Front. Mar. Sci. Frontiers in Marine Science Front. Mar. Sci. 2296-7745 Frontiers Media S.A. 10.3389/fmars.2026.1761306 Methods High-resolution mapping of ocean floor based on synthetic aperture technique ZhangLili 1 † Conceptualization Data curation Formal analysis Funding acquisition Investigation Methodology Project administration Resources Software Supervision Validation Visualization Writing – original draft Writing – review & editing LiuJiaqi 2 † Conceptualization Data curation Formal analysis Funding acquisition Investigation Methodology Project administration Resources Software Supervision Validation Visualization Writing – original draft Writing – review & editing FengJing 3 Conceptualization Formal analysis Methodology Supervision Validation Writing – review & editing Writing – original draft WuPeng 4 * Conceptualization Formal analysis Funding acquisition Project administration Resources Validation Visualization Writing – review & editing Writing – original draft XuZhiping 5 Formal analysis Methodology Resources Writing – review & editing Supervision Continuing Education Center, Shandong Labor Vocational and Technical College, Jinan, China Department of Information Engineering, Shandong Labor Vocational and Technical College, Jinan, China Department of Basic Physical Education, Shandong Labor Vocational and Technical College, Jinan, China Technician Department, Shandong Labor Vocational and Technical College, Jinan, China School of Ocean Information Engineering, Jimei University, Xiamen, China Correspondence: Peng Wu, wp20262019@163.com

†These authors have contributed equally to this work and share first authorship

 24 02 2026 03 03 2026 2026 13 1761306 05 12 2025 30 01 2026 10 01 2026 Copyright © 2026 Zhang, Liu, Feng, Wu and Xu. 2026 Zhang, Liu, Feng, Wu and Xu https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

To improve the imaging performance of multireceiver synthetic aperture sonar system, this paper proposes a new imaging methodology. The presented approach exploits the idea of Loffeld’s bistatic formula (LBF) that is an accurate spectrum. Using the presented method, the bistatic phase of all receiver dataset is compensated, and the interleaving arrangement related to the compensated datasets is carried out. After the interleaving arrangement, the SAS datasets are imaged based upon chirp scaling. The outcomes of our method are presented, and performance comparisons between our method and back projection algorithm are further carried out. The comparisons show that the recommended approach can gain good results. Besides that, the recommended approach is much more time-saving than traditional approaches.

 focusing method focusing performance imaging sonar LBF multireceiver system The author(s) declared that financial support was not received for this work and/or its publication. section-at-acceptance Ocean Observation Introduction

An underwater optical image is limited due to energy loss and murky water. Therefore, the optical image is not often used by underwater engineers (Oyaei et al., 2022; Zhang et al., 2024b; Diaw et al., 2022). The acoustic in water can travel to a farther range compared to optic. Consequently, sonar (Lee et al., 2022; Zhang et al., 2024a) based on acoustic is widely used in water. In practice, underwater engineering can benefit from sonar images. Currently, there are three imaging sonar offering underwater acoustic image. We call the first one the multibeam sonar (Wang et al., 2022a; Zhang et al., 2022d; Debese et al., 2012; Tan et al., 2019) that is based on Mills’ array. The second one is the side-view sonar (Wang et al., 2022b; Xu et al., 2023; Yang and Liu, 2022; Wong et al., 2021; Shang et al., 2022). Generally speaking, the resolution of the multibeam sonar (Bu et al., 2022) and side-view sonar (Polap et al., 2022) is strongly related to the sonar array length. In practice, the aperture of both sonars is limited. Due to this reason, the resolution is also limited, and it is inversely proportional to the array length. Furthermore, an increase in target range would lead to a worse resolution of the sonar system. In summary, higher performance can be gained at a close range using large array, while low resolution at a far range is obtained with the same array. Because of these limitations, the synthetic aperture sonar (SAS) (Wu and Yan, 2021; Huang and Yang, 2022) providing a range- and frequency-independent image is presented by underwater acoustic engineers. The basic idea of this sonar is that the data sampled by SAS (Zhang et al., 2019d) is coherently processed. That is to mean that a large virtual array can be synthetized. Therefore, high resolution in the along-track dimension can be obtained. Up to now, multiple receivers (Zhang et al., 2022a, 2021; Yang, 2024; Zhang et al., 2024d) are utilized. With this operation, spatial sampling is increased to relax the instantaneous sampling. Consequently, maximum imaging range in the range direction and high resolution in the along-track direction can be obtained at the same time. However, it is not easy to develop imaging processors because traditional processing methods based on Fourier domain are studied based on the monostatic SAS system (Wang et al., 2015; Gough, 1986), which transmits and receives the signal with the same transducer.

