Correlating Synthetic Aperture Radar (CoSAR) - IEEE Xplore

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING

1

Correlating Synthetic Aperture Radar (CoSAR) Paco López-Dekker, Senior Member, IEEE, Marc Rodriguez-Cassola, Francesco De Zan, Gerhard Krieger, Fellow, IEEE, and Alberto Moreira, Fellow, IEEE

Abstract—This paper presents the correlating synthetic aperture radar (CoSAR) technique, a novel radar imaging concept to observe statistical properties of fast decorrelating surfaces. A CoSAR system consists of two radars with a relative motion in the along-track (cross-range) dimension. The spatial autocorrelation function of the scattered signal can be estimated by combining quasi-simultaneously received radar echoes. By virtue of the Van Cittert–Zernike theorem, estimates of this autocorrelation function for different relative positions can be processed by generating images of several properties of the scene, including the normalized radar cross section, Doppler velocities, and surface topography. Aside from the geometric performance, a central aspect of this paper is a theoretical derivation of the radiometric performance of CoSAR. The radiometric quality is proportional to the number of independent samples available for the estimation of the spatial correlation, and to the ratio between the CoSAR azimuth resolution and the real-aperture resolution. A CoSAR mission concept is provided where two geosynchronous radar satellites fly at opposing sides of a quasi-circular trajectory. Such a mission could provide bidaily images of the ocean backscatter, mean Doppler, and surface topography at resolutions on the order of 500 m over wide areas. Index Terms—Bistatic radar, ocean currents, sea level, sea surface, synthetic aperture radar.

I. I NTRODUCTION

A

GENERAL assumption allowing synthetic aperture radar (SAR) [1], [2] imaging is that the observed scene does not change during the aperture time. Even for motion detection techniques, such as ground moving target indication or alongtrack interferometry (ATI), it must be assumed that the target remains coherent during some time. This condition is generally assumed to be true for land scenes (although defocusing due to the wind-driven motion of the tree crowns is visible in highresolution SAR images). In contrast, it may be stated that SAR is not particularly well suited to the observation of fast decorrelating objects in general and of water surfaces in particular. One straightforward argument to support this statement is that the achievable azimuth resolution is limited by the coherence time of the surface. In the case of airborne or low Earth orbit (LEO) SAR systems, the coherence time of the sea surface

Manuscript received May 21, 2014; revised November 4, 2014, May 21, 2015, and September 18, 2015; accepted October 17, 2015. The authors are with the Microwaves and Radar Institute, German Aerospace Center (DLR), 82234 Oberpfaffenhofen, Germany (e-mail: Francisco. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2498707

still allows for a relatively long coherent aperture, yielding a resolution good enough for many applications that require relatively coarse final product resolutions (on the order of 0.1 to 1 km). In these situations, the much finer nominal resolution can still be exploited to generate large numbers of independent looks. For higher orbits, as the angular velocity of the sensor with respect to the target decreases, the achievable azimuth resolution will degrade until collapsing to that of a real aperture radar. A second limitation becomes apparent when SAR observations are compared to the radar observations made, for example, by fixed coastal radars [3]. Indeed, those allow the observation of the spatiotemporal statistics of the radar echoes, for example, in the form of spatially varying Doppler spectra [4]. In this sense, aside from the normalized radar cross section (NRCS) σ0 , both single-channel and ATI-capable SAR systems are limited to the estimation of the first moment of these Doppler spectra, which are derived from the Doppler centroid anomaly [5] or the along-track interferometric phase [6], respectively. For the sake of scientific rigor, it is worth pointing out that the retrieved mean Doppler cannot be directly translated into a surface current component: It provides an NRCS weighted average of the radial velocities, where the coupling between NRCS and velocity modulations by the underlying wave field results in strong sea-state-dependent biases. This paper discusses a novel radar imaging concept which we call a correlating SAR (CoSAR) [7] that is capable of generating high-resolution images of some statistical properties of fast decorrelating surfaces. In particular, we are interested in observing the NRCS, the mean Doppler shift, and a cross-track interferometric phase associated to the mean surface height (MSH). The basic idea is to operate two or more physically separated radars (which may share a common transmitter) with a relative motion so that their azimuth (cross-range) separation varies with time. Pairs of echoes acquired at each instant of time and relative position can be combined to produce estimates of the spatial autocorrelation function of the received signal. Estimates of this autocorrelation function for different positions can then be combined to obtain high-resolution images of some statistical properties of the scene, including estimates of the space-varying Doppler spectrum. This approach to imaging, which follows from the Van Cittert–Zernike theorem [8], [9], is generally used for imaging radiometers [10]–[12], radiotelescope arrays [13], and imaging mesoscale-stratospheretroposphere radars [14]–[16]. In particular, we are interested in a CoSAR system consisting of two geosynchronous spacecraft with a relative motion around a nominal geostationary position. Such a configuration would allow the observation of the ocean surface at moderate

0196-2892 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING

resolution, with regional coverage, and high temporal sampling (two observations per day). Aside from allowing the estimation of the Doppler spectrum of the surface, the configuration discussed will also yield a cross-track interferometric phase, from which ocean topography could be derived. The resulting observational capabilities of the ocean surface would be unique in terms of the temporal and spatial sampling, the nature of the measurement (short-time averaged Doppler spectra), and the range of geophysical quantities simultaneously addressed (surface wind, surface currents, and MSH). Like in any radar system, in CoSAR, range resolution results from the ability to discriminate different range delays, and it is inversely proportional to the transmitted pulse bandwidth. This results in two fundamental differences with respect to synthetic aperture radiometers. The first is the fact that resolution is provided already in one dimension before any aperture synthesis. The second is that, in synthetic aperture radiometers, independent samples are acquired at the rate given by the receiver bandwidth, so the number of independent samples available is virtually unlimited. In contrast, in CoSAR, the amount of independent samples available to estimate the spatial autocorrelation function is limited. Therefore, where performance analyses of synthetic aperture imaging radiometers emphasize the achievable resolution and hardware related errors [17], in this paper, we focus on the radiometric error associated to this finite number of available independent samples. Section II of this paper discusses the signal model, including the assumptions made and the achievable resolutions. This is followed, in Section III, by an outline of how CoSAR image formation could be implemented, and a discussion of the wavenumber-domain interpretation of CoSAR imaging supported by some simple examples. A central part of this paper is the discussion of the radiometric quality in Section IV. Section V outlines a possible practical implementation of a geostationary CoSAR mission. Finally, Section VI summarizes the main findings of this paper and discusses the way forward. II. S IGNAL M ODEL A. Surface Backscattering The backscattering of a monochromatic single polarization radar signal on a surface can be described by a complex scattering coefficient s(x, y, ts ) which is, generally speaking, spatially and slow-time varying. Here and throughout this paper, slow time ts refers to time scales that are large compared to the pulse repetition interval (PRI). Likewise and as usual in the SAR literature, fast time (tf ) will refer to the two-way travel time of the radar signal from the radar to the (x, y) location. Since we are primarily interested in radar observations of the ocean surface, we may assume that the scattering coefficient decorrelates quickly, so s(·) should be treated as a multidimensional random process. We will assume that it is a complex zero-mean locally homogeneous temporally ergodic process, which is usually valid over some temporal scales. This random

field is characterized (at least partially) by its second-order statistics. In particular, we assume E [s(x + Δx, y + Δy, ts + τ ) · s∗ (x, y, ts )] = Rτ (x, y, τ ) · δ (Δx − vx (x, y) · τ, Δy − vy (x, y) · τ )

(1)

where E[·] is the expected value operator, δ(·) is the Dirac delta function, and Rτ (·) is a space-varying temporal autocorrelation function. The Dirac-delta in the spatial autocorrelation term expresses mathematically the assumption of a fully developed speckle. We have also introduced a horizontal surface velocity vector vs with components vx (·) and vy (·), which will shift the peak of the local autocorrelation linearly with time. We can anticipate that this velocity term will introduce a Doppler frequency (or phase) in the received signal. Strictly speaking, also the term Rτ (·) should be shifted accordingly as a result of this horizontal velocity. By ignoring this dependence, we are assuming that Rτ (·) is spatially smooth. We may express this smoothness mathematically as ∂Rτ (x, y, τ ) ∂Rτ (x, y, τ ) ∂Rτ (x, y, τ ) · vx + · vy (2) ∂x ∂y ∂τ which allows ignoring any translation of the surface scattering statistics for temporal lags τ of interest. This temporal autocorrelation can be expressed as Rτ (x, y, τ ) = σ0 (x, y) · γτ (x, y, τ )

