Figure 1 shows a submerged target crossing the bistatic source–receiver line, the length of which is represented by L. The length of the incident acoustic-path from the source to the target is denoted by , and that of the scattered acoustic-path from the target to the receiver is denoted by . The target crosses the source–receiver line along a straight track at a constant velocity v. The intersection angle between the target track and the source–receiver line is denoted by α, meanwhile the crossing-point on the source–receiver line is labeled as C. The range from point C to the receiver is denoted by . In the process of the target crossing, the bistatic angle β changes with the target position.

Fig. 1.

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	Fig. 1. Diagram of a submerged target crossing the source–receiver line (top-down view).

If the target crosses the source–receiver line at time , then the distance from the target to the source–receiver line at an arbitrary moment t can be expressed by . Therefore, and can be obtained in accordance with simple geometric relationships

(1)

The acoustic-path length for the forward scattered field is

(2)

It is seen from Eqs. (1) and (2) that the acoustic-path of the forward scattered field is related to the target motion time. These formulas from which the received field aberration and the Doppler shift are derived constitute the basis of the theoretical analysis in this context.

According to Fig. 1, application of the law of cosine generates the bistatic angle β

(3)

The arrival-time difference of the forward scattered field referred to the direct blast is obtained from Eq. (1)

(4)

For the target with a given shape illuminated by an incident wave with a known frequency, the scattering pattern is dominated by the bistatic angle β. At the receiver, the interference effect between the forward scattered field and the direct blast is dominated by . Therefore, the received acoustic-field is the coherent sum of the forward scattered field and the direct blast ,

(5)

The Doppler shift of the forward scattered field is defined as^[13]

(6)

(7)

Equation (7) indicates that when the target is just located at the crossing point ( , the corresponding Doppler shift is exactly zero. Moreover, the cross-term for (an oblique crossing) makes the Doppler shift curve more complex than that case for (a vertical crossing). It is seen from Eqs. (5) and (7) that aberration in the received field and Doppler shift in the forward scattered field exist simultaneously when a submerged target is crossing the source–receiver line, and both the received acoustic field and the forward scattered Doppler shift can be expressed as functions of the target motion time.

As shown in Fig. 2, an aluminum spherical shell is illuminated by a spherical wave emitted from a point source located at S which is apart from the shell center. The origin of the coordinate system is located at the shell center. The spherical shell is filled with air. The inner and outer radii of the spherical shell are denoted by a and b, respectively. The angle θ is called the scattering angle, where θ = 0 corresponds to the forward scattering and corresponds to the backward scattering. The scattering angle and the bistatic angle satisfy . The field point located at P records the received acoustic field.

Fig. 2.

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	Fig. 2. Diagram of forward acoustic scattering from an aluminum spherical shell illuminated by a spherical wave.

According to Eqs. (1) and (3), the incident and the scattered acoustic fields can be expressed as series of spherical functions^[14]

(8)

(9)

(10)

Let , L = 14 m, m, , and the aluminum spherical shell crosses the source–receiver line at a constant velocity m/s. The intersection angle α is chosen to be , , , and respectively. The aberration in the received field and the Doppler shift for the forward scattered field are calculated from Eqs. (10) and (7), and are shown together in Fig. 3. The longitudinal coordinate represents the target motion time. The latitudinal coordinates in the upper panel and in the lower panel represent the normalized received acoustic-field level (in dB) and the forward scattered Doppler shift (in Hz).

Fig. 3.

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	Fig. 3. Simultaneous detection output of the received acoustic-field aberration and the forward scattered Doppler shift for an aluminum spherical shell with (a) , (b) , (c) , and (d) .

