An Operational Processing Framework for Spaceborne SAR Formations

Abstract

The paper proposes a flexible and efficient wavenumber domain processing scheme suited for close formations of low earth orbiting (LEO) synthetic aperture radar (SAR) sensors hosted on micro-satellites or CubeSats. Such systems aim to generate a high-resolution image by combining data acquired by each sensor with a low pulse repetition frequency (PRF). This is usually performed by first merging the different channels in the wavenumber domain, followed by bulk focusing. In this paper, we reverse this paradigm by first upsampling and focusing each acquisition and then combining the focused images to form a high-resolution, unambiguous image. Such a procedure is suited to estimate and mitigate artifacts generated by incorrect positioning of the sensors. An efficient wave–number method is proposed to focus data by adequately coping with the orbit curvature. Two implementations are provided with different quality/efficiency. The image quality in phase preservation, resolution, sidelobes, and ambiguities suppression is evaluated by simulating both point and distributed scatterers. Finally, a demonstration of the capability to compensate for ambiguities due to a small across-track baseline between sensors is provided by simulating a realistic X-band multi-sensor acquisition starting from a stack of COSMO-SkyMed images.

1. Introduction

The new space economy has introduced a significant shift in the paradigm of frequent observations, where public agencies and private companies are building constellations of multiple small satellites. Synthetic aperture radar (SAR) plays a unique role thanks to its all-weather monitoring and fine sensitivity to motion and deformations. Several mini SARs have already been launched in low earth orbiting (LEO) orbit, while others are expected to follow [1]. The present systems provide high resolution, in the meter or sub-meter range, and good imaging capabilities, despite their compact dimensions (an overall mass of less than a hundred kilograms). Nevertheless, achieving a high-resolution wide-swath (HRWS) acquisition requires a large antenna and high power, increasing the minimum size of the sensor [2]. There is a growing interest in compact formations, where few sensors cooperate, acting as a single SAR. This concept, introduced over twenty years ago [3], has become increasingly popular thanks to the attractive advantages of distributing resources over small, lightweight systems, besides the intrinsic robustness, flexibility, and scalability of a fractioned mission.

Among the many concepts proposed and widely studied [4,5,6,7,8,9], we address those compact formations made of N satellites, where ambiguous acquisitions are joined to generate a single, high-quality, high-resolution image. Such constellations are the basis for CubeSat missions [4,5,10].

The system studied here is the one sketched in Figure 1a. One satellite is transmitting, and all are receiving in a single-input-multiple-output (SIMO) configuration. All the sensors should ideally follow on the same track—as seen from an Earth-fixed reference so that a single SAR image could be created by coherently combining the N acquisitions. The result would be a gain of N in the signal-to-noise-ratio (SNR) and the possibility of imaging over a swath N times larger than the one achievable by a single sensor. To that aim, the relative sensor displacements along track $\mathrm{\Delta }{x}_n$ should comply with the anti-DPCA (displaced phase center antenna) condition [7] $:$

$$\mathrm{\Delta }{x}_n=\frac{2v_s}{N\cdot f_{PRF}}(n-1)+\frac{v_s}{f_{PRF}}k,$$

(1)

where $v_s$ is the velocity of each spacecraft, $f_{PRF}$ is the pulse repetition frequency, and $n=(1,\dots ,N)$ is the index of the sensor. Note that the second term in (1) is not strictly needed to fulfill the anti-DPCA condition; however, an integer number ($k$) of samples are added to allow safe distancing between the satellites and avoid collisions. $k$ should be kept moderately small, such that all sensors observe almost the same Doppler spectrum. The logic of Equation (1) is explained in Figure 1b, which shows the sampling grids, along-track, of N = 3 sensors. The circles mark the positions of the sensors’ phase centers when echoes are collected by each sensor. The upper diagram reports the sampling grid that results by joining the monostatic phase center from sensor S₁ and the equivalent ones made by the bistatic pairs S₂-S₁ and S₃-S_1. The effective grid is regularly sampled at 1/N of the original pulse repetition interval (PRI), allowing the generation of a single SAR image where N − 1 ambiguities are removed [3,6,8,9,11,12].

