1 Introduction

The Vector Hydrophone (VH) is widely used to remotely detect underwater targets (Sherman and Butler, 2007; Wilson, 1988; Gordienko, 2014; Yang, 2009; 2013), and especially to detect low-frequency targets (Abraham, 2006; Zou and Nehorai, 2009; Korenbaum and Tagiltsev, 2015; Gür, 2013). Various types of VH-based systems have become preferred design options. Accurate measurements of VH self-noise provides an important basis for evaluating the performance of a detection system utilizing a VH (McEachern et al., 2006; Gerald et al., 2006; Yang, 2012; Guo et al., 2016; Guo et al., 2015), since the ability to acquire weak signals is determined by the VH self-noise level (Fang et al., 2014; Gabrielson, 1991; 1993; 1996).

Compared to sound-pressure hydrophones, VHs are sensitive not only to sound pressure signals, but also to vibration or acceleration signals (Yuan, 2002). Therefore, the results are significantly influenced by ambient background noise when measuring VH self-noise. It is pretty difficult to make a valid evaluation of the VH self-noise level based on results obtained by direct measurement. In this paper, measuring the self-noise of VHs by using Dual-channel Transfer Function Method (DTFM) is proposed. First, the underlying principles of DTFM is analyzed. Then, the numerical simulations are performed to analyze the influence of ambient background noise and the sensitivities of VHs on the measurement results. Finally, a small laboratory water tank based experiment is performed to the VH by using DTFM.

2 Theory of vector hydrophone self-noise measurement

In measurement of the self-noise level of a VH, the output signal consists of both the hydrophone self-noise and the ambient environmental background noise. The VH self-noise, is that contributed by the internal sensor and preamplifier (Gabrielson, 1993; 1996), which arise from mechanical and electronic thermal noise, (Levinzon, 2000; 2004; 2012; Woollett, 1962; Young, 1977). Ambient background noise comprises sounds in the air and vibrations transmitted from the measuring platform and suspension system. We placed two VHs in the same test site, for which the self-noise were uncorrelated and the self-noise of each VH was also uncorrelated with the ambient background noise. The ambient background noise received by the two VHs were generated by the same noise source. Therefore, the uncorrelated self-noise signals can be eliminated by determining the cross-correlation of the output signals of the two VHs and the power spectrum of ambient background noise can be amputated. One of the VHs' power spectrum of the self-noise can be obtained from the output power spectrum of a hydrophone and the power spectrum of the ambient background noise subtraction. Fig. 1 shows the block diagram of the DTFM measuring method.

As shown in Fig. 1, the noise power spectral density of the two VHs are ${S_{{X_1}}}\left(\omega \right)$ and ${S_{{X_2}}}\left(\omega \right)$, the power spectral density of the ambient background noise is ${S_N}\left(\omega \right)$, and the transfer functions of the two VHs are ${H_1}\left(\omega \right)$ and ${H_2}\left(\omega \right)$, respectively. The self-power spectral densities of the output signals are ${S_{{Y_1}}}\left(\omega \right)$ and ${S_{{Y_2}}}\left(\omega \right)$, respectively. The corresponding cross-power spectral density is ${S_{{Y_1}{Y_2}}}\left(\omega \right)$. The relationship between the above parameters is expressed as follows:

${S_{{Y_1}}}\left(\omega \right) = {\left| {{H_1}\left(\omega \right)} \right|^2}\left[ {{S_{{X_1}}}\left(\omega \right) + {S_N}\left(\omega \right)} \right]$

(1)

${S_{{Y_2}}}\left(\omega \right) = {\left| {{H_2}\left(\omega \right)} \right|^2}\left[ {{S_{{X_2}}}\left(\omega \right) + {S_N}\left(\omega \right)} \right]$

(2)

${S_{{Y_1}{Y_2}}}\left(\omega \right) = {H_1}\left(\omega \right)H_2^*\left(\omega \right){S_N}\left(\omega \right)$