When it comes to fast SAS image formation approaches (Zhang et al., 2019b, 2017b), the primary task lies in the deriving of spectrum (Zhang and Yang, 2019; Zhang et al., 2019a, 2023a). The multireceiver type owns a common transmitter and many receivers. Hence, the transmitter and each receiver is regarded as a bistatic SAS, and the system has lots of bistatic SASs. Consequently, we can simply calculate the analytical spectrum based on bistatic type that owns two square-rooted equations. Based on the square of two-round slant range (Geng et al., 2008; Zhang et al., 2013a), a square-rooted-equation considered equivalent to a two-round range is obtained. Then, it is easy to deduce the spectrum. However, spectrum accuracy is limited by the limited terms approximation. The primary range with accurate expression is firstly approximated by series approximation based on Taylor (Zhang et al., 2014b; Neo et al., 2008), Legendre polynomials (Wang and Li, 2010; Zhang et al., 2019c), and Chebyshev polynomials (Clemente and Soraghan, 2012; Zhang et al., 2022c). Adopting the series reversion approach, PTRS can also be analytically presented. In general, PTRS is very complicated, and the imaging algorithms are hard to develop. That is to say, further approximations would be adopted to deduce the focusing algorithms. Currently, multireceiver SAS focusing algorithms based on monostatic configuration are widely used as these methods just preprocess the multireceiver SAS data firstly. Then, traditional focusing approaches of monostatic configuration can be directly applied to preprocessed data. Up to now, the phase center approximation (PCA) methodology (Gough and Hayes, 2005; Zhang et al., 2023b; Bonifant et al., 2000; Zhang and Yang, 2021) is the main processor used to preprocess the multireceiver SAS data, and then traditional imaging algorithms are directly adopted. The main characteristic of both methods lies in the fact that the multireceiver SAS PTRS can be separated into two terms. The first term is named multireceiver deformation (MD) phase, and the remaining expression is named the monostatic factor. However, the PCA is characterized by a large computation load. Besides that, it is indispensable to correct the space-variant errors. Hence, the imaging performance is degraded.

Here we offer a multireceiver chirp scaling (CS) associated with Loffeld’s bistatic formula (LBF) (Zhang et al., 2024c), which still has the MD and monostatic factors. Using our method, the first step is to compensate the MD term for every dataset of the receiver. After handling this stage, the equivalent monostatic SAS datasets are obtained by combining all datasets in sequence. Compared to back projection (BP) method, the efficiency would be largely improved as the multiplication rather than interpolation is used by our method. The simulations and lake-trail results further verify the efficiency and effectiveness of our method.

 Echo model

The configuration of the multireceiver system in Figure 1 has M receivers and a single transmitter. All receivers and transmitters are in a linear array. In azimuth, u·κ shows the position of the transmitter. RT(r;κ)=r2+(uκ)2 shows the distance from transmitter to target. Here κ and u denote the slow time and sonar carrier velocity. The slow time mainly means that the sonar movement is along the moving direction. u·τi shows the movement from signal emitting and the reception of target echo. RRi(r,di;κ)=r2+(uκ+di+uτi)2 shows the i-th receiver range between the target and this receiver. Here di shows the length of the i-th bistatic SAS. τi shows the precise time from signal emitting and the reception of target echo (Xu and Chen, 2022; Bonifant, 1999; Zhang et al., 2018). This time is denoted by

System configuration.

