} (k))}{2 J_{0} (iu)} I_{33} = \frac{J_{0} (P_{d} (k)) + \cos 2 χ J_{2} (P_{d} (k))}{2 J_{0} (iu)} I_{12} = I_{21} = I_{24} = I_{42} = \frac{i \cos χ J_{1} (P_{d} (k))}{J_{0} (iu)} I_{13} = I_{31} = I_{43} = I_{34} = \frac{i \sin χ J_{1} (P_{d} (k))}{J_{0} (iu)} I_{23} = I_{32} = - \frac{\sin χ \cos χ J_{2} (P_{d} (k))}{J_{0} (iu)}

(10)

where

\frac{(kd \sin θ_{0} - iu \sin γ)}{\sqrt{(kd \sin θ_{0} - iu \sin γ)^{2} + (kd \cos θ_{0} - iu \cos γ)^{2}}} = \sin χ

(11)

\frac{(kd \cos θ_{0} - iu \cos γ)}{\sqrt{(kd \sin θ_{0} - iu \sin γ)^{2} + (kd \cos θ_{0} - iu \cos γ)^{2}}} = \cos χ

(12)

P_{d} (k, γ, u, θ_{0}) = \sqrt{- u^{2} + (kd)^{2} - 2 iukd \cos (γ - θ_{0})}

(13)

The Green’s function in horizontally stratified media is¹⁵

g (k) = \frac{σ (z_{s})}{2 π} \sum_{n} \frac{Ψ_{n} (z_{s}) Ψ_{n} (z)}{k^{2} - k_{n}^{2}}

(14)

where k_n and

Ψ_{n} (z)

represent the eigenvalue and eigenfunction of the n-order normal mode, respectively. After substituting equations (10) and (14) into equation (8), using these relation between Bessel function and Hankel function

J_{n} (k) = [H_{n}^{1} (k) + H_{n}^{2} (k)] / 2

, equation (8) then converts to an integral about k in

(- \infty + \infty)

. So the integral can be evaluated on complex plane of k. There are two branch points

k_{1} = - k_{n}^{*}

and

k_{2} = k_{m}

in the upper half complex plane of k. By using the residue theorem, the various elements in the CSD function are finally revealed (the deducing process is shown in Appendix 2)

W_{11} = i \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{H_{0}^{(1)} (P_{d} (k_{m})) + H_{0}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(15)

W_{22} = i \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} (\frac{k_{m}^{2} H_{0}^{(1)} (P_{d} (k_{m})) + k_{n}^{2} H_{0}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})} - \cos 2 χ \frac{k_{m}^{2} H_{2}^{(1)} (P_{d} (k_{m})) + k_{n}^{2} H_{2}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})})

(16)

W_{33} = i \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} (\frac{k_{m}^{2} H_{0}^{(1)} (P_{d} (k_{m})) + k_{n}^{2} H_{0}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})} + \cos 2 χ \frac{k_{m}^{2} H_{2}^{(1)} (P_{d} (k_{m})) + k_{n}^{2} H_{2}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})})

(17)

W_{44} = i \sum_{nm} \frac{Ψ'_{n} (z_{1}) Ψ'_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{H_{0}^{(1)} (P_{d} (k_{m})) + H_{0}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(18)

W_{12} = - \cos χ \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{k_{m} H_{1}^{(1)} (P_{d} (k_{m})) + k_{n}^{*} H_{1}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(19)

W_{13} = - \sin χ \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{k_{m} H_{1}^{(1)} (P_{d} (k_{m})) + k_{n}^{*} H_{1}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(20)

W_{14} = \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ'_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{H_{0}^{(1)} (P_{d} (k_{m})) + H_{0}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(21)

W_{23} = - i \sin χ \cos χ \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{k_{m}^{2} H_{2}^{(1)} (P_{d} (k_{m})) + k_{n}^{2} H_{2}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(22)

W_{24} = i \cos χ \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ'_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{k_{m} H_{1}^{(1)} (P_{d} (k_{m})) + k_{n}^{*} H_{1}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(23)

W_{34} = i \sin χ \sum_{nm} \frac{Ψ_{n} (z_{1}) Ψ'_{m} (z_{2}) Ψ_{n}^{*} (z_{s}) Ψ_{m} (z_{s})}{J_{0} (iu)} \frac{k_{m} H_{1}^{(1)} (P_{d} (k_{m})) + k_{n}^{*} H_{1}^{(2)} (P_{d} (k_{n}^{*}))}{(k_{m}^{2} - k_{n}^{2})}