(3)

with σ0 (·) being the space-varying real-valued NRCS and γτ (·) being a complex-valued temporal coherence function (with γτ (x, y, 0) = 1). Note that, in addition to the Doppler shift that will result from the surface velocity explicitly included in the model, the derivative of the phase of γτ (x, y, τ ) with respect to τ at τ = 0 will introduce an additional Doppler term. Physically, one could relate this second term to the wind-related Doppler bias (anomaly) present in radar data of the ocean surface. It is worthwhile returning to the ergodicity assumption. By definition, this allows us to drop the slow-time dependence (ts ) in (2). Physically, this implies the assumption that the scattering statistics of the surface do not change during the temporal scales of interest, i.e., the CoSAR integration time Tint . A more relaxed interpretation is that the quantity of interest is some temporal average of some instantaneous statistics. This interpretation accommodates situations in which the surface statistics will be advected by the surface current. This may happen, for example, in the case of a low backscattering area resulting from an oil spill. As pointed out in the introduction, the purpose of the CoSAR technique presented in this paper is to provide estimates of the autocorrelation function Rτ (·). In particular, we are interested in imaging the NRCS and the mean Doppler shift. In addition, we will see later that a CoSAR system may typically include a cross-track baseline that will provide sensitivity to the MSH. B. Radar Signal In the following paragraphs, we derive a model for the joint second-order statistics of the echoes received by two radars observing simultaneously the same scene. This is schematically

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. LÓPEZ-DEKKER et al.: CORRELATING SYNTHETIC APERTURE RADAR (CoSAR)

3

justify ignoring the horizontal velocity altogether in (2) and considering that we have a surface autocorrelation function Rτ (·) that includes all the contributions to the Doppler phase. The last term introduces a phase that depends on the positions of the two radars and on the height of the surface. Since the positions of both radars can be parametrized as a function of slow time ts , (5) can be written as Γ (Δ p(ts ), tf,1 , tf,2 , τ ) (x, y, ts , tf,1 , tf,2 ) · Rτ (x, y, τ ) = W · e−2j·k0 ·ˆr2 ·vs (x,y)·τ · e−2j·k0 Δr(x,y,ts,τ ) dx dy.

Fig. 1. Illustration of CoSAR acquisition geometry, with two radar systems moving in opposite directions in the azimuth (cross-range) direction.

illustrated in Fig. 1. The monostatic echo, in baseband, for the ith radar sensor can be written as vi (ts , tf ) = Wi (x, y, ts , tf )s(x, y, ts )e−2j·k0 ri dx dy (4) where Wi (·) is a weighting function that combines the beam pattern and the pulse waveform on the surface, k0 = 2π/λ0 is the wavenumber corresponding to the carrier frequency, and ri represents the range from the ith radar antenna, at its slow-timedependent position pi (ts ) to the scatterer at position (x, y). Note that range shift and range migration are implicitly included in Wi (·). If the scattering coefficient does not depend on time, then it can be estimated from the samples of the received signal vi (·). This situation is that of a regular SAR system. We can now use (2) and (4) to derive the cross-correlation function of the echoes received by a pair of radar systems as illustrated in Fig. 1 Γ(ts , tf,1 , tf,2 , τ ) = E [v2 (ts + τ, tf,2 ) · v1∗ (ts , tf,1 )] (x, y, ts , tf,1 , tf,2 ) · Rτ (x, y, τ ) = W · e−2j·k0 ·ˆr2 ·vs (x,y)·τ · e−2j·k0 (r2 −r1 ) dx dy.

(5)

(·), is a combined weighting function that The first term, W depends on the two range (or fast) time positions and the beam patterns. In general, this weighting function will only be nonvanishing when the range delays tf,1 and tf, are such that the radar echoes correspond to overlapping areas on the ground (more or less overlapping or crossing iso-range lines; see Fig. 2). The second term represents the geophysical information of interest, which reduces to the space-varying NRCS if τ is set to zero. The third term describes a Doppler shift associated to the projection of the surface velocity in the line-of-sight direction. It is worth noticing that this phase term cannot be distinguished from a phase term linearly dependent on τ in Rτ (·). This could

(6)

The term Δr(x, y, ts , τ ) is the CoSAR equivalent to the range history in a regular SAR system. For the geometries considered, where the separation between the two radars is much smaller than the slant range to the imaged area, the differential range depends on the difference of the radar position vectors, Δ p, so that this cross-correlation takes the form of a spatial autocorrelation function. The CoSAR differential range history is mathematically similar to the general case of the range history of a bistatic SAR [18]. Therefore, theory and processing approaches studied in the context of bistatic SAR may be reused in our context. C. Resolution CoSAR resolution can be understood analogously to that of SAR, thinking in terms of iso-range and iso-Doppler lines. This is illustrated in Fig. 2 for the canonical CoSAR geometry shown in Fig. 1. We have assumed two radar systems moving at 1 ms−1 in almost opposite directions (170◦ relative headings), flying at 1-km altitude above a flat surface. A wavelength of 1 m has been assumed for Doppler calculations. The left panel shows the iso-range (black) and iso-Doppler (blue) lines for the two radars. In the right panel, the individual iso-Doppler lines have been substituted by contour lines showing the Dopplerfrequency differences (delta-Doppler). From a point of view of resolution, we may ignore one of the sets of iso-range lines. Resolution emerges from the gradient of the range to one of the two radars and the gradient of the delta-Doppler. Mathematically, range and azimuth resolution can be derived in a way analogous to that of a bistatic system, for example, following [19]. Since we consider quasi-parallel LOS vectors, the range resolution vector field will be roughly that of a single radar δrg (x, y) =

∇ri (x, y)

c0 ∇ri (x, y) 2Br 2

(7)

where the operator ∇ represents the gradient in the x−y plane and Br is the pulse bandwidth. In order to derive the azimuth resolution, we can identify a delta-Doppler frequency fΔD (x, y, ts ) = −

2 ∂ Δr(x, y, ts ) λ ∂ts

(8)

and write the azimuth (synthetic aperture) resolution as δraz (x, y) =

1 ∇fΔD (x, y, ts ) 2 T (x, y) ∇fΔD (x, y, ts ) int

(9)



Fig. 2. (Left) For the canonical acquisition geometry shown in Fig. 1, with iso-range (in black, units in meters) and iso-Doppler (in blue, units in hertz) lines for the two radars forming the CoSAR system (solid and dashed lines, respectively). (Right) Iso-range (as in left panel) and iso-delta-Doppler (colored) lines. (a) SAR. (b) CoSAR.

where Tint is the integration time. A canonic stripmap equivalent for a CoSAR system is not easy to define, so this integration time will be the result of an end-to-end design process. Once the azimuth direction has been identified, the corresponding resolution can be approximated as δraz ≈

λ0 · R0 2 · Δvaz · Tint

(10)

with Δvaz being the relative velocity of the radars in the azimuth direction and R0 being the slant-range distance between the radars and the imaged area. Here, we can recognize the product of the relative velocity and the integration time, Δvaz · Tint , as the CoSAR synthetic aperture length. III. C O SAR I MAGING A. Back-Projection Algorithm The simplified signal model in (6) quickly suggests an approach to CoSAR imaging, following the back-projection algorithm [20] that is often used as the gold standard for SAR processing due to its accurate image reconstruction. The idea is to obtain estimates of Γ(·) for a range of relative positions. The phase term resulting from the delta-range history will be a fast-varying term that allows the application of the principle of stationary phase so that estimates of the signal of interest can be obtained by multiplying the estimates of Γ(·) by a complex conjugated reference function and integrating over slow time ˆ τ (x, y, τ ) = Cp R

ˆ (ts , tf,1 (x, y, ts , τ )) , tf,2 (x, y, ts , τ )), τ ) Γ · e2j·k0 Δ˜r(x,y,ts ,τ ) dts . (11)

The first term inside the integral is the estimate of Γ(·) for the two back-projected range-delay positions corresponding to a decorrelating scatterer at the reference position (x, y), slow time ts , and time lag τ . The integral is normalized with a constant Cp , which can be shown (see Appendix A) to be given by Cp = ∇fΔD (x, y) .

(12)

Algorithmically, (11) expresses that, for each point on the ground, (x, y), slow time ts , and time lag τ , we look for the corresponding range-time samples in the two raw-data sets. Subsequently, an estimate Γ(·) is obtained by multiplying the samples from the first raw-data set by the complex conjugate of the corresponding samples of the second raw-data set. After compensating the phase with the geometrically calculated reference phase, we integrate (sum) over slow time. Particularizing to τ = 0 will result in an estimate of the NRCS, σ ˆ0 (x, y). It is interesting to highlight some properties of the CoSAR focused images. ˆ τ (x, y, 0)) should be a positive 1) The estimated NRCS (R real number. A nonzero phase would be the result of noise, an insufficient number of independent samples for the estimation of Γ(·), and, most notably, trajectory knowledge errors and an erroneous assumed surface height. Let us consider this last contribution in more detail. If the two phase centers forming the CoSAR system were moving on a common trajectory but just in different directions, the zero-lag CoSAR image should provide intensities equivalent to those obtained by SAR processing of the data acquired at a single phase center (let us ignore temporal decorrelation for this discussion). In general, we do not require the two phase centers to follow the same trajectories. In this general case, we could understand the


5

Fig. 3. Illustration of how the azimuth-wavenumber components for a particular resolution cell are sampled at different instants of time. The vectors kxy,i represent the 2-D wavenumber of the complex scattering field (s(x, y)) sampled at a particular time for each radar, with kx,i being the corresponding azimuthwavenumber components. The cross-correlation Γ(·) samples the surface correlation function (R(x, y, τ )) at the azimuth wavenumber Δkx = kx,2 − kx,1 . The acquisition geometry considered in this figure consists of two platforms flying in the same direction, but with the relative motion required to form the CoSAR aperture. (a) t1 . (b) t2 .