The along-track oversampling is particularly relevant for satellites with a compact antenna, such as for CubeSATs [4,5]. These sensors can carry an antenna of limited size, which requires a high PRF to avoid azimuth ambiguities. However, increasing the PRF in a wide-swath acquisition is constrained by range ambiguities. A compact SIMO formation can solve the well known PRF trade-off [2], which adds to the implicit advantages of flexibility, scalability, robustness, and cost-effective features of formations and constellations. Beyond SIMO, several other concepts have been proposed where all the sensors are transmitting and receiving, i.e., multiple input multiple output (MIMO) configurations, which would keep the ambiguity mitigation property while extending the power gain from the factor N to a factor N² [7,12,13,14,15].

This paper addresses the SIMO case, but generalization to the MIMO concepts would be straightforward. Without loss of generality, we will focus on a formation with N = 3 satellites, whose parameters are detailed in

Table 1. Symbol list (left) and values of system and processing parameters (right).

Parameter	Symbol	Parameter	Symbol	Value
Central angular frequency	ω0	Wavelength	$\lambda$	$0.0312\hspace{0.17em}\mathrm{m}$
Azimuth	x	Carrier frequency	$f_0$	$9.6 \mathrm{G}$Hz
Slow time	τ	Bandwidth	$B$	100 MHz
Fast time	t	Antenna length	$L_a$	3 m
Squint angle	ψ	Mean slant range	$r$	640 km
Speed of light	$c$	Incidence angle	$\theta$	30°
Baseband range wavenumber	$k_r^{\prime }$	Velocity	$v_s$	$7650$ m/s
Azimuth wavenumber	$k_x$	Swath depth (ground range)		30 km
Sampling angular frequency	$k_{xs}$	Pulse Repetition Frequency	$f_{PRF}$	$2.2$ kHz
		Azimuth Resolution		1.5 m
		Ground Range Resolution		3 m
		Synthetic aperture	${\mathrm{L}}_{\mathrm{s}}$	6.6 km

. The X-band is adopted by most modern small SAR constellations, providing wide bandwidths while using compact power devices and small antennas.

Processing the set of N undersampled raw datasets into a fine-resolution single look complex (SLC) image is straightforward if the anti-DPCA condition is met. It would be enough to generate the upsampled data by properly delaying and interleaving, according to Figure 1b, performing a bistatic to monostatic compensation, and proceeding with conventional SAR focusing. However, condition (1) is met only with some accuracy in the general case. It is still possible to generate a good quality SLC image by a proper wave–number domain multichannel inversion described in [6,7,8,10,12]. The correction is usually applied to the range-compressed data (RGC) before proceeding to the azimuth focusing, as shown in Figure 2a. The recombination in the raw data domain is the most natural [7], but it is neither correct nor desired in many real scenarios. First, one may wish to distribute raw data processing, for example, onboard each satellite. Secondly, but not less importantly, multichannel recombination requires knowledge of the system’s timing and position with high precision. Such accuracy can be obtained by estimating the parameters from the focused data before merging the channels.

Alternatively, one could first upsample and focus each RGC data. The obtained ambiguous images can then be used to estimate and perform local corrections. Finally, the corrected datasets are merged to cancel ambiguities. The efficient implementation of this idea, depicted in Figure 2b, is the aim of this paper.

As for the bulk focusing of every single image and the multichannel recombination, wavenumber domain methods are mainly adopted for LEO orbiting systems, thanks to their simplicity and efficiency. However, those methods were developed for rectilinear tracks, and their adaptation to the curved orbit limits their phase preservation [16,17,18]. To cope with that, we discuss a full numerical wavenumber processor attaining high efficiency and quality. In particular, the phase preservation, analyzed with the methods provided in [19,20], fulfills the most rigorous requirements [21,22]. Moreover, these methods are checked by simulating realistic scenarios made by both distributed and point targets.

The paper is organized as follows: in the next section, the methods for processing multichannel SIMO data are reviewed and critically discussed. Then, Section 3 details the processing framework in Figure 2b and proposes two efficient implementations: the numeric monochromatic (NM) approach and the numeric chirp-Z (NCZ) one, comparing their quality versus efficiency, with particular attention to their phase preservation. Section 4 details results achieved with simulations of point and distributed targets and shows an example of mitigating ambiguities due to an unknown across-track baseline. The motivation of the processor is discussed, together with an analysis of the efficiency and quality of results. After conclusions, two appendixes with details on the numerical wave number domain processor are given.

