Passive localization using horizontal dual-array acoustic intensity striations

The formation of interference striations is closely related to channel structure and target location, which has attracted extensive attention from scholars at home and abroad. In this paper, the passive localizing method using observed interference striations of horizontal dual-arrays will be studied. The relationship between interference phenomena and target motion is discussed firstly, then the basic principle of the passive positioning method by using horizontal dual-array acoustic intensity striations is derived mathematically. Meanwhile, it can be seen that the shape of the interference striations is related to the parameters, which reflects the motion parameters of the target. This provides mathematical theory support for extracting the the target motion state parameters carried out by Hough transformation according to these acoustic interference striations.The simulation results show that the proposed method can effectively obtain the targets’ range. Meanwhile, comparing the ranging results of different groups of parameters, we can see that the more accurate the target motion parameters are, the smaller the estimated target distance error is. Finally, the performance of the algorithm under different noise conditions is discussed. The simulation results show that the error of extracted parameters is larger under the condition of low SNR, which leads to a larger passive positioning error. However, at high SNR, the range estimation performance is good, within 10%.


Introduction
Due to the requirements and advantages of military concealment, passive ranging has always been a research hotspot of modern sonar technology, especially for multi-level countermeasures of submarines, torpedoes and other targets, because if there is no sufficient accuracy of the underwater threat target position prediction, its probability of success is very low.
At present, the passive ranging methods of underwater target include focusing beam-forming method Matched Field Processing (MFP) and Target Motion Analysis (TMA), As the hotpots of passive sonar signal processing, MFP and TMA have received extensive attention and research.However, because MFP is affected by factors such as calculation amount, model mismatch and noise, it is difficult to enter the use stage at present.As one of the main functions of sonar system, TMA method has been studied extensively and deeply in the world.It is a relatively mature passive ranging method [1][2][3][4][5][6][7][8].
In addition, Rouseff et.al modeled waveguide invariants as a distribution [9], and Baggeroer discussed the problem of using seismic streamers to estimate the distribution of interference invariants [10].Thode et al. applied waveguide invariants to matching field positioning [11].Spain and Kuperman applied waveguide invariants to spectrogram analysis of sea test data of shallow sea waveguides varying with distance and Angle [12].It is worth mentioning that American scholar Sunwoog Lee et al proposed array invariant method for passive positioning, which does not require Marine environment information and requires less computation [13].
Huang Guanqing proposed a method for calculationg the slope of the interference stripes by using Soble sharpening operator to detect pattern edge, simulation result shows the method can get a good performace [14].Four methods of passive ranging based on double elements model were proposed, the simulation results show that the four algorithm's performance is up to the target heading angle [15].WANG Chao estimated parameters from interference striations under no information on the most approaching point, the simulation and sea-trial results show that the method can get an accurate result of parameter estimation [16].In order to solve the problem that the wavedide invariant method requires a large amount of computation, a high precision passive ranging algorithm based on waveguide invariant is studied [17].

The relationship between interference striations and motion state
In order to analyze the notching properties, we assume that I is the observed intensity, which is a function of distance r and frequency w.Line satisfaction The slope of the interference fringe can be called For the purpose of the analysis of equation ( 2), the normal wave model is cited, where the depth of the source and receiver are respectively sum, then the sound pressure can be expressed as Where: m  represents the modular function and m  is the corresponding horizontal wave number.
Then the intensity can be expressed as: According to the above formula, it can be calculated as: Where: mn Q  is a number of modal slowdowns of phase velocity, mn S  is a number of modal slowdowns of group velocity.According to the definition of the Russian scholar β invariant, it can be calculated: Integrated formula (2), ( 5), (6), and finally can be considered In an ideal ocean waveguide, 1   In a waveguide environment weakly dependent on distance, the sound intensity can be expressed as The wavenumber differential mn   is related to distance and orientation.In this case, the sound intensity level can be expressed as: Ignoring the correlation between modal amplitude and distance and orientation, the above formula can be expressed as: (11) As ( ) Using equation (12), equation ( 11) can be expressed as (13) Considering that the area sub-item after the above formula is related to frequency, it can be obtained: (14) Using the properties of equation ( 12), waveguide invariants related to distance and orientation can be obtained: By integrating the formula (15), you get If the Marine environment is assumed to be independent of distance, then when the underwater broadband sound source moves in this environment, it can be approximated as: Consider the wideband source moving in a straight line at constant speed v. Its distance from the receiving array can be expressed as: Where: 0 r is the nearest passing distance of the target from the receiving array, 0 t is the time corresponding to the distance 0 r between the target and the receiving array, and 0  is the frequency corresponding to the nearest passing distance of the target.Substituting equation (18) into equation ( 17) yields: ,equation(20)can be abbreviated as When the square term in parentheses in the equation (20) tends to 1, the first order Taylor expansion of the equation can be obtained(22) From a mathematical point of view, equation ( 22) can be represented as a cluster of parabolas, and while 1   , equation ( 21) can be represented as a cluster of straight lines.This also explains the cause of the parabolic and linear phenomena of the spectrum diagram when the data is received.
As prior information, 0  and t0 can be obtained directly from the interferogram,  and t are variables of the image space.At the same time, it can be seen from equation (20) that the shape of the interference striations is related to the parameters 0 r v ,while v and 0 r reflects the motion parameters of the target.The formula shows that there is a corresponding mapping relationship between interference striation phenomenon and target motion.This provides mathematical theory support for extracting the parameters reflecting the target state from the interference striations.