CoSAR image as being equivalent to the interferogram that could be formed with the two SAR images obtained for each phase center. For disjoint trajectories, part of the compensated phase in CoSAR processing may be understood as flat earth (and topographic) phase removal. For approximately parallel trajectories, a reference height ˆ τ (x, y, 0) error will result in an interferometric phase in R that may be used to estimate the correct height. 2) Multilooking (i.e., averaging of independent samples) is done before CoSAR imaging, during the estimation of Γ(·), so that CoSAR image products are, in principle, speckle free. The impact of having a finite number of independent samples available for the estimation of Γ(·) is treated in depth in Section IV. The required estimate of Γ(·) is a second-order moment of an ergodic process. It can therefore be estimated by averaging independent realizations of the product of the two signals. The most straightforward source of independent samples is to average independent range bins. Thus, if the range resolution of the individual radars is a factor Nr higher than that of the desired product, we can average this number of realizations.1 The second source of independent realizations is a bit more intricate. Our cross-correlation function Γ(·) has a low-pass frequency spectrum with respect to slow time ts . Insofar as the ergodicity assumption holds, the only cause to a slow-time dependence of Γ(·) is the relative motion of the CoSAR antennas. In other words, the slow-time spectrum is given by the deltaDoppler spectrum. From a point of view of CoSAR imaging, we need only samples of Γ(·) at the rate given by the deltaDoppler bandwidth BΔD . Considering the Doppler bandwidth BD,i of the individual signals vi (ts , tf ), we could average Na ≈

BD,1 + BD,2 BΔD

(13)

1 Strictly speaking, the effective number of independent samples depends on the spectral power density of the product signal.

independent realizations of the product of the radar signals in the estimates of Γ(·). Here, we have assumed that the Doppler bandwidth of the product is the sum of the two individual bandwidths, which will generally be an overestimate. The Doppler bandwidth of each channel may be approximated as BD,i ≈

1 vaz,i +2· τca Laz

(14)

where vaz,i is the velocity of each radar in the azimuth direction, Laz is the effective antenna length, and τca is the coherence time associated to γτ (·). The first term is the intrinsic Doppler bandwidth associated to the observed surface, while the second one can be recognized as the Doppler bandwidth of a regular SAR system. We see that the decorrelation of the surface provides a natural mechanism to generate independent realizations of the product of both radar signals. In addition, we observe that a common motion component would increase the individual Doppler bandwidths without affecting the delta-Doppler bandwidth, hence providing a controlled mechanism to generate independent samples. Note that, for a canonical CoSAR system with no common motion, without the temporal decorrelation of the surface, Na is equal to 1. B. Interpretation and Spectral Support It is worth spending a few words interpreting the meaning of (6) and (11). Both expressions represent basically a Fourier transform pair in the CoSAR azimuth direction [which is given by δraz in (9)]. Thus, for each relative position of the two radars, the function Γ(·) represents a Fourier component of the signal of interest, Rτ (x, y, τ ), in the azimuth direction. By varying the relative positions of the two radars, we collect different Fourier components of this signal which are used later to reconstruct the space-domain signal. This is illustrated in Fig. 3, which shows how a given resolution cell is sampled, in the wavenumber domain, at two instants



Fig. 4. Azimuth NRCS profiles corresponding to scenes with (left) one or (right) two targets estimated using (blue solid line) CoSAR processing and (red dashed line) SAR processing. The upper panels show the case of coherent scatterers, while the lower panels correspond to quickly decorrelating ones. (a) 1 coherent target. (b) 2 coherent targets. (c) 1 decorrelating target. (d) 2 decorrelating targets.

of time. At each time, the vectors kxy,i represent the 2-D wavenumber component of the complex scattering field (s(x, y)) that is being sampled by each radar, and kx,i are the corresponding azimuth-wavenumber components. The cross-correlation Γ(·) samples the surface correlation function (R(x, y, τ )) at the azimuth wavenumber given by the difference of the previous wavenumbers Δkx = kx,2 − kx,1 . Note that, in this particular illustration, the two radars have a common velocity term so that, in a given integration time, the SAR aperture could be much larger than the CoSAR one. With this understanding and since Rτ (x, y, τ ) is generally a low-pass signal in the spatial domain, it should be clear that the CoSAR acquisition geometry should allow the sampling of the small wavenumber components. In order to avoid filtering out (i.e., failing to acquire) the essential low-frequency components, the interfering wavenumber projected on the x−y surface, Δk xy , should vanish for some time t0 (x, y) within the CoSAR integration time Δkxy (x, y, t0 (x, y)) = 2k0 ∇Δr (x, y, t0 (x, y)) = 0.

(15)

This condition is only met if the position (x, y) and the two radars are all contained in a straight line (for example, if the two radars share the same position in space). In the ground-range direction, this can be relaxed by thinking in terms of range

spectral shift and common band filtering [21] so that only the cross-range component of the interfering wavenumber needs to vanish. This can be expressed as ∇Δr (x, y, t0 (x, y)) × ∇ri (x, y, t0 (x, y)) = 0

(16)

with × representing the cross product. This condition can be used to determine the region that can be CoSAR imaged at any instant of time for a given set of trajectories. It has already been pointed out that the existence of such a cross-track baseline renders the system sensitive to topography. This sensitivity does not come for free: A cross-track baseline will cause a range spectral shift [21]. This implies a loss of common range bandwidth, with the associated imperative of performing common-band filtering, and requires accommodating this spectral shift in the transmit pulse bandwidths. In the case of very large relative spectral shifts, the natural solution would be to transmit pulses that are adequately frequency shifted. C. Point Target Examples Before addressing the critical topic of radiometric quality, this section discusses the results of a few 1-D simulations, whose results are shown in Fig. 4. The general simulation procedure is described in the Appendix, and the detailed acquisition geometry is described later in Section IV-D. In all four panels, the blue solid line and the red dashed lines show the CoSAR and


7

SAR estimated profiles, respectively. The SAR profiles have been obtained processing the simulated received signal for one of the two radars forming the CoSAR system. A straightforward back-projection algorithm has been used also for the SAR processing. Both the CoSAR and the SAR processing have been done without any azimuth tapering. Fig. 4(a) shows the estimated intensity profile (given as dimensionless NRCS) for a single coherent point target. Although this is not the case of interest for CoSAR, it is interesting to illustrate a few issues. Although, in both cases, the single-lookcomplex estimated profile would correspond to a sinc function, in the case of the SAR image, the square of the absolute value of this profile needs to be taken in order to retrieve the intensity. As a result, the same tapering leads to a better azimuth resolution but higher sidelobes in the CoSAR case. We also see that CoSAR can result in negative estimates of the intensity. While this may seem unphysical and, therefore, undesirable, it allows CoSAR to provide unbiased estimates of the NRCS. Nevertheless, in practice, some tapering would be applied to reduce sidelobe energy. Fig. 4(b) serves to illustrate the fundamental limitation of CoSAR: It is not well suited to image discrete point targets. Here, two coherent-point targets have been simulated, which are well discriminated in the case of the SAR profile. For the CoSAR case, the result includes not only the two targets of interest but also what could be described as an intermodulation artifact. The amplitude and phase of this artifact depend on the relative positions of the targets and the phase of the corresponding scattering coefficients. The appearance of this type of artifacts is a consequence of the nonlinearity introduced in the estimation of Γ(·). Fig. 4(c) corresponds to a single decorrelating target. More specifically, this target is modeled as a line of random scatterers all at the same azimuth position, but extending 50 range bins. For these particular results, the targets have a coherence time equal to Tint /30. For the SAR profile, the 50 range bins have been incoherently averaged after SAR processing, whereas for the CoSAR profile, the independent range bins were used in the estimation of Γ(·). As expected, in the SAR profile, the target appears smeared out in azimuth since the effective resolution is now given by the coherence time and not by the integration time. In contrast, the CoSAR profile retains almost perfectly the expected sinc-like shape. Similarly, Fig. 4(d) shows the results for two decorrelating targets. A key observation to be made is that the resulting profile can be very well described as the addition of two sincfunctions. The implication is that, if the signal fits the signal model, with enough averaging, the end-to-end CoSAR system behaves like a linear system with respect to the signal of interest, i.e., Rτ (x, y, τ ).

estimated by substituting the expected value operator in (5) by a sample average

IV. R ADIOMETRIC E RROR B UDGET A. Range Averaging This section analyzes the radiometric quality of a CoSAR system. We start by considering the error in the estimation of the spatiotemporal autocorrelation. This autocorrelation is

ˆ (Δ Γ p(ts ), tf,1 , tf,2 , τk ) 1 = v2 [ns + k, mf,2 ] · v1∗ [ns , mf,1 ] (17) Nr mf,1 mf,2