(24)

Constant term is ignored for the various elements in the above equations, but this does not affect the spatial correlation characteristics of noise field. Since media are generally absorbent, eigenvalue is generally a complex number, in the form of $k_{m} = K_{m} + i α_{m}$ , $K_{m}, α_{m} > 0$ , $α_{m}$ denotes the attenuation coefficient of the m-order normal mode. In general, $K_{m} >> α_{m}$ , therefore, the denominator can be reduced where $m \neq n$ denotes cross term of different modes with rapid phase oscillation, when there are enough normal modes propagated in water, and $α_{m}$ is far less than minimum interval of eigenvalues, the contribution of cross term modes can be ignored after summation, then the noise field reduces to an incoherent sum over the normal modes, and at this point, it is only necessary to consider the contribution of $m = n$ . The above equations can be simplified as

W_{11} = \frac{1}{2} \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{0} (P_{d} (K_{m}))}{K_{m} α_{m}}

(26)

W_{22} = \frac{1}{4} \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{K_{m} (J_{0} (P_{d} (K_{m})) - \cos 2 χ J_{2} (P_{d} (K_{m})))}{α_{m}}

(27)

W_{33} = \frac{1}{4} \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{K_{m} (J_{0} (P_{d} (K_{m})) + \cos 2 χ J_{2} (P_{d} (K_{m})))}{α_{m}}

(28)

W_{44} = \frac{1}{2} \sum_{m} \frac{Ψ'_{m} (z_{1}) Ψ'_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{0} (P_{d} (K_{m}))}{K_{m} α_{m}}

(29)

W_{12} = \frac{i}{2} \cos χ \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{1} (P_{d} (K_{m}))}{α_{m}}

(30)

W_{13} = \frac{i}{2} \sin χ \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{1} (P_{d} (K_{m}))}{α_{m}}

(31)

W_{14} = \frac{- i}{4} \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ'_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{0} (P_{d} (K_{m}))}{K_{m} α_{m}}

(32)

W_{23} = \frac{- 1}{4} \sin χ \cos χ \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{K_{m} J_{2} (P_{d} (K_{m}))}{α_{m}}

(33)

W_{24} = \frac{1}{4} \cos χ \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ'_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{1} (P_{d} (K_{m}))}{α_{m}}

(34)

W_{34} = \frac{1}{4} \sin χ \sum_{m} \frac{Ψ_{m} (z_{1}) Ψ'_{m} (z_{2}) | Ψ_{m} (z_{s}) |^{2}}{J_{0} (iu)} \frac{J_{1} (P_{d} (K_{m}))}{α_{m}}

(35)

For the case $u = 0$ , $P_{d} (K_{m}) = K_{m} d$ , the model was simplified as the vector spatial correlation model of isotropic ambient noise field, which is identical with the model presented in Huang and Guo,²¹ proving the model given in this paper is correct indirectly. Usually, we pay more attention to the spatial correlation coefficient of ambient noise vector field, the correlation coefficient is defined as follows

Γ_{ij} (d, z_{1}, z_{2}) = \frac{W_{ij} (d, z_{1}, z_{2}, z_{s})}{\sqrt{W_{ii}^{2} (0, z_{1}, z_{1}, z_{s})} \sqrt{W_{jj}^{2} (0, z_{2}, z_{2}, z_{s})}}

(36)

As mentioned above, the horizontal directivity of any ambient noise field always can be approximated by the sum of several Von Mises distributions: $Ω_{\sum} (φ) = \sum Ω_{l} (φ)$ , where $Ω_{l} (φ) = a_{l} \exp [u_{l} \cos (φ - γ_{l})] / J_{0} ({iu}_{l})$ . The parameter a₀ represents the level of the azimuthally uniform component of the wind-generated noise. a_l is the ratio of the peak height to the isotropic level. So the correlation function of total noise field can be expressed as

Γ_{ij Σ} (d, z_{1}, z_{2}) = \frac{\sum_{l} a_{l} W_{ij}^{l} (d, z_{1}, z_{2}, z_{sl})}{\sqrt{(\sum_{l} a_{l} W_{ii}^{l} (0, z_{1}, z_{1}, z_{sl}))^{2}} \sqrt{(\sum_{l} a_{l} W_{jj}^{l} (0, z_{2}, z_{2}, z_{sl}))^{2}}}

(37)

where

W_{ij}^{l}

denotes the CSD function of ambient noise field corresponding to noise component described by the

l th

Von Mises distribution.