Diagram showing a rectangular array with labeled heights and segments along the azimuth axis, arrows pointing from the array to a blue star labeled r representing the target, and range-related distances labeled RT and RRi. τi={u(uκ+di)+c(uκ)2+r2}  2+(υ2−u2)(2uκdi+di2)υ2−u2+u(uκ+di)+υ(uκ)2+r2υ2−u2

The system kept on moving along the azimuth direction. The precise time shown in Equation 1 denotes the propagation duration between the signal transmission and reception (Zhang et al., 2017a).

The term υ denotes the propagation speed of sound in water. An approximation τi≈2ur/υ is often used to replace with the precise time due to its complicated expression. The broadband signal is often emitted by the transmitter. For simplicity, the chirp signal is used by the transmitter in this paper. The echo of the i-th bistatic SAS (Zhang, 2024; Huang and Zhong, 2021) is

ssi(τ,κ)=q(τ−Ri(r,di;κ)υ)exp{−j2πνcRi(r,di;κ)υ}

In Equation 1, q(τ) is the emitted chirp waveform. τ denotes the fast time, which mainly stands for the propagation of sound in water. νc denotes the center frequency. Ri(r,di;κ)=RT(r;κ)+RRi(r,di;κ) denotes the double-round slant range, and it is variant with the target range, baseline, and slow time. It is worth noting that the echoed waveform shown in Equation 1 ignores the influence of beampattern (Zhang and Ying, 2022).

 Presented method Multireceiver SAS PTRS

In general, PTRS is the most important factor for multireceiver SAS fast imaging algorithms. In practice, it is just the two-dimensional (2D) Fourier transformation with respect to Equation 1. We conduct fast Fourier transformation (FFT) in the range dimension. Then, Equation 1 can be reformulated as

Ssi(ντ,κ)=Q(ντ)exp{−j2πνcRi(r,di;κ)υ}exp{−j2πντRi(r,di;κ)υ}

Conducting the FFT with respect to Equation 2 in the azimuth dimension, we can get

SSi(ντ,νκ)=Q(ντ)∫−0.5Ts+0.5Tsexp{−j2πνcRi(r,di;κ)υ}exp{−j2πντRi(r,di;κ)υ}exp{−j2πνκκ}dκ=Q(ντ)∫−0.5Ts+0.5Tsexp{−j2π(νc+ντ)RT(r;κ)+RRi(r,di;κ)υ}exp{−j2πνκκ}dκ=Q(ντ)∫−0.5Ts+0.5Tsexp{−j2π[νcRT(r;κ)+RRi(r,di;κ)υ+ντRT(r;κ)+RRi(r,di;κ)υ+νκκ]}dκ=Q(ντ)∫−0.5Ts+0.5Tsexp{−j2π[ξT(ντ,κ)+ξRi(ντ,κ)]}dκ

with

ξT(ντ,κ)=2π[(νc+ντ)RT(r;κ)υ+0.5νκκ]ξRi(ντ,κ)=2π[(νc+ντ)RRi(r,di;κ)υ+0.5νκκ]

Ri(r,di;κ) is very complicated, and it is impossible to calculate the integral shown in Equation 3. Ts is the synthetic aperture time. Q(ντ) stands for the frequency-domain expression of the emitted chirp waveform. ντ is the temporal frequency, while νκ denotes the Doppler frequency. The phase of the emitter is ξT(ντ,κ). The symbol ξRi(ντ,κ) indicates the phase associated with the receiver.

Adopting the theory of stationary phase (Zhang et al., 2022b), Equation 5 can be further transferred into two equations.

dξT(ντ,κ)dκ=(νc+ντ)u2κυr2+(uκ)2+0.5νκκ=0dξRi(ντ,κ)dκ=(νc+ντ)u(uκ+di+uτi)υr2+(uκ+di+uτi)2+0.5νκκ=0

The points of stationary phase related to Equation 6 are shown as shown in Equation 7.