∈ Sr

where ns is the slow-time sample corresponding to ts , k is the slow-time sample offset corresponding to the temporal lag τk , and the averaging is done over the Nr -element set of range (fast-time) samples Sr . Analytical expressions of the probability density functions of the phase and amplitude of the sampleaveraged (multilooked) correlation can be found in [22]. In the following paragraphs, we derive approximate expressions for the case of large Nr . In this case, by virtue of the central limit theorem, the real and imaginary parts of the estimation should be jointly Gaussian distributed. Under the assumption that the signal of interest is locally homogeneous and temporally ergodic, it is obvious that this sample average is an unbiased estimator of the true autocorrelation function ˆ p(ts ), tf,1 , tf,2 , τk ) = Γ(Δ E Γ(Δ p, tf,1 , tf,2 , τk ). (18) We now turn our attention to the autocorrelation estimation error ˆ − Γ(·). δΓ(·) = Γ(·)

(19)

The power of the estimated signal can be derived using Reed’s moment theorem for complex Gaussian processes [23], yielding 1 ˆ 2 E Γ(·) = |Γ(·)|2 + E |v1 (·)|2 · E |v2 (·)|2 (20) Nr where, for the sake of compactness and readability, the dependences on fast and slow time are omitted. The first term in (20) is the power of the signal of interest itself. Therefore, under the assumption that the estimation error and the signal of interest are uncorrelated, the second term can be identified as the power of the estimation error 1 E |δΓ(·)|2 = (21) E |v1 (·)|2 · E |v2 (·)|2 Nr the product of the range-dependent power of the signals received by both radars. Assuming identical radars, the echo power can be expressed as the sum of the autocorrelation function at zero spatial and temporal lag and the noise power E |vi (·)|2 = Γ(0, tf,i , tf,i , 0) + N (22) where the signal-to-noise ratio (SNR) is identified as SNR =

Γ(0, tf,i , tf,i , 0) . N

(23)

Combining (21)–(23) results in

2 SNR + 1 1 E |δΓ(·)|2 = |Γ(0, tf,i , tf,i , 0)|2 . (24) Nr SNR A couple of observations are in place. The estimation error δΓ(·) is complex valued. Even under the assumption that the



central limit theorem holds and that both the real and imaginary parts are jointly Gaussian random variables, this does not imply that δΓ(·) is circularly Gaussian. To show that this is indeed not the case, let us consider cases with different coherences, with the coherence given by γ (Δ p(ts ), tf,1 , tf,2 , τk ) =

Γ (Δ p(ts ), tf,1 , tf,2 , τk ) . Γ(0, tf,1 , tf,2 , 0)

(25)

From (24), it follows that the power of δΓ(·) does not depend on the coherence. However, as the coherence approaches unity, the estimation phase error ˆ − ∠Γ(·) ∠Γ(·)

(26)

will quickly converge to zero. This implies that, as the coherence increases, the estimation error δΓ(·) will be in phase or in counterphase with the actual estimate. Thus, for high coherences, all the power of δΓ(·) is concentrated in the in-phase component, ˆ affecting only the amplitude of the Γ(·). In contrast, for low coherences, δΓ(·) will tend to be circularly Gaussian, with the power equally distributed among the phase and quadrature components. This behavior is analyzed in detail in [22]. B. Azimuth Processing We want to eventually estimate the error after the CoSAR back-projection operator, which is obtained by substituting the integral in (11) by an equivalent summation ˆ [ns , mf,1 (x, y, ns ), mf,2 (x, y, ns ), k] ˆ τ (x, y, τk ) = Cp Γ R PRF n

ities and, at the same time, low enough to avoid range ambiguities. We will consequently ignore ambiguities and address separately the two other sources of error. Let us denote the windowed CoSAR image by Rτ,W (·), i.e., the result of (28) assuming δΓ(·) = 0. The CoSAR error due to autocorrelation estimation errors (δΓ(·) = 0) will be δRτ,W [τk ] =

Cp δΓ [ns , mf,1 [ns ], mf,2 [ns ], k] PRF n s

· Wp [ns ] · e2j·k0 Δ˜r[ns ]

where, for the sake of notational compactness, the explicit dependence on the (x, y) position has been dropped. Since it has been assumed that the autocorrelation estimates are unbiased (E[δΓ(·)] = 0), the CoSAR image will be unbiased, i.e., E [δRτ,W [τk ]] = 0.

can be simplified if the samples of δΓ(·) are uncorrelated, which is, generally speaking, not the case. The individual radar signals vi [ns , mf,i ] will decorrelate in the slow-time domain due to the azimuth motion of the radars and, if present, the temporal decorrelation of the surface. We will assume a simplified model in which the radar signal remains coherent during a slow-time interval τca , corresponding to

ˆ τ (x, y, τl ) − Rτ (x, y, τl ) δRτ (x, y, τl ) = R

Nca = τca · PRF

(27)

where Wp [·] is a processing window, and the general dependence on the 2-D, (x, y), position of the imaged resolution is explicitly shown. The term Cp /PRF ensures a correct amplitude scaling, under the assumption that max(|Wp [·]|) = 1. Note that the fast-time indexes mf,i are not necessarily integer numbers, representing an implicit interpolation in that domain. The CoSAR imaging error (28)

results from the following. 1) The spatiotemporal autocorrelation estimation error (δΓ(·)). 2) The processing window. The main impact of this window is the limitation of the integration time (or, equivalently, the processed delta-Doppler bandwidth), which limits the azimuth resolution. High azimuth-wavenumber components of R(x, y, τl ) are filtered out. 3) Potentially, range and azimuth ambiguities. Azimuth ambiguities result from aliasing from the delta-Doppler frequency components. For the type of mission concept considered in this paper, the expected delta-Doppler bandwidths would be small enough to allow accommodating a PRF high enough to minimize azimuth ambigu-

(30)

The derivation of the power of the CoSAR imaging error (31) E |δRτ,W [τk ]|2

s

· Wp [x, y, ns ] · e2j·k0 Δ˜r(x,y,ns )

(29)

(32)

samples. After this coherence time, the samples are assumed to be fully independent. We can now write (29) as δRτ,W [τk ] =

Cp δΓca [ns , mf,1 [ns ], mf,2 [ns ], k] PRF n s

· Nca · Wp [ns · Nca ] · e2j·k0 Δ˜r[ns ·Nca ]

(33)

with δΓca [ns , mf,1 [ns ], mf,2 [ns ], k] ≈

1 Nca

(ns +1)Nca −1

δΓ [na , mf,1 [na ], mf,2 [na ], k] . (34)

na =ns ·Nca

Note that, in a noise-free scenario and with perfect correlation between the samples averaged, this operation would not reduce the power of this error signal and could be replaced by the down-sampling of the signal. In reality, however, this operation decreases the contribution to the error from thermal noise. It can be shown that the power of this averaged error is given by |Γ(0, tf,1 , tf,1 , 0)|2 E |δΓca [·]|2 = Fn · Nr

(35)


9

TABLE I PARAMETERS U SED IN 1-D C O SAR S IMULATIONS

with a noise degradation factor

Nca · SNR + 1 2 Fn = . Nca · SNR

(36)

Since the values summed in (33) are a sequence of uncorrelated random variables, the power can be straightforwardly derived Nca · C 2 p 2 · |Wp [ns ]|2 E |δRτ,W [·]|2 = 2 E |δΓca [·]| PRF ns (37) where we can identify the effective integration time 1 · |Wp [ns ]|2 . (38) Ti = PRF n s

Finally, we use the quality measure used for spectral power estimators [24], the ratio between the square of the estimate and the variance of the estimation error 2 ˆ τ (·) E R . Q= (39) E |δRτ [·]|2 Manipulating the previous results, the quality of the windowed CoSAR image can be expressed as 2

QW =

2

Nr T i |δraz (x, y)| |Rτ (x, y, τ )| · · 2 . Fn τca W (x, y, ts , tf,1 , tf,2 ) · σ0 (x, y)dx dy (40)