When $d \neq 0, z_{1} = z_{2}$ , equations (26) to (35) represent the CSD between arbitrary two components of sound pressure and vibration velocity. It can be seen that the horizontal correlation structure of noise field relies on many parameters, but we can still obtain the following conclusions through observing those analytic expressions:

For horizontally isotropic noise field, the horizontal CSD functions only depend on the horizontal separation d, which is decided by the horizontally uniform distribution characteristics of noise source.

When noise field is horizontally non-isotropic ( $u > 0$ ), $P_{d} (k, γ, u, θ_{0})$ is codetermined by parameters $d, γ, θ_{0}$ . At the moment, the horizontal CSD functions not only depend on horizontal separation d, but also relies on the relative location of noise source and receiving array.

When $u = 0$ , $W_{23}$ is proportional to $\cos φ \sin φ$ . In other words, when $θ_{0} = π / 4$ , the correlation between the two components of horizontal vibration velocity is the strongest.

When $d = 0, z_{1} \neq z_{2}$ , equations (26) to (35) represent the vertical correlation between arbitrary two components of sound pressure and vibration velocity on two vertically aligned hydrophones. When $d = 0$ , $P_{d} (k_{m}) = iu$ , $W_{11}$ , $W_{14}$ and $W_{44}$ are completely independent of φ and u. In other words, these three scales will not be impacted by the horizontal non-uniformity of noise field. However, the vertical correlation properties of other components are susceptible to the non-uniformity of noise field.

Simulating calculation of the correlation of noise vector field

The model geometry is shown in Figure 1. With a layer of water overlying a semi-infinite bottom, the water depth is H, the density and sound speed of the water and the bottom given by $σ, c$ and $σ_{b}, c_{b}$ , respectively, attenuation in the water and the bottom are expressed using $α, α_{b}$ . The noise sources are assumed to be uncorrelated and located 0.5 m below the surface, they are equivalent to sources at the surface with $\cos θ$ directionality. The receiver array is located 10 m below the surface.

Figure 1.

The model geometry showing the ocean waveguide.

An impact analysis of Von Mises parameters on pressure spatial correlation of noise field

First, we present three cases which illustrate the various parameters of Von Mises distribution on the spatial correlation of noise field. In all three cases the sound frequency is 500 Hz. Figure 2(a) shows the pressure horizontal correlation function as γ varies from 0 to $π / 2$ , while $θ_{0} = 0$ , $u = 10$ and $a = 1$ , as illustrated in the figure, while noise components originating from an endfire orientation with respect to the sensors result in strong sound pressure horizontal correlation; the sound pressure horizontal correlation power originating from off endfire orientations is relatively weak and attenuates rapidly. This is consistent with the conclusion drawn in Roux and Kuperman.⁹

Figure 2.

The impact of Von Mises distribution parameters on the correlation characteristics of sound pressure. (a) Influence of bearing of the peak parameters γ, γ varies from 0 to $π / 2$ , $θ_{0} = 0$ , $u = 10$ and $a = 1$ ; (b) influence of angular width parameters u, u varies from 0 to $100$ , $θ_{0} = 0$ , $γ = 0$ and $a = 1$ and (c) influence of angular width parameters u, u varies from 0 to $100$ , $θ_{0} = π / 4$ , $γ = 0$ and $a = 1$ .

Figure 3.

Polar schematic of superposition of Von Mises components, each directional feature is characterized by bearing of the peak parameters γ, an angular width parameter u and a relative power parameter a, feature 0–feature 3 represent isotropic wind-generated noise, a distant storm from direction $γ_{1} = 0$ , coastal wave breaking from direction $γ_{1} = π / 2$ and a ship at bearing $γ_{1} = 5 π / 4$ , relative intensities are $a_{0} = 1$ , $a_{0} = 0.5$ , $a_{2} = 0.5$ , $a_{3} = 10$ , respectively. A schematic of the summation of components is provided in the upper left (box).