The maximum aberration is observed occurring at the time the target is crossing the source–receiver line due to the fact that the forward scattered energy has the peak value at this moment. The farther the target departs from the source–receiver line, the weaker the aberrations are. Meanwhile, the Doppler shift changes from positive to negative when the target crosses the source–receiver line and is exact zero when the target locates exactly at the cross-point. Moreover, the patterns of the field aberration and the Doppler shift are related to the target track. For the vertical crossing case, both the acoustic-field aberration and the Doppler shift reveal their symmetries of the crossing time as shown in Fig. 3(c), whereas the patterns become complicated as the track deviates from the vertical case.

An experiment was conducted in the anechoic tank (20 m×8 m×7 m), as shown in Fig. 4(a). The transmitter, the target, and the hydrophone were deployed at the same depth of 3 m. The hydrophone was about 14.2 m apart from the transmitter. An aluminum spherical shell (shown in Fig. 4(b)) was used as the target, of which the inner radius m and the outer one m. In the experiment, the target crossed the source–receiver line along two tracks (labeled as A and . The two tracks shared the same length but the opposite orientations. The target motion time for each track was about 20 s.

Fig. 4.

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	Fig. 4. (color online) Diagrams of the experiment: (a) top-down view layout, (b) target physical map.

The target velocity v was fixed to 0.35 m/s, which was kept constant due to the fixed power system. The source center frequencies were chosen to be 30 kHz and 50 kHz. The Doppler shifts generated in the tank experiment represent the cases in which the target crosses the source–receiver line at a velocity of 5 knots illuminated by signals centered at 2.1 kHz and 3.0 kHz, respectively. The duration of the transmitting was about 65 s, which was three times longer than the duration of target motion.

Since the continuous wave is transmitted, the field aberration in the receiver can be directly represented by the envelope of the received field. The Hilbert transform is employed herein to extract the envelope in the received acoustic field. Figure 5(a) and 5(b) present the measured envelopes in tracks A and B for the source frequency 30 kHz, while Figure 5(c) and 5(d) show the corresponding envelopes for 50 kHz. Meanwhile, the theoretical envelopes are also given in Fig. 5 by solid blue lines. These theoretical curves are calculated based on the vertical crossing for simplicity. The parameters used in the calculation are consistent with those in the experiment.

Fig. 5.

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	Fig. 5. (color online) Normalized acoustic-field aberrations in (a) track A, (b) track B for the source frequency 30 kHz; and in (c) track A, (d) track B for 50 kHz.

In Fig. 5(a), the measured envelope exhibits an approximated symmetry due to the fact that the target trajectory is approximately symmetric about the crossing-point. The closer the target is to the source–receiver line, the larger the fluctuation is. The magnitude of the aberration is around 4 dB. A similar phenomenon is observed from Fig. 5(b), except that the environment-induced fluctuations are relatively large. After the source frequency increases to 50 kHz, two variations are observed in the envelope. The first one is the enhanced aberration magnitude; the other is the shortened aberration duration. The reason is that the forward scattered energy beam is narrowed in width while strengthened in magnitude. Therefore, the magnitude of the aberration for 50 kHz is about 5 dB, which is a little larger than that for 30 kHz. The measured envelopes agree with the theoretical curves at aspects of aberration duration, pattern, and aberration magnitude during the period of the target crossing the source–receiver line. The maximum discrepancy between the measured and the theoretical envelopes is observed at the moment corresponding to the crossing-point, at which the interference effect between the forward scattered field and the direct blast is sensitive to factors such as target's practical depth and time-delay difference. These factors are difficult to be measured exactly in the experiment. It should be noted that the sound absorption is ignored in the theoretical calculation.