C. Discussion In the following paragraphs, we discuss the interpretation and implication of (40). The numerator in the last term represents the square of the intensity of the signal of interest integrated over the resolution cell. With everything else left equal, improving the azimuth resolution leads to a degraded quality. The denominator is, normalization factors aside, the square of the power of the range-compressed signal. For the estimation of the NRCS (τ = 0) and for a homogeneous scene, the entire fraction is the square of the ratio of the azimuth resolution and the width of the antenna footprint in the azimuth direction. Clearly, this ratio will tend to be much smaller than one. The noise degradation factor will play a small role as long as the product Nca · SNR is at least on the order of 5 to 10, after which the factor quickly converges to unity. The quality scales with the number of independent range samples, which can be easily increased by either degrading the product resolution or improving the range resolution. The ratio Ti /τca is the number of independent azimuth samples processed. Increasing the integration time will only improve the quality if the azimuth resolution is kept constant. For a given CoSAR system trajectory, this would not be the case; in fact, in that case, the quality factor becomes inversely proportional to Ti . Slower relative motions, however, will improve the product quality. The product quality can also be improved if the surface decorrelation time τca is shortened. As mentioned in Section III, fast decorrelation could be forced by introducing a

common motion of the spacecraft. To avoid possible pitfalls, it is worth noting that, by construction, the coherence time cannot result in a number of independent samples smaller than one. In practice, this implies that it is lower bounded by the PRI (τca ≥ PRI). D. One-Dimensional Performance Simulations In order to validate the consistency of the theory and for the purpose of illustration, this section discusses the results of a series of CoSAR simulations. A detailed discussion on how the simulations have been implemented is provided in the Appendix. The simulations correspond to the geosynchronous mission scenario discussed in Section V. The parameters used in the simulations are listed in Table I. A narrow quasi 1-D strip of 110 km aligned in the (instantaneous) azimuth direction is simulated. In the range dimension, a number of range bins are simulated in order to provide the necessary independent samples. In azimuth, the scene statistics (NRCS, Doppler velocity, and MSH) are assumed to vary smoothly with some characteristic length scales. However, the scene statistics are assumed to be range independent. This simplification implies an assumption that the range resolution is good enough to provide the required number of range samples within the aforementioned length scales. All the results shown in this section correspond to a fixed set of NRCS, Doppler velocity, and MSH profiles so that the effect of changing several parameters can be appreciated. Fig. 5 shows the results of three realizations of the simulation, where the integration times (Tint ) were 300, 100, and 600 s, respectively. We show first the 300-s case since the resulting nominal resolution is the closest to the smallest length scale considered (1 km). The results show the amplitude and phase of the expected (blue lines) and the estimated (red lines) surface correlation function Rτ (x, τ ) for temporal lags of zero (left plots) and one PRI, respectively. The expected phase for the correlation is a linear combination of the MSH profile and the Doppler velocity with scaling factors depending on the height of ambiguity and the temporal lag E [∠Rτ (x, τ )] = 2π ·

MSH(x) vD (x) · τ . + 4π · hamb λ

(41)



Fig. 5. Examples of (blue lines) true and (red lines) CoSAR-estimated amplitudes and phases for Rτ (·) at (left) τ = 0 and (right) τ = 1/PRF. From top to bottom, the results shown correspond to integration times of 300, 100, and 600 s. (a) Tint = 300 s. (b) Tint = 100 s. (c) Tint = 600 s.

This illustrates how the surface topography related phase is always present, but also that the combination of estimates for different lags provides enough information to separate the topographic and the Doppler velocity related phases. The amplitude of the expected correlation is the NRCS multiplied by the corresponding interferometric coherence. For an ocean surface, this coherence will be close to one if the significant wave height is small compared to the height of ambiguity. Under this condition, the amplitude of the zero-lag correlation should provide a good estimate of the NRCS. We now turn our attention back to the results in Fig. 5 and make several general observations. 1) The estimated amplitude and phase profiles clearly follow the expected ones.

2) The estimation errors tend to be larger near the local minima of the expected amplitude. This happens because the errors are the results of self-interference, whose strength depends on the mean NRCS level. 3) Reducing the integration time results in smoother and less noisy-looking estimates. Since this filters out higher frequency components of the signal of interest, too short integration times result in the introduction of windowing errors. 4) The errors in the estimates of the amplitudes and phases for the τ = 0 and τ = 1/PRF cases appear to be quite independent. This is expected and implies that, for the estimation of the geophysical parameters of interest, it would be useful to combine estimates of the correlation of as many temporal lags as possible.


11

Fig. 6. Statistics of the Rτ (·) estimation error at τ = 0 for integration times of (left) 100 s and (right) 300 s. The plots show the real and imaginary parts of the mean error computed using 20 realizations, and the corresponding standard deviation. In addition, the dashed line shows the theoretical standard deviation of the error according to (37). (a) Tint = 100 s. (b) Tint = 300 s.

Fig. 6(a) shows some statistics of the Rτ (·) estimation error at τ = 0 for integration times of 100 s (left) and 300 s (right). The plots show the real and imaginary parts of the mean error computed using 20 realizations, and the corresponding standard deviation. In addition, the theoretical standard deviation of the error according to (37) is also shown. As expected, the standard deviation of the error estimated from the simulations matches the theoretical expression. Considering now the mean of the errors, it is clear that, for the short integration time, the filtering out of the high-frequency components of the signal of interest results in systematic errors. This effect is usually present in observation systems with a spatial resolution coarser than the signal of interest. The Tint = 300 s case illustrates that, aside from windowing effects, the CoSAR image is unbiased. Thus, the self-interference can be modeled as zero-mean additive noise, with a power level that depends essentially on the mean power of the radar echoes and the number of independent looks. For nonzero lags, the phase errors obtained in these examples are a small fraction of the actual phase excursion. This would represent Doppler velocity errors of a few cm s−1 . Nevertheless, the retrieval of the MSH and Doppler velocity profiles would typically combine estimates of Rτ (·) for different temporal lags. A final product error budget analysis is beyond the scope of this paper. V. M ISSION C ONCEPT A. Geometry and Resolution In the previous sections, we have introduced and analyzed the CoSAR concept. The purpose of this section is to provide an outline of a possible CoSAR mission concept to observe some properties of the ocean surface, aiming at the retrieval of ocean winds, surface topography, and surface currents. A complete mission design and analysis is beyond the scope of this paper. From an implementation point of view, the most salient CoSAR challenge is to find a mechanism to provide the desired relative motion between two radars. Considering quasi-straight trajectories, the CoSAR imaging opportunities are limited to the instants of time at which the two trajectories cross. For example, for any two-spacecraft LEO configuration, CoSAR

TABLE II S OME O RBIT AND S YSTEM PARAMETERS OF AN E XAMPLE G EOSYNCHRONOUS C O SAR M ISSION

imaging could be possible at most two times per orbit. An interesting alternative is circular trajectories where the two radars are always at opposing ends of the same circle. In this situation, considering the instantaneous direction of motion, the two radars are permanently crossing each other. Table II gives some key parameters for the mission concept considered, consisting of two radar satellites in geosynchronous orbits centered at 160◦ E, to cover the oceans in Southeast Asia and the Australian East Coast. Quasi-circular trajectories (with respect to the Earth’s surface) can be achieved with geosynchronous orbits by choosing an appropriate combination of inclination and eccentricity vector (eccentricity and argument of perigee). In the proposed scenario, both spacecraft would have a small eccentricity of 0.0005, with an argument of perigee of 90◦ , and an orbital inclination of 0.05◦ [25], [26]. The spacecraft would fly in identical orbits but with a 180◦ relative phase, corresponding to a 12-h delay. The inclination and eccentricity vector have been chosen such that the resulting mean distance between the two spacecraft is around 100 km (the actual mean value is 85 km). This provides a reasonable balance between height sensitivity and spectral shift. With these parameters, the mean velocity of the spacecraft, in an Earth-centered Earth-fixed (ECEF) coordinate system, is only 3.05 ms−1 . The ECEF orbit is a vertically tilted ellipse, with an almost circular ground projection. We have exemplarily assumed the use of X-band since it provides a large allocated bandwidth and reasonable expected NRCS values over the oceans for large incident angles. A



Fig. 7. Incident angle, CoSAR azimuth resolution, and magnitude of range specral shift after a 12-h observation period and a 300-s integration time. (a) Incident angle. (b) Cross-range (azimuth) resolution. (c) Range spectral shift.

Fig. 8. NRCS as a function of incident angle for (solid lines) up-wind and (dashed lines) cross-wind according to the SASS2 (Ku-band), XMOD2 (X-band), and CMOD5 (C-band) geophysical model functions. The left and right panels correspond to wind speeds of 4 and 8 ms−1 , respectively.

300-s CoSAR integration time has been also arbitrarily set for illustration purposes. Fig. 7 illustrates the CoSAR performance. First, the incident angle is shown, as this would limit the observable region. The second panel shows the cross-range (azimuth) CoSAR resolution obtained after a 12-h observation, with the assumed 300-s integration time for each geographical location. The resulting azimuth resolution is in the 300–450-m range for all the observable region. The last panel shows the corresponding spectral shift in range. The values, in the range of 0.5–0.8 rad m−1 , are significant and require, for adequate performance, either pulse bandwidths on the order of 100 MHz or a range adaptive frequency offset of tens of megahertz in order to provide spectral overlap. These values correspond to heights of ambiguity between 12 and 16 m, thus providing very high sensitivity to surface topography. The configuration discussed could be adapted to other wavelengths simply by scaling the orbital inclination and the eccentricity with the wavelength. From the point of view of imaging, it seems to be possible to downscale the trajectory and simultaneously increase the integration times. However, this goes at the cost of the cross-track baseline. B. Sensitivity The second main mission design challenge is to provide adequate sensitivity. Fig. 8 shows the NRCS as a function of

incident angle for up-wind and cross-wind conditions according to the SASS2 (Ku-band) [27], XMOD2 (X-band) [28], and CMOD5 (C-band) [29] geophysical model functions. Note that the XMOD2 model seems to exaggerate the wind-direction dependence on the NRCS and also deviate from the SASS2 and CMOD5 trends for large incident angles, so it seems more reasonable to assume that the X-band NRCS will be somewhere between the SASS2 and CMOD5 predictions. Since the area covered by the mission increases more or less quadratically with the maximum incident angle, it is clearly desirable to push this maximum incident angle as far as possible, for example, to 50◦ . At these incident angles, an adequate noise equivalent sigma zero (NESZ) should be in the −25 dB to −30 dB range. As an example, let us consider the instrument system described in Table II, featuring a 15-m-diameter unfurlable reflector antenna and 1-kW average transmit power (this is a high value, but only about a factor of two more than what the Sentinel-1 system already provides [30]). Digital beamforming (DBF) capabilities are implicitly assumed in order to access a 500-km swath [31], [32]. Since the assumed use of DBF techniques decouples the swath width from the antenna dimensions, the achievable swath width scales more or less linearly with the available RF power if all other system and performance parameters are kept constant. A moderate single-look groundrange resolution of 50 m is set, from which the incidentangle-dependent pulse bandwidth is derived. For SAR imaging purposes, due to the very low ECEF velocities, PRF values of