2. Multichannel Processing of Data from Compact SIMO Formation

The usual approach for multichannel recombination is shown in Figure 2a [8,9,23,24]. The set of N range compressed datasets is transformed into the azimuth wavenumber domain ($k_x$) and then merged to a wideband spectrum by the following matrix operator, which is applied to each sample in the Doppler spectra:

$$\begin{gathered} \mathit{V}=\mathit{G}\cdot \mathit{D} \\ \left[\begin{array}{c}{V}_1\left(k_x\right)\\ V_2\left(k_x+k_{xs}\right)\\ V_N\left(k_x+N{k}_{xs}\right)\end{array}\right]=\left[\begin{array}{ccc}{g}_{11}& \dots & \dots \\ \dots & g_{nm}& \dots \\ g_{N1}& \dots & g_{NN}\end{array}\right]\left[\begin{array}{c}{D}_1\left(k_x\right)\\ \dots \\ D_N\left(k_x\right)\end{array}\right] \end{gathered}$$

(2)

where D is the [N,1] column vector formed by taking one spectral sample for each data set, G is the reconstruction matrix formulated, for example, in [8,9,12], and V is a [N,1] vector with unfolded spectral contributions.

The effect of the multichannel recombination on two-dimensional data spectra is demonstrated in Figure 3. In particular, Figure 3a shows the ideal wideband spectrum of the fine-resolution image to be reconstructed. The undersampling of each acquisition, caused by the low PRF, creates a N-time spectral folding along k_x, as shown in Figure 3b. The generation of the full-resolution single look complex image can be split into the following steps:

• The data acquired by the n-th sensor are upsampled by first transforming into the Doppler domain and then replicating (mosaicking) N-times to form the complex spectrum shown in Figure 3c.
• Each of the N spectral replicas is weighted by the term $g_{n,m}$ (n being the image, and m it the replica) according to (2).
• All N images are summed together, obtaining a single RGC with a broad Doppler spectrum.
• The data are focused, obtaining the final fine-resolution ambiguity-free image.

The approach we propose inverts steps 3 and 4 to obtain a set of focused images before compensating for the ambiguities. Loosely speaking, we first focus all the data on the fine grid of Figure 1b, but we do not sum them until residual knowledge errors are estimated and compensated.

The equivalence of the two approaches stems from the linear superposition are suggested in Figure 4 As an example, let us formulate the multichannel recombination (2) by feeding only the first acquisition (and the others being null):

$$V\left(k_x\right)=\left[\begin{array}{c}{V}_1\left(k_x\right)\\ V_2\left(k_x+k_{xs}\right)\\ V_N\left(k_x+N{k}_{xs}\right)\end{array}\right]=\left[\begin{array}{ccc}{g}_{11}& \dots & \dots \\ \dots & g_{nm}& \dots \\ g_{N1}& \dots & g_{NN}\end{array}\right]\left[\begin{array}{c}{D}_1\left(k_x\right)\\ 0\\ 0\end{array}\right]=\left[\begin{array}{c}{D}_1\cdot g_{11}\\ D_1\cdot g_{21}\\ D_1\cdot g_{N1}\end{array}\right]$$

(3)

Equation (3) can be interpreted as a weighted combination of the N-times mosaicked Doppler spectra of the first image, where the weights are the elements $g_{1,n}\left(k_x\right)$ of the matrix G, represented in Figure 3c. Then, by switching the weighted combination (3) and the focusing kernel, we obtain the alternative procedure of Figure 2a and Figure 4, where data are first oversampled and focused and then recombined as post-processing.

3. Efficient and Flexible Wavenumber Domain Implementation

The objective of the section is to define a numeric wavenumber domain (WD) implementation of the entire procedure in Figure 2b, which adequately copes with the curved orbit.

