Bistatic Formula Research Articles

An omega‐k algorithm for multireceiver synthetic aperture sonar

AbstractIn this Letter, the authors present a novel imaging algorithm for multireceiver synthetic aperture sonar (SAS). The Loffeld's bistatic formula (LBF) including quasi‐monostatic (QM) and multireceiver deformation (MD) phases is first reformulated into range‐variant and range‐invariant phases. After compensating the range‐invariant phase, the subsequent steps are the extended Stolt mapping for each bistatic SAS. Based on simulations, the presented method can also provide high‐resolution result which is similar to that of back projection (BP) algorithm.

Efficient imaging method for multireceiver SAS

With the Loffeld's bistatic formula, the point target reference spectrum (PTRS) of multireceiver synthetic aperture sonar (SAS) can be decomposed into the quasi monostatic term and bistatic deformation (BD) term. The former term is analogous to the monostatic SAS PTRS while the latter one describes the bistatic phase introduced by the bistatic displacement between the transmitter and receiver. The multireceiver SAS echo signal can be transformed into monostatic SAS equivalent data after correcting the BD term. Based on the range sub-block processing method, this data transformation is conducted by using the preprocessing step. Then, fast imaging processors based on monostatic SAS system can be directly exploited to process the monostatic SAS equivalent data. In this paper, the authors mainly focus on the application of non-linear chirp scaling algorithm. At last, the simulations and real data are used to validate the presented method. The processing results show that the imaging performance of presented method is nearly close to that of traditional method. Besides, the imaging efficiency is highly improved compared to the phase centre approximation (PCA) method.

Multireceiver SAS Imagery Based on Monostatic Conversion

To use monostatic based imaging algorithms for multireceiver synthetic aperture sonar, the monostatic conversion is often carried out based on phase centre approximation, which is widely exploited by multireceiver SAS systems. This article presents a novel aspect for dealing with the multireceiver SAS imagery, which still depends on the idea of monostatic conversion. The approach in this article is based on Loffeld's bistatic formula that consists of two important terms, i.e., quasi monostatic and bistatic deformation terms. Our basic idea is to preprocess the bistatic deformation term and then incorporate the quasi monostatic term into an analogous monostatic spectrum. With this new spectrum, traditional imaging algorithms designed for monostatic synthetic aperture sonar can be easily exploited. In this article, we show that Loffeld's bistatic formula can be reduced to the same formula as spectrum based on phase centre approximation when certain conditions are met. Based on our error analysis, the maximum error magnitude of PCA method is about 1 rad, which would noticeably affect the SAS imagery. Fortunately, the error magnitude of presented method can be always kept within ${{\rm{\pi }} \mathord{/ {\vphantom {{\rm{\pi }} 4}} \kern-\nulldelimiterspace} 4}$ . It means that Loffeld's bistatic formula provides a more accurate approximation of the spectrum compared to that based on phase centre approximation. After that, this article develops a new imaging scheme and presents imaging results. Based on quantitative comparisons, the presented method well focuses multireceiver SAS data, and it provides better image compared to phase centre approximation method.

Imaging Algorithm for Multireceiver Synthetic Aperture Sonar

This paper presents a range migration algorithm for the multireceiver synthetic aperture sonar (SAS) based on Loffeld’s bistatic formula, which consists of the quasi monostatic term and the bistatic deformation (BD) term. It is expected that the multireceiver data can be easily focused by the monostatic SAS imaging algorithm after compensating the BD term. In this paper, the BD term is first decomposed into the range variant phase and the range invariant phase. At this point, the BD term is compensated via two steps. One step is the compensation of the range variant phase through the range-dependent sub-block processing in the 2-D frequency domain. The other one is the arrangement of the multireceiver data in a sequential order in the 2-D time domain. After these operations, we obtain the monostatic SAS equivalent data, which is viewed as the input of the traditional range migration algorithm. The processing results of the simulated data and the real data show that the proposed method can work well for the focusing of the multireceiver SAS data. More importantly, the presented method dramatically improves the imaging efficiency without loss of performance since the complex multiplication and the data arrangement are only exploited by the presented method.