The basic principle of dual-array passive positioning using interference striations
Based on the observation of interference striations in the previous chapter, this section uses the dual array model to passively locate the target according to the interference striations observed by the two array elements.
Assuming that there are two array elements (or "equivalent sound centers" of two sub-arrays), power spectrum analysis is performed on the received signal at each discrete t time, that is, a section parallel to the horizontal axis (frequency is the f axis) on the LOFAR diagram is obtained, and the time history of the section is synthesized into the LOFAR diagram.When the signal-to-noise ratio is high, the clear LOFAR pattern of interference striations can be measured by a single hydrophone.For the remote target, when the SNR is low,the array signal processing and LOFAR analysis of the tracking beam output are necessary to obtain a clear LOFAR graph.The only difference in the treatment of these two cases is that there is no matrix processing gain, and the other principles are essentially the same.Without loss of generality, it is assumed that the signal-to-noise ratio of the target signal is infinite, that is, noise is not considered.The interference striations of each dual-array can be used to estimate the motion parameters of the target ( Assuming that the distance between the two sensors is known, the velocity of the target can be calculated as: When the target velocity is calculated, the sum of the nearest passing distances of the two sensors can be calculated from .If sensor 1 is used as the reference point, the distance of the target can be estimated as:

Basic principle simulation
The basic principle of positioning is briefly introduced, and the feasibility of the positioning algorithm is verified by simulation research.
The Pekeris is the simplest model and is taken as example to illustrate the problem.The Pekeris waveguide structure is composed of a constant velocity seawater layer and a constant velocity seabed half space.The specific parameters are shown in Figure 2. Simulation conditions are as follows: ocean depth is 100m,target depth is 10m,receiving array depth is 85m, seabed parameter information is as follows: sound speed is 1650m/s, density is 2.0g/cm 3 ,attenuation is 0.1dB/λ.The motion of the target is shown in Figure 1.From about 12km, the target moves from far to near towards the ship, and then away from the ship.The target speed is 25m/s, and the nearest passing distance of the target through sensor 1 and sensor 2 is 2km and 2.15km respectively.By taking the HOUGH transform, the motion parameters( CPA t )of the target can be solved, as shown in Table 1 and Table 2 Once the above parameter values are extracted, the distance of the target can be solved according to equations ( 24)-( 27).The virtual black lines in Figure 3 and Figure 4 are the lines drawn by the extraction parameter descendant (22).According to the target motion parameters extracted by the interference striations of sensor 1 and sensor 2, the appropriate target motion parameters are selected for target ranging.The measured target distance is shown in Figure 5  .As shown in table 1 and table 2, five measurements of motion parameters are given.In order to further obtain more accurate parameter estimation, the mean and median methods are used to synthesize the results of five estimates shown in table 3,and the corresponding ranging errors are analyzed, shown in Figure 7.
The simulation results show that the proposed method is effective without considering the background interference.Meanwhile, comparing the ranging results of the two groups of parameters, we can see that the more accurate the target motion parameters are, the smaller the estimated target distance error is.

Performance comparison simulation under different SNR conditions
On the basis of the previous chapter, further simulation is done in this chapter in order to discuss the performance of the algorithm under different noise conditions.Here, the signal-to-noise ratio is taken successively as -10dB,0dB,6dB,12dB,and20dB.Other conditions are the same as in the previous chapter.The simulation results are shown in Figure 8-12.It can be seen from the simulation results that under the condition of low SNR (such as -10, 0, 6dB), the interference striations are very fuzzy and it is not easy to propose parameters.When the SNR is sufficient (for example, greater than 12dB), the observed striations are obvious and the parameters can be easily extracted.Figure 13 shows the range estimation error under different SNR conditions.Obviously, the method can get an accurate result of range estimation under high SNR, within 10%.However, the ranging results are poor at low SNR.

Conclusion
It can be seen that the shape of the interference striations is related to the parameters, which reflects the motion parameters of the target, this provides mathematical theory support for extracting the parameters reflecting the target state from the dual-array acoustic intensity striations.In this paper, the relationship between interference phenomena and target motion is discussed firstly, then the basic principle of the passive positioning method by using horizontal dual-array acoustic intensity striations is derived mathematically.The simulation results show that the proposed method can effectively obtain the targets' range.Meanwhile, comparing the ranging results of different groups of parameters, we can see that accurate extraction of target motion parameters is the key, the more accurate the target motion parameters are, the smaller the estimated target distance error is.The impact of the noise on the method is discussed lastly, simulation results show that the method can get an accurate result of parameter estimation under high SNR, the target distance estimation is smaller than 10%.

1 CPA r and the nearest passing distance between the target 2 CPA r , 1 CPA t and 2 CPAtFigure
Figure Motion state.
The interference striations observed by sensor 1 and sensor 2 are shown in Figure3and Figure4.The true values of the target motion parameters observed by Sensor 1 and 2 are as follows:

Figure 3
Figure 3 The striations observed by Sensor 1.Figure 4. The striations observed by Sensor 2.

Figure 4 .
Figure 3 The striations observed by Sensor 1.Figure 4. The striations observed by Sensor 2.

Figure 9 .
Figure 9.The striations observed in case of SNR=0dB.

Figure 10 .
Figure 10.The striations observed in case of SNR=6dB.

Figure 13 .
Figure 13.The ranging error result under different SNR It has been proved in the literature that equation (19) is also suitable for mathematical interpretation of LOFAR striations obtained from analysis of beamforming output data.For the range-independent waveguide environment, equation (19) can be abbreviated as

Table 2 .
Estimates of each motion parameter at ω0＝520Hz from Sensor 2.

Table 3 .
Estimates of each motion parameter.
Figure 7.The ranging error results of different groups of parameters obtained by the mean and median methods .