3.1. Implementation of Curved Orbit Focusing and Upsampling

WD focusing kernels were derived for rectilinear tracks, and their adaptation to the curved orbit case introduces phase errors and biases [16,22,25]. Several alternatives were developed in the literature based on modified, extended chirp-scaling approaches [26,27,28]. However, these approaches are not really needed for the case discussed here, whereas fast and accurate enough kernels can be formulated by numerical methods [19,25,29]. More specifically, let us start from the classical approach for straight orbit derived in Appendix A. The focused scene can be expressed in the azimuth wavenumber and range domain as from (A9):

$$\begin{gathered} U\left(k_x,r^{\prime }\right)=\int U_c\left(k_x,\omega \right)\cdot \exp \left(-j{\beta }_0\left(k_x\right)\cdot r^{\prime }\right)\cdot \exp \left(j\omega \frac{2r^{\prime }}{c}\cdot {\beta }_1\left(k_x\right)\right)d\omega \\ {U}_c\left(k_x,\omega \right)=D\left(k_x,\omega \right)H_{c0}^{*}\left(k_x,\omega \right)\cdot \exp \left(-j\frac{\omega +{\omega }_0}{c/2}{t}_0\right) \end{gathered}$$

(4)

where $r_0$ is the slant range of the target closest to the sensor, $r^{\prime }=r-r_0$ is the relative slant range, $D\left(k_x,\omega \right)$ is the two-dimensional-FT of the RGC data, $H_{c0}\left(k_x,\omega \right)$ is the two-dimensional-FT of the impulse response of a target at a range $r_0$, and ${\beta }_0$,${\beta }_1$ are two parameters originated by the series expansion of the exact operator (A20).

To generalize (4), the kernel $H_{c0}$ should be evaluated numerically for the curved orbit, as shown in Appendix B. One can easily check that doing so (4) would provide the proper focus for those targets located at the closest approach, $r_0$, which corresponds to $r^{\prime }=0$.

The role of the parameters ${\beta }_0$,${\beta }_1$ is then to extend the depth of focusing to targets at any range $r^{\prime }=r-r_0\ge 0$, within some approximation. To find the best values of those parameters, we first evaluate the spectrum of the target at range $r^{\prime }$, which can be expressed as $H_c^{*}\left(k_x,\omega ,r\prime \right)=H_{c0}\cdot \exp \left(j{\varphi }_d\left(k_x,\omega ,r\prime \right)\right)$, and then we impose the fit:

$$\exp \left(j{\varphi }_d\left(k_x,\omega ,r\prime \right)\right)\simeq \exp \left(r^{\prime }\cdot \left({\beta }_0\left(k_x\right)+{\beta }_1\left(k_x\right)\cdot \frac{2\omega }{c}\right)\right).$$

(5)

In particular, two numerical methods are proposed here: the numerical monochromatic (NM) and the numerical chirp-Z transform (NCZ), summarized in Figure 5 and Figure 6. They differ in the fitting accuracy of (5). The fastest method, the NM, imposes ${\beta }_1$ to be unitary, and then implements (4) by an inverse FT. The slowest, but most accurate, (NCZ), accounts for non-unitary ${\beta }_1$ by implementing (4) as an inverse chirp-zeta transform (CZT) at the cost of three fast Fourier transforms (FFT) [30].

The coefficients ${\beta }_0\left(k_x\right),{\beta }_1\left(k_x\right)$ are evaluated as the best approximation of (5):

$${\varphi }_d({\omega ,k}_x,r^{\prime })\simeq r^{\prime }\cdot \left({\beta }_0\left(k_x\right)+{\beta }_1\left(k_x\right)\cdot \frac{2\omega }{c}\right),$$

(6)

To this aim, we estimate the phase surface numerically ${\varphi }_d$ for M ≥ 2 range bins ($r_m$) as described in Appendix B, obtaining, for each k_x, a matrix ${{\mathsf{\Phi }}_k=\{\varphi }_d({\omega }_n,r_m^{\prime };k_x)\}$ of size [N,M]. Then, the vector B = ${\left[{\beta }_0{\beta }_1\right]}^T$ is retrieved by the least square fitting of (6):

$$\begin{gathered} {\mathsf{\Phi }}_k\simeq \mathit{A}\cdot \mathit{B}\cdot {\mathsf{\Delta }}_r^T \\ \mathit{A}=\left[\begin{array}{cc}1& 1\\ \dots & \dots \\ \dots & \frac{4\pi f_s}{c\cdot N}n\\ \dots & \dots \\ 1& \frac{4\pi f_s}{c}\left(N-1\right)\end{array}\right];\mathit{B}=\left[\begin{array}{c}{\beta }_0\left(k_x\right)\\ {\beta }_1\left(k_x\right)\end{array}\right];{\mathsf{\Delta }}_r^T=\left[\begin{array}{ccc}0& \dots & \left(M-1\right)d_r\end{array}\right] \end{gathered}$$