κT=−υrνκ2u(νc+ντ)1u2−[υνκ2(νc+ντ)]2κRi=−υrνκ2u(νc+ντ)1u2−[υνκ2(νc+ντ)]2−diu−2rυ

Conducting the quadratic approximation to Equation 4, we obtain

ξT(ντ,κ)≈ξT(ντ,κT)+ξ˙T(ντ,κT)(κ−κT)+12ξ¨T(ντ,κT)(κ−κT)2ξRi(ντ,κ)≈ξRi(ντ,κRi)+ξ˙Ri(ντ,κRi)(κ−κRi)+12ξ¨Ri(ντ,κRi)(κ−κRi)2

In Equation 6, ξ˙T(ντ,κT) and ξ˙Ri(ντ,κRi) are the first derivative of ξ˙T(ντ,κ) and ξ˙Ri(ντ,κ), respectively. ξ¨T(ντ,κT) and ξ¨Ri(ντ,κRi) are the second derivative. With the theory of stationary phase, we know that ξ˙T(ντ,κT)=0 and ξ˙Ri(ντ,κRi)=0. Adopting Equations 6 and 3, we get

SSi(ντ,νκ)=Q(ντ)∫−0.5Ts+0.5Tsexp{−j2π[ξT(ντ,κ)+ξRi(ντ,κ)]}dκ=Q(ντ)exp{−j[ξT(ντ,κT)+ξRi(ντ,κRi)]}Ii(ντ,νκ)

where I_i is shown in Equation 10.

Ii(ντ,νκ)=∫−0.5Ts+0.5Tsexp{−j2[ξ¨T(ντ,κT)(κ−κT)2+ξ¨Ri(ντ,κRi)(κ−κRi)2]}dκ

Based on the tedious algebra, Equation 8 is reformulated as

Ii(ντ,νκ)=∫−0.5Ts+0.5Tsexp{−j2[ξ¨T(ντ,κT)·ξ¨Ri(ντ,κRi)ξ¨T(ντ,κT)+ξ¨Ri(ντ,κRi)(κT−κRi)2+[ξ¨T(ντ,κT)+ξ¨Ri(ντ,κRi)](κ−κi*)2]}dκ

In Equation 11, κi* is computed by using the theory of stationary phase, and it is given by Equation 12.

κi*=ξ¨T(ντ,κT)κT+ξ¨Ri(ντ,κRi)κRiξ¨T(ντ,κT)+ξ¨Ri(ντ,κRi)

Therefore, we achieve Equation 9, which is shown as

SSi(ντ,νκ)=exp{−jπ4}Q(ντ)exp{−jΩ(ντ,νκ)}exp{−j2Ωi(ντ,νκ)}

Ω(ντ,νκ) and Ωi(ντ,νκ) are the monostatic equivalent expression and bistatic deformation expression, and they are indicated by

Ω(ντ,νκ)=ξT(ντ,κT)+ξRi(ντ,κRi)=−πνtdiu−πνκ2rυ+4πυr(νc+ντ)2−νκ2υ24u2 Ωi(ντ,νκ)=ξ¨T(ντ,tT)·ξ¨Ri(ντ,κRi)ξ¨T(ντ,tT)+ξ¨Ri(ντ,κRi)(κT−κRi)2=πυ·(di+2rυu)2(νc+ντ)2·[(νc+ντ)2−νκ2υ24u2]3/2r

Inspecting Equation 11, we can obtain the monostatic equivalent datasets after correcting the bistatic deformation term.

Equation 13 is the foundation of the deduced algorithm. However, the deduction of Equation 13 includes some approximations. Therefore, we present the error produced by the approximations. The SAS center and bandwidth frequencies are 16,000 and 20,000 Hz, respectively. The system works with 3 m/s. The size of the receiver and receiver-array in the azimuth axis is 0.04 and 2.64 m, respectively. The repetition interval of emitted ping is 440 ms. The spectrum based on the numerical methodology is the criterion (Zhang et al., 2018), and the corresponding phase is denoted by Ωnu, which yields the error

ΔΩi=Ωnu−Ω(ντ,νκ)−Ωi(ντ,νκ)

From Equation 16, the error is getting larger with the incensement of the i-th bistatic SAS length. To simplify the error, the error of the largest length of the M-th bistatic SAS is focused. Adopting Equation 16, the error is shown in Figure 2. Figure 2a is the error of close target located at 50 m. Figure 2b is the error of far target at 300 m. In general, the maximum error is not larger than π8 (Xu et al., 2013). The error shows that the accuracy of the presented spectrum is sufficient for imaging.