Figure 2(b) and (c) demonstrates the influence on the pressure horizontal correlation associated with angular width parameters u varying from 0 to 100, $a = 1$ , $γ = 0$ , while noise components originating from endfire orientation ( $θ_{0} = 0$ ) and diagonal orientation ( $θ_{0} = π / 4$ ) with respect to the sensors. As a side note, the parameter λ in horizontal axis is wavelength. It can be seen that the horizontal non-uniformity of noise field has a very significant impact on the correlation structure of ambient noise field. Combined with equation (29), it is not difficult to realize that the horizontal correlation structure of noise field relies largely on the structure of $Y_{m} (d, u)$ , $(Y_{m} (d, u) = J_{0} (P_{d} (K_{m})) / J_{0} (iu))$ . With the order of normal mode increasing, $Y_{m} (d, u)$ monotonically decreases, so the spatial correlation structure of noise field is determined mainly by $Y_{m} (d, u)$ structure corresponding to low order normal mode. When receiving array lie on x axis ( $θ_{0} = 0$ ), with angular width parameters u increases from 0 to 100, $Y_{m} (d, u)$ corresponding to lower modes monotonically decreases. Therefore, the horizontal correlation coefficient amplitude of pressure field also changes monotonously with u. As $θ_{0}$ increases to $π / 4$ , the $Y_{m} (d, u)$ corresponding to lower modes meets a contrary sign around $u = 10$ , so the correlation coefficient of sound pressure reaches its minimum at around $u = 10$ .

Simulation analyses on spatial correlation of horizontal directivity vector field

We use the test case previously modelled in Walker.¹³ Figure 3 presents an example of the approach applied in a shallow ocean Pekeris waveguide. The horizontal sound distribution model incorporates various directivity features consistent with a realistic ocean soundscape. One feature (labelled feature 0) arises from the canonical scenario involving wind-generated noise uniformly distributed throughout the surface (u₀ = 0) whose relative intensity is 1. Other features (labelled features 1–3, respectively) include horizontally directive contributions typical of a distant storm from direction $γ_{1} = π$ , coastal wave breaking from direction $γ_{2} = 2 π / 3$ and a ship at bearing $γ_{3} = 5 π / 4$ , whose relative intensity is 0.5, 0.5 and 10, respectively. The bearing of the peak parameters $γ_{n}$ can be achieved according to the position of the ambient noise, while the remaining Von Mises parameters, that is to say, the angular width parameters u_n, and relative amplitude, a_n, were adjusted to give the best agreement with the time coherence data. Detailed demonstrations can be found in Walker and Buckingham¹⁶ and Barclay and Buckingham.¹⁷

Figure 4(a) to (d) shows the horizontal correlation coefficients of $p v_{x} v_{y} v_{z}$ , respectively. As seen in Figure 4(a), when the noise fields are horizontally isotropic (corresponds to feature 0), the correlation properties of sound pressure attenuate rapidly with the increase of receiving spacing. Feature 1 and feature 2 have the same directive characteristics, with a difference that the corresponding noise source positions are different. Storm noise is mainly distributed in the endfire direction of horizontal receiving array, while coastal wave breaking noise is primarily distributed in the broadside direction of horizontal receiving array. It can be seen that the impact of storm noise is much higher than coastal wave breaking noise. This result shows that the spatial position of noise source has a very significant impact on the horizontal correlation structure of the sound pressure field. Compared to other noise sources, ship radiated noise intensity is the strongest. Although the ship are not distributed in the endfire direction of horizontal receiving array, the correlation structure of sound pressure subjected to total noise sources is quite similar to the correlation structure under the action of ship radiated noise. This also means that in the presence of strong source, the correlation structure of sound pressure relies largely on the strong source. When the intensity of sources is almost equal, the correlation characteristics of sound pressure depend on the relative intensity of various noise sources and their relative position to the receiving array.

Figure 4.

The horizontal correlation coefficients of noise vector field for the individual directional of wind-generated noise, distant storm, coastal wave breaking and a ship. (a) Horizontal correlation coefficient of pressure p, (b) horizontal correlation coefficient of velocity v_x, (c) horizontal correlation coefficient of velocity v_y and (d) horizontal correlation coefficient of velocity v_z.

Compared with the correlation of p, the correlation of $v_{x} v_{y}$ is more susceptible to the directional components of noise source. So the correlation of $v_{x} v_{y}$ may contain more information about noise source position. Therefore, using the cross-correlation of sound pressure and vibration velocity may lead to a more accurate DOA estimation result relative to traditional methods. Since the intensity of noise radiated from ship is much stronger than that of other noise sources, the correlation property of total ambient noise field is largely similar to correlation of feature 3.

An impact analysis of noise source depth on pressure spatial correlation of noise field

For wind-generated noise source, it is generally assumed that a plane of monopoles lying immediately beneath the pressure-release ocean surface, which forms negative images of the monopoles to create, in effect, a surface layer of dipoles. In this case, the spatial correlation of wind-generated noise field is barely related to noise source depth; however, when such noise sources as storms and ships, etc. are considered, since noise sources are not always distributed near sea surface, they cannot be approximately regarded as a monopole. How does the noise source depth affect the correlation characteristic of noise field? Simulation result is presented below.