(7)

where $f_s$ is the sampling frequency and $d_r$ is the step of the slant-range grid assumed in (6). The solution is:

$$\mathit{B}={\left({\mathit{A}}^T\cdot \mathit{A}\right)}^{-1}\cdot {\mathit{A}}^T\cdot {\mathsf{\Phi }}_k\cdot \frac{{\mathsf{\Delta }}_r^T}{{\mathsf{\Delta }}_r^T\cdot {\mathbf{\Delta }}_r}={\mathit{A}}^{†}\cdot \frac{{\mathsf{\Phi }}_k{\mathsf{\Delta }}_r^T}{{\sum }_m{m}^2{d}_r^2}$$

(8)

where ${\mathit{A}}^{†}$ is the pseudoinverse of $\mathit{A}$.

In the NM approach, the procedure simplifies, since we impose β₁ = 1. Therefore ${\beta }_0$ is retrieved by fitting (6) near the end of the processing block, $r_M$:

$${\beta }_0=\frac{{\varphi }_d\left({\omega }_0,k_x,{\mathrm{r}}_{\mathrm{M}}\right)}{r_M}-\frac{2\left(\omega +{\omega }_0\right)}{c}$$

(9)

Notice that, for the straight orbit, the numeric approach defaults to the classical one [16,31,32], as results from (A4) and (9):

$${\beta }_0=\sqrt{{\left(\frac{\omega +{\omega }_0}{\frac{c}{2}}\right)}^2-k_x^2}-\frac{2\left(\omega +{\omega }_0\right)}{c}\simeq -\frac{k_x^{2}c}{4{\omega }_0}$$

(10)

3.2. Phase Preservation

The phase preservation of the focusing approach is evaluated here by considering a typical orbit of a LEO SAR at 500 km altitude. Specifically, we started from a TLE file of Capella Space 4, whose main orbit parameters are listed in

Table 2. Kepler orbit element assumed for simulations and performance evaluation TLE form Capella-4. RAAN stands for the right ascension of the ascending node.

Semi-Major Axis	Eccentricity	Inclination	RAAN	Arg of Periapsis
6892.2 km	8.2 × 10−3	97.5°	112.3°	307.16°

. A set of state vectors (SVs) was generated with the MATLAB aerospace toolbox and used for simulation and focusing. We remind our readers that the interest here is to show the capability of the processor to cope with the orbit curvature.

The phase preservation can be checked by evaluating, for each range, the error in fitting (6):

$${{\varphi }_{err}\left(r^{\prime },k_x,\omega \right)=\varphi }_d-\left(r^{\prime }\cdot \left({\beta }_0\left(k_x\right)+{\beta }_1\left(k_x\right)\cdot \frac{2\left(\omega +{\omega }_0\right)}{c}\right)\right)$$

(11)

That phase error, evaluated for k_x corresponding to 0.6° squint, has been represented in Figure 7 as a function of frequency and range. We assume a processing block of 5 km in the slant range, which does not introduce any limitation, since it is much larger than the range migration (~8 m). A block-wise implementation is not only needed for quality, but also desired for efficiency and parallelism. NCZ has a negligible peak phase error, less than 5 mrad, whereas, for the NM version, the error is noticeable, but it is less than the acceptable limit of 1 radian [17,32].

Phase bias is even more critical than the peak error since slight values could lead to visible artifacts in the interferometric products [33]. Such bias is one of the major limitations in implementing a processor defined for rectilinear geometries, even for very small blocks, hindering an efficient implementation. The bias can be evaluated by integrating the phase error (11) as a function of range:

$${\varphi }_{bias}\left(r^{\prime }\right)=\angle \iint \exp (j{\varphi }_{err}(r^{\prime },k_x,\omega ))d{k}_{x}d\omega ,$$

(12)

Results are shown in Figure 8 for the block size of 5 km for both methods. The bias is kept within ~2 mrad in both cases, which fulfills the very stringent conditions of the CEOS offset test [21].