Error. (a) close range. (b) Far range.

Two 3D surface plots labeled (a) and (b) display phase error in radians as a function of azimuth in Hertz and range in kilohertz, using a color gradient from blue to red. Plot (a) shows a larger range of phase error values than plot (b), indicating a greater magnitude of phase error across the same axes. Multireceiver SAS CS algorithm

From Equations 13 and 14, we find that the CS algorithm (Wang et al., 2009; Zhang et al., 2014c; Callow et al., 2009; Zhang et al., 2013c) based on monostatic SAS can be directly applied after correcting Eq. (15).

Based on Equation 15, it was found that Equation 15 is a function of ντ, νt, and r. It means that Equation 15 is characterized by range variance. Considering the characteristic, the data partitioning approach (Xu et al., 2013) is exploited. Here we list the detailed steps.

For each receiver dataset, 2D Fourier transformation is carried out, and each receiver dataset is partitioned into Y components along the range axis. Zero-padding is conducted to protect the component edge datasets. The minimum size of zero-padding should be the pulse duration. In the spectral domain, the compensation factor for the y-th component is given by

Hsub(ντ',νκ;ry)=exp{jπυ·(di+2ryυv)2(νc+ντ')2·[(νc+ντ')2−(ντ')2υ24u2]3/22ry}

In Equation 13, ντ' is the instantaneous frequency related to the component (Zhang et al., 2013b). ry means the middle target range. The maximum error of this component is usually not larger than π8 (Xu et al., 2013). This condition is shown as

|πυ·(di+2ryυv)2(νc+ντ')2·[(νc+ντ')2−(ντ')2υ24u2]3/22ry−πυ·[di+2(ry±0.5W)υv]2(νc+ντ')2·[(νc+ντ')2−(ντ')2υ24u2]3/22(ry±0.5W)|<π8

The term W in Equation 18 is the sub-block size. After getting this size W, the block number is deduced.

The step shown in Equation 17 is performed for each component. Then, the padded zero at the sub-block edge is removed, and all sub-blocks are combined into the entire data. Following this process, the deformation component shown in Equation 15 is completely corrected. In the azimuth frequency domain, the receiver data within each ping are arranged in order to obtain the datasets corresponding to a large synthesized array, and this step is called the interleaving arrangement. Then, the first term in Equation 14 is compensated. The remaining steps focus on the correction of the square-rooted phase shown in Equation 14. The second-order expansion of the square-rooted phase shown in Equation 14 is denoted by

ϕ=4πυr(νc+ντ)2−νκ2υ24u2≈4πrλ1−υ2νκ24u2νc2+4πrυ1−υ2νκ24u2νc2ντ−2πrλυ2(11−υ2νκ24u2νc23−11−υ2νκ24u2νc2)ντ2

Currently, chirp scaling is carried out in the azimuth frequency domain, and the correcting factor is written as Equation 20.

Hscaling(τ,νκ;rs)=exp{jπγe(νκ;rs)Cs(νκ)(τ−2r(νt;rs)υ)2}

where C_s, gama_e and r are shown in Equation 21, Equation 22 and Equation 23, respectively.

Cs(νκ)=11−υ2νκ24u2νc2−1 γe(νt;rs)=1γ−2λrsυ2νt24u2νc2υ2(1−υ2νt24u2νc2)3/2 r(νκ;rs)=rs(Cs+1)

Here, Cs(νκ) is the chirp scaling factor. γe(νκ;rs) is the improved chirp rate, and γ represents the rate of chirp waveform. Equation 22 can be deduced based on the last expression in Equation 19 and the rate of chirp waveform. rs represents the center range. After this compensation, the variance of range migration trajectories is corrected, and this step is named differential range cell migration correction (RCMC).