Extended Range Doppler Algorithm for Multiple-Receiver Synthetic Aperture Sonar Based on Exact Analytical Two-Dimensional Spectrum

For the imaging of multiple-receiver synthetic aperture sonar (SAS) under the nonstop-hop-stop case, an exact analytical solution for the two-dimensional (2-D) frequency spectrum is usually difficult to obtain because of the existence of the double-square-root (DSR) term in the range history equation. Several approximate solutions for the 2-D spectrum have been derived to focus the multiple-receiver SAS data. In this paper, the range history geometry for multiple-receiver SAS is newly constructed to meet the demand of the derivation of an analytical 2-D spectrum. According to the geometry-based bistatic formula (GBF) method, a quasi-analytical 2-D spectrum with an unknown variable named half quasi-bistatic angle (HQBA) is reviewed. Based on the method of series reversion (MSR), the fourth order equation with respect to the HQBA is solved and an analytical HQBA is obtained. After substituting the analytical HQBA into the quasi-analytical 2-D spectrum, a completely analytical 2-D spectrum is obtained and the problem of the 2-D spectrum derivation caused by the DSR term is solved successfully. In this paper, an extended range Doppler (RD) algorithm based on the derived 2-D spectrum is proposed for focusing the multiple-receiver SAS data under the non stop-hop-stop case. The results of simulation and real experiment data imaging have validated the effectiveness and accuracy of the proposed algorithm.

An Omega-K Algorithm for Translational Invariant Bistatic SAR Based on Generalized Loffeld's Bistatic Formula

In this paper, an omega-K imaging algorithm to focus the raw data of translational invariant (TI) bistatic synthetic aperture radar (BSAR) is proposed. The method utilizes a point target reference spectrum of generalized Loffeld's bistatic formula (GLBF). Without the bistatic deformation term, GLBF is the latest development of Loffeld's bistatic formula. It is comparable in precision with the method of series reversion (MSR), but it has a much simpler form than MSR and a similar form to a monostatic case. Based on the spatial linearization of GLBF, the Stolt transformation relationship is derived. The method can consider the linear spatial variation of Doppler parameters, which is always ignored in previous publications about bistatic omega-K algorithms. This method can handle the cases of TI BSAR with high squint angles and large bistatic degrees. In addition, a compensation method for the phase error caused by the linearization is discussed. Numerical simulations and experimental data processing verify the effectiveness of the proposed method.

New analytical imaging algorithm for general case airborne bistatic SAR

This paper focuses on the general case (GC) airborne bistatic synthetic aperture radar (SAR) data processing, and a new analytical imaging algorithm based on the extended Loffeld's bistatic formula (ELBF) is proposed. According to the bistatic SAR geometry, the track decoupling formulas that convert the bistatic geometry to the receiver-referenced geometry in a concise way are derived firstly. Then phase terms of ELBF are decomposed into two independent phase terms as the range phase term and the azimuth phase term in a new way. To get the focusing result, the bistatic deformation (BD) term is compensated in the two-dimensional (2-D) frequency domain, and the space-variances of the range phase term and the azimuth phase term are eliminated by chirp scaling (CS) and chirp z-transform (CZT), respectively. The effectiveness of the proposed algorithm is verified by the simulation results.

A Generalized Omega-K Algorithm to Process Translationally Variant Bistatic-SAR Data Based on Two-Dimensional Stolt Mapping

In translationally variant (TV) bistatic synthetic aperture radar (BSAR), 2-D spatial variation is a major problem to be tackled. In this paper, a generalized Omega-K imaging algorithm to deal with this problem is proposed. The method utilizes a point target reference spectrum of the generalized Loffeld's bistatic formula (LBF) (GLBF). Without the bistatic-deformation term, GLBF is the latest development of LBF. Similar to the monostatic case, it has a much simpler form than other point target reference spectra. Based on the spatial linearization of GLBF, the Stolt mapping relationship is derived. Different from the traditional Omega-K algorithms for monostatic SAR and translationally invariant BSAR, this approach uses a 2-D Stolt frequency transformation. Through this transformation, the method can deal with the 2-D spatial variation. It can also consider the linear spatial variation of Doppler parameters, which is usually not considered in the previous publications on bistatic Omega-K algorithms. This method can handle the cases of TV-BSAR with different trajectories, different velocities, high squint angles, and large bistatic angles. In addition, a compensation method for the phase error caused by the linearization is discussed. Numerical simulations and experimental data processing verify the effectiveness of the proposed method.