(3)

The output self-power spectral density, ${S_{{Y_1}}}\left(\omega \right)$ and ${S_{{Y_2}}}\left(\omega \right)$, and the cross-power spectral density of two VH output signals ${S_{{Y_1}{Y_2}}}\left(\omega \right)$ can be obtained from the direct measurement. After some algebra, ${S_{{X_1}}}\left(\omega \right)$ and ${S_{{X_2}}}\left(\omega \right)$ can be expressed as:

${S_{{X_1}}}\left(\omega \right) = \frac{{{S_{{Y_1}}}\left(\omega \right)}}{{{{\left| {{H_1}\left(\omega \right)} \right|}^2}}} - \frac{{{S_{{Y_1}{Y_2}}}\left(\omega \right)}}{{{H_1}\left(\omega \right)H_2^*\left(\omega \right)}}$

(4)

${S_{{X_2}}}\left(\omega \right) = \frac{{{S_{{Y_2}}}\left(\omega \right)}}{{{{\left| {{H_2}\left(\omega \right)} \right|}^2}}} - \frac{{{S_{{Y_1}{Y_2}}}\left(\omega \right)}}{{{H_1}\left(\omega \right)H_2^*\left(\omega \right)}}$

(5)

If the transfer function of the two VHs are known, the self-noise power spectral density of the VHs can be calculated independently of the ambient background noise.

Regarding the output signal of two VHs, the coherence is defined as follows:

${\gamma ^2} = \frac{{{{\left| {{S_{{Y_1}{Y_2}}}\left(\omega \right)} \right|}^2}}}{{{S_{{Y_1}}}\left(\omega \right){S_{{Y_2}}}\left(\omega \right)}}.$

(6)

If the measured transfer functions of the hydrophones in the experiment are similar and the output noise spectrum of the VHs are approximately equal, then the VH self-noise power spectral densities can be approximately expressed as follows:

${S_{{X_1}}}\left(\omega \right) \approx \frac{{{S_{{Y_1}}}\left(\omega \right)}}{{{{\left| {{H_1}\left(\omega \right)} \right|}^2}}}\left({1 - \gamma } \right)$

(7)

${S_{{X_2}}}\left(\omega \right) \approx \frac{{{S_{{Y_2}}}\left(\omega \right)}}{{{{\left| {{H_2}\left(\omega \right)} \right|}^2}}}\left({1 - \gamma } \right).$

(8)

The equivalent self-noise spectral density of the VH can be obtained by using the coherence function.

3 Numerical simulations of DTMF

In practice, the sensitivities (transfer function) of the two hydrophones will not be the same and the strength of the ambient background noise will also have an impact on the calculation results. In this section, numerical simulation is employed to analyze the factors that may affect the DTFM calculation result and the optimal DTFM conditions for practical applications is explained.

3.1 Ambient background noise inteferences

As explained above, the self-noise of the two VHs are uncorrelated, their magnitudes are the same, and their sensitivities (transfer functions) are also the same. We define the ratio R = VH self-noise/ambient background noise in dB(${\rm{re}}.{\rm{ }}1{\rm{m/}}{{\rm{s}}^{{\rm{ - 2}}}}{\rm{/}}\sqrt {{\rm{Hz}}}$). The ambient background noise and the two VHs' self-noises are also uncorrelated.

The Figs. 2(a)–2(d) show the self-noise results of the vector hydrophone 1# with different self-noise to ambient background noise ratio defined above. The bule dashed line indicates the direct measurement self-noise, the red line denotes the self-noise obatined by DTMF and the black line is ture self-noise of the vector hydrophone in simulations.

Based on the numerical simulation results, we can draw the following conclusions:

1) DTFM can effectively reduce ambient background noise in the interference strength conditions described;

2) The real self-noise of VH can be obtained by using the DTFM when the SNR (VH self-noise/ambient background noise) is greater than -6 dB. This method should not be used when the SNR is less than -6 dB.