In the range and azimuth frequency domain, we simultaneously accomplish the bulk RCMC and range compression. The correcting term is shown as Equation 24.

H rc&rcmc ( ν τ, ν κ; r s)=exp{jπ ν τ 2 γ e ( ν κ ; r s ) [ 1 + C s ( ν κ ) ] }·exp{j4π C s ( ν κ ) r s υ ν τ}

In the azimuth frequency domain, the last step aims to correct the error that is developed by the scaling stage, and the corresponding function used for the correction is shown as Equation 25.

Hac&pc(ντ,νκ;rs)=exp{j4πrβλ}·exp{−jπνκ2rυ}×exp{−j4πυ2γe(νκ;rs)Cs(νκ)[1+Cs(νt)](r−rs)2}

Based on the processing steps discussed, Figure 3 exhibits the imaging scheme of the discussed methodology.

Flowchart of our method.

Flowchart illustrating the processing steps for receiver data in a multi-channel imaging system, starting with sub-block segmentation, followed by iterative 2D FFT, mixing, and range IFFT for each receiver, merging via data interleaving, then applying range FFT, mixing, IFFT, and azimuth IFFT to produce the final imaging result.

The proposed methodology depends on the FT, IFT, and multiplication. For N sampling points, the floating point operations based on FT/IFT are 5Nlog2N. For a single complex multiplication, the floating point operations are six. These can be found in Cummings’ work (Cumming and Wong, 2005). For the SAS datasets, P pings are emitted. Along the azimuth direction, the data size is La=PM. Along the range direction, the data size is Lr=LsubY. Lsub is the data size of each sub-block. Based on Figure 3, the computation load of the discussed methodology is

O=10LaLrlog2(Lsub)+5LaLrlog2(P)+24LaLr+10LaLrlog2(Lr)+5LaLrlog2(La)

From Equation 26, the discussed methodology can significantly save the processing time.

 Experiments

This section concentrates on the validation of the presented method based on simulations and real datasets.

Main results of the presented method

Here we firstly discuss the results of the main steps based on the suggested methodology. In order to evaluate the imaging performance of the suggested methodology, it is supposed that the imaging area contains a single target. The azimuth position is 15 m, while its position in range axis is 151 m. The SAS center and bandwidth frequencies are 16,000 and 20,000 Hz, respectively. The system works with 3 m/s. The size of the receiver and receiver-array in the azimuth axis is 0.04 and 2.64 m, respectively. The repetition interval of emitted ping is 440 ms. These parameters are directly identical to those in Section 3.1.

For the first receiver, the real component of the received echo after the range compression is shown in Figure 4a. In practice, the SAS echo is still the discrete sampling based on the repetition interval. Along the azimuth axis, the receiver data shown in Figure 4a is under-sampling.

Echo after arrangement without processing. (a) after range compression. (b) 2D spectrum.

Two scientific data visualizations labeled as (a) and (b) are shown side by side. Panel (a) is a heatmap displaying values along range in meters and azimuth in meters, featuring a curved pattern with varying colors on a light green background. Panel (b) is a heatmap using range in kilohertz and azimuth in hertz, with a dense rectangular cluster of bright yellow and red tones against a dark blue background. Both plots highlight spatial or spectral data distributions for analysis.

After applying 2D FT to the data shown in Figure 4a, we can obtain the spectrum, and Figure 4b exhibits the spectrum magnitude. Since the data of each receiver is sampled with the time, repetition interval, the maximum azimuth Doppler bandwidth is just the reciprocal of the repetition interval. Figure 4b further enhances this conclusion. According to the signal theory, the signal bandwidth which is less than the sampling rate leads to a non-overlapped spectrum. When the bandwidth of the signal is larger than the sampling rate, the corresponding spectrum of the sampled signal would be overlapped.

With the suggested method, the monostatic analogous signal would be acquired, and Figure 5a exhibits this waveform. It should be noted that the component width used by this experiment is 5.7 m. After the interleaving arrangement, a monostatic analogous signal is acquired. From Figure 4, we find that the azimuth signal is continuous. Figure 5b is the magnitude of the spectrum corresponding to the spatial signal in Figure 4a. The improvement is evident when analyzing Figures 3b and 4b.