3.3. Multichannel Recombination

The multichannel recombination, discussed in Section 2, requires the knowledge of the inverse matrix G(k_x) that can be computed as in [24]:

$$\mathit{G}(k_x)={\mathit{H}\left(k_x\right)\cdot \left({\mathit{H}}^{\mathit{*}}\left(k_x\right)\cdot \mathit{H}\left(k_x\right)+k_w{I}_N\right)}^{-1},$$

(13)

where ${\mathit{I}}_N$ is the [N,N] identity matrix, and $k_w$ is the Wiener parameter that rules the trade-off between ambiguities suppressions and SNR. The [N,N] matrix $\mathit{H}$ is made by pure phase terms, $H_{nm}$, that implement the correction between the effective bistatic system made by the n-th sensor and the transmitter and the monostatic equivalent used for focusing (A1). For the rectilinear track [8,11,12]:

$$H_{nm}\left(k_x,\omega \right)=\exp \left(-j\frac{{\omega }_0}{c}\frac{{\left(x_n-x_m\right)}^2}{4\cdot {\mathrm{r}}_{\mathrm{p}}}\right)\cdot \exp \left(j\frac{x_n+x_m}{2}{k}_x\right),$$

(14)

where $x_n$ is the along-track position of the n-th receiver, and $x_m$ is the transmitter’s position. The first exponential in (14) compensates the bistatic path excess w.r.t the equivalent monostatic sensor positioned at ($x_n+x_m$)/2. For a target located in ($x_p,r_p$), the path difference amounts to:

$$\begin{array}{l}\mathsf{\Delta }R=R_{bis}\left(S_n,S_m,P\right)-2R_{mono}\left(S_{\frac{n+m}{2}},P\right)\\ =\sqrt{r_p^2+{\left(x-x_n-x_p\right)}^2}+\sqrt{r_p^2+{\left(x-x_m-x_p\right)}^2}-2\sqrt{r_p^2+{\left(x-\frac{x_n+x_m}{2}-x_p\right)}^2}\end{array}$$

(15)

$$\simeq \frac{{\left(x-x_n-x_p\right)}^2}{2r_p}+\frac{{\left(x-x_m-x_p\right)}^2}{2r_p}-2\frac{{\left(x-\frac{x_n+x_m}{2}-x_p\right)}^2}{2r_p}=\frac{{\left(x_n-x_m\right)}^2}{4{\cdot r}_p}.$$

(16)

This term is a delay that is approximated for a monochromatic system by:

$$H_{nm}\left(\mathsf{\omega }\right)=\exp \left(-j\frac{\omega +{\omega }_0}{\mathrm{c}}\mathrm{\Delta }R\right)\simeq \exp \left(-j\frac{{\omega }_0}{\mathrm{c}}\mathrm{\Delta }R\right)=\exp \left(-j\frac{{\omega }_0}{c}\frac{{\left(x_n-x_m\right)}^2}{4{\cdot r}_p}\right),$$

(17)

that is consistent with (14). We point out that this term is so slowly varying with range that it can be kept constant for the whole processing block of 5 km (see Section 3.3). It would change the phase term only by a few parts per million, according to the parameters in Table 2, even for a large sensor separation of 1 km.

The second term in (14) is the along-track shift needed to align the focused image to the equivalent monostatic sensor located in $(x_m+x_n)/2$. This shift is not correct for the curved orbit.

The simplest way to adapt the recombination to the actual case is to replace (14) with the residual operator:

$$H_{nm}\left(k_x,\omega ,r_p\right)\simeq \int \exp \left(\frac{j\omega }{c}\left(R_{nm}\left({\omega }_0,r_p\right)-2R_m\left({\omega }_0,r_p\right)\right)\right)\cdot \exp \left(-j{k}_{x}x\right)\cdot dx,$$

(18)

where $R_{nm}$ is the bistatic hodograph computed for the n-th sensor and the receiver, assuming a target at mid-range $r_p$, and 2R_m is the monostatic one. The spectrum (18) is computed as described in the Appendix B, but assuming ω=ω₀ and approximated to the first order in $k_x$. It is important that all hodographs are computed by assuming the same azimuth, as well as slant range (x,r) reference, which would implicitly compensate for the along-track shifts of the different sensors.