The received field consists of both the direct blast and the forward scattered field, in which the two components interfere totally with one another. The simulated Doppler shift in Fig. 3 is relatively small, of which the magnitude decreases as the target approaches the source–receiver line. In the spectrum of the received acoustic-field, the direct blast corresponds to the main-lobe, whereas the forward scattered spectrum must be contained in the side-lobes due to the Doppler-shift. Therefore two problems should be overcome to extract the forward scattered Doppler shift: one is that the strength of the forward scattered field is much smaller than that of the direct blast; the other is that the Doppler shift contained in the forward scattered field is small and is sensitive to the target position. A hybrid scheme combining the sliding Blackman window function with fast Fourier transform (FFT) is proposed to extract the Doppler shift. The time domain expression and the corresponding spectrum for the Blackman window are given by^[15]

(11)

(12)

(13)

The shape and the amplitude spectrum of a Blackman window function consisting of 100 samples are presented in Figs. 6(a) and 6(b), respectively. Compared with the commonly used rectangular window function, the side-lobe level of the Blackman window is significantly lowered to −58 dB. In data processing, the window length is chosen to be 10 s, providing a frequency (or Doppler) resolution of 0.1 Hz. The side-lobe low level is guaranteed by the characteristics of the Blackman window.

Fig. 6.

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	Fig. 6. Description of the Blackman window function: (a) the shape, (b) the amplitude spectrum.

There are about 273 data segments truncated by the Blackman window sliding along the total data stream with a step length of 0.2 s. The normalized amplitude spectrum (in dB) for every data segment is then stacked in accordance with the observation time, forming the final Doppler–time image. Figure 7(a) and 7(d) depict the Doppler–time images corresponding to Figs. 5(a)–5(d), respectively. The longitudinal coordinate corresponds to the observation time, which includes the target motion time.

Fig. 7.

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	Fig. 7. (color online) Comparisons of the measured Doppler shift (color contour) with the theoretical one (the dashed-dotted line) in (a) track A, (b) track B for 30 kHz; and in (c) track A, (d) track B for 50 kHz.

The zero-Doppler straight striation is related to the direct blast, since the relative positions of the source and the receiver are fixed. The Doppler striation varying with observation time is exactly caused by the forward scattered field. The measured Doppler striation agrees with the theoretical curve in magnitude, duration time, and tendency. Unlike the theoretical curve, the energy distribution along the Doppler striation is not uniform. When the target is far from the source–receiver line, the Doppler shift is large in magnitude whereas the corresponding scattered field is weak in intensity. When the target is near or located at the source–receiver line, the Doppler shift is around zero, but the scattered field is rather intensive. This phenomenon can be easily explained by the directionality of the forward scattered energy. This kind of Doppler-extraction approach is only suitable for the continuous wave.

It is seen from the above theoretical analysis, simulations, and experimental verifications that when a submerged target crosses the source–receiver line, the received acoustic-field aberration and the forward scattered Doppler shift are simultaneously induced and can be represented by functions that share the same argument – the target motion time. Simultaneous application of the approaches proposed in Subsections 3.2 and 3.3 to the same received acoustic field generates the joint representation of the acoustic-field aberration and the Doppler shift, where they span the same observation time. Figure 8 shows the simultaneous detection results, corresponding to those in Figs. 5 and 7. When there is no target (in this case the target is static), only fluctuations caused by the environment are observed in the envelope. Correspondingly, there is only the zero-Doppler straight striation in the Doppler–time image. As the target approaches and crosses the source–receiver line, the aberrations on the envelope are strengthened. At the same time, a Doppler striation arises and decreases with the target motion time. After the target departs from the source–receiver line, the aberrations are weakened and then disappear. The received envelope recovers to the state before the target crossing. In the Doppler image, the Doppler striation crosses the zero-Doppler straight line and converts from positive to negative. Finally, the Doppler striation vanishes after the target stops moving, and only the zero-Doppler straight line exists.

Fig. 8.

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	Fig. 8. (color online) Simultaneous detection results of the received acoustic-field aberration and the forward scattered Doppler shift in (a) track A, (b) track B for 30 kHz; and in (c) track A, (d) track B for 50 kHz.

In the published literature upon forward scattering detection, results only provide direct or indirect information on the received acoustic-field aberration. The simultaneous detection is beneficial to enhance the reliability on target detection, for providing not only the aberrations in the received field but also the Doppler shift of the forward scattered field.