Echo after arrangement based on the presented method. (a) After range compression. (b) 2D spectrum.

Panel (a) displays a heatmap with a color gradient indicating intensity, showing a curved, high-intensity region centered at approximately 151 meters range and 15 meters azimuth. Panel (b) presents a heatmap with a rectangular, high-intensity area centered at zero kilohertz range and zero hertz azimuth, surrounded by lower intensity in blue.

Based on the CS algorithm, the datasets in Figures 4 and 5 are processed individually. Figure 6 exhibits the contours along the azimuth axis. After directly imaging the echoed datasets, the results seriously suffer from the false targets. The focusing ability can be substantially enhanced by utilizing the suggested strategy.

Azimuth contours after focusing.

Line graph compares amplitude in decibels versus azimuth in meters for two methods: without preprocessing (blue line) and presented method (red line with markers). The presented method shows lower amplitude side lobes, indicating improved noise suppression. Comparisons with BP algorithm

We consider a target located at close range. Its coordinates in 2D space domain are (50 m, 5 m). Via pulse compression, we can get the real part of the original signal collected by the receiver array. The original signal following the compression along the range axis is displayed in Figure 7a. Figure 7b exhibits the datasets according to the methodology used for this job. In comparison with Figures 7a, b shows that our technique effectively eliminates the bistatic property of original signal.

Signal of target at close range. (a) Original signal after pulse compression in range dimension. (b) Signal after interleaving arrangement.

Two false-color heatmaps compare radar intensity with range on the x-axis and azimuth on the y-axis. Panel a shows a curved signal, while panel b displays a straighter, more centered signal. Both have intensity concentrated along the center vertical region against a light green background. Labels beneath indicate (a) and (b).

According to the BP algorithm (Zhang and Yang, 2022; Zhang et al., 2014a), the objects are recovered initially. Figure 8a provides the processing outcome. We are able to obtain the focusing result depicted in Figure 8b using the method mentioned.

Processing results for close objects. (a) BP. (b) Suggested method.

Two grayscale heatmaps labeled (a) and (b) display intensity in decibels for azimuth versus range in meters, each with a vertical color bar on the right ranging from zero to negative forty-five decibels. Both heatmaps show a central peak with similar cross-shaped patterns and axis labels indicating range and azimuth measurements.

Based on Figure 8b, we find that the target at close range is well focused with our method. The azimuth slices corresponding to the BP and the presented methods are shown in Figure 9. It can be seen in Figure 9 that the BP algorithm possesses fewer improvements of performance than that with our method. Generally, the suggested solution almost matches the quality of the BP method.

Azimuth contours for close target.

Line graph comparing amplitude in decibels versus azimuth in meters for two methods: BP algorithm (blue dashed line) and presented method (solid red line). The presented method shows a higher peak amplitude around 5 meters.

Then, the focusing of the far target is carried out. The location of the object in range is 300 m. The location of the object along the azimuth axis is 18 m. Figure 10a shows the original signal after pulse compression, while Figure 10b illustrates the datasets after the interleaving arrangement. According to our solution, the signal of multireceiver SAS is successfully enforced into monostatic datasets for far targets.

Signal of target at far range. (a) Original signal after pulse compression in range dimension. (b) Signal after interleaving arrangement.

Two color-coded heat maps compare radar return intensity across range versus azimuth in meters, labeled (a) and (b); both display a similar curved, multicolored signature on a green background, with axes labeled “Range (m)” and “Azimuth (m)”.

Figure 11a illustrates the focusing results with the BP algorithm (Zhang and Yang, 2022), and those of the recommended methodology are deployed in Figure 11b. According to the focusing outcome in Figure 11, the target at the far range can also be focused well with our method.

Processing results for the target at far range. (a) Result with BP algorithm. (b) Result with the presented algorithm.

Panel (a) and panel (b) each show grayscale heatmaps of radar signal intensity in decibels over range and azimuth axes, both with a central bright point and surrounding gradients, accompanied by vertical color bars.