4. Results

The quality of the overall focusing and multichannel reconstruction as post-processing has been tested by simulating a realistic scenario of point and distributed targets acquired by a sensor moving along the 500 km LEO orbit described in Table 2. The parameters of the system and mission are listed in Table 1. We have generated two sets of simulated data:

• a set of nine-point targets, spanning an entire processing block of 5 km in slant range and five footprints in azimuth;
• a set of distributed targets spanning a block of 1 km in slant range and three footprints in azimuth.

A complex random noise was added to the RGC data in both cases. We have considered a relative sensor distance of about 150 m, with two different configurations:

• the optimal sensor displacements, foreseen by the anti-DPCA condition (1), for which perfect ambiguities-free recombination could be achieved;
• shifting the second and third sensors, respectively, by 50 cm and −50 cm w.r.t the case above. So doing, the equivalent monostatic phase centers are shifted by ±25 cm, a significant fraction of the fine image resolution L_a/2 = 1.5 m. This condition stresses the capability of the inversion to remove ambiguities as much as possible.

The matrix inversions (13) have been computed by assuming the Wiener parameter $k_w=0.3$. The most efficient NM version was used for focusing, as it did not lead to appreciable artifacts, in concordance with the results of Section 3.3, even considering a squint angle of 0.2° and the largest block of 5 km.

4.1. Point Target Analysis

The set of nine-point targets is represented in Figure 9, which draws the amplitude of one of the N = 3 range-compressed, subsampled images. Stars mark the zero-Doppler position of the targets. After focusing and multichannel recombination, the end-to-end impulse response function along range and azimuth is drawn in Figure 10. A significant oversampling has been applied to assess resolution and sidelobes. However, almost no difference is found in the impulse response functions of all targets, where the nominal slant range and azimuth resolutions of ~1.5 m were measured. The plots of Figure 10 refer to the non-ideal sensor displacements. Still, no difference could be appreciated to the ideal one, similar to the behavior close to the target position.

The suppression of ambiguities by the multichannel combination is evaluated by the energy of the focused impulse response, integrated over range and azimuth, for the target in the middle of the scene in Figure 11. The top panel of Figure 11 refers to the ideal along-track positioning of the sensors, in which only the ambiguities coming from the 3 m SAR antenna are visible. In the bottom panel, the wrong positioning of the two receiving sensors, by 0.5 and −0.5 m w.r.t the optimal case, is considered. The positional error causes 2 × (N − 1) = 4 additional ambiguities. The ambiguities’ energy level is evident but comparable to the one of the fine resolution SAR. Note that the same results are obtained when performing the combination before and after focusing, as seen in Figure 11b,c. This qualifies the whole processing chain and the positioning error.

4.2. Distributed Area Target Analysis

Point target simulation provides all the input to evaluate the end-to-end performance regarding resolution, sidelobes, and ambiguities. However, distributed targets help to analyze ambiguity structures and their suppression. To that aim, we have generated a set of patches of different shapes distributed over the whole image. The map of the simulated distributed scene is provided in Figure 12a, representing the ground truth. The overall scene size is that of a processing block, 1 km × 12 km (slant range, azimuth), where the small area patches result from half a million-point targets overall. The range-compressed, azimuth-defocused data are shown inb, referred to as one of the N = 3 acquisitions. The illumination pattern of the wide-aperture antenna is visible, extending for the entire footprint and at least one sidelobe.

The amplitude of data after focusing and multichannel recombination as post-processing, following the entire chain in Figure 2b, is represented in Figure 13. Multi-looking over local windows of 3 × 3 samples was performed. In the case of perfect spacing (Figure 13a), only the ambiguities due to the 3 m antenna can be seen. The amplitude of such ambiguities w.r.t the target is smaller than −25 dB, which is comparable to the typical values of spaceborne SAR systems, such as Sentinel-1 [34].

In the other case, Figure 13.b also shows the residual ambiguities from the improper multichannel recombination. They are at the same level of those due to the SAR antenna, and therefore they are acceptable in a SAR mission.

4.3. A Realistic Scenario: The Impact of Across-Track Baselines

So far, we have assumed the same track for all the sensors in a fixed Earth reference. Yet, even the best design and control of the formation flying would leave some residual offset between sensor tracks, as shown, for example, in [35,36,37]. We then assumed that the tracks of the second and the third satellites are displaced respectively by 1 m and −1 m in the direction perpendicular to the line-of-sight, w.r.t the track of the first sensor. For simplicity, we keep this offset constant within a few synthetic apertures, lasting a couple of seconds.

