Underwater reverberation often hinders the effectiveness of adaptive methods in active target localization with snapshot-deficient conditions. To overcome this challenge, a knowledge-aided reverberation covariance-based approach is proposed to maintain high resolution while reducing sidelobe levels. Using the aided reverberation covariance computed from the reverberation model, the knowledge-aided sample covariance matrix is constructed and used to decrease reverberation and compensate for snapshot deficiency. Simulations show that the proposed approach can localize targets with improved resolution and reduce reverberation levels in low signal-to-reverberation ratio situations, manifesting its potential to enhance adaptive processing reliability for active target localization.

## 1. Introduction

Active target localization often involves detecting and localizing targets with their echoes after being illuminated by actively transmitted pulses. In a shallow-water reverberation environment, the low repetition rate of pulses limits the total number of snapshots available for target localization. This limitation is imposed by the number of pings (i.e., snapshots in slow time) and the number of snapshots (in fast time) of target returns in a single ping, leading to snapshot-deficient problems,^{1} which is similar to the issues encountered in passive cases.^{2–4} The adaptive approaches, such as the minimum variance distortionless response (MVDR), suffer from performance degradation, as the “true” signal-interference covariance is difficult to appropriately construct using the limited snapshots.

The snapshot-deficient problem is commonly addressed using the diagonal loading (DL) technique and subspace methods.^{3,5–7} These approaches approximate a full-rank covariance matrix using reduced-rank methods and assume that the noise is uncorrelated. The reverberation presented in active problems often violates that assumption and its covariance presents as a non-diagonal matrix with many non-zeros on the off-diagonal entries. Thus a diagonal matrix is not enough to effectively represent the reverberation covariance. For the receivers, the reverberation of the rough bottom is the sum of contributions from the involved scattering patches, and the spatial covariance among the receivers should be constructed by taking an average over a sufficient number of snapshots statistically.^{8} Such adverse conditions and limitations finally hinder the effectiveness of those methods in active target localization and result in performance degradation.

To address the challenge of computing the signal-reverberation covariance from snapshot-deficient measurements, we turn to the knowledge-aided formalism to incorporate the *a priori* reverberation covariance into the sample covariance matrix (SCM). The knowledge-aided technique has been successfully applied in space-time adaptive processing^{9,10} and passive localization^{11} to overcome snapshot deficiency. The covariance computed from the physical model is employed as the knowledge priors. In Ref. 11, a knowledge-aided covariance was generated using a surface noise model to overcome snapshot-deficient measurements and reduce passive localization ambiguity. Inspired by this approach, we employ a similar knowledge-aided technique to address active target localization problems challenged by incoherent reverberation caused by bottom roughness scattering.

By combining the simulated reverberation covariance with SCM using a weighted coefficient,^{10,11} the knowledge-aided SCM is constructed and then used for the MVDR processor, named as knowledge-aided minimum variance distortionless response (KA-MVDR). With the well-reconstructed signal-reverberation covariance, KA-MVDR maximizes the signal-to-interference-plus-noise ratio (SINR) and thus improves target localization. Additionally, KA-MVDR can adapt to the single snapshot scenario by empirically choosing the weighting and tuning the output SINR. The proposed KA-MVDR processor is evaluated with bearing estimation and further applied to the bearing-time record (BTR) analysis to localize targets in the spatiotemporal domain.

For coherent reverberation, the dominant multipath arrivals returned by the environment may pose a similar snapshot-deficient problem when capturing them directly in SCM. This problem can be addressed by estimating the arrival times and bearings as the priors used to reconstruct the knowledge-aided covariance. In active problems, predicting these arrivals is feasible due to the ease of determining the positions of the source and the receiving array. Using the provided knowledge-aided covariance, the impact of incoherent or coherent reverberation can be mitigated adaptively by KA-MVDR, without requiring the used *a priori* covariance to fully match the data.

## 2. The KA-MVDR processor in a reverberant environment

**w**of MVDR is solved by minimizing the optimization problem as

*N*-element uniform linear array with an element spacing

*d*, the array steering vector is defined by $ v ( sin \u2009 \varphi ) = [ 1 , e jkd \u2009 sin \u2009 \varphi , \u2026 , e jkd ( N \u2212 1 ) \u2009 sin \u2009 \varphi ] T$, where

*k*is the wavenumber and $ \xb7 T$ denotes the transpose operation. Then

**w**has the form as follows:

^{12}

**R**is usually unknown and replaced by SCM $ R x = ( 1 / L ) \u2211 l = 1 L x l x l H$, where

*L*is the number of snapshots and $ \xb7 H$ denotes the Hermitian transpose. $ x l \u2208 \u2102 N \xd7 1$ is the

*l*-th snapshot measurement. Two challenges arise when the number of snapshots is insufficiently large. First, $ R x$ is rank-deficient, which results in instability when performing the matrix inverse, and second, $ R x$ does not adequately incorporate the complete knowledge of reverberation. The first challenge can be mitigated by applying the DL technique. It involves adding a small amount

*ε*to the diagonal of $ R x$, resulting in the weight vector of the diagonalized minimum variance distortionless response (DL-MVDR) processor given by

*a priori*, $ R KA$ can be calculated as

*α*and

*β*are weighting coefficients and $ R x$ is employed as a substitute of $ R s$.