Focusing bistatic forward-looking synthetic aperture radar based on modified Loffeld’s bistatic formula and chirp scaling algorithm

A focusing solution for bistatic forward-looking synthetic aperture radar (BFSAR) is presented. Forward-looking imaging is highly desirable in some potential applications, such as self-landing in bad weather, military surveillance, and navigation. Unfortunately, monostatic synthetic aperture radar reaches its limit when it is used in a forward-looking configuration. BFSAR can provide a high-resolution image in the forward-looking direction. However, due to the special forward-looking geometry, many proposed methods of deriving a bistatic point target reference spectrum (BPTRS) cannot handle the BFSAR data well. A modified Loffeld's bistatic formula (MLBF) for a forward-looking configuration is proposed first; it can get an accurate BPTRS of BFSAR. Then, a chirp scaling algorithm (CSA) based on MLBF is derived. CSA without interpolation allows high performance. Numerical simulations show that the proposed focusing solution can handle the BFSAR data well and achieve a high- resolution focused image. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI. (DOI: 10.1117/1.JRS.8.083586)

An Analytical Imaging Algorithm for Airborne Parallel Translational Variant SAR Data Focusing

For focusing airborne parallel translational variant SAR, an analytical imaging algorithm based on the extended Loffeld bistatic formula (ELBF) is proposed. With the bistatic geometry, point target (PT) position offsets to the transmitter track are expressed in terms of PT position offsets to the receiver track. Then phase terms of spectrum are divided into range and azimuth phase terms. The bistatic deformation (BD) term is compensated in 2-D frequency domain, and space variances of range and azimuth phase terms are eliminated by chirp scaling (CS) method and chirp z-transform (CZT) respectively. The proposed imaging algorithm is verified by simulations.

BISTATIC FORWARD-LOOKING SYNTHETIC APERTURE RADAR IMAGING BASED ON THE MODIFIED LOFFELD'S BISTATIC FORMULA

Bistatic forward-looking SAR (BFSAR) has many potential applications, such as self- landing in bad weather and military detection. Therefore, BFSAR receives considerable attention recently. The imaging algorithms for BFSAR are the di-culties of the study. The original Lofield's Bistatic Formula (LBF) can handle most general bistatic SAR conflgurations well. But in some complex bistatic geometries, such as high squint or forward-looking cases, the performance of LBF is degenerated. Some extended LBF (ELBF) methods have been developed, which improve the performance of LBF in some special geometries, but still not the forward-looking conflguration. In this paper, we modify the LBF method and try to solve the instantaneous azimuth frequencies of transmitter and receiver directly. Then, we can obtain a bistatic point target reference spectrum (BPTRS), which is accurate enough for forward-looking conflguration. A range Doppler algorithm (RDA) based on this BPTRS is derived. Finally, simulations validate the accuracy of the modifled Lofield's Bistatic Formula (MLBF) and efiectiveness of imaging algorithm.

Using omega‐ K algorithm for bistatic synthetic aperture radar image formation based on modified extended Loffeld's bistatic formula

Bistatic synthetic aperture radar (BSAR) as a way of Earth remote sensing has been developed considerably in recent years, both theoretically and practically due to its unsubstitutable services. BSAR frequency-domain processing algorithms are efficient ways of image formation in comparison to ideal two-dimensional matched filtering and to relatively accurate and time-consuming time-domain algorithms. Among these frequency-domain algorithms, omega-K is the most precise. The starting and key step of frequency-domain algorithms is the derivation of bistatic spectrum. Recently, a new bistatic spectrum is reported, which is probably the latest and the most modified version of Loffeld's bistaic formula (LBF), maintaining its accuracy even in azimuth-variant configurations with high squint angles. So far, this spectrum has only been used within range-Doppler algorithm to process BSAR data. The authors investigate the possibility and results of applying this modified version of LBF as a basis for omega-K algorithm. Two approaches, based on Stolt interpolation and inverse scaled Fourier transform, are examined and their effectiveness in general azimuth-variant geometry is validated through several simulations. The proposed implementations show higher performance in terms of image quality measurements as compared to extended LBF-based implementations.

A New Look at Loffeld's Bistatic Formula in Tandem Configuration

A new way of looking at Loffeld's bistatic formula (LBF) is presented for tandem configuration in this letter. The factors that affect the precision of the spectrum are obtained through the comparison with another analytical one. It has been proved that the cosine of the half bistatic angle plays a more important role respect to the other factor, which is whether the baseline to range ratio is equal to the tangent of the half bistatic angle or not. As long as the cosine of the half bistatic angle is very close to one, the LBF spectrum is of high quality, and it does not have a direct influence by the length of the baseline (baseline-to-range ratio) or the size of the squint angle. The factors that affect the precision of the spectrum are discussed in detail through simulated experiments.