The across-track baseline results in an unwanted multiplicative phase screen, different for each channel, which interferes with the correct multichannel recombination and prevents suppression of ambiguities. The proper compensation of this effect is quite complicated, and it is indeed one of the major objectives of the present literature [12,38,39]. However, for small baseline and smooth topography, first-order mitigation could be performed as suggested in [6,39,40], which consists in estimating the residual phase screen, given the precise knowledge baselines and the elevation of the scene’s center, and removing it prior to the multichannel recombination.

We simulated the compensation of the across-track baseline in two scenarios. First, the conventional method was employed, as described in Figure 2a. Since the phase screen was essentially along the range, the compensation task can be performed on the range compressed images. However, the specific orbit knowledge for near real-time applications is limited to several centimeters [41]. We assumed a 1 cm error in the knowledge of both normal and perpendicular directions. Such lack of precision affects the accuracy of the computation of the phase screens and then hinders the quality of the multichannel combined image. The result of the conventional processing is shown in Figure 14a: the quality is compromised by the uncompensated ambiguities, which spread throughout the image.

The figure was generated by simulating a set of N = 3 multichannel data. A complex reflectivity map, obtained by multi-temporal averaging over 80 COSMO-SkyMed images, was used with a DEM and sensor’s orbit to simulate the source RGC data.

The procedure in Figure 2b was implemented by upsampling and focusing separately on the three datasets. This allowed for a fine, data-driven baseline estimation with a sub-millimetric accuracy, as from [42,43,44]. The refined baselines are then used to compute the precise topographic phase screens shown in Figure 14c,d. The phase variation is tiny, yet its compensations lead to the final-multichannel image, shown in Figure 14b, which is almost free from ambiguity clutter.

4.4. Efficiency

The proposed approach, which first upsamples and then merges the N ambiguous focused images, is clearly less efficient than the classical merge and upsample procedure. To assess performance, a set of prototype Matlab codes for the different methods has been profiled and optimized to evaluate and compare their efficiency. In particular, we have considered the numerical monochromatic and the numerical chirp-Z approaches and measured the computing time on a modern eight-core desktop PC. The time reported in

Table 3. Computing times, in seconds, measured for the MATLAB prototypes of the various processing blocks.

Upsampling, Focusing and Multichannel Combination			Multichannel Combination with Upsampling and Focusing
	NM	NCZT	NM	NCZT
Data Focusing	5.1 (×3)	22(×3)	12	29
Multichannel recombination	1.1

refers to focusing one range compressed data block as in Figure 9, approximately 5000 × 9400 samples. The cost of the “multichannel recombination” in the bottom left of the table has been estimated by assuming the recombination as a fast convolution implemented in ten blocks along the azimuth. Notice that the recombination is essentially a fractional resampling, as from (14), requiring a very short operator in the along-track space. In the most favorable case, where condition (1) is perfectly met, one could interleave the focused data at no cost.

From the table, one notices that the conventional approach, on the right column, is the most efficient, where the NM method is approximately 2.5 times faster than the NCZT one. However, if we implement the proposed approach, where the images are up-sampled and focused on-the-fly, by each sensor, then the overall computing time is half of the usual approach.

5. Conclusions

A new processing framework has been proposed, suited for compact SAR formations where acquisitions from different sensors are combined into a unique full-resolution image, such as SIMO or MIMO SAR. The processing scheme is based on oversampling and focusing on each acquisition, while leaving the multichannel reconstruction as the last step. This procedure is useful to integrate the estimation and compensation of impairments acting on a local space-varying scale, similar to the effect of topographic phases.

A very efficient, full numerical implementation of monostatic focusing and multichannel recombination has been discussed. The result is a phase-preserving processor whose efficiency is compared to a couple of bidimensional FT. The scheme has been validated with point and simulated targets generated by assuming the orbit of a typical LEO SAR flying at 500 km altitude. The method has been shown effective for the mitigation of ambiguities due to unknown across-track baselines between sensors by simulating a realistic SAR scene, starting from X-band Cosmo-SkyMed data.