^{10}and rewrite it as

*α*can be determined by minimizing the mean square error between $ R KA$ and the full rank covariance

**R**using at least two independent snapshots.

^{10,11}With Eq. (6), we define the adaptive weight $ w KA$ of KA-MVDR as follows:

*a priori*statistical knowledge of reverberation $ R r$ into SCM, the proposed KA-MVDR can approximate the “true” signal-interference covariance

**R**better than DL-MVDR, which exploits only the orthogonal noise space $ \epsilon I$, thereby achieving performance improvement.

*α*in Eq. (8) has been observed to control over the confidence level of the knowledge-aided priors $ R r$ vs $ R x$, and subsequently affects the SINR. By tuning the weight

*α*, KA-MVDR can retain robustness in snapshot-deficient conditions with a simple formulation. Furthermore,

*α*can be empirically chosen to adjust SINR when only a single snapshot is available by knowing that it represents leverage between the priors and data. It alleviates the requirement of at least two independent snapshots,

^{11}thereby making the proposed approach suitable for more common snapshot-deficient conditions.

## 3. Simulation analyses of active target localization in reverberation

### 3.1 Shallow-water reverberation environment

The simulation configuration for active target localization in reverberation involves a shallow-water environment and a horizontal linear array (HLA), as shown in Fig. 1. The HLA consists of *N* = 96 elements with a uniform spacing $ d = 0.4167 \u2009 m$. The sound speed profile and geo-acoustic parameters in Fig. 1(a) are based on the Asia Seas International Acoustics Experiment,^{13} with a rough bottom between a $ 103 - m$ water column and sediment. The source and HLA are deployed at a depth of $ H = 40 \u2009 m$, with the source $ 500 \u2009 m$ away from the array's first hydrophone.

The target returns are generated using the KRAKEN normal mode model,^{14} assuming a point-like target is present.^{15} The source waveform used here is a linear frequency modulation (LFM) pulse that sweeps from 1750 to $ 1850 \u2009 Hz$ in a 1-s pulse duration. The source level is normalized to the source's 1-m position. Since we primarily evaluate the target localization performance in reverberant environments, the noiseless data are generated and used in simulations.

The time series of the reverberation field is generated using the OASSP module in OASES^{16} with environmental parameters shown in Fig. 1. The frequency interval used for the spectrum of the reverberation field is 0.5 Hz. The phase velocity range is set to $ [ 1200 , \u2212 1200 ] \u2009 m / s$, where the negative velocity is included to exploit the negative wavenumber spectrum. Only the reverberation caused by bottom roughness scattering (i.e., incoherent reverberation) is considered and used for the following data generation. The roughness parameters are shown in Fig. 1(b), where the root-mean-square (RMS) roughness of the interface is 0.05 m, the correlation length (CL) is 1 m, and the spectral exponent of the roughness power spectrum is set to 2. The reverberation covariance $ R r$ is generated using the OASS module with the same simulation configuration. OASS is the physical model (also provided by OASES) used to compute the spatial statistics of scattering in waveguides with one-dimensional rough interfaces.^{16}

### 3.2 Bearing analysis

In this section, the target localization performance of KA-MVDR is investigated using the bearing estimation. Bearings are searched on a uniform grid in $ [ \u2212 1 , 1 ]$ with a 0.001 bearing interval. Noting that the $ \u2212 3 - dB$ resolution of the beam width in this configuration is $ \Delta \u2009 sin \u2009 \varphi 3 - dB \u2248 0.0211$. The input signal-to-reverberation ratio (SRR) is defined by $ SRR = 10 \u2009 log 10 P t / P r$, where $ P t$ and $ P r$ are the power of the received target return and the mean background reverberation, respectively.