13

C. Further Aspects

Fig. 9. Comparison between achievable SAR NESZ and single pulse CoSAR NESZ for the system described in Table II.

around 1 Hz would suffice. However, in order to estimate the Doppler velocities of the surface, the PRI needs to be similar or smaller than the coherence time of the surface. Here, a PRF of 200 Hz has been assumed. For regular SAR processing of static scenes, the resulting azimuth oversampling would provide coherent integration gain. The resulting SAR NESZ is shown in Fig. 9 (red line), with values around −50 dB across the range of incident angles considered. That the small ECEF velocities, which lead to very long dwell times for a given azimuth resolution, result in high sensitivity despite the very large slant ranges is a well-known fact [33]. In the CoSAR case, coherent integration is limited by the decorrelation of the surface, but the high PRF contributes to the required provision of a large number of independent samples. Thus, for the purpose of calculating the CoSAR SNR, the NESZ is upper bounded by the single pulse NESZ (blue line in Fig. 9). In this example, an SNR from 25 to 0 dB can be achieved for incident angles ranging from 15◦ to 50◦ . As reflected in (36), the actual sensitivity will be better by a factor given by the effective number of azimuth samples coherently integrated. The main conclusion of this discussion is that the required sensitivity can be achieved with state-of-the-art system specifications. It is interesting to note a CoSAR specific tradeoff. Compared to a SAR system flying in the same orbit, CoSAR has very high power requirements. This is because, in CoSAR, a large number of pulses need to be correlated and averaged in order to suppress the self-interference. The number of independent samples required depends on the desired performance, but in any case, it is proportional to the ratio between the azimuth dimension of the beam-pattern footprint and the CoSAR azimuth resolution. Thus, the number of required independent samples can be reduced by making the antenna longer, even while keeping constant the total antenna area. This reduction of required independent samples can be translated directly into a reduction of the average power requirement, an improvement of the SNR, or an increase of the swath. For example, keeping the effective area of the reflector considered in the example constant, but going from a circular shape to an elliptic reflector with a ratio of major to minor axis equal to 2, could extend the swath from 500 to 1000 km without sacrificing radiometric performance.

Since Tomiyasu’s discussion of a geosynchronous SAR mission [33], a variety of embodiments of the idea have been discussed. One of the major recognized challenges is the issue of dealing with the time-varying atmospheric phase screen (APS) [34]. In the CoSAR scenario, since the information used is the correlation of pairs of samples acquired practically simultaneously, the APS affecting a given point on the ground will be common for both radars and therefore cancel out. There could still be some residual phase errors resulting from spatial gradients of the APS, but these should be expected to be very small given the coarse target resolutions. It has already been pointed out that the PRF is not driven by the motion of the radars, but rather by the decorrelation time of the scene. This invites us to think about a CoSAR timing subtlety. If each radar transmits its own pulses, the radar echoes will not be truly simultaneous and a zero-lag CoSAR image would not be possible. There are two possible solutions to this problem. 1) Use a PRF high enough to allow the assumption that there is no temporal decorrelation between consecutive pulses. This may lead to problematic timing constraints and range ambiguities. To some extent, these timing constraints can be relaxed if the two radars transmit orthogonal waveforms, which would be the case if a spectral shift larger than the pulse bandwidth is compensated on transmit. 2) Operate bistatically, using a common transmitter (one of the two CoSAR radars, or a dedicated transmitter). Another technical challenge would be the synchronization of the two radar instruments, both at a pulse level and, in particular, at carrier frequency and phase level. Like other aspects of CoSAR, this synchronization issue is common to that of bistatic radar systems, for which extensive literature is available [35]–[38]. Depending on the pulse bandwidth used and the spectral shift associated to the cross-track baseline, it may be possible to operate both radars simultaneously, each in monostatic mode, using nonoverlapping frequency bands. This would remove the need for precise pulse and phase synchronization, although small center frequencies still would result in range-delay-associated phase differences. If the system is operated bistatically, then some explicit synchronization mechanism would be needed. Options range from the exchange of synchronization pulses like in TanDEM-X [36], a continuous wave synchronization loop [38], or the use of calibration targets or transponders. Finally, it is worth pointing out that, for a geosynchronous, almost geostationary, mission as the one proposed, the acquired data could be directly streamed to a ground station, relaxing some spacecraft system and ground-segment requirements and allowing for real-time processing of the acquired data. VI. O UTLOOK This paper has presented the theoretical foundations for CoSAR, a novel radar imaging concept. We have attempted to hint at the plausibility of a CoSAR mission, providing a



specific but rough mission concept and analysis. It is clear that a geosynchronous mission concept requiring two relatively large spacecraft is associated to considerable costs. Further in-depth studies are therefore needed before the theoretical exercise discussed here can be translated into a real mission proposal. A CoSAR system would deliver unique observation capabilities of the ocean surface. Compared to spaceborne realaperture scatterometers, a CoSAR system would provide much higher product resolution, a characterization of the Doppler spectrum, and, moreover, wide swath altimetric information. In terms of resolution, CoSAR products would be similar to that of highly multilooked SAR products, but with the aforementioned additional information. CoSAR products are inherently timeaveraged quantities. This is unique and, in many cases (e.g., for pure scatterometric products), an advantage. It also provides a combination of regional coverage and temporal sampling (two observations per day for a two-spacecraft system) that can only be achieved with a constellation of LEO radar systems. We can identify several lines for future research. The first one is the experimental validation of the concept. This involves designing an experiment and pairing it with an appropriate study object: CoSAR assumes a surface that decorrelates quickly, but where some space-varying statistical properties remain constant during some time. Carefully crafted experiments using pairs of airborne radars seem the most promising experimental approach. In this paper, we have used some simple models to describe our 1-D ocean surface. In order to really understand the merits of a CoSAR mission, these models should be substituted by realistic 2-D models reflecting how the observable quantities really vary in space and in time. Another interesting topic is how to combine CoSAR with SAR imaging. This may be necessary in order to remove artifacts generated by nondecorrelating targets within the scene, as it would happen, for example, when imaging coastal areas. Finally, in this paper, we have presented CoSAR in the context of the observation of ocean surfaces. Indeed, quickly decorrelating ocean surfaces seem to be a good match to the CoSAR concept. Nevertheless, this focus may also be linked to the authors’ current scientific interests. It may well be that the concepts discussed will find application in totally unrelated fields. For example, the CoSAR concept could be applied in the context of spaceborne or ground-based meteorological radars.

This normalization constant has been verified with the numerical simulations discussed throughout this paper.

A PPENDIX A N ORMALIZATION OF C O SAR P ROCESSING

A PPENDIX B O NE -D IMENSIONAL C O SAR S IMULATION

This appendix provides an outline of how the normalization constant (12) can be derived. To simplify, we may start by choosing a reference ground coordinate system with an x ˜-axis orthogonal to the iso-delta-Doppler lines and ignore the corresponding y˜-axis. The CoSAR focusing integral (11) can be written as

For the sake of reproducibility, the following paragraphs provide some detail on how the 1-D simulations discussed have been implemented.

ˆ τ (˜ x, τ ) = Cp R

ˆ (ts , tf,1 (˜ Γ x, ts , τ ), tf,2 (˜ x, ts , τ ), τ ) · e2j·k0 Δ˜r(˜x,ts ,τ ) dts . (42)

We intend to cast this integral as an inverse Fourier transform, for which the phase of the exponential term should take the form 2πj · k · x ˜. To achieve this goal, the phase term can be approximated by the second-order Taylor expansion with respect to the variables x ˜ and ts . This expansion will have a constant term, two terms depending linearly on x ˜ and ts , two terms with a quadratic dependence x ˜ and ts , and one term that depends on both variables of interest. The first five terms do not provide resolution and are uninteresting. We may, without loss of generality, ignore them. The cross term is given by j · 2 · k0

∂2 Δ˜ r (˜ x, ts , τ )˜ x · ts . ∂x ˜∂ts

(43)

The time derivative of the differential range multiplied by a factor 2/λ0 is the delta-Doppler frequency so that the previous expression can be written as 2πj

∂ fD (˜ x, ts , τ ) · x˜ · ts . ∂x ˜

(44)

We now make the variable change k=

∂ fD (˜ x, ts , τ ) · ts ∂x ˜

(45)

∂ fD (˜ x, ts , τ ) · dts . ∂x ˜

(46)

with dk =

Setting the normalization constant ∂ Cp = fD (˜ x, ts , τ ) ∂x ˜

(47)

leaves (42) as a normalized inverse Fourier transform, as desired. For a general ground coordinate system, we have to replace the x ˜-derivative by the magnitude of the gradient Cp = ∇fΔD (x, y) .