3.2 Sensitivity (transfer function)

As mentioned in section 2, the self-noises of the two VHs are uncorrelated, their magnitudes are the same, and we maintain the ideal ambient background noise level (0 dB). We define the ratio R = VH sensitivity #1/VH sensitivity #2 in dB. The ambient background noise and the two VHs self-noise are also uncorrelated.

The Figs. 3(a)–3(c) show the self-noise results of the vector hydrophone 1# and the Figs. 3(d)–3(f) indicate the self-noise results of the vector hydrophone 2# with different sensitivity ratio defined above. The blue dashed line indicates the direct measurement self-noise, the red line denotes the self-noise obtained by DTMF and the black line is true self-noise of the vector hydrophone in simulations.

According to the Fig. 3(a) and Fig. 3(d), if the sensitivities (transfer functions) of the two hydrophones are equal, then both self-noise of the VHs can be measured by using the DTFM.

The self-noise of VH with lower sensitivity can be obtained by using the DTFM when the ratio R is greater than -2 dB based on the simulations from Fig. 3(b) and Fig. 3(c), and the other situations, Fig. 3(e) and Fig. 3(f), cannot be measured. The overall simulations results indicate that the DTFM cannot be used when the defined ratio R is less than -2 dB.

4 Laboratory experiment

The self-noise level of the data acquisition hardware must be low enough to ensure that the VH self-noise signal can be acquired. The lowest self-noise level of VH can be measured, as well as the lower limit of system, is determined by the self-noise level of the data acquisition hardware. In this study, we used a high-precision data acquisition recorder with a wide dynamic range (type 3052, B & K Company) to acquire dual-channel noise. The actual test result for the equivalent input noise spectral density was 41.5 nV/√Hz@100 Hz (shorted at the input terminals with 50-Ω resistance).

As in the numerical simulations described in section 3, we used two VHs with the same sensitivities (less than 0.5 dB in band) and the same self-noise levels to perform the self-noise measurement experiment using the DTFM. The two VHs were hung sufficiently close enough but not tough with each other and kept in the same direction, then placed them in a small water tank, as shown in Fig. 4. The data was acquired during a period of low ambient background noise and was processed using the DTFM. Fig. 5 shows the obtained results for VH #1, in which the blue dashed line is the noise power spectral density result by direct measurement and the red solid line is the equivalent noise acceleration power spectral density (0 dB = 1 g√Hz) calculated by the DTFM.

From the Fig. 5, we can see that the noise measurement results are higher in the low frequency band below 100 Hz than the direct measurement results, which is due to the low-frequency vibration interference from the ground into the water through the measurement platform (the small water tank). In the high-frequency band above 100 Hz, there are line spectra, which are caused by the suspension of the VH. Compared with the direct measurement results, the calculated VH self-noise levels are significantly reduced by the DTFM at lower frequencies (less than 100 Hz) and the line spectra is removed by the DTFM calculation at high frequency to obtain a flat self-noise spectrum, based on the good correlation of the suspension of the two vector sensors. The equvilent noise acceleration spectral density measured by DTFM is 72.8 ng/√Hz@161 Hz, and the direct measurement result is 187.8 ng/√Hz@161 Hz, which is also reflect the DTFM is effectual for self-noise measurement of the VHs.

5 Conclusions

In this paper, we used DTFM to numerically simulate the self-noise of a VH in the presence of ambient background noise. Our experimental measurement results in a small laboratory water tank show that we can use DTFM calculations to reduce ambient background noise and more accurately measure self-noise levels by the use of two well-matched VHs. The DTFM provides an effective approach for accurately measuring the self-noise of a VH.

However, there are some drawbacks when using DTFM. First, this method requires that the consistency between the pair of VHs be as good as possible, which is not easily achieved in practice. Secondly, the measurement cost increases by the use of two VHs. Therefore, improvements in the DTFM should be considered and a new self-noise measurement method be developed in future research.