^{17}which matches the normalized replica vectors $ v ( sin \u2009 \varphi i )$ on the

*i*-th bearing to the measurement as

*K*is the total number of discretized bearing grids. The DL-MVDR output is obtained by applying the weight $ w DL$ to SCM as

Using Eqs. (9)–(11), we first examine the bearing estimation for a single target at a frequency of $ 1800 \u2009 Hz$. The sole incoherent reverberation data are employed to generate the measurements, and the knowledge-aided covariance is computed based on the sea-bottom roughness parameters. Two different SRRs at $ 10 \u2009 dB$ and $ \u2212 5 \u2009 \u2009 dB$ are used to evaluate three approaches. For DL-MVDR, we set $ \epsilon = 0.01 \xd7 Tr ( R x )$, where $ Tr ( \xb7 )$ denotes the trace operation of a matrix. For KA-MVDR, *α* is empirically set to 0.08. The actual bearing of the single target is $ sin \u2009 \varphi = \u2212 0.35$. With the bistatic active geometry depicted in Fig. 1, reverberation arrives at the array from all directions and presents strong levels near the source direction (the endfire direction in this case), which can significantly interfere with the target echo. Therefore, it is essential to suppress the reverberation for target localization.

Figure 2 shows that KA-MVDR can appropriately suppress the reverberation levels, especially in the direction of the endfire and at a high SRR condition, e.g., 10 dB shown in Fig. 2(a). When SRR decreases to $ \u2212 5 \u2009 \u2009 dB$, as shown in Fig. 2(b), KA-MVDR can outperform DL-MVDR with an approximately 9 dB lower reverberation level. DL-MVDR noticeably degrades and behaves more akin to Bartlett using $ \epsilon = 0.01 \xd7 Tr ( R x )$, meaning that the sole DL operation is insufficient to decrease the reverberation appropriately.

With regard to the resolution, KA-MVDR exhibits a decreased beam width of approximately 0.002 when measured from the $ \u2212 3 - dB$ points in the zoom-in inset of Fig. 2(a). Bartlett and DL-MVDR show approximately 9- and 2-times beam width, respectively. Therefore, KA-MVDR can achieve better localization resolution than DL-MDR while maintaining the simplicity of constructing the adaptive weight vector. Even when SRR is low, as shown in Fig. 2(b), it still retains a narrower beam width than DL-MVDR. The results demonstrate the effectiveness of KA-MVDR in reverberant scenarios.

The performance improvement of KA-MVDR is attributed to the use of the *a priori* reverberation covariance which improves the SINR of the knowledge-aided SCM. Evaluated with Eq. (8), $ R KA$ shows an improvement of $ SINR$ about 10 dB as compared to $ R x + \epsilon I$, where the added diagonalized matrix $ \epsilon I$ is deemed the interference component. The $ SINR out$ is employed to evaluate the processor's ability to background interference reduction, which is estimated by steering the array in the target's direction and summing the outputs in other directions to obtain the output power of interference. Compared to DL-MVDR, KA-MVDR shows an increased $ SINR out$ by about 4.6 dB. Consequently, KA-MVDR shows lower output reverberation levels than Bartlett and DL-MVDR in low SRR conditions.

Figure 3 compares the TRR performance under different SRRs. In high SRR conditions, all processors can localize the target with a high TRR level. Both MVDR-based processors achieve better TRR levels than Bartlett. In particular, KA-MVDR outperforms DL-MVDR because it provides *a priori* reverberation information When SRR decreases, the TRR level of DL-MVDR decreases and approaches that of Bartlett. Both of them can achieve a TRR level larger than $ 0 \u2009 dB$ if SRR is greater than $ \u2212 8 \u2009 \u2009 dB$. For KA-MVDR, the required SRR is approximated at $ \u2212 12 \u2009 \u2009 dB$.

### 3.3 BTR analysis

To achieve target localization with their arrival times and bearings, the proposed KA-MVDR is applied to the BTR analysis and used for active target localization. The BTR analysis may potentially be employed to localize targets with spatially coherent multipath reflections through their two-dimensional structures in the spatial-temporal domain. The time series of reverberation and targets are generated from their received frequency spectrum with a sampling rate of $ 8192 \u2009 Hz$. They are transformed with 16 384 samples via the inverse fast Fourier transform (FFT) separately and then summed together to obtain 2-s receiving data. Four targets with different SRRs are employed, and their parameters are specified in Table 1. The positions of targets are determined and showcased as “ $ ( t , \u2009 sin \u2009 \varphi t )$,” where *t* denotes the received time in seconds.