Figure 5(a) shows the correlation function when the depth of storm noise source varies from 0.5 to 10 m. As can be seen from the figure, when the depth of noise source changes within the range of a wavelength near sea surface, the horizontal correlation of noise field fluctuates obviously, but as the noise source depth continues to increase, almost no change happens to the horizontal correlation of noise field (because most realistic noise sources are far away from the floor of ocean, the case that noise sources near bottom is irrespective here). To get a better look at the impact of noise depth, Figure 5(b) gives the horizontal correlation function for storm noise source is equal to 1, 3, 5 m. When storm centre depth varies from 1 to 3 m, the amplitude of horizontal correlation coefficient increases by an estimated 0.2, as storm centre depth increases further, the horizontal correlation coefficient of sound pressure was nearly unchanged.

Figure 5.

The horizontal correlation as a function of the noise source depth. (a) Horizontal correlation of p as storm noise depth changes from 0.5 to 10 m and (b) horizontal correlation coefficient of p for storm noise depth $z_{s} = 1, 3, 5 m$ , respectively.

Conclusions

By introducing Von Mises distribution to describe the horizontal non-uniformity of noise field, a spatial correlation model for horizontally non-isotropic noise vector field was presented. Compared with the existing models, the model in this document can consider not only wind-generated noise source, but can also storm noise source, ship noise source, biological noise source, etc. Furthermore, this model can forecast the cross-correlation properties of sound pressure and vibration velocity, which may lead to a better performance of DOA. So this model can achieve more widespread applications than traditional ambient noise models. The main conclusions of this paper are shown as follows:

In a horizontal isotropic noise field, sound pressure and vertical vibration velocity are completely uncorrelated to the two components of horizontal vibration velocity, and the horizontal correlation of sound pressure relies only on the distance among receiving points. However, in a horizontal non-isotropic noise field, correlation between sound pressure and the three orthogonal components of vibration velocity is non-zero; moreover, the horizontal correlated characteristics of noise field are not only related to the horizontal distance but also susceptible to the character of noise source.

When components originating from an endfire orientation with respect to the sensors result in strong sound pressure horizontal correlation, the sound pressure horizontal correlation power originating from off endfire orientations is relatively weak and attenuates rapidly.

When strong noise source exists, the correlation properties of the total noise field will rely largely on the spatial correlation properties of strong source. However, when the noise sources are comparable in intensity, the correlation properties of the total noise field will depend on the relative intensity of the various noise sources and their relative position to receiving array.

For horizontally stratified homogeneous media, the horizontal correlation of noise field will fluctuate obviously when the noise source depth varies in a wavelength range, but if noise source depth continues to increase, the horizontal correlation of noise field has no obvious variation.

Due to space limitations, this paper focuses on the impact of horizontal non-uniformity of noise field on the spatial correlation of sound pressure field, while little simulation was done about the impact of noise field horizontal non-uniformity on spatial correlation of vibration velocity field. However, the CSD function expressions for combination-type three-dimensional vector hydrophone was presented in this paper, with those analytic expressions, the cross-correlation properties of sound pressure and vibration velocity of arbitrarily directive noise field can be analysed easily. The results of this paper are useful for the design and performance analysis of vector sensors systems and array processing algorithms. Besides, the model might prove useful as a tool in the analysis of experimental efforts aimed at Green’s function extraction,⁹ wave front extraction¹⁰ and passive target location.¹² All in all, this model has wide applications prospect on engineering application.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science and Technology Foundation of State Key Laboratory, China (Grant No.9140C200103120C2001), the Natural Science Foundation of GuangDong province under the contract 2014A030310256.

Appendix 1: The derivation process of I ( k )

Appendix 2: The derivation process of CSD matrix

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A spatial correlation model for the horizontal non-isotropic ocean ambient noise vector field

Abstract

Keywords

Introduction

Derivation of the spatial properties of noise vector field

Simulating calculation of the correlation of noise vector field

An impact analysis of Von Mises parameters on pressure spatial correlation of noise field

Simulation analyses on spatial correlation of horizontal directivity vector field

An impact analysis of noise source depth on pressure spatial correlation of noise field

Conclusions

Footnotes

Declaration of conflicting interests

Funding

Appendix 1: The derivation process of I ( k )

Appendix 2: The derivation process of CSD matrix

References