(48)

A. Observation Geometry and Radar Trajectories It is assumed that the instantaneously illuminated area and the CoSAR aperture are short enough to assume a flat Earth and, in particular, straight-line radar trajectories. The straight trajectory approximation should be considered a reasonable simplification as long as the integration time is small compared


15

to the orbit period (24 h). The two radars are assumed to move with opposing velocity vectors so that the position of the ith phase center at each instant of time is given by

nt Tint v i Δ · − (49) pi [nt ] = (−1) + p0,i 2 PRF 2

In our simulations, the coherence time of the scattering coefficient (τs ) is assumed to be constant for the entire scene.

The raw data are generated in a straightforward manner by coherently combining the echoes corresponding to all the scatterers. To do that, first, the ranges from the radar to each grid point are computed

with nt = 0, . . . , Tint · PRF − 1 where the t = 0 positions, p0,i , are calculated according to a given incident angle and a given spectral shift. Note that this spectral shift translates to a small incident angle difference. Without loss of generality, we take the x-axis as the azimuth dimension, setting to zero the y-component of the relative velocity vector (Δv [1] = 0) and the x-component of the t = 0 positions ( p0,i [0] = 0). The radars are assumed to have an idealized beam pattern, with uniform gain within the beamwidth and zero gain outside it. B. Scene Generation The scene considered is quasi-one dimensional. It is aligned in the azimuth direction (x-axis), and its length is determined by the length of the illuminated area. We consider only short integration times so that the illuminated area remains virtually constant. The scene is defined in terms of three azimuth-dependent statistics: NRCS, Doppler velocity, and MSH. Each of these quantities is modeled as a low-pass random process, characterized by some standard deviation and some length scale. NRCS, Doppler velocity, and MSH remain constant throughout the simulation. To set up the simulation, an azimuth grid is defined with a spacing Δx several times smaller than the expected azimuth resolution. The grid is extended in the y-axis in order to simulate a number of range bins. It is assumed, however, that the statistics of the scene do not depend on this y-dimension. This is a reasonable assumption as long as the distance spanned in this direction is smaller than the smallest length scale considered. A scatterer is placed at each grid point. The scattering coefficients are modeled as circular Gaussian complex random variables, independent in space. Therefore, for each grid point, the scattering coefficient is given by

(50) s[nx , ny , nt ] = Δx · NRCS [nx ] · s0 [nx , ny , nt ] where s0 is modeled as an order 1 autoregressive process in the time domain s0 [nx , ny , nt ] = ρ · s0 [nx , ny , nt − 1] · ej2π·fD [nx ]

+ 1 − |ρ|2 · z[nx , ny , nt ] (51) with z[·] being a circularly Gaussian 3-D white noise signal. This model corresponds to an exponential temporal autocorrelation function so that the correlation coefficient ρ is given by ρ = e−1/(PRFτs ) .

C. Raw Data Generation and Processing

(52)

ri [nx , nt ] = pi [nt ] − (Δx · (nx − Nt /2), 0, h[nx])t (53) where the superscript t indicates the transpose operator. The raw data are then calculated as vi [ny , nt ] =

s[nx , ny , nt ]e−j·2π·ri [nx ,nt ] .

(54)

nx

Here, the phase associated to the range difference between different range bins is ignored, which amounts to assuming a perfect flat-earth phase correction. In the absence of strong topography, this seems a reasonable simplification. The CoSAR processing follows the steps discussed throughout Section IV, taking into account that, in the quasi 1-D case analyzed, the phase histories are considered but any range-cell migration is ignored.

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their time and comments, which have helped improve this paper technically and from a presentation point of view. The genesis of the correlating synthetic aperture radar concept may be traced back to a casual conversation between the first author and A. Camps, back in 2000, in Amherst, Massachusetts. R EFERENCES [1] C. W. Sherwin, J. P. Ruina, and R. D. Rawcliffe, “Some early developments in synthetic aperture radar systems,” IRE Trans. Mil. Electron., vol. MIL-6, no. 2, pp. 111–115, Apr. 1962. [2] W. M. Brown, “Synthetic aperture radar,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-3, no. 2, pp. 217–229, Mar. 1967. [3] D. E. Barrick, M. W. Evans, and B. L. Weber, “Ocean surface currents mapped by radar,” Science, vol. 198, no. 4313, pp. 138–144, Oct. 1977. [4] K. W. Gurgel, G. Antonischki, H. H. Essen, and T. Schlick, “Wellen Radar (WERA): A new ground-wave HF radar for ocean remote sensing,” Coastal Eng., vol. 37, no. 3/4, pp. 219–234, Aug. 1999. [5] B. Chapron, F. Collard, and F. Ardhuin, “Direct measurements of ocean surface velocity from space: Interpretation and validation,” J. Geophys. Res., Oceans, vol. 110, no. C7, 2005, Art. ID C07008. [6] R. M. Goldstein, H. A. Zebker, and T. P. Barnett, “Remote sensing of ocean currents,” Science, vol. 246, no. 4935, pp. 1282–1285, Dec. 1989. [7] P. Lopez-Dekker, F. De Zan, M. Rodriguez-Cassola, and G. Krieger, “Correlating SAR (CoSAR): Concept, performance analysis, and mission concepts,” in Proc. IEEE IGARSS, Melbourne, VIC, Australia, Jul. 2013, pp. 1–4. [8] P. H. van Cittert, “Die wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica, vol. 1, pp. 201–210, 1934. [9] F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica, vol. 5, pp. 785–795, Aug. 1938.


[10] C. S. Ruf, C. T. Swift, A. B. Tanner, and D. M. LeVine, “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. Geosci. Remote Sens., vol. 26, no. 5, pp. 597–611, Sep. 1988. [11] D. M. Le Vine, “Synthetic aperture radiometer systems,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2228–2236, Dec. 1999. [12] F. T. Ulaby, D. G. Long, W. J. Blackwell, C. Elachi, and K. Sarabandi, Microwave Radar and Radiometric Remote Sensing. Ann Arbor, MI, USA: Univ. Michigan Press, Nov. 2013. [13] A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio Astronomy. New York, NY, USA: Wiley, Nov. 2008. [14] W. Pfister, “The wave-like nature of inhomogeneities in the e-region,” J. Atmos. Terrestrial Phys., vol. 33, no. 7, pp. 999–1025, Jul. 1971. [15] E. Kudeki and R. F. Woodman, “A poststatistics steering technique for MST radar applications,” Radio Sci., vol. 25, no. 4, pp. 591–594, 1990. [16] B. L. Cheong, M. W. Hoffman, R. D. Palmer, S. J. Frasier, and F. J. López-Dekker, “Pulse pair beamforming and the effects of reflectivity field variations on imaging radars,” Radio Sci., vol. 39, no. 3, p. 13, Jun. 2004. [17] C. S. Ruf, “Error analysis of image reconstruction by a synthetic aperture interferometric radiometer,” Radio Sci., vol. 26, no. 6, pp. 1419–1434, Nov. 1991. [18] M. Rodriguez-Cassola, P. Prats, G. Krieger, and A. Moreira, “Efficient time-domain image formation with precise topography accommodation for general bistatic SAR configurations,” IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 4, pp. 2949–2966, Oct. 2011. [19] G. P. Cardillo, “On the use of the gradient to determine bistatic SAR resolution,” in Proc. IEEE AP-S, Merging Technol. Dig. Antennas Propag. Soc. Int. Symp., 1990, vol. 2, pp. 1032–1035. [20] D. C. Munson, Jr., J. D. O’Brien, and W. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE, vol. 71, no. 8, pp. 917–925, Aug. 1983. [21] F. Gatelli et al., “The wavenumber shift in SAR interferometry,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 4, pp. 855–865, Jul. 1994. [22] J. S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 5, pp. 1017–1028, Sep. 1994. [23] I. Reed, “On a moment theorem for complex Gaussian processes,” IRE Trans. Inf. Theory, vol. 8, no. 3, pp. 194–195, Apr. 1962. [24] J. Proakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4/E. Upper Saddle River, NJ, USA: Pearson Educ., Sep. 2007. [25] O. Montenbruck and E. Gill, Satellite Orbits: Models, Methods, and Applications. Berlin, Germany: Springer-Verlag, 2000. [26] J. R. Wertz and W. J. Larson, Space Mission Analysis and Design. Dordrecht, Netherlands: Springer-Verlag, Sep. 1999. [27] F. J. Wentz, S. Peteherych, and L. A. Thomas, “A model function for ocean radar cross sections at 14.6 GHz,” J. Geophys. Res., Oceans, vol. 89, no. C3, pp. 3689–3704, 1984. [28] F. Nirchio, “XMOD2—An improved geophysical model function to retrieve sea surface wind fields from COSMO-SkyMed x-band data,” Eur. J. Remote Sens., vol. 46, pp. 583–595, Jul. 2013. [29] H. Hersbach, A. Stoffelen, and S. de Haan, “An improved c-band scatterometer ocean geophysical model function: CMOD5,” J. Geophys. Res., vol. 112, p. 18, Mar. 2007. [30] R. Torres et al., “GMES Sentinel-1 mission,” Remote Sens. Environ., vol. 120, pp. 9–24, May 2012. [31] G. Krieger et al., “Advanced concepts for ultra-wide-swath SAR imaging,” in Proc. IEEE 7th EUSAR, Jun. 2008, pp. 1–4. [32] A. Freeman et al., “SweepSAR: Beam-forming on receive using a reflector-phased array feed combination for spaceborne SAR,” in Proc. IEEE Radar Conf., May 2009, pp. 1–9. [33] K. Tomiyasu and J. L. Pacelli, “Synthetic aperture radar imaging from an inclined geosynchronous orbit,” IEEE Trans. Geosci. Remote Sens., vol. GE-21, no. 3, pp. 324–329, Jul. 1983. [34] J. R. Rodon, A. Broquetas, A. Monti Guarnieri, and F. Rocca, “Geosynchronous SAR focusing with atmospheric phase screen retrieval and compensation,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 8, pp. 4397–4404, Aug. 2013. [35] T. Johnsen, “Time and frequency synchronization in multistatic radar. Consequences to usage of GPS disciplined references with and without GPS signals,” in Proc. IEEE Radar Conf., 2002, pp. 141–147. [36] G. Krieger and M. Younis, “Impact of oscillator noise in bistatic and multistatic SAR,” IEEE Geosci. Remote Sens. Lett., vol. 3, no. 3, pp. 424–428, Jul. 2006.