Target Number . | 1 . | 2 . | 3 . | 4 . |
---|---|---|---|---|

Position $ ( t , \u2009 sin \u2009 \varphi )$ | $ ( 0.43 , \u2212 0.35 )$ | $ ( 0.52 , \u2212 0.75 )$ | $ ( 0.49 , \u2212 0.36 )$ | $ ( 0.55 , 0.2 )$ |

SRR [dB] | −7.2 | −3.6 | −2.9 | −7 |

Target Number . | 1 . | 2 . | 3 . | 4 . |
---|---|---|---|---|

Position $ ( t , \u2009 sin \u2009 \varphi )$ | $ ( 0.43 , \u2212 0.35 )$ | $ ( 0.52 , \u2212 0.75 )$ | $ ( 0.49 , \u2212 0.36 )$ | $ ( 0.55 , 0.2 )$ |

SRR [dB] | −7.2 | −3.6 | −2.9 | −7 |

The short-time Fourier transform (STFT) is first applied to the matched filtering (MF) outputs of array time series to obtain a sequence of data segments. The MF outputs are analyzed starting at 0.3 s with a length of 0.6 s and divided into 0.02-s length segments with a step size of 0.005 s. For each segment, the snapshot consists of a vector of Fourier coefficients at $ 1800 \u2009 Hz$ transformed via an 8192-point FFT. Unlike accumulating all snapshots to obtain a single SCM, we construct SCMs with a single snapshot in each segment separately. By doing so, the influence of reverberation is limited to the vicinity of the target.^{18} The target localization is thus localized using one snapshot in each segment.

Figure 4 shows the BTR localization using these three methods. Using Bartlett, one finds that weak targets may exhibit comparable output levels as reverberation, as illustrated in Fig. 4(a). Similar levels are observed at the position of the target 2 as those in the background, making it difficult to distinguish the target from reverberation. When targets exhibit stronger strengths than the background at the positions they are present, they can be localized using Bartlett, for example, targets 1, 3, and 4 are localized at $ ( 0.4225 , \u2212 0.344 ) , \u2009 ( 0.4775 , \u2212 0.354 )$, and $ ( 0.5425 , 0.198 )$, respectively. However, the localization presents wide beam width and high sidelobe levels. In addition, reverberation manifests as an arc shape along the time and bearing axes, indicating that it significantly hinders target localization in both the temporal and spatial domains. The strongest reverberation is associated with the forward scattering and appears in the range of $ sin \u2009 \varphi = \u2212 0.92$ to −1. Localizing targets near that region can be considerably influenced by the strong reverberation outputs.

The BTR localization results using DL-MVDR and KA-MVDR are shown in Figs. 4(b) and 4(c), respectively. The strong targets can be localized using DL-MVDR, yet the high reverberation level still hinders the weak target localization, such as target 2. Compared with Bartlett, the mean BTR output levels are reduced by approximately 6 dB. However, reverberation is still strong, and the localization resolution does not improve significantly.

As expected, KA-MVDR achieves the target localization with improved resolution and reduced reverberation levels, as shown in Fig. 4(c). It enhances the discrimination between the targets and the background, and four targets are correctly localized at $ ( 0.4175 , \u2212 0.344 ) , \u2009 ( 0.5075 , \u2212 0.727 ) , \u2009 ( 0.4775 , \u2212 0.354 )$, and $ ( 0.5475 , 0.198 )$ despite the minor localization errors. The normalized bearing outputs at the arrival times of targets are compared and showcased in Figs. 4(d)–4(g). These comparisons demonstrate that the reverberation level can be reduced using KA-MVDR, and the target can be localized with an improved TRR level than using Bartlett and DL-MVDR. In addition, we find that the most prominent output of reverberation is at $ ( 0.3425 , \u2212 0.971 )$, which coincides with the arrival time and bearing of the forward scattering.

## 4. Conclusion

This study presents an MVDR-based knowledge-aided approach, KA-MVDR, for snapshot-deficient active target localization in the presence of reverberation. Due to the limitation of the number of snapshots, calculating the spatial covariance of reverberation through statistical averaging over multiple available snapshots is challenging. The proposed KA-MVDR overcomes the limitation by leveraging the accurate knowledge of the *a priori* reverberation covariance computed from the reverberation model and thus reconstructing the signal-reverberation covariance. With the well-reconstructed covariance, KA-MVDR maximizes the output SINR and enhances active target localization. The performance improvement is validated through the simulation results involving bearing estimation and BTR localization, suggesting that the proposed approach is a promising solution for real-sea active target localization applications. Future work will focus on investigating the impact of the reverberation covariance mismatch and potential limitations posed by the coherence of multipath reflections of targets in reverberation.

## Acknowledgments

This research was supported partially by the Science and Technology on Sonar Laboratory (Fund Grant No. 6142109KF2018) and partially by the Natural Science Foundation of China (Grant No. 62171410).

## REFERENCES

*Optimum Array Processing: Part IV of Detection, Estimation and Modulation Theory*