A NEW LOOK AT THE POINT TARGET REFERENCE SPECTRUM FOR BISTATIC SAR

Focusing bistatic synthetic aperture radar (SAR) data in frequency domain requires two-dimensional (2D) point target reference spectrum (PTRS). Lofield's bistatic formula (LBF) and the Method of Series Reversion (MSR) have been introduced recently to compute PTRS of bistatic SAR. In this paper, flrstly we generalize the original LBF (OLBF) by introducing the Doppler contribution functions of transmitter and receiver. Thus, OLBF and its derivatives (e.g., extended LBF) can be viewed as special forms of the generalized LBF with constant Doppler contributions. Based on this, secondly the ideal LBF (ILBF) with no computing error, except the error resulting from the principle of stationary phase, is also presented. The ILBF reveals that the theoretical PTRS of bistatic SAR consists of only two monostatic terms, but it does not include bistatic deformation term in comparison with OLBF. It supplies us with a target when we deduce the PTRS for bistatic SAR. Finally, to get the precise analytical PTRS for general bistatic SAR, an approximated ILBF (AILBF) is proposed. It expresses the Doppler contributions of the transmitter and receiver as power series and can approach the ILBF very well. AILBF can keep the precision as MSR and inherit a simple form from LBF. In addition, error limit for the validity of bistatic PTRS is also given. The results in this paper can be used to develop imaging algorithms for extreme bistatic (e.g., spaceborne/airborne) and high squint (e.g., bistatic forward-looking) cases.

Exact analytical two-dimensional spectrum for bistatic synthetic aperture radar in tandem configuration

An exact analytical solution for the two-dimensional (2-D) point target spectrum is usually difficult to obtain because of the existence of a double-square-root (DSR) term in the bistatic range equation. Some approximate solutions for the 2-D spectrum have been derived and used in order to focus bistatic synthetic aperture data. A geometry-based bistatic formula (GBF) method was used for obtaining a quasi-analytical form of a bistatic 2-D spectrum by Zhang (2007). Although the GBF method cannot completely solve the problem of the DSR term, it provides a novel aspect for dealing with this problem. In this study, based on the quasi-analytical spectrum, the authors change the signal expression space and transform the eight-order polynomial equation in terms of the slow time into the four-order polynomial equation in terms of the half quasi-bistatic angle (HQBA) for the tandem configuration. Then the DSR-term problem is successfully solved and a corresponding exact analytical bistatic 2-D spectrum is obtained. It is proved that the spectra acquired by Rocca's smile operator, Loffeld's bistatic formula (LBF) and the method of series reversion (MSR) are equivalent to this proposed analytical spectrum when certain conditions are met, and that this exact analytical spectrum is the most accurate among them.

RANGE DOPPLER ALGORITHM FOR BISTATIC SAR PROCESSING BASED ON THE IMPROVED LOFFELD'S BISTATIC FORMULA

This paper presents a new range Doppler algorithm (RDA) for bistatic synthetic aperture radar (SAR) processing in a general conflguration based on a bistatic point target reference spectrum: the improved extended Lofield's bistatic formula (ILBF). The ILBF spectrum is proved to be comparably accurate with the spectrum derived using the method of series reversion (MSR). Based on the expansion of the ILBF spectrum, a new bistatic RDA is developed to process the azimuth invariant and variant bistatic SAR data. Compared with existing bistatic RDA, the new algorithm has a simpler formulation and is able to cope with moderate or high squint bistatic SAR data. The simulated data in the azimuth invariant and variant bistatic conflgurations are used to validate the new algorithm.

Focusing of Tandem Bistatic-Configuration Data With Range Migration Algorithm

A bistatic range migration algorithm (RMA) based on an exact analytical bistatic point-target (PT) spectrum in the tandem configuration is proposed in this letter. For the conventional geometry-based bistatic formula method, the derived spectrum is only a quasi-analytical one because a variable called half-quasi-bistatic angle (HQBA) is not exactly analytically expressed. The key step of the proposed algorithm is to deduce an analytical HQBA in the tandem configuration, and thus, an exact analytical closed-form PT spectrum is acquired. Based on this analytical spectrum, a bistatic RMA is presented. It is demonstrated that this algorithm can handle the bistatic tandem configuration with extremely large baseline/range ratio and can also be applied to wide-swath imaging.