[37] P. López-Dekker, J. J. Mallorqui, P. Serra-Morales, and J. Sanz-Marcos, “Phase synchronization and Doppler centroid estimation in fixed receiver bistatic SAR systems,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 11, pp. 3459–3471, Nov. 2008. [38] J. C. Merlano-Duncan, J. J. Mallorqui, and P. López-Dekker, “Carrier phase synchronisation scheme for very long baseline coherent arrays,” Electron. Lett., vol. 48, no. 15, pp. 950–951, Jul. 2012.

Paco López-Dekker (S’98–M’03–SM’14) was born in Nijmegen, The Netherlands, in 1972. He received the Ingeniero degree in telecommunication engineering from Universitat Politica de Catalunya (UPC), Barcelona, Spain, in 1997, the M.S. degree in electrical and computer engineering from the University of California, Irvine, CA, USA, in 1998, under the Balsells Fellowship, and the Ph.D. degree from the University of Massachusetts, Amherst, MA, USA, in 2003, for his research on clear-air imaging radar systems to study the atmospheric boundary layer. From 1999 to 2003, he was with the Microwave Remote Sensing Laboratory, University of Massachusetts. In 2003, he was with Starlab, which is a privately held company, where he worked on the development of GNSS-R sensors. From 2004 to 2006, he was a Visiting Professor with the Department of Telecommunications and Systems Engineering, Universitat Autonoma de Barcelona. In March 2006, he joined the Remote Sensing Laboratory, UPC, where he conducted research on bistatic synthetic aperture radar (SAR) under a fiveyear Ramon y Cajal grant. At the university, he taught courses on signals and systems, signal processing, communications systems and radiation, and guided waves. Since November 2009, he has been leading the SAR Missions Group at the Microwaves and Radar Institute, German Aerospace Center, Wessling, Germany. His current research is focused on the study of future SAR missions and novel mission concepts.

Marc Rodriguez-Cassola was born in Barcelona, Spain, in 1977. He received the Ingeniero degree in telecommunication engineering from the Universidad Publica de Navarra, Pamplona, Spain, in 2000 and the Ph.D. degree in electrical engineering from the Karlsruhe Institute of Technology, Karlsruhe, Germany, in 2012. From 2000 to 2001, he was a Radar Hardware Engineer with the Centre d’Etudes des Environnements Terrestre et Planires/Centre National de la Recherche Scientifique, Saint Maur des Fosses, France. From 2001 to 2003, he was a Software Engineer with Altran Consulting, Germany. Since 2003, he has been with the Microwaves and Radar Institute, German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he has been working on airborne and spaceborne synthetic aperture radar (SAR) system analysis and data processing. His current research interests encompass radar signal processing, innovative high-precision SAR imaging algorithms, and radar system analyses and applications.

Francesco De Zan received the Master’s degree in telecommunication engineering and the Ph.D. degree from Politecnico di Milano, Milano, Italy, in 2004 and 2008, respectively. During his Ph.D. studies, he worked on extending permanent scatter interferometry to decorrelating targets and contributed to European Space Agency studies in preparation for Sentinel-1. Since 2008, he has been with the German Aerospace Center (DLR), Wessling, Germany, first at the Microwaves and Radar Institute and then at the Remote Sensing Technology Institute. He has been involved in studies for numerous future interferometric SAR missions and in calibration activities of the TanDEM-X interferometer. His research interests include algorithms and theoretical bounds for shift estimation and SAR interferometry and interpretation of phase and coherence signatures.


17

Gerhard Krieger (M’04–SM’09–F’13) received the Dipl.Ing. (M.S.) and Dr. Ing. (Ph.D.) degrees in electrical and communication engineering from the Technical University of Munich, Munich, Germany, in 1992 and 1999, respectively. From 1992 to 1999, he was with the Ludwig Maximilians University, Munich. Since 1999, he has been with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany, where he is currently the Head of the Radar Concepts Department of the Microwaves and Radar Institute. He is a Systems Engineer of the TanDEM-X mission and serves as a Lecturer at the University of Erlangen, Erlangen, Germany. Dr. Krieger received several national and international awards, including the W.R.G. Baker Prize Paper Award from the IEEE Board of Directors and the Transactions Prize Paper Award of the IEEE Geoscience and Remote Sensing Society. In 2014, he was the Technical Chair of the European Conference on Synthetic Aperture Radar. Since 2012, he has been an Associate Editor of the IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING.

Alberto Moreira (M’92–SM’96–F’04) received the B.S.E.E. and M.S.E.E. degrees from the Aeronautical Technological Institute (ITA), SJosos Campos, Brazil, in 1984 and 1986, respectively, and the Eng. Dr. degree (with honors) from the Technical University of Munich, Munich, Germany, in 1993. From 1996 to 2001, he was the Chief Scientist and Engineer with the SAR Technology Department, German Aerospace Center (DLR), Oberpfaffenhofen, Germany. Under his leadership, the DLR airborne SAR system has been upgraded to operate in innovative imaging modes like polarimetric SAR interferometry and SAR tomography. Since 2001, he is the Director of the Microwaves and Radar Institute at DLR and a Full Professor with the Karlsruhe Institute of Technology, Karlsruhe, Germany, in the field of microwave remote sensing. His DLR’s Institute contributes to several scientific programs and projects for airborne and spaceborne SAR missions like TerraSAR-X, TanDEM-X, SAR-Lupe, and SAR-Lupe follow-on as well as Sentinel-1a/b, BIOMASS, and Tandem-L. The mission TanDEM-X, led by his Institute, has successfully started the operational phase in December 2010 and is generating a global high-resolution digital elevation model of the Earth with unprecedented accuracy. He is the initiator and Principal Investigator for this mission. His professional interests and research areas encompass spaceborne radar end-to-end system design, analysis and operation, innovative microwave techniques and system concepts, signal processing, and remote sensing applications. He is the author or coauthor of more than 350 publications in international conferences and journals and 7 book chapters and is the holder of 18 patents in the radar and antenna field. Prof. Moreira has served as the President of the IEEE Geoscience and Remote Sensing Society (GRSS) in 2010. He is the recipient of several international awards, including the IEEE AESS Nathanson Award (1999) for the “Young Radar Engineer of the Year,” the IEEE Kiyo Tomiyasu Field Award (2007), the IEEE W.R.G. Baker Award from the IEEE Board of Directors (2012), and the IEEE GRSS Distinguished Achievement Award (2014). He was the founder and Chair of the GRSS German Chapter (2003–2008) and served as Associate Editor for the IEEE GRS Letters (2003–2007) and for the IEEE TGRS (since 2005). He and his colleagues received the GRSS Transactions Prize Paper Awards in 1997, 2001, and 2007 and the GRSS Letters Prize Paper Award in 2015. Since 2012, he has served as the Principal Investigator for the Helmholtz Alliance “Remote Sensing and Earth System Dynamics,” comprising 18 research institutes and 30 associated international partners.