The qualities of focusing can be examined using the azimuth contours. Figure 12 illustrates the contours along the azimuth axis. Adopting Figures 11 and 12, it can be seen that the recommended methodology owns a high-quality capacity of reconstruction.

Contours along the azimuth axis for far target.

Line graph comparing amplitude in decibels versus azimuth in meters for two approaches: BP algorithm (blue dashed line) and presented method (red solid line). Both methods show a peak near 18 meters with the presented method reaching a higher amplitude.

In Table 1, the peak sidelobe ratio (PSR) and integrated sidelobe ratio (ISR) are computed to assess the handling quality. Both parameters are defined as Equation 27 and Equation 28, individually.

PSR and ISR of focusing methodologies.

Methods Close targets Far targets
PSR/dB ISR/dB PSR/dB ISR/dB
BP algorithm -14.66 -9.11 -14.89 -10.17
Presented method -14.38 -9.06 -14.62 -10.15
PSR=10log10ZsZ−3dB ISR=10log10∑Z−∑−3dBZ∑−3dBZ

Here Zs and Z-3dB are the intensities of the peak side lobe and the main lobe, respectively. The term Z means the intensity of the azimuth contours after focusing.

For the target at close range, the PSR difference between our methodology and the BP methodology is nearly 0.28 dB, while the ISR discrepancy is about 0.05 dB. Both parameters corresponding to both methodologies are almost identical. For the far target, the sidelobe levels of the suggested approach are closely comparable to those of the BP algorithm. As a result, our strategy is very powerful to cope with the SAS datasets.

 Water floor data imaging results

We utilize the water floor data to assess the effectiveness of the recommended methodology. The processing results adopting the recommended solution (Zhang et al., 2014d) and the BP (Zhang and Yang, 2022) are shown in Figure 13. The water floor data imaging results with both methodologies indicate that the recommended solution gains the equivalent outcomes compared to those of the BP methodology (Zhang and Yang, 2022). As a result, the proposed strategy could accurately cope with the original datasets. The imaging outcomes adopting the water floor data illustrate that the recommended solution has the ability to handle the SAS datasets across the entire area.

Imaging results of water floor data. (a) Presented method. (b) BP.

Two adjacent side-by-side radar images labeled (a) and (b) display terrain with contour-like bright lines indicating elevation or structure against a mottled dark brown background. Both have azimuth on the vertical axis from zero to fifty meters and range on the horizontal axis from sixty to two hundred twenty meters, showing similar structural details for comparison.

Adopting the same computer, Table 2 indicates the focusing time of both methodologies. Based on the comparisons of processing time, we find that the recommended solution can highly speed up the imaging efficiency. It shows that the presented method has the potential to efficiently focus the SAS data.

Imaging time.

Time Recommended methodology BP algorithm
Processing time (s) 3 8933
Conclusions

With this job, a focusing solution is established. With our method, we firstly correct the bistatic phase for each multireceiver SAS data. After this step, all receiver datasets are processed by exploiting the interleaving arrangement. We then gain the datasets similar to the monostatic formation. The CS algorithm which is characterized by high efficiency is applied to focus the datasets. Based on simulations, we firstly show the main step results of our method. Then, we compare the focusing performance and efficiency between our method and the conventional method. The comparisons indicate that our method can achieve the same results based on the BP algorithm. Furthermore, our method is much more efficient than the conventional methods.

 Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

 Author contributions

LZ: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. JL: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. JF: Conceptualization, Formal Analysis, Methodology, Supervision, Validation, Writing – review & editing, Writing – original draft. PW: Conceptualization, Formal Analysis, Funding acquisition, Project administration, Resources, Validation, Visualization, Writing – review & editing, Writing – original draft. ZX: Formal Analysis, Methodology, Resources, Writing – review & editing, Supervision.

 Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Correction note

This article has been corrected with minor changes. These changes do not impact the scientific content of the article.

 Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Edited by: Lin Liu, Ministry of Natural Resources, China

Reviewed by: Pan Huang, Weifang University, China

Peixuan Yang, Lanzhou Jiaotong University, China