Extending Loffeld's bistatic formula for the general bistatic SAR configuration

This study derives an accurate bistatic point target reference spectrum based on a zeroth-order polynomial model. The spectrum contains only two hyperbolic square root terms that are very analogous in form to the analytical monostatic spectrum. The new formulation can be considered as an improvement of the Loffeld's bistatic formula (LBF) and allows it to handle a wider range of bistatic configurations. The original LBF works well only in the case where the contributions of transmitter and receiver to the total Doppler modulation are approximately equal. An earlier paper on the extended LBF (EBLF) uses time bandwidth product (TBP) to weight the azimuth phase modulation from each platform. However, this extension is valid only in the low squint bistatic geometry. Both LBF and ELBF are expanded up to the quadratic term to derive an approximate bistatic spectrum; however, they do not show a good focusing performance in the more complex bistatic geometry, for example, the high squint case. This is due to the inaccurate individual time-Doppler correspondences. In this study, a norm in Euclidean space is defined to derive the optimal individual time-Doppler correspondences. Using the accurate individual time-Doppler correspondences, a zeroth-order polynomial model can be used to readily derive a more accurate bistatic spectrum. Two simulation experiments in the high squint geometry are used to verify the accuracy of this new formulation. The first experiment uses a parallel-track spaceborne/spaceborne configuration, and the second experiment involves an orthogonal-track airborne/airborne case.

Focusing Spaceborne/Airborne Hybrid Bistatic SAR Data Using Wavenumber-Domain Algorithm

This paper focuses on the bistatic synthetic aperture radar (SAR) data processing in a spaceborne/airborne hybrid bistatic configuration. Due to the extreme differences in platform velocities and slant ranges, the airborne system operates in the inverse sliding-spotlight mode, while the spaceborne system works in the sliding-spotlight mode to achieve a tradeoff between azimuth scene size and azimuth resolution. In this extreme bistatic configuration, our original bistatic formula shows a limitation of accurately describing the bistatic point-target reference spectrum, owing to the assumption of equal contributions of transmitter and receiver to the total Doppler spectrum. We extend our previous formula using the weighting operation where the weighting factor is the ratio of the azimuth time-bandwidth product (TBP) of the platform to the total azimuth TBP. In this paper, the bistatic-deformation and azimuth-dependent range-cell-migration terms were removed with phase multiplications performed blockwise in range-azimuth subsections. The remaining quasi-monostatic term shows the characteristic of the conventional monostatic SAR besides an additional azimuth-scaling term. For the monostatic characteristic, any precision monostatic SAR processing algorithms can handle. In this paper, we prefer the wavenumber-domain algorithm (also known as Omega-K), since it can accurately correct the range dependence of the range-azimuth coupling, as well as the azimuth-frequency dependence. For the azimuth-scaling term, an inverse scaled Fourier transformation is performed to correct it. Finally, a hybrid spaceborne/airborne simulation experiment is conducted to validate the proposed processing procedure.

Chirp-Scaling Algorithm for Bistatic SAR Data in the Constant-Offset Configuration

This paper discusses the processing method for bistatic SAR data in the constant-offset configuration. The constant-offset configuration is also known as the azimuth stationary or invariant configuration where transmitter and receiver follow each other, moving on identical velocity vector. In this paper, the proposed processing method for bistatic SAR data is based on Loffeld's bistatic formula that consists of two terms, i.e., the quasi-monostatic (QM) term and bistatic-deformation (BD) term. Our basic idea is to linearize the aforementioned two terms and then incorporate the BD term into the QM term to obtain an analogous monostatic spectrum. Based on the new spectrum, any efficient 2-D frequency or range-Doppler domain processor can easily be employed to process the bistatic data, where the Doppler phase parameters of the processor need to be adjusted. In this paper, we concentrate on the application of chirp-scaling-algorithm (CSA) processor. In addition, a bistatic-motion error model is developed where the position deviations of the two platforms are simplified as the bistatic slant-range displacement in the zero Doppler plane. Using this model, the monostatic motion-compensation technique is applied and integrated into CSA to compensate the trajectory deviations of transmitter and receiver. Finally, real and simulated data are used to validate